mara markinson dissertation - academic commons

The Teaching and Learning of Geometric Proof:

Roles of the Textbook and the Teacher

Mara P. Markinson

Submitted in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

under the Executive Committee of the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2021

Abstract

The Teaching and Learning of Geometric Proof:

Roles of the Textbook and the Teacher

Mara P. Markinson

Geometric proof-writing is a widely known cause of stress for secondary school students and

teachers alike. As the textbook is the primary curricular tool utilized by novice teachers, a two-

part qualitative study was conducted to determine (a) the types of proofs presented in a typical

high school geometry textbook and (b) teachers’ preparedness and confidence to teach proof and

proving. I conducted a qualitative analysis of the selected textbook based on its presentation of

proofs and proof tasks, and then used said analysis to inform the creation of a five-question

content assessment on proof, which was administered to 29 preservice and in-service secondary

mathematics teacher participants. During the administration of the assessment, I interviewed

each participant regarding their thought processes, as well as their knowledge, beliefs, and

preparedness to teach proof and proving. The data were analyzed using a qualitative coding

system to categorize participants’ responses to the interview questions according to their beliefs

and attitudes, as well as issues with mathematical language and content that they encountered.

The qualitative analyses indicated that the selected textbook largely underemphasizes the role of

proof in the secondary school geometry curriculum, and that most participants are largely

underprepared to teach proof at the secondary level. Participants expressed sentiments about the

nature of proof and proving, verifying trends from the literature and providing the impetus for

future study. The findings support that more studies are needed to analyze the intersection

between curricular knowledge and content knowledge for secondary mathematics teachers.

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Table of Contents

List of Tables ............................................................................................................................vii

List of Figures ............................................................................................................................ ix

Acknowledgements ..................................................................................................................... x

Dedication .............................................................................................................................. xiii

Chapter I: Introduction ................................................................................................................ 1 Need for the Study ........................................................................................................... 1 Purpose of the Study ........................................................................................................ 5 Procedure and Research Questions .................................................................................. 5

Research Question 1 ............................................................................................ 5 Research Question 2 ............................................................................................ 6

Chapter II: Literature Review ...................................................................................................... 8 The Nature of Proof ....................................................................................................... 10

General Remarks About the Nature of Proof ...................................................... 10 Defining Proof ................................................................................................... 13 The Importance of Proof .................................................................................... 15 Students’ Conceptions of Proof .......................................................................... 17

The Recent History of Proof in Geometry Curricula ...................................................... 20 History of Proof and Proving in School Mathematics ......................................... 20 How Expectations of Students Change Over Time ............................................. 22

The Psychology of Proof and Proving............................................................................ 23 Van Hiele Levels ............................................................................................... 23 Mindset .............................................................................................................. 24 Truth-Seeking .................................................................................................... 25 Math Anxiety ..................................................................................................... 27

The Effects of Teacher Content Knowledge on Student Learning and Achievement ...... 28 Findings of Studies Regarding Teachers’ Knowledge of Geometry................................ 30 Beliefs and Opinions of Teachers .................................................................................. 33

Teachers’ Biases and Beliefs .............................................................................. 33 Teachers’ Acceptance of Varying Proof Methods............................................... 37 Differences in Pedagogical Methods .................................................................. 38

Teaching Proof and Proving .......................................................................................... 39 Role of the Teacher ............................................................................................ 40 Challenges in Teaching Proof and Proving ......................................................... 40

The Role of Textbooks in the Teaching and Learning of Proof and Proving ................... 45 The Importance of the Textbook ........................................................................ 45 Common Criticisms of the Exposition of Proof in Geometry Textbooks............. 47 The Frequency of Proof Tasks in Geometry Textbooks ...................................... 51

Concluding Remarks ..................................................................................................... 52

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Chapter III: Methodology .......................................................................................................... 54 Research Design ............................................................................................................ 54

Role of the Researcher ....................................................................................... 55 Selection of Participants ..................................................................................... 55

Data Collection and Analysis ......................................................................................... 56 Confidentiality ................................................................................................... 56 Data Collection .................................................................................................. 57 Data Collection Instruments ............................................................................... 58

Data Analysis ................................................................................................................ 70 Textbook Data Analysis ..................................................................................... 70 Interview and Content Assessment Data............................................................. 71

Chapter IV: Results and Discussion for Research Question 1 .................................................... 75 General Remarks ........................................................................................................... 75

Ambiguity in PRTs and GPTs Due to Verbiage ................................................. 77 Frequency of PRTs and GPTs in End-of-Section Exercises ................................ 80 Frequency of PRTs and GPTs in End-of-Chapter Problem Sets .......................... 82 Frequency of PRTs and GPTs in End-of-Chapter Cumulative Review Problems 82

Interpreting the Lowest and Highest Percentages of GPTs ............................................. 84 Analysis of a Subset of GPTs ........................................................................................ 88 Missed Opportunities for Proof Tasks ............................................................................ 97

A Missed Opportunity for an Entry-Level Proof Task ........................................ 97 A Missed Opportunity for a Construction-Based Proof Task .............................. 98 A Missed Opportunity for a GPT ....................................................................... 98

Chapter V: Results and Discussion for Research Question 2 ................................................... 100 Content Assessment Item 1 .......................................................................................... 101

General Remarks ............................................................................................. 101 Specific Findings Based on Codes ................................................................... 101

Content Assessment Item 2 .......................................................................................... 107 General Remarks ............................................................................................. 107 Specific Findings Based on Codes ................................................................... 109




Chapter VI: Summary, Conclusions, and Recommendations ................................................... 135 Summary ..................................................................................................................... 135 Conclusions ................................................................................................................. 137 Recommendations ....................................................................................................... 141

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References .............................................................................................................................. 146

Appendix A: Informed Consent .............................................................................................. 154

Appendix B: Textbook Data .................................................................................................... 158

Appendix C: Interview Questions............................................................................................ 215

Appendix D: Content Assessment Data ................................................................................... 218

Appendix E: Frequency of PRTs and GPTs in End-of-Section Exercises................................. 336

Appendix F: Frequency of PRTs and GPTs in End-of-Chapter Problem Sets .......................... 339

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List of Tables

Table 1 Sections With 0% PRTs and Non 0% GPTs.................................................................. 81

Table 2 Cumulative Review Problems at the Conclusion of Each Chapter ................................. 83

Table D1 Instances of Beliefs and Attitudes Code for Content Assessment Item 1 .................. 218

Table D2 Instances of Issue of Correspondence Between Substance and Notation Code for

Content Assessment Item 1 ..................................................................................................... 222

Table D3 Instances of Mathematical Language Code for Content Assessment Item 1.............. 223

Table D4 Instances of the Expressing Understanding or Self-Doubt Code for Content

Assessment Item 1 .................................................................................................................. 227

Table D5 Instances of Pure Mathematical Issue Code for Content Assessment Item 1 ............. 243

Table D6 Participants’ Reason(s) for Option Selection for Content Assessment Item 2 ........... 257

Table D7 Instances of Beliefs and Attitudes Code for Content Assessment Item 2 .................. 271



Table D9 Instances of Expressing Understanding or Self-Doubt Code for Content Assessment

Item 2 ..................................................................................................................................... 274

Table D10 Instances of Mathematical Language Code for Content Assessment Item 2 ............ 281

Table D11 Instances of Pure Mathematical Issue Code for Content Assessment Item 2 ........... 285

Table D12 Instances of Beliefs and Attitudes Code for Content Assessment Item 3 ................ 290




Item 3 ..................................................................................................................................... 296

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Item 4 ..................................................................................................................................... 316





Item 5 ..................................................................................................................................... 332



Table E1 Practice Problems at the Conclusion of Every Section .............................................. 336

Table F1 Review Problems at the Conclusion of Each Chapter................................................ 339

Table F2 Cumulative Review Problems at the Conclusion of Each Chapter ............................. 340

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List of Figures

Figure 1 Question 1: Introductory Problem ............................................................................... 60

Figure 2 Question 2: Proof 1, Option 1 ...................................................................................... 61

Figure 3 Question 2: Proof 1, Option 2 ...................................................................................... 62

Figure 4 Question 3: Proof 2 ..................................................................................................... 64



Figure 7 Cumulative Review GPT: Question 14 ........................................................................ 89



Figure 10 Cumulative Review GPT: Question 17 ...................................................................... 93




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Acknowledgements

I am grateful to so many individuals for their support, encouragement, and unyielding

belief in me. I wish to express gratitude to those who made sure I never gave up on my dreams.

To my dissertation defense committee: Dr. Alexander Karp, Dr. J. Philip Smith, Dr. Erica

Walker, Dr. Ann Rivet, and Dr. Alan Sultan, thank you for your feedback and guidance. Your

support has made completion of this project possible.

To Dr. Alexander Karp: Thank you for being my sponsor and for the opportunities you

have provided throughout my time in the program. Your guidance through all three research

seminars was instrumental in getting this dissertation off the ground.

To Dr. J. Philip Smith: Thank you for reminding me that my talent in mathematics has

nothing to do with how fast I can solve problems; for boosting my confidence; for encouraging

and entertaining my MANY mathematical questions; for checking in on me and pushing me to

finish this project.

To my study participants: Thank you for your willingness to participate and share your

thought processes with me. Your responses motivated the majority of this work, and further

studies to come.

To Dr. Cara Weston-Edell and Ms. Karen Crosby: Thank you. You listened to me rant,

answered my questions, helped me with APA formatting, and reminded me that the work I’ve

done is of high quality and worth celebrating.

To my mentors, Dr. Alice Artzt, Dr. Frances Curcio, Dr. Theresa Gurl, Dr. Eleanor

Armour-Thomas, Dr. Alan Sultan, Dr. Fern Sisser, Dr. Alex Ryba, Ms. Sara Gant, Mrs. Naomi

Weinman, Mr. Howard Weinman, Mr. Julio Penagos, Mr. Ben Sherman, Ms. Cathy Wilkerson,

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Mr. Manfred Korman, Mr. David Bantz, and Mr. Anthony Cromer: You believed in me even

when I didn’t, you pushed me to keep improving, and you all provided opportunities for growth

that have changed my life in substantial ways. I am eternally thankful for your mentorship.

To Dr. Alice Artzt: You are a rare gem. My dream to follow in your footsteps has been

actualized, and none of it would be possible without you. You have quite literally altered the

course of my life, and working alongside you is a rare privilege every day. I am so lucky to have

you in my corner, and I will always work my hardest to pay your mentorship forward.

To Ms. Sara Gant: Without you, it is certain that none of this would have been possible.

You were the patient, firm, and clear high school teacher I needed to make me fall in love with

mathematics. I am so grateful for your mentorship and sincere friendship.

To my students at all levels: For providing the impetus for this work and continually

motivating me to become a better educator.

To my dear friends, near and far, for your support, patience, and understanding. You

know who you are. I love you all.

To C and M: No matter how far away you were, I felt you rooting for me every step of

the way. Now I’m really the school’s freaky genius girl. This one is definitely going in the scrap.

I love you both.

To Joe, my B.C.: Thank you (“a lot”) for recognizing when I needed either space or

encouragement in this endeavor, and making sure I had both; for just listening; for pushing me

gently; for making sure I laughed, ate, slept, and laughed some more; for validating me; for

understanding me; for just “getting it”. Most importantly, thank you for being you. I love you.

To my aunts, uncles, and cousins: Thank you for your cheerleading and positivity. Your

encouragement, love, and support have not gone unnoticed. You always made me feel special

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and kept me laughing. Aren’t you relieved that you no longer have to ask me how school is

going? I love you all.

To the one and only Grandpa Bernie: You’ve been waiting for this day for quite some

time. I hope I have made you proud. I am so thankful that you are here to witness this on behalf

of my grandmothers, Miriam, Rose, and Lillian, and my grandfather Murray, whom I know

would have also been so proud of me. I love you.

To my siblings, Andrew, Spencer, and Serena, and your partners Karla, Jessica, and

Joseph: Thank you for the laughs, love, and support you’ve provided as I have ventured through

this arduous process. Andrew, now that I’m done with my “long essay,” we can get together for a

“homburger” (if you’ll “be respectful to my business and to me”). Spencer, I know you can’t

believe the girl who “couldn’t do Math B” is now on the other side of this degree program. I

wouldn’t be here without you. Weenie, my best friend—the one who named me “Mara Math”—

you are my heart. I’m so grateful to have finished this in New Paltz with you by my side. I love

you all.

To my parents, Alice Finkel-Markinson and Bryan Markinson: You have been there

every step of the way, reiterating your belief that I would finish this degree even when I wasn’t

so sure. You taught me the importance of being a lifelong learner and encouraged me to pursue

each of my dreams. I wouldn’t be the person I am today without you. Thank you for everything.

I love you.

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Dedication

To Courtney Shayna Lee

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Chapter I: Introduction

Need for the Study

In most countries, a course in Euclidean geometry is a cornerstone of any student’s high

school mathematics learning experience. In the United States, the inclusion of a geometry course

in the high school curriculum has been justified “primarily on the grounds that it provides a

context in which students can encounter and learn the ‘art of mathematical reasoning’” (Herbst &

Brach, 2006, p. 74). This viewpoint was supported by Hodge and Frick (2009), who quoted

Usiskin’s two reasons why it is important to teach geometry:

1. Geometry “uniquely” connects mathematics with the real outside world.

2. Geometry “uniquely” enables ideas from other areas of mathematics to be “pictured.”

(p. 28)

Hodge and Frick (2009) demonstrated support for Herbst and Brach’s (2006) idea that

geometry exposes students to the art of mathematical reasoning by asserting that

geometry is important not only to the world around us, but also to other areas of mathematics. For instance, understanding the distributive property can be illustrated to a student using area models. Geometry can be used to concretely illustrate . . . abstract concepts. (p. 29)

Despite the widespread teaching of geometry in this country, “US students’ geometry

performance remains far below the performance of most of their international peers” (Unal et al.,

2009, p. 999). Widespread research supports this claim. For example, in the 71st National

Council of Teachers of Mathematics (NCTM) yearbook, Battista (2009) reported that “a great

majority of students in the United States have inadequate understanding of geometric concepts

and poorly developed skills in geometric reasoning, problem solving, and proof” (p. 97).

Additionally, Usiskin (as cited in Harel & Sowder, 1998) studied 99 U.S. geometry classes and

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found that a mere 31% were proof-competent. Moreover, “28% of students couldn’t do a simple

triangle congruence proof” (Harel & Sowder, 1998, p. 235).

Antink (2010) shared data about U.S. teenagers lagging behind those of other nations in

mathematics in general, and explains that as mathematics concepts get harder in courses such as

geometry, U.S. students regress as compared to students from other nations (p. 10). Despite an

overall lack of performance in geometry, Fawcett (1938) claimed that the educational value of

geometry is undebatable and is not what is called into “serious question, [but rather whether] . . .

desirable results are actually achieved through the usual course in this subject” (p. 7). It is

therefore necessary to look closely at who is teaching geometry, how geometry is being taught

(including the textbooks and curricular materials that are being used), and how that is affecting

(or has the potential to affect) student learning outcomes. Teachers’ levels of content mastery,

beliefs, misconceptions, and curricular materials all contribute to the happenings in geometry

classrooms.

Although “proof is considered the basis of mathematical understanding and is essential

for developing, establishing, and communicating mathematical knowledge” (Cirillo & Herbst,

2011–2012, p. 11), the writing of geometric proofs is a widely known cause of stress for high

school geometry students. Proofs can be daunting for students who have little to no experience

writing them. Teachers also have limited experience writing high school geometry proofs, in

some cases. According to Herbst (2002),

Having high school students prove geometrical propositions became the norm in the United States with the reforms of the 1890’s—when geometry was designated as the place for students to learn the “art of demonstration.” (p. 1)

As students prove theorems, they are learning to think like mathematicians by making

sense of problems and being perseverant in solving them. Herbst and Brach (2006) discussed the

value of proofs in the geometry classroom, asserting, “the activity of proving theorems is central

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to the work of mathematicians” (p. 73). Proofs are high cognitive demand tasks that take getting

used to, and teachers must encourage students to push through the difficulty.

In order to improve the teaching and learning of geometric proofs, the struggles that

teachers and students encounter with regard to proofs needs to be examined. Just as students

have difficulty grappling with the concept of proof, teachers have difficulty moving away from

the traditional two-column proof format which dominates geometry proofs (Cirillo & Herbst,

2011–2012). Over time, the types of proofs in geometry as well as the roles of geometry students

in proving have changed (Barbin & Menghini, 2014; Barbin & Rogers, 2016; Cirillo & Herbst,

2011–2012). “These changes have had an effect on the presentation of geometry in school text-

books and also on the training of teachers” (Barbin & Rogers, 2016, p. 1). Challenges within

proving can arise due to gaps in teachers’ content knowledge as well as varying degrees of

difficulty of propositions that need to be proven. Categorizing proofs required of high school

geometry students based on their cognitive demand and the prerequisite knowledge and skills

they require can be a first step towards improving preservice and novice teachers’ and students’

attitudes towards, and mastery of, proofs in the high school geometry curriculum.

According to Zaslavsky et al. (2012),

Just as it is unrealistic to expect students to see a need for proof without purposeful and focused actions by the teacher, it is unrealistic to expect teachers to be able to [teach proofs meaningfully and successfully] without appropriate preparation and support. (p. 226)

Zaslavsky et al. (2012) asserted that the facilitation of proving activities in a classroom

places “strong demands on teachers in terms of the required mathematical knowledge and degree

of confidence as well as the challenging and time-consuming task of instructional design” (p.

226).

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In 1986, Shulman proposed a theoretical framework for studying the “domains and

categories of content knowledge in the minds of teachers” (p. 9). Shulman broke down teachers’

knowledge into three categories: “(a) subject matter content knowledge, (b) pedagogical content

knowledge, and (c) curricular knowledge” (1986, p. 9). Subject matter content knowledge refers

to a teacher’s “amount and organization of knowledge” (Shulman, 1986, p. 9), as well as a

teacher’s understanding not only of that something is so, but also why it is so. Pedagogical

content knowledge refers to

the particular form of content knowledge that embodies the aspects of content most germane to its teachability [and] . . . includes an understanding of what makes the learning of specific topics easy or difficult. (Shulman, 1986, p. 9)

Shulman asserted that it is within the area of pedagogical content knowledge that “research on

teaching and on learning coincide most closely” (1986, p. 10).

Curricular knowledge is “the knowledge of . . . curriculum materials for a given subject

or topic” (Shulman, 1986, p. 10).

The curriculum and its associated materials are the materia medica of pedagogy, the pharmacopeia from which the teacher draws those tools of teaching that present or exemplify particular content and remediate or evaluate the adequacy of student accomplishments. (Shulman, 1986, p. 10)

Since textbooks are the primary curricular tool used by beginning teachers to inform their

instruction, research about how textbooks present subject matter must be conducted in tandem

with research about teachers’ content knowledge. Aslan-Tutak and Adams (2015) asserted,

The limited number of research projects focused on knowledge of geometry for teaching concludes that beginning teachers are not equipped with necessary content and pedagogical content knowledge of geometry, and it is important to address this issue in teacher education. (p. 303)

There is no shortage of pre-existing research focusing distinctly on curriculum or

teachers’ preparedness; however, although these elements are intimately connected, studies

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examining both in conjunction are scarce. The present study examined the roles of both the

textbook and teachers’ preparedness in the teaching and learning of geometric proof.

Purpose of the Study

The purpose of this study was twofold: The first purpose was to investigate and describe

the types of proof tasks offered in a typical high school geometry textbook. The second purpose

was to explore preservice and novice secondary mathematics teachers’ proof-writing abilities

and beliefs about geometry (particularly proofs), in order to propose ideas about how to improve

the teaching and learning of geometric proofs.

Procedure and Research Questions

I recruited preservice and novice secondary mathematics teacher participants for the

study via email, social networking, and in-person requests. The preservice participants were

undergraduate or graduate students recruited from mathematics teacher preparation college or

university programs in the penultimate year or the final year of their programs. The novice

participants were in-service teachers in their first or second years of teaching. The target number

of total participants was 30.

The study was guided by the following research questions:

1. What kinds of proof tasks and proofs are offered in a typical high school geometry

textbook?

2. How prepared and how confident are preservice and novice secondary mathematics

teachers to teach proofs?

Research Question 1

Research Question 1 (What kinds of proof tasks and proofs are offered in a typical high

school geometry textbook?) was addressed qualitatively. I chose a common core-aligned high

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school geometry textbook, which I have anonymized in this report ([Textbook Author], 2016),

and analyzed it with a focus on proofs. The selected textbook was analyzed based on the types of

proofs (regarding both form and content) that are most heavily emphasized, and the amount of

proving that students are expected to do within the proof tasks. First, I made a collection of all of

the tasks and prompts that require students to prove geometric propositions or explain a

geometric idea. Next, I composed commentary regarding content, type (statement-reason,

paragraph, induction, contradiction, etc.) and level of complexity/ rigor for each exercise. I

tallied the instances of tasks that explicitly instruct students to prove, versus those that prompt

students to “explain,” “show,” “demonstrate,” or “justify,” and computed the percentages of

genuine-proof tasks versus proof-related tasks in the section exercises, end-of-chapter exercises,

and cumulative chapter exercises. Last, I drew conclusions about the kinds of proofs in the

textbook, and what is required of students, based on the overarching ideas elicited by the

analysis. The findings from the textbook analysis were used to inform the design of the content

assessment I created, as well as the interview protocol, both which were used to answer Research

Question 2.

Research Question 2

To answer Research Question 2 (How prepared and how confident are preservice and

novice secondary mathematics teachers to teach proofs?), qualitative methods were used.

Preservice and novice teacher participants were administered a content assessment intended to

assess their knowledge of how to justify geometric ideas and prove geometric propositions. All

participants were administered the assessment individually, while I asked clarifying questions

regarding the participants’ work and thought processes, and discussed the assessment items with

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each participant. I asked questions regarding the participants’ feelings and attitudes while

completing each assessment item.

Initially, the interviews were planned to be conducted separately, after the administration

of all content assessments, and the target number of participants to be interviewed was 5–10. The

interview questions drafted by I addressed the preservice and novice teachers’ feelings and goals

when completing proof exercises as well as teaching proofs to high school geometry students.

Questions targeted participants’ feelings about doing proofs, the necessity of proofs, why

proving is important, what is important to prove, students’ ability to prove, their preparation and

confidence to teach proofs, and, if they were in a geometry student teaching or teaching

placement, their opinions on how their assigned curriculum presents proofs. As the

administration of the content assessments progressed and participants were asked to think out

loud about their thought processes, the interview questions were tied in to the content assessment

before, between, and after the five assessment items. Thus, all 29 of the participants ended up

being interviewed because their answers to my questions provided insight regarding what,

specifically, was troubling them about the content assessment items. Additionally, I asked the

participants questions about their mathematical histories, backgrounds in geometry, and

collegiate mathematics coursework. Each administration of the content assessment and its

corresponding interview was audio-recorded and transcribed, and I analyzed the data to draw

connections between each participant’s responses to the interview questions and their content

performance, feelings, and beliefs.

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Chapter II: Literature Review

A major focus of historical and contemporary geometry curricula is the idea of proof. In

1998, Harel and Sowder reported that NCTM had re-emphasized the learning of proof and

“recommended that proof should be taught to all students [after] only the ablest students” (pp.

234–235) seemed to be making strides in proof and proving. High school students’ notorious

struggles with proof are well documented in the literature. According to Battista (2009),

“Students rarely see the point of proving” (p. 103). Harel and Sowder (1998) explained that

struggles with proving are not unique to high school students, but additionally university

students, whose instructors assume “on the part of entering students a general understanding of

proof and its roles in mathematics” (p. 275). Hoyles and Healy (2007) supported the

aforementioned ideas, stating, “Despite numerous attempts to teach students to prove conjectures

in the context of geometry, there is overwhelming evidence of persistent confusion and

misunderstandings” (p. 81). Since the notion of proof is at the core of the geometry curricula,

difficulty with understanding the nature and expectations of proof is a major reason that

performance in geometry is so poor.

A wide variety of factors impacts students’ learning and understanding of proof and its

nature, as well as their ability to prove. Two of these factors are teacher content knowledge and

beliefs, and the exposition of proof in students’ course textbooks. What specific roles do the

classroom teacher and the classroom textbook play, and how do they work together or against

each other to support or hinder student achievement? It is important to consider the nature of

proof, its history in geometry courses, and the psychology of proof and proving in order to

determine how these factors contribute (whether positively or negatively) to the teaching and

learning of proof via teachers and textbooks. Only once these aspects are examined and studied

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can improvements be made to the state of teaching and learning of proof in high school

geometry.

This literature review discusses the knowledge and beliefs that elementary and secondary

teachers have regarding geometry and the teaching of geometry, as well as the role of textbooks

in high school geometry courses, with a specific focus on proof. Since much research exists

regarding elementary and preservice secondary teachers, but little exists about novice secondary

teachers (Cirillo, 2008), this literature review and the subsequent study also focus on novice

secondary teachers (in their first or second years of teaching secondary mathematics).

A look at teachers’ knowledge and beliefs about proofs is helpful because “many

mathematics educators have criticized . . . approaches to teaching proof, . . . arguing that [poor

student performance and outlook] are inevitable consequences of these approaches” (Harel &

Sowder, 1998, p. 235). First, the nature of proof (including students’ conceptions of proof), its

psychology, and its history in the geometry curricula are discussed. Then, effects of teacher

content knowledge (and competence in proving) on student learning and achievement are

discussed. Next, the results of pre-existing studies regarding teachers’ knowledge of geometry

(specifically proof) are summarized. Moreover, the beliefs and opinions of teachers regarding

their content knowledge, their feelings about proof, and the teaching of geometry are considered.

The role of textbooks is discussed and previous studies involving high school geometry

textbooks are reported. Finally, suggested topics for further research are provided in the

conclusion.

The sources selected for inclusion in this literature review were chosen because they

either supported or refuted the work of another researcher, or provided a fresh perspective. Both

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national and international research are discussed to fully conceptualize the state of the teaching

and learning of proof, and to make suggestions for new research.

The Nature of Proof

General Remarks About the Nature of Proof

Most mathematicians and university mathematics majors would state that the idea of

proof is at the heart of mathematics. The fact that something so important and central to

mathematics can cause such difficulty for students at all levels speaks to the complexity of

proof’s nature. Otten (2009), in his review of literature on proof education, stated that

“traditional instruction fails to instill in students an understanding of the nature of proof” (p. 11).

Whereas many mathematical problem-solving tasks presented to students involve the

application of an isolated concept with a prescribed set of steps, writing a proof requires the

application and synthesis of ideas and can be approached from a variety of ways. Proof is a broad

concept, of which “a narrow view . . . neither reflects mathematical practice nor offers the

greatest opportunities for promoting mathematical understanding” (Hanna & de Villiers, 2012, p.

3). Students are used to checklists which determine whether they have met certain criteria for

solving a mathematics problem or exercise. For example, when solving a quadratic equation,

students might be used to making sure the following things have happened: Make one side equal

to zero, factor the quadratic expression, set each factor equal to zero, and solve the resulting

linear equations. Although students might not have a formal understanding of why they are

performing these steps, a lack of conceptual understanding in this case does not prevent them

from arriving at an answer to the question. The certainty that comes with the repetitive nature of

such problems makes students feel secure and prefer to engage in mathematics of this type, over

proofs, which are not as prescribed, comfortable, repetitive, or predictable. Whereas exercises of

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the type described previously only require students to follow a prescribed process, proof requires

that students “[relate] organization of knowledge to understanding [in a] . . . deep unity”

(Mariotti, 2006, p. 176). Hsieh et al. (2012) added further insight regarding why proof is in deep

contrast to common exercises:

It is impossible to list with certainty all the factors necessary for the successful construction of a mathematics proof; however, they certainly include cognition of the necessary theories to be used, the ability to employ sequencing steps according to accepted logical rules, and the competence to use accepted mathematics registers to express the steps to be conveyed. (p. 280)

Hoyles and Healy (2007) supported the views of Hsieh et al. (2012) and Mariotti (2006),

and added their perspective regarding psychological components of proof and proving, as well as

why teachers must be aware of the nature of proof:

Proving in mathematics is undoubtedly a complex process. It not only involves logical and deductive argument coordinated with visual or empirical evidence and mathematical results and facts, but is also influenced by intuition and belief, by perceptions of authority and personal conviction, and by the social norms that regulate what is required to communicate a proof in any particular situation. The failure of traditional geometry teaching in schools stemmed at least partly from a lack of recognition of this complexity. (pp. 81–82)

I discuss the “perceptions of authority and personal conviction” and the “social norms

that regulate what is required to communicate a proof in any particular situation” that Hoyles and

Healy (2007) referenced in subsequent sections of this literature review. Previous research makes

clear the differences in cognitive demand and the need for conceptual understanding between

solving mathematics exercises that most students are used to, and writing mathematical proofs.

Another factor which makes proof-writing complex is the diversity of methods and

acceptable solutions for any given proof task. Due to the different types of proofs that can be

written (inductive, deductive, proof by contradiction), there is usually not one correct place to

start a proof. This contrasts with most other problems that students have encountered in

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mathematics up to geometry. Netz (1999) explained how the lack of rules in the order of proof-

writing contributes to this complexity:

Where are the starting-points? Everywhere in the proof. The only general rule is that the first assertion in a proposition tends to be a starting-point, while the last assertion tends not to be. Otherwise, the position of starting-points is flexible . . . We begin to see something about the global structure of proofs . . .: proofs do not reuse over and over again materials, which have been presented earlier, once and for all. Instead, whatever is required by the proof is brought in at the moment when it is required. The introduction of new material for deductive manipulation goes on through the length of the proof. (p. 170).

Netz (1999) nicely summarized the complexity of what is required when one engages in a

proof. Deductive proof tasks require multiple levels of cognition—not only must students

identify and apply relevant prior knowledge, but also navigate between given and inferred

statements and reasons as they journey through a proof.

Deduction, in fact, is more than just deducing. To do deduction, one must be adept at noticing relevant facts, no less than combining known facts. The eye for the obviously true is no less important than the eye for the obvious result and, as is shown by the intertwining of starting-points and argued assertions, the two eyes act together. (Netz, 1999, p. 171)

This quotation speaks to the complexity of what students must do when they are proving,

and alludes to the fact that teachers need to be able to think flexibly about students’ proof plans

and responses.

Confining proof to geometry, which is an all-too-common practice, adds to the difficulty

that students have. “Proof was confined to a single course for nearly the entire twentieth century

and remains so confined in many schools today” (Otten, 2009, p. 2). Rather than having

students’ ideas about proof develop as they mature mathematically and are exposed to various

proof activities, geometry is the sole place where proof takes place in many classrooms, schools,

and districts. Otten’s (2009) claim supports NCTM’s (2000, as cited in Mariotti, 2006)

philosophy that “reasoning and proof are not special activities reserved for special times . . . but

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should be [an] ongoing part of classroom discussions, no matter what topic is being studied” (p.

173). “The study of proof should not be considered as a course which a pupil begins at a certain

point in his secondary school experience and which he completes at the end of a given time”

(Fawcett, 1938, p. 119). Instead, Fawcett (1938) has advocated for the gradual introduction to,

and engagement with, proof, over the course of a student’s educational career. Coming to truly

understand the objectives of proof, as well as its nature, takes time, but “teachers of mathematics

have . . . assumed that this . . . can best be [done] through the study of demonstrative geometry”

(Fawcett, 1938, p. 120). Fawcett pointed out that these views have effectively served to isolate

the idea of proof, “whereas [proof] may well serve to unify the mathematical experiences of the

pupil” (1938, p. 120). Although Fawcett’s work was published in 1938, the NCTM’s (2000)

Principles and Standards still raised the same concerns. This points to a continuing need to study

the teaching and learning of proof.

A famous quotation from Euclid, considered the father of geometry, supports these ideas:

“There is no royal road to geometry” (as cited in Saito, 2017, p. 50). Euclid used this response to

his student, who asked to be taught shorter methods to learn geometry. Although a historical

quotation, the prevalence of it remains in geometry education today, specifically in proof

instruction. There are no shortcuts to learning what proof is and how to prove. The process of

engaging in proof is so misunderstood by students who are often unaware of the purpose or

power of what they are doing. In order to alleviate these issues, understandings of what proof is,

why it is important, and what students’ conceptions of proof are must be developed.

Defining Proof

Although it is described as “the professional currency of mathematics” (Otten, 2009, p.

2), part of what makes proof so complex is its ill-definition. According to Cabassut et al. (2012),

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“No explicit general definition of a proof is shared by the entire mathematical community” (p.

169). They explained that mathematicians

stray from a formal definition of proof when explaining what one is [and that] . . . consequently, it is difficult to explain precisely what a proof is, especially to one who is a novice at proving. (Cabassut et al., 2012, pp. 186–187)

Many researchers and authors have attempted to define proof by depicting the actions or

characteristics that constitute proving. Some examples are Hanna and Jahnke (2002, as cited in

Reid & Knipping, 2010), who categorized proof as “a guide to the intelligent exploration of

phenomena” (p. 35); Reid and Knipping (2010), who stated that “all proofs have three

characteristics: they must be deductive, convincing, and at least semi-formal” (p. 37); and Harel

and Sowder (1998), who asserted that “proofs are first and foremost convincing arguments” (p.

237). Further, they stated:

By “proving” we mean the process employed by an individual to remove or create doubts about the truth of an observation. The process of proving includes two subprocesses: ascertaining and persuading. Ascertaining is the process an individual employs to remove her or his own doubts about the truth of an observation. Persuading is the process an individual employs to remove others’ doubts about the truth of an observation. Central to [proving] is the question How are conjectures rejected or rendered into facts? (Harel & Sowder, 1998, p. 241, original in italics)

Cabassut et al.’s (2012) discussion of the definition of proof again speaks to proof’s

complexity. They asserted that “formal definitions of proof cover the meaning of the notion only

incompletely, whereas mathematicians are convinced that, in practice, they know precisely what

a proof is” (Cabassut et al., 2012, p. 170). In educational settings, students are often encouraged

to explain concepts in order to ensure their own understanding of them. The fact that there is no

agreed-upon definition of a mathematical proof suggests that the recipe for a proof is intricate

and dependent on what is accepted as proof in a particular setting. This in turn speaks to the

difficulty of teaching proof in a manner that is clear and consistent with mathematical standards

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(and, further, acquiring a clear picture of what the “standards” of proof are), as implied by the

“social norms” referenced in Hoyles and Healy’s (2007) aforementioned comments.

Lakatos (1963a, 1963b, 1963c, 1964) wrote perhaps one of the most well-known works

on proof. Lakatos published several influential papers, later combined in a book, Proofs and

Refutations. Lakatos’s work shed light on the fact that mathematics is not a rigid, fixed body of

facts and formulas as generally seen by the layman, but rather an ever-changing body of

knowledge that moves forward as new conjectures come about. The notion of “proof” was

challenged as Lakatos brought to light how dependent proofs are on definitions, special cases,

and counterexamples. Lakatos (1963a) defined a proof as a “thought experiment, which suggests

a decomposition of the original conjecture into subconjectures or lemmas” (p. 10). The pedagogy

conveyed in Lakatos’s work suggests that he believed in a discovery approach to teaching and

learning proof, wherein students consider the implications of a given conjecture.

Lakatos’s (1963a) definition of proof is similar to that of Fischbein’s (1999), whose

definition relies on students posing a conjecture (an “intuitive interpretation”) and using their

prior knowledge to determine its validity. Fischbein described mathematical proof as a

“structural schema (i.e., a ‘behavioral-mental [device] which [makes] possible the assimilation

and interpretation of information and the adequate reactions to various stimuli’)” (1999, p. 11).

Both Lakatos and Fischbein supported the idea that when asking students to prove geometric

propositions, it is important to elicit the information from them rather than prescribe a set of

instructions they cannot consistently rely on due to the diversity of what needs to be proven.

The Importance of Proof

Why is proof regarded as so important in a student’s mathematics learning experience?

There is a myriad of benefits to learning proof which extend beyond the geometry classroom.

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Well-known mathematics education researchers share their views of the importance of proof in

the existing literature. Fawcett (1938) cited the “clear and forceful statement of Professor C. B.

Upton” (p. 5), who asserted that teaching demonstrative geometry gives “certain ideas about the

nature of proof, [which] furnish[es] pupils with a model of all their life thinking” (p. 5). Other

scholars have agreed that learning mathematical proof teaches students how to reason in other

subjects and areas of life, and state additional important reasons for learning proof. According to

Grabiner (2012),

There are a number of functions for proof in the classroom: explanation, verification, discovery, systematisation, and intellectual challenge . . . in order to understand the properties of the non-visible, the non-intuitive, or the counterintuitive, mathematicians need logic; we need proof. (pp. 163–164)

Furthermore, Hanna et al. (2012) asserted that the importance of proof to the field of

mathematics has played a part in keeping proof in the school mathematics curriculum. Proof is

“essential to maintaining the connection between school mathematics and mathematics as a

discipline” (p. 444). Schoenfeld (2009) agreed, asserting that

proofs are necessary. Unless you have one, you run the risk of being fooled by a false pattern. . . . If you think something might work, you try to prove it—you try to work out why it should be the way you think it is. Sometimes you succeed, sometimes you don’t—but even in failure you may come to a better understanding of the phenomenon you’re trying to make sense of. That is, proving is a form of mathematical sense making. (p. xiv)

Schoenfeld’s (2009) description of the importance of proofs relates to Harel and

Sowder’s (1998) aforementioned ideas about ascertaining. Hanna (2007) also mentioned the

importance of proof as verification, and stated that

proof deserves a prominent place in the curriculum because it continues to be a central feature of mathematics itself, as the preferred method of verification, and because it is a valuable tool for promoting mathematical understanding. (p. 3)

One of the biggest contradictions regarding proof is its importance in the field of

mathematics compared with the all-too-often distaste for its teaching and learning. Considering

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students’ ideas about proof sheds light on some of the problems associated with its teaching,

learning, and psychology.

Students’ Conceptions of Proof

As Schoenfeld (2009) noted,

A proof is typically taught as something to be memorized, and proving comes to be understood by students as a ritual that confirms what they already know to be true, rather than as a means of developing understanding. (Schoenfeld, 2009, p. xv)

In the 71st NCTM Yearbook, Battista (2009) cited his 1995 work with Clements, which

established that “students do not conceive of proof as establishing validity but instead see it as

conforming to a set of formal rules that are unconnected with their personal mathematical

activity” (p. 103). The sentiments expressed here explain the disconnect between the perceived

purpose and the actual purpose of proof, and prevent students from meaningfully engaging in the

process of proving. Mariotti (2006) supported the views of the aforementioned authors and added

that in addition to serving the purpose of validating the truth of a conjecture, “proof has to

contribute more widely to knowledge construction. If this is not the case, proof is likely to

remain meaningless and purposeless in the eyes of students” (p. 198).

Harel and Sowder (1998) referred to the work of two colleagues, Fischbein and Kedem,

who in 1982 observed high school students who still wanted to check a few cases after producing

a proof of a proposition. This showed that the students were still uncertain about the truth value

of the proposition and that they did not understand the true nature of mathematical proof, which

is that it does not necessitate further verification (Harel & Sowder, 1998). Weber (2017)

discussed this study, as well as a study with similar findings by Schoenfeld (1989), in which

students tried to “construct geometric figures they [had] just proven were impossible to

construct” (Weber, 2017, Slide 20). Weber reported findings from other studies, including that

students will produce one example and call it a proof; students might think proofs apply only to

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one case; students believe proofs must be written in two columns and contain mathematical

symbols (and that proofs cannot be written in paragraphs); students trust what their teachers say,

but not necessarily their own reasoning; and “students are utterly perplexed by the seemingly

arbitrary rules associated with proof” (2017, Slide 20).

This viewpoint was further supported by Schoenfeld (1985), who said:

As a result of instruction that focuses heavily on writing results in specific ways, and grading procedures that penalize students for not expressing otherwise correct answers in those ways, students can come to believe that ‘being mathematical’ means no less—and no more—than expressing oneself via the prescribed forms. (p. 369)

In a similar vein, Fawcett (1938) reported Christofferson’s findings after analyzing a

geometry test given to students in Ohio:

The reasons given by pupils for statements often seem to disregard entirely the thought of the situation. Often it seems that it is mere habit that dictates the response, not a thought process. Pupils have often used the various theorems as reasons and with satisfaction. They seem in some cases to have used them so often without meaning that they give them as so many memorized nonsense syllables. (p. 7)

Harel and Sowder (1998) reported Usiskin’s view of students’ impression that geometry

class is “the only place where they are exposed to the idea of proof” (p. 236). Usiskin suggested

that their views were distorted for several reasons. One reason is that mathematicians explore

new phenomena rather than prove “obvious propositions, as is often done in geometry classes”

(Usiskin, as cited in Harel & Sowder, 1998, p. 236), and “mathematicians do not write proofs in

two columns” (p. 236). Otten (2009) supported Usiskin’s point, stating:

Instruction based around the two-column format has a tendency to turn proof into a procedure or something to be memorized (NCTM, 2000), which is the antithesis to the true nature of proof and does not lead to the positive modes of thought that are the primary goal of proof instruction. (p. 5)

Usiskin (1980) added that students miss out on understanding the true nature of proof

because they are too often only taught them in geometry—students do not even have the

awareness that there are many types of proofs and that proofs are used in different subjects (such

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as proving trigonometric identities). Usiskin recommended to get rid of proofs of trivialities in

geometry curricula and allow informal proof-writing in geometry classrooms, which would value

students’ reasoning over sets of rules that make the concept feel forced and foreign to them.

Usiskin’s recommendations to decrease the importance of formality and instead emphasize the

“aspects of communication and social processes” (Harel & Sowder, 1998, p. 236) associated

with proof were backed by a variety of mathematics education researchers. Additionally, Goddin

et al. (2014) reported the words of Ogilvy, which further support a more informal environment

for learning proof: “To avoid the catastrophe of an uninspired and uninspiring geometry course

we will beg the forgiveness of the mathematicians, skip the formalities and take our chances with

the rest” (p. 22).

Otten et al. (2011) summed up the concerns raised in this section by posing the question,

“Do students [even] realize that a proof would be an effective response to an ‘explain’ exercise?”

(p. 353). In order to improve students’ conceptions of proof and help them internalize its

overarching goals and purpose, “mathematics educators need to understand students’

perspectives on the need for proof and which situations, tasks, and knowledge encourage that

need in students” (Zaslavsky et al., 2012, p. 219).

As mathematics educators, our goals for students’ understanding of proof are well stated

by Fawcett (1938, as cited in Reid & Knipping, 2010, p. 41):

A student understands the nature of deductive proof when he understands:

(1) The place and significance of undefined concepts in proving any conclusion.

(2) The necessity of clearly defined terms and their effect on the conclusion.

(3) The necessity of assumptions or unproved propositions.

(4) That no demonstration proves anything that is not implied by the assumptions.

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Weber (2017) paraphrased this sentiment by setting a goal of “proving as convincing:

Students should see proof as the most convincing type of mathematical justification and should

seek out proofs to gain conviction” (Slide 8). In order to improve the teaching and learning of

proof, efforts must be directed at improving students’ conceptions of proof.

The Recent History of Proof in Geometry Curricula

History of Proof and Proving in School Mathematics

According to Hoyles and Healy (2007), “Proof has traditionally appeared in school

mathematics in exercises involving the formal confirmation of Euclidean geometry theorems” (p.

81). Although the presence of proof has been a constant in geometry for centuries, there have

been developments and changes over time to proof forms, proof standards, and proof’s

prominence in the curriculum. “Not just styles of proof, but standards of proof, change”

(Grabiner, 2012, p. 158). For example, Netz’s (1999) book The Shaping of Deduction in Greek

Mathematics “shows how manuscript diagrams are different from what we see in today’s

editions” (Saito, 2017, p. 49). Saito (2017), after analyzing ancient texts, explained that much

less writing was included in proof in ancient times and that a facilitator or teacher likely

accompanied the studying of the text to fill in the gaps using speech versus writing. Regarding

this finding, Saito said, “Generally speaking, it is obvious that oral communication was much

more important in ancient times than it is today” (2017, p. 59).

Hanna et al. (2012) explained, “Proof has not enjoyed the same degree of prominence in

mathematical practice in all periods and contexts, and . . . standards of rigour have changed over

time” (p. 444). As recently as 1950, mathematical proof in U.S. classrooms was “confined to

geometry. Very few students acquired the notion that the same deductive structure . . . also

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applied to arithmetic, algebra, and all areas of mathematics” (Kinsella, 1965, as cited in Pruitt,

1969, p. 37).

According to Hanna (2007), “Evidence of Lakatos’ thinking can be heard quite clearly in

the curriculum guidance developed . . . by the National Council of Teachers of Mathematics” (p.

11). This refers to the fact that Lakatos’s work and others like it which attack the idea that

mathematics can be predictable and foolproof, led to a suggested downgrade in the role of proof

by NCTM in the 1990s. Mathematics educators “tended to accept Lakatos’ principal insight the

critique of mathematical results by others has been the motive force in the growth of

mathematical knowledge” (Hanna, 2007, p. 9). This led to contradictory feelings in mathematics

educators who agreed with Lakatos’s suggestions involving the social aspects of proof and the

benefits of informal proof instruction, but disagreed with a movement towards less formal proof

in the curriculum. Hanna expressed her disagreement with downplaying formal mathematics and

argued that proof continues to have value in the classroom and advocated for its emphasis in

curricular standards:

Over the past thirty years or so proof has been relegated to a less prominent role in the secondary mathematics curriculum in North America. This has come about in part because many mathematics educators have been influenced . . . to believe that proof is no longer central to mathematical theory and practice, and that . . . its use in the classroom will not promote learning. As a result many educators appear to have sought relief from the effort of teaching proof by avoiding it altogether. (2007, p. 3)

Responding to this downplay of proof in the 1990s version of the NCTM Standards,

Hanna (2007) asserted that

proof deserves a prominent place in the curriculum because it continues to be a central feature of mathematics itself, as the preferred method of verification, and because it is a valuable tool for promoting mathematical understanding. (p. 3)

The 2000 NCTM standards re-emphasized the role of proof across the grade levels. “In

the International Congress on Mathematics Education ICME 13 [2016] there were reports that in

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many countries an emphasis on proof is reappearing in mathematics curricula” (Meinerz &

Doering, 2017, p. 95). Weber (2017) supported this claim, stating that “proof plays an important

role in policy and curricula. Proof is a practice in the Core Curricula Content Standards in

Mathematics and a practice in NCTM’s Principles and Standards” (Slide 5).

How Expectations of Students Change Over Time

According to Harel and Sowder (1998), “The view of what constitutes an acceptable

mathematical proof has had many turning points” (p. 239). Pruitt (1969) reported that “early

geometry textbooks contained no exercises at all” (p. 4). Exercises began to appear “after the

middle of the nineteenth century [after which] . . . their number gradually increased in textbooks

published” (Pruitt, 1969, p. 4). This change in emphasis speaks to the shifting of expectations of

students. Traditionally, proofs were the heart of textbooks and there was a lack of emphasis on

exercises. Over time, this essentially swapped and there have been studies (such as Pruitt’s) that

study the frequency of proof tasks in textbooks. Some of these studies are discussed later in this

literature review.

Today, students are most commonly taught to write proofs today in two columns, rather

than in paragraphs or other forms. Whereas writing a two-column proof is not a practice upheld

by mathematicians, statement-reason became the norm “in order for proof to be accessible to all

students” (Herbst, 2002, as cited in Otten, 2009, p. 5). Many students leave high school not

having experienced any other proof types and this impacts their conceptions of what a proof is

and of the constricting “rules” previously discussed.

Researchers and mathematics educators historically have conflicting opinions on what

separates argumentation from proof.

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In mathematics education, in recent years it has become customary to use the term “argumentation” for reasoning which is “not yet” proof and the term “proof” for mathematical proof proper. (Cabassut et al., 2012, p. 171)

What, though, constitutes “mathematical proof proper”? Whereas some teachers will

accept written arguments in the forms of sentences, others might consider this approach

“informal” and demand their students provide a reason for every statement in a two-column

format. Wohlhuter (1996) found that “teachers were not in complete agreement as to what

[proof] content needed to be studied in the secondary geometry classroom” (p. 323), and cited an

example wherein one teacher in the study solely addressed two-column proofs, whereas another

presented multiple proof formats.

A lack of consistent standards leaves teachers and students bewildered when it comes to

determining if they have produced proofs. This discrepancy, coupled with the evolution of

textbooks over time, is discussed in the subsequent sections.

The Psychology of Proof and Proving

In order to improve students’ conceptions of proof and proving, and to determine exactly

why students—and teachers—generally have a poor outlook on the teaching and learning of

proof, it is necessary to understand the psychology behind the concept. The nature and

complexity of proof, paired with its high cognitive demand and the difficulties that teachers have

teaching proof make students’ outlooks about proof all the more difficult to analyze. Varying

psychological factors impact conceptions of proof. This section of the literature review speaks to

the psychology of proof and proving.

Van Hiele Levels

Most mathematics educators are familiar with the work of van Hiele, whose model of

geometric thinking describes van Hiele’s position on how students learn geometry. Vojkuvkova

(2012) described the van Hiele levels: Level 0: Visualization, Level 1: Analysis, Level 2:

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Abstraction, Level 3: Deduction, and Level 4: Rigor (p. 72). “Many studies all over the world

demonstrated that van Hiele theory can help improve geometric understanding” (Vojkuvkova,

2012, p. 75). A critical component of van Hiele’s theory is that of students’ progression through

the levels in sequential order. That is, a student cannot skip to level 3 without first passing

through levels 0, 1, and 2. This poses difficulties for students and teachers—due to teachers’ lack

of awareness or identification of students’ van Hiele levels, they often teach beyond the van

Hiele level that a student is presently at. Battista (2009) cited a statistic that

more than 70 percent of U.S. students begin high school below van Hiele Level 2 even though only students who enter at Level 2 or higher have a good chance of becoming competent with proof by the end of the course. (p. 98)

Otten (2009) also found that most students enter high school geometry at Level 0 or

Level 1 (p. 9). Otten said, “With regard to proof, the van Hiele theory suggests that students need

to be reasoning in the third or fourth level in order to be successful in a deductive geometry

course” (2009, p. 7). Otten further explained that it would be unfair to expect students to be able

to learn how to prove and “consider necessary and sufficient conditions” (2009, p. 8) without

having successfully achieved the objectives of the prior van Hiele levels. Shaughnessy and

Burger (1985, as cited in Otten, 2009) further pointed out that “miscommunication often

occurred because students were reasoning at different levels than the teacher, so perceptions and

the use of language were different” (p. 9).

Mindset

According to Selden (2012), “Students need to learn more than logic in order to become

successful provers” (p. 400). Students need to exhibit grit, persistence, and an understanding of

the goal of proof in order to develop a convincing and correct argument that proves a proposition

or conjecture. Mariotti (2007) supported this viewpoint, suggesting, “The evolution of a

justification in a proof is not expected to be simple and spontaneous” (p. 301). However, once

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students have developed sound reasoning skills and used those skills to learn the art of proof,

they will have acquired skills that can be used in areas of life outside of mathematics: the ability

to see different perspectives of an argument, the ability to persevere through a challenge, and the

ability to reason logically. Siu (2012) summed up these sentiments by commenting that a benefit

of learning how to prove that is often overlooked is “its value in character building” (p. 439).

As mentioned in the previous paragraph, a large part of becoming a successful prover is

developing the patience and persistence to persevere through discomfort in mathematical

problem solving. Harel and Sowder (1998) reported that in one of their teaching experiments,

“many of the students read the problem only once and haphazardly began manipulating the

symbolic expressions involved in the problem, with little or no time spent on comprehending the

problem statement” (p. 251). This behavior likely stems from procedural mathematical

experiences that students have relied on in the past, and speaks to the need for students to

understand what it actually means to engage in the process of proving.

Truth-Seeking

Harel and Sowder (1998) provided an interesting suggestion regarding the teaching and

learning of proof and proving, which highlights an important psychological aspect of proof. They

asserted that when humans think something is true, they seek to prove its truth. On the contrary,

if humans think something is false, they seek information to justify its falsehood. This

description of natural human reasoning contradicts students being told what to prove by a teacher

or a textbook. If students were rather asked to first make a conjecture about whether they think

something is true or false, and then determine a plan to prove their supposition, their attitudes

towards proof and its necessity might improve (Harel & Sowder, 1998, p. 242).

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Once the need for proof becomes apparent to students, teachers can slowly introduce the

rules and conventions of proof which so often turn students off from the beginning. “Students’

judgments of an argument [are] influenced by its appearance in the form of a mathematical

proof—the ritualistic aspects of proof—rather than the correctness of the argument” (Martin &

Harel, 1989, as cited in Harel & Sowder, 1998, p. 246). The need for motivation to prove, as well

as a smooth introduction to the conventions of proof and proving, was supported by Balacheff

(1991, as cited in Reid & Knipping, 2010), who asserted, “In order to teach mathematical proof

successfully, the major problem seems to be that of how to negotiate the acceptance by the

students of new rules” (p. 48).

The mission of proving a conjecture is a search for a priori reasoning (derived from

deduction, rather than observation or experience). According to Weber and Dawkins (2017, as

cited in Weber, 2017), leading students to value this type of reasoning is crucial in helping

students understand proof, and “only then can students appreciate the norms that are in place to

support those values” (Slide 139). Weber (2017) further explained that “if proof is only in place

to increase psychological confidence, then students are rational to prefer non-deductive evidence

such as Geometer’s Sketchpad demonstration” (Slide 139). Since proofs are usually new abstract

experiences for students in high school geometry, it is reasonable for students to prefer a more

concrete demonstrative explanation of a conjecture’s truth or falsity. “Saying that a proof can

always provide psychological certainty more than other types of evidence is unrealistic” (Weber,

2010, as cited in Weber, 2017, Slide 130).

For students, negotiating new symbolic representations as well as unfamiliar proof

structures can seem convoluted and unnecessary when simultaneously trying to understand what

a conjecture is asserting and seeking to determine its truth. This viewpoint was supported by the

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work of Johnson-Laird, who validated the frustrations of students when first learning how to

prove, asserting that “formal logic . . . is not a model for how people make inferences” (Johnson-

Laird, 1975, as cited in Durand-Guerrier et al., 2012b, p. 369). Durand-Guerrier et al. (2012a)

spoke to the balance between exploring and formal proving that must be achieved by students in

order for them to produce a proof that “conforms to specific cultural constraints involving both

logical and communicative norms in the classroom and in the mathematical community” (p.

350).

The sentiment about cultural constraints in the previous paragraph was echoed by

Mariotti (2006), who summarized the role of organization when one seeks to determine the truth

of a conjecture: “Organization becomes functional to understanding, which consequently

becomes strictly tied to the constraints of acceptability and validation shared within a given

community” (p. 176). Mariotti’s (2006) words relate to the psychology of proof and proving

because the “standards of acceptability” (p. 176) she referred to cause much of the performance

anxiety that teachers and students have when it comes to constructing and sharing their

mathematical proofs.

Math Anxiety

Math anxiety is a widely studied phenomenon in mathematics education. While this

literature review does not focus on math anxiety, it would be remiss to not mention the role that

math anxiety can play in students’ and teachers’ psychology of proof and proving. According to

Schoenfeld (1985),

Mathematics anxiety is so well known that it hardly needs comment. Faced with mathematical situations, some people simply freeze; others do whatever they can to avoid situations that threaten to involve the use or discussion of mathematics. (p. 198)

It is important to recognize that many teachers and students fear the process of proving

because it is yet another unknown concept in mathematics. Harel and Sowder (2009) reported

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that many undergraduate students (even mathematics majors) fear proof and “believe they can’t

do it” (p. 283). The point must be raised that many of these mathematics majors go on to become

mathematics teachers. If teachers of geometry fear proof and are unsure of their own capabilities

to write proofs, there are sure to be negative impacts on their teaching of proof, and subsequently

their students’ learning. University-level instructors, then, need be as aware as secondary

mathematics teachers about the struggles students face with proof. Otherwise, they have the

tendency to “proceed without well-articulated, principled guidance for their pedagogy” (Harel &

Sowder, 2009, p. 287).

It is important for teachers at all levels to be aware of their own math anxiety, and be

sensitive to the potential math anxiety of their students. Teachers often unintentionally pass on

their anxiety to their students, and for a topic fraught with restlessness such as proof, it is helpful

to frame it in a way that will set students up for success. “People’s judgments in certain

situations may depend less on the objective reality of the situations than on the way the situation

is framed psychologically” (Schoenfeld, 1985, p. 153).

The Effects of Teacher Content Knowledge on Student Learning and Achievement

Much of the literature supports the notion that teachers’ knowledge of the content is a

leading factor in determining students’ success. It is for this reason that “in recent years,

teachers’ knowledge of the subject matter has attracted increasing attention from policymakers”

(Hill et al., 2005, p. 371). Hill et al. (2005) “found that teachers’ mathematical knowledge for

teaching positively predicted student gains in mathematics achievement” (p. 399). This view has

been supported by Merrill et al.’s (2010) work, which asserted that “[improving] teacher’s [sic]

subject matter knowledge . . . could promote student academic achievement in mathematics and

science” (p. 21). Further, Gorlewski and Gorlewski (1997) asserted that if

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the teacher understands the central concepts, tools of inquiry, and structures of the discipline(s) he or she teaches, [then he or she can create] . . . learning experiences that make these aspects of the discipline accessible and meaningful for learners to assure mastery of the content. (p. 51)

Ly and Malone (2010) validated the aforementioned research, establishing that

effectively learning geometry is the result of the teacher’s quality (p. 367). Maxedon (2003)

agreed, describing teachers’ content knowledge as “of the most important influences on what

students learn” (p. 93). Ng (2011) also supported that students’ progress and skill development is

related to teachers’ content knowledge (p. 152).

Quantifying teachers’ knowledge has been done in varying and contrasting ways. Some

researchers measure knowledge based on courses taken, degrees held, or results of tests.

However, other researchers believe that mathematical knowledge for teaching goes beyond what

teachers know about their college mathematics courses and rather should be judged by teachers’

ability to “understand and use subject-matter knowledge to carry out the tasks of teaching” (Hill

et al., 2005, p. 372). Gorlewski and Gorlewski (2012) supported this notion, describing content

knowledge as a critical factor within a teacher’s control that “connects teacher effort with student

achievement” (p. 51). The authors explained that a teacher’s content knowledge does not only

refer to the ability to do the mathematics required of the students, but rather his or her “deep,

ongoing, passionate engagement” with mathematics (Gorlewski & Gorlewski, 2012, p. 52).

Although teacher certification systems are in place in the United States, the high demand

for mathematics teachers paired with the relatively low supply of candidates has contributed to

the quality of preservice candidates being compromised (Ng, 2011, p. 161). This quality refers

not only to a candidate’s content knowledge, but also his or her ability to create an interactive

learning environment that supports the development of students’ potential (Maxedon, 2003, p.

17). Unal et al. (2009) and Hodge and Frick (2009) have supported Maxedon’s opinion. Hodge

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and Frick found that when the preparation is lacking for a mathematics teacher, it affects his or

her teaching and his or her students’ learning (2009, p. 35), and Unal et al. asserted that “pre-

service teachers’ geometric thinking levels influenced students’ difficulty or insight” (2009, p.

1001).

Based on the prevalent themes in the reviewed literature, it can be soundly concluded that

teacher content knowledge has much to do with teacher effectiveness and student learning. A

multitude of studies has been conducted regarding teachers’ content knowledge of geometry in

particular, and the findings of said studies are presented in the next section of this literature

review.

Findings of Studies Regarding Teachers’ Knowledge of Geometry

This section summarizes the findings of research studies in the areas of teachers’

knowledge of, and preparedness to teach, Euclidean geometry. Both Hill et al. (2005) and Ng

(2011) studied teachers’ preparedness in terms of their mathematical knowledge for teaching.

That is,

the mathematical knowledge used to . . . [explain] terms and concepts to students, [interpret] students’ statements and solutions, [judge] and [correct] textbook treatments of particular topics, . . . and [provide] students with examples of mathematical concepts, algorithms, and proof. (Hill et al., 2005, p. 373)

Hill et al.’s (2005) findings included that U.S. teachers at large lack imperative

knowledge for teaching mathematics effectively and that 12% of teachers reported never having

taken a mathematics content or mathematics teaching methods course (p. 391). Ng’s (2011)

study revealed “teaching requires a specialized form of mathematical content knowledge that is

not intertwined with knowledge of pedagogy, students, curriculum, or other non-content

domains” (p. 152). Ng predicted that “pre-service and in-service primary teachers’ content

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knowledge of geometry is particularly poor” (2011, p. 151). Ng shared an account of a 1997

study by the Ministry of Education and Culture, which revealed that

the mathematics teachers of junior high schools comprehended only 77% of the mathematics content in the curriculum. Some teachers who did not understand the mathematics content prescribed in the curriculum postponed the teaching of difficult concepts until the end of the academic term. Furthermore, due to teachers’ lack of content mastery, teaching progress was often so slow that the postponed concepts were not taught at all. (2011, p. 154)

This “postponement” leading to omitted topics from the curriculum can impede students’

progress in future courses. If teachers had more of a grasp on the content knowledge necessary

for successful teaching, they would be more able to create meaningful learning tasks and engage

students in the curriculum, rather than leave those topics out altogether. Herbst and Brach (2006)

conducted a study that examined the work of a geometry teacher in relation to the students’

learning activities. The results found that “tasks that demand high levels of cognitive activity

from students create tensions on the work of the teacher, who needs to ensure engagement and

sustain those demands” (Herbst & Brach, 2006, p. 115). In high school geometry, where a large

focus of the curriculum is on proofs, the situation of teaching students to write proofs and

supporting their work forces teachers to be accountable for mastery of the content and a deep

level of understanding (Herbst & Brach, 2006, p. 107). Proof-writing is more open-ended and

unpredictable than other topics in the secondary mathematics curriculum, due to the large variety

of methods that students can use to prove geometric propositions. This makes students’

misconceptions less predictable, and therefore puts more emphasis on the necessity of the

classroom teacher to be confident with his or her ability regarding the material to be taught and

learned.

Maxedon (2003) studied elementary teachers’ knowledge of the connections between

learning geometry and child development. She found that “as a whole, teachers do not

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demonstrate knowledge of the mechanisms of intellectual development in geometry because they

did not recognize the accomplishments in geometry that young children bring to school”

(Maxedon, 2003, p. 263). Teacher preparation programs should, therefore, teach students the

implications of human development linked to the geometry curriculum and geometry curriculum

decision-making (Maxedon, 2003, p. 263). Teacher education about the van Hiele Levels can

help teachers at all levels to become more in-tune with their students’ skill-levels and

achievements in geometry.

Sabey (2009) administered a test to determine teachers’ readiness to teach geometric

concepts. She found that although most teacher participants in her study were able to use the

Pythagorean theorem and the distance formula, they exhibited a lower set of skills for problems

involving similarity, types of quadrilaterals, and constructions (Sabey, 2009, p. 109). Hill (2007)

conducted a similar study in which assessments were administered to teachers and teacher

candidates. “The findings [of this study] suggest that for a significant portion of the population, a

great deal of content remains to be learned” (Hill, 2007, p. 110).

According to Browning and Garza-Kling (2009), the notion of teachers’ content

knowledge is alluded to by explaining a fundamental concept that teachers have been found to

struggle with:

When we consider the ways that we classify and define shapes, describe such transformations as rotations, look for rotational symmetry in objects, or discuss how such objects as lines on the plan or in space relate to one another, we are struck with how fundamental the idea of angle is and how widely it is used throughout geometry and later in trigonometry and calculus. The fact that many of our preservice elementary and middle school teachers appear to struggle to articulate what an angle is, or to describe what a degree measure is, is disconcerting. (p. 137)

Clearly, fundamental concepts such as that of “angle” come into play early in a student’s

mathematical career and set the stage for success with many related topics later on. Teachers’

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mastery of the content and comfort with definitions is of utmost importance if students are to

gain sufficient comprehension of the building blocks of geometry.

The aforementioned studies and excerpts have contributed to what is known about

teachers’ content knowledge and preparedness to teach geometry. Furthermore, some of the

studies addressed the beliefs and opinions of teachers about their roles as geometry teachers.

Beliefs and Opinions of Teachers

This section of the literature review discusses the importance of teachers’ recognizance of

their own biases and beliefs about geometry (and more specifically, proof and proving), their

ability to teach it, and their opinions regarding its importance. Further, it elaborates on how

different pedagogical methods have been shown to ameliorate or exacerbate students’ struggles

with geometry.

Teachers’ Biases and Beliefs

According to Parsons (1993), “Teachers develop their beliefs about what mathematics is

from their experiences as students in classrooms” (p. 12). Parsons referred to a 1984 study by

Thompson that investigated the relationship between teachers’ beliefs about the nature of

mathematics and their teaching styles, and concluded that teachers’ beliefs play a significant role

in teachers’ instructional practices. Teachers’ confidence with the material they are teaching is

transferrable to students. A self-efficacious teacher is likely to instill confidence in his or her

students, while a teacher doubting his or her ability to solve geometry problems is likely to shy

away from challenges and pass that trait on to his or her students, as well. Merrill et al. (2010)

examined teachers’ “perceptions of their . . . preparedness” (p. 21) to teach mathematics and

found that it was necessary to provide professional development to “improve geometric and

trigonometric knowledge and skill for mathematics teachers” (p. 20) so that teachers would feel

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more confident. Teachers’ perceptions of their ability and readiness to teach the content have a

positive relationship with student achievement in the classroom.

Geometry can be quite challenging, and proofs can be daunting for students who have

little to no experience writing them and for teachers who have limited experience teaching them.

The open-minded attitudes of teachers and students, therefore, are essential for learning.

The need for an encouraging, positive classroom learning environment is [therefore] regarded as of prime importance by . . . teachers in their efforts to assist their students in improving their achievement in geometry. (Ly & Malone, 2010, p. 367)

Although proofs are high cognitive demand tasks that take getting used to, teachers must

encourage students to push through the difficulty. As students construct proofs, they are learning

to think like mathematicians by making sense of problems and persevering in solving them,

which satisfies one of the Standards for Mathematical Practice set forth by the NCTM. Herbst

and Brach (2006) discussed the value of proofs, asserting, “the activity of proving theorems is

central to the work of mathematicians” (p. 73).

If educators consistently saw the value inherent in the teaching and learning of geometry,

it is quite possible that attitudes towards the subject would drastically increase. According to

Maxedon (2003), the study of geometry is important for life skills development, kinesthetic

development, perceptive abilities, spatial abilities, and logical thinking. “It has, however,

historically been omitted or treated as an optional topic in early childhood mathematics

curricula” (Maxedon, 2003, p. 20). This is due to most teachers’ unease with geometry topics

and their resulting hesitance to try new methods to aid student understanding. Most teachers

possess “a fear of moving out of their comfort zone of teaching traditional mathematics” and

“lack the confidence to use [new research-based methods] in the classroom” (Merrill et al., 2010,

p. 28). The teacher participants in Sears’s (2012) study “expressed a preference towards teaching

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proof using the two-column representation” (p. 176), “[desired] to make mathematics easy” (p.

xix) and promoted memorization of proof procedures rather than conceptual understanding.

Teachers wanted students to be comfortable with mathematical tasks posed. They ensured that their instructional practices aligned with textbook examples and that students were able to complete homework assignments very easily. Additionally, teachers valued the formulas and rules provided on the back of the textbook. The teachers in the study frequently assured students that the tasks posed were easy and accessible. (Sears, 2012, p. 183)

Wohlhuter (1996) found that “teachers’ general comments about whether they enjoyed

geometry [as students] suggested their experiences as students influenced their decision making

as teachers of geometry” (pp. 331–332). Moreover, Wohlhuter found that teachers’ beliefs about

geometry were related to the aspects of geometry that they accentuated in their courses and that

teachers “relied on their textbooks for making decisions” (1996, p. 342). Like Wohlhuter, Sears

(2012) found that teachers’ own experiences as learners played a part in the frequency of their

deviations from their assigned textbooks.

Ng (2011) found that the teachers in his study self-reportedly had the lowest level of

confidence about teaching geometry (p. 151). Some teachers blamed their lack of confidence on

little to no classroom experience. There exists a widespread belief that more years of teaching

experience leads to better quality of teaching. However, Ng’s study showed that “the relationship

between years of teaching experience and teachers’ mathematical knowledge for teaching

geometry score was not linear” (2011, p. 158). In fact, “teachers who had taught for a longer

period of time tended to have lower mathematical knowledge for teaching geometry scores.”

This finding contradicts Hill’s (2007) results that teachers with more experience displayed better

mathematical knowledge for teaching. Therefore, it is not clear as to whether there exists a

relationship between years of teaching experience and confidence in teaching geometry at any

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level. Studying groups of teachers within the same few years of teaching experience can help

shed light on whether a relationship holds in this area.

Maxedon (2003) studied the perceptions of early childhood educators regarding the goals

of geometry in early childhood and what those educators knew about selected geometry topics in

the curriculum. In interviews with the teachers, she found that

all of the teachers themselves had difficulty with terminology and with verbalizing their thinking about geometric content and concepts. In some cases, the teachers were incorrect when responding to direct questions about terminology. . . . Most of the teachers were uncertain about the state’s goals for geometry for young children. (Maxedon, 2003, pp. 243–246)

This complacency on the part of the teachers is demonstrative of a lack of commitment to

improving their teaching of geometry. This could be due to a variety of factors, such as low

confidence, low preparedness, math anxiety, or a lack of interest. It is quite possible, however,

that the teachers were not inclined to give their maximal efforts to mastering the teaching of

geometry because they were not adequately trained in the most effective teaching methods.

Much research exists that reports whether teachers believe in the importance of proof in

students’ learning of mathematics. Weber (2017) reported that teachers think “proof constitutes

an important objective of mathematics education” (Slide 7). This perspective contradicts the

findings of Knuth, as reported by Cabassut et al. (2012), that “although the roles that the teachers

attached to proof in secondary-school mathematics seemed promising, their beliefs about the

centrality of proof were limited” (p. 177). According to Knuth (as cited in Cabassut et al., 2012),

some teachers think that proof should only be a central idea of advanced courses and, more

specifically, for students who plan to enter math-related fields. Furinghetti and Morselli (2009, as

cited in Lin, Yang, Lo, et al., 2012) found that in-service secondary mathematics teachers

believed that Euclidean geometry was “the most suitable domain for the teaching of proof” (p.

337).

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Another factor to investigate is teachers’ beliefs in their students’ mathematical abilities,

and how those beliefs impact their classroom practice. According to Lin, Yang, Lo, et al. (2012),

“Teachers may value the role proofs play for practicing mathematicians, and yet teach proof only

by rote learning because they do not believe their students can create proofs” (p. 337). Due to the

fact that “teachers’ values and beliefs regarding mathematics and the kind of mathematics they

bring to the classroom” (Lin, Yang, Lo, et al., 2012, p. 334) infiltrate their classroom practice,

“continued research on beliefs about proof that focuses not only on detecting beliefs but also on

understanding their origins seems highly necessary” (Cabassut et al., 2012, p. 187).

Teachers’ Acceptance of Varying Proof Methods

In addition to holding beliefs regarding the importance of proof in the school curriculum,

“teachers may also hold beliefs regarding proof methods, which in turn may influence their

evaluation of students’ proofs” (Lin, Yang, Lo, et al., 2012, p. 335). For example, teachers may

or may not accept visual proofs, paragraph proofs, or proofs by contradiction or counterexample.

From classroom to classroom, what is or is not demonstrated and accepted by the teacher can

change drastically. This is demonstrative of how “knowledge and beliefs not only affect practice;

they affect each other” (Lin, Yang, Lo, et al., 2012, p. 336), and in turn, they affect students.

Teachers’ decisions about which methods to use in their classrooms can stem from what is

prescribed by their specific textbook, as well as from their own biases and beliefs about what

students are capable of doing. Teachers who struggled with proofs as students might “believe

that formal proofs are . . . inaccessible and thus inappropriate for students” (Lin, Yang, Lo, et al.,

2012, p. 336). Since “teachers’ knowledge and beliefs rest on their past experiences as teachers

and students . . . it is important to give teachers new experiences to build on” (Lin, Yang, Lo, et

al., 2012, p. 336). These new experiences can take the form of professional development

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sessions, graduate-level methods courses, or collaboration between mathematics education

researchers and mathematics educators.

Differences in Pedagogical Methods

The pedagogical ideas conveyed in Lakatos’s (1963a, 1963b, 1963c, 1964) papers

suggest that he believed in a discovery approach to teaching and learning proofs. Both Lakatos

(1963a, 1963b, 1963c, 1964) and Fischbein (1999) supported the idea that when asking students

to prove geometric propositions, it is important to elicit the information from them rather than

prescribe a set of instructions they cannot consistently rely on due to the diversity of what needs

to be proven. Cirillo and Herbst (2011–2012) referred to Lakatos’s work, and agreed that

“[involving] students in solving problems, conjecturing, writing conditional statements to prove,

and then explaining and verifying their conjectures can provide students with more authentic

opportunities to engage in mathematics” (p. 28). The question arises as to which instructional

methods are most impactful in teaching students how to prove geometric theorems. Moreover,

how preservice and novice teachers are being prepared to teach proof writing must be

considered.

Parsons (1993) conducted a study to

investigate the teaching of geometry by pre-service elementary teachers to determine what relationship exists between a teacher’s beliefs about geometry, the teacher’s geometric knowledge, and the teacher’s lesson crafting. . . . Analysis of the data indicated that a pre-service teacher’s van Hiele level was an important influence on lesson crafting. (p. v)

Teachers asked questions and designed activities at their van Hiele levels, as they were

uncomfortable with higher level questions and tasks. The study also found that many teachers

described their teaching methods as constructivist, but they were actually observed to be more

traditional “chalk and talk.” The results of Parsons’ s (1993) study suggested that teachers’

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knowledge of geometry need improve, and that geometry instruction should take a constructivist

approach.

Teachers will have an easier time implementing constructivist lessons in their classrooms

if they feel more confident with the material. This is due to the fact that in a constructivist

approach, students will be figuring out the material on their own and the teacher may have to

handle unanticipated misconceptions or inquiries.

Some pre-service teachers may question why they need to take the advanced mathematics courses they are required to complete as part of their undergraduate education since they will most likely not be directly teaching any of the content they are learning. However, it is the background knowledge and history the teachers are gaining when they are taking these classes. For instance, the pre-service mathematics teachers’ future students may inquire about why they are learning how to write proofs or where the Pythagorean theorem originated. It is expected that the classes the pre-service mathematics teachers complete as part of their undergraduate education will help them accurately and knowledgably answer such questions. (Hodge & Frick, 2009, p. 31)

Teachers’ comfort with the material they are teaching and their beliefs about their

abilities to teach it have clear impacts on the classroom environment, the delivery and instruction

of the material, and student learning.

Teaching Proof and Proving

“How you teach proof depends on what you mean by ‘proof’ and what you think proofs

are for” (Reid & Knipping, 2010, p. 211). This quotation reiterates the complex nature of proof

and pairs with that complexity the decision-making processes of teachers when teaching students

how to prove. The role of the teacher is critical in students’ learning of proof, because “much of

a student’s achievement in writing geometry proofs is due to factors within the direct control of

the teacher and the curriculum” (Senk, 1989, as cited in Otten, 2009, p. 12). Moreover, due to the

psychology of proof and proving previously discussed, there are unique challenges associated

with teaching these concepts. This section of the literature review discusses both the role of the

teacher and some of said challenges.

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Role of the Teacher

According to Schoenfeld (1985), “Many of the counterproductive behaviors we see in

students are learned as unintended by-products of their mathematics instruction” (p. 374).

Although this statement can be said about any level of mathematics instruction, the complexity

of teaching students how to prove places even greater demands on teachers to clearly

communicate and accomplish the objectives of their instruction. The teacher’s role is to guide

“mathematical argumentation as a zigzag between conjectures and refutations” (McClain, 2009,

p. 222). Hanna (2007) supported this sentiment, asserting that the teacher “must judge when it is

worthwhile insisting on more careful proving to promote the elusive but most important

classroom goal of understanding” (p. 14). The teacher’s role as a facilitator of students’ proofs

comes with distinct challenges.

Challenges in Teaching Proof and Proving

This section of the literature review addresses a set of unique challenges associated with

the teaching of proof and proving. Each identified challenge is supported by quotation(s) from

reports of pre-existing research.

Challenge 1. Teachers have difficulty with determining what can be left out and what

must be included in formal proof-writing, and it is therefore challenging to convey standards,

conventions, and expectations to students. Reid and Knipping (2010) explained that because

proofs written by mathematicians are often semiformal (without including justifications for every

step), “this raises a problem for teachers and students for whom it is difficult to know what steps

are required and what can be omitted, and what assumptions one can make without justification”

(p. 213).

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Challenge 1 was also addressed by Netz (1999, as cited in Reid & Knipping, 2010), who

“discusses the omission of references to theorems and assumptions in classical Greek proofs. He

calls the set of theorems and assumptions that can be used without comment the ‘tool-box’” (p.

213). The uniformity of standards has to be questioned when it is considered that teachers of

unique classroom communities are put in positions to make their own rules regarding what their

students must include in proofs.

Challenge 2. Teachers are faced with the instructional decisions of including and

omitting certain proof methods in their instruction. Students in high school geometry are often

solely taught the statement-reason format. There are pros and cons to this approach. Students

who enter high school geometry underprepared for the rigor and cognitive demand of proof

writing might benefit from two-column proofs as an entry-point. In this vein, “two-column

proofs as a support for teaching developed in the context of classrooms as teachers struggled to

find a way to teach proof that was possible given their circumstances” (Reid & Knipping, 2010,

p. 217). However, according to Reid and Knipping (2010):

Herbst (2002) describes some aspects of teaching two-column proofs that may contribute to students not learning proof well through this approach. He describes how the form of the two column-proof and the teaching practices associated with it serve to make the task of writing a proof easier for the students to do and for the teacher to teach. However, they also lead to the teacher providing significant guidance to the students resulting in a division of labor that could be described by saying the teacher proves and the students write down the proof. (pp. 216–217)

Teachers are seldom observed teaching paragraph proofs, flowchart proofs, or proofs by

contradiction in high school classrooms. This not only limits the students’ exposure to genuine

proof activities, but also limits the teachers’ ability to foster a truly constructivist classroom

environment. The question arises as to which methods, if any, should be included in a high

school geometry course, and whether this should be standardized in the wake of an initiative like

the common core.

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Challenge 3. Teachers are often stumped when deciding what is worth proving and

which proofs are meaningful for their students to engage in. On the one hand, only proving

obvious trivialities makes it likely that students will not come to understand the purpose of

proving; on the other hand, proving all unknown results brings with it a significantly harder

starting point for students who have little-to-no proof-writing experience. Reid and Knipping

(2010) explained that “the decision [on the part of the teacher] as to . . . what to prove must be

made in order to develop students’ understanding of mathematical concepts and ability to apply

them” (p. 215). Further, they asserted that

in cases where the proof brings a new understanding of the concepts involved, the proof is useful. Assumptions that have unexpected implications will need to be made explicit in exploring those implications. Assumptions that would seem obvious and trivial to the students if made explicit can safely be left implicit. (Reid & Knipping, 2010, p. 215)

This speaks to the necessity of teachers being able to distinguish between what their

students already know, and what the students need to explore. This clearly places heavy demands

on the teacher in the areas of sequencing of content, assessing students’ prior knowledge, and

being confident enough with the material to have a flexible mindset regarding what unique

groups of students need proof of, and what can be left alone.

Challenge 4. The teacher of the course must have a sound and confident mastery of the

content. While this necessity applies to any teacher of any subject, those certified in secondary

mathematics education are not preparing to teach in a specific course. For a course like geometry

which contains dense concepts like proof, it is difficult to say exactly what a teacher should

know to be deemed “prepared” to teach the subject. According to Reid and Knipping (2010),

“Research must address the question of what teachers should know about proof in order to

successfully teach it” (p. 222). Browning and Garza-Kling (2009) referred to “deep roots” of

geometry knowledge that students must achieve in order to “reach higher levels of geometric

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understanding” (p. 138). This is widely accepted and supported by the concept of the van Hiele

levels; however, how can students progress through levels of geometric understanding without

teachers who have mastered the levels themselves? Teachers of geometry must be ready to

answer questions from students regarding the different methods that can be used to prove any

given geometric proposition, and be able to come up with questions in reaction to students’

specific lines of reasoning. McClain (2009) referred to this readiness as “a sense of knowing-in-

action” (p. 222) that takes place when a teacher “attempts to capitalize on opportunities that

emerge from students’ arguments” (p. 222). McClain asserted that

the teacher must have a deep understanding of the mathematics under discussion [which] . . . requires decision making-in-action concerning the pace, sequence and trajectory of discussions in order to ensure that topics under discussion move the mathematical agenda forward. (2009, p. 222)

Fernandez (2005, as cited in Cabassut et al., 2012) supported this point, asserting that

“ideas that surprise and challenge teachers are likely to emerge during instruction” (p. 178). In

these situations, teachers need to be able to reason on-the-spot, which requires confidence in

their content knowledge. This points to the lack of predictability in geometry teaching and

emphasizes the sizeable demands on the geometry classroom teacher.

Moreover, geometric understanding is heavily rooted in definitions. “Mathematical

definitions are very concise, contain technical terms, and require an immediate synthesis into a

sound concept image” (de Villiers et al., 2009, p. 189). Definitions pose challenges to both

students and teachers. In geometry, the definitions are almost all new and have not been studied

in the previous courses such as pre-algebra and algebra. In order to help students develop correct

concept meanings, teachers must be well versed in geometric definitions and not only commit

them to memory, but also understand the special cases or exclusions inherent in geometric

definitions.

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Challenge 5. The geometry classroom teacher is tasked with the difficult role of helping

students understand the complex nature of proof. According to Cabassut et al. (2012), “There

exist no easy explanations of what proof and proving are that teachers could provide to their

pupils” (p. 170). Teachers of geometry need to determine how to explain to their students what

proof is, why it is needed, and what constitutes mathematical proof. In the common case of

students who think that demonstration via example is proof of a concept, the teacher faces the

challenge of adjusting the students’ beliefs in a sensical and sequential manner. “Teachers need

to help . . . students understand that . . . an examination of cases does not constitute evidence for

the truth of a . . . claim; that is, proof” (Lin, Yang, Lee, et al., 2012, p. 314).

Cirillo (2008) conducted a longitudinal, three-year case study which examined “how

[one] new teacher, [using a conventional textbook], learns to teach proof to high school geometry

students” (p. xi). The subject of Cirillo’s study voiced that creating “a plan for proof cannot be

systematized in the same way that much of arithmetic and algebra can. Therefore, there is a

major shift in the expectations of . . . students when they are asked to do a proof” (2008, p. 101).

This finding is supported by much of the literature on proof and proving and circles back to the

challenge of helping students come to understand the nature of proof.

Challenge 6. Teachers of geometry must know, and be able to teach their students, how

to read geometrically:

Reading geometrically involves connecting objects with real-world contexts, recognizing if the diagram is to be taken as drawn or as an abstract representation of [objects], interpreting with deduction, mentally redrawing, and interpreting conventional markings. . . . [The question is raised] of how mathematics educators can help students understand how to read diagrams and navigate the transitions between the different expected ways of reading. (Dieteker et al., 2017, p. 378)

Battista (2009) also addressed the challenge of reading geometrically, specifically with a

focus on diagrams:

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When teachers use diagrams to represent formal concepts, students reason about the diagrams, not the formal concepts. Consequently, students often attribute irrelevant characteristics of a diagram to the geometric concept it is intended to represent. For instance, students might not recognize right triangles in nonstandard orientations because they have unintentionally abstracted a particular orientation as an attribute of such triangles. (p. 97)

It is difficult to teach students to “be careful” with diagrams. Diagrams are a commonly

used tool in geometry and can be useful in representing given information, but it is important to

not unknowingly associate concepts that seem to appear in a diagram with a geometric concept.

The appearances of diagrams in textbooks can also be misleading to students and cause them to

associate incorrect characteristics with a geometric concept. This difficulty, and the role of

textbooks in general, is discussed in the next section.

The Role of Textbooks in the Teaching and Learning of Proof and Proving

The textbook that a teacher uses, or is assigned to use, has the potential to have a

tremendous impact on the teacher’s practice and, consequently, the students’ learning. This

section of the literature review discusses the importance of textbooks, common criticisms about

the exposition of proof in geometry textbooks, and quotations from published research on the

frequency of proof tasks in geometry textbooks.

The Importance of the Textbook

When teachers are assigned to teach geometry, their first inquiries are often about which

textbook their students will be using. “Mathematics textbooks play a critical role for the

construction of mathematical knowledge through the ordering, presentation and explanation of

mathematical concepts and problems and by providing solutions to those problems” (O’Halloran,

2017, p. 25).

“[The textbook] is an educative technology, which the teacher adopts and manipulates

with the objective of mediating the students learning” (Nasser & Aguilar Júnior, 2017, p. 112).

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For a preservice or novice teacher, the textbook often provides a source of comfort because it is

something seemingly concrete to rely on in a time otherwise fraught with uncertainty. “[The

textbook] acts as the main support material for the teacher in the preparation of the lessons.

Textbooks can help the teachers, since they act as guides for . . . educational practice” (Nasser &

Aguilar Júnior, 2017, p. 108). How a teacher uses a textbook, however, is of critical importance

to students’ learning, as not all textbooks adequately address the standards to be taught or present

concepts in ways that are understandable to students. “Textbooks are ‘a vehicle for learning

mathematics’, where ‘the only other vehicle of comparable importance is the teacher’” (Usiskin,

2013, as cited in O’Halloran, 2017, p. 25). Just as all textbooks are different, all teachers are

different, and therefore “the different features of [a] textbook ‘may act as constraints and

affordances in different situations and for different teachers’” (Lloyd, 2008, as cited in Cirillo,

2008, p. 112). Sears (2012) asserted that quality proof tasks in geometry textbooks are scarce,

and “when enacting these tasks teachers primarily complete the proofs for students during whole

class instruction” (p. 206). In order for instruction based on a course textbook to be effective, a

balance between what is provided in the textbook and the teacher’s adaptations due to

pedagogical decision-making must be achieved.

Truelove (2004) conducted a study that aimed to “clarify the nature of the beliefs of

secondary level geometry teachers regarding proof and explore some of the factors that may

affect teachers’ beliefs” (p. 5). One of Truelove’s research subquestions was, “Are there

differences in geometry teachers’ conceptions of proofs with regard to approach of textbook?”

(2004, pp. 5–6). Truelove found that teachers “believed in the use of a variety of instructional

activities in proof instruction [and that the type of textbook used] . . . proved to be statistically

significant in teacher opinions” (2004, p. 73). Specifically, teachers who were assigned to use

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deductive texts were more likely to implement a variety of proof tasks, since deductive texts

“incorporate a more central focus on proof activities than what is . . . found in other types of

geometry texts” (Truelove, 2004, p. 73).

Despite what their publishers may boast, not all textbooks are actually designed to foster

the Standards for Mathematical Practice set forth by NCTM. According to Dietiker and Richman

(2017), some textbooks “add only superficial features that appear to be consistent with reform

[and others] . . . are structured throughout to provide students with the experiences that reforms

are designed to promote” (p. 197). Nissen (2000) analyzed textbooks to examine their exposition

of transformation geometry. He found that the textbooks studied were not in compliance with the

2000 Standards, although they claimed to be. Nissen stated that there would be “a need for a

radical rewriting of any textbook surveyed in [his] study” (2000, p. 228) in order for them to

actually represent the standards. Sears (2012) agreed with this sentiment and suggested that

[textbook] developers should revise the tasks they include in their textbooks so that students have more opportunities to create original proofs and experience tasks of a higher-level of cognitive demand. [Moreover,] . . . textbooks need to reduce the excessive amounts of proof tasks which only require students to fill in blanks or identify missing links. (p. 205)

Teachers need to be aware of whether their assigned textbook presents and develops

content in accordance with the standards, in order to help them “make the design and

pedagogical decisions that shape [their] daily instruction” (Dietiker & Richman, 2017, p. 197).

Common Criticisms of the Exposition of Proof in Geometry Textbooks

Although publishers promote grandiose images of their textbooks as all-inclusive

resources which align perfectly to the standards and can be used without adaptation, in reality,

mathematics educators often experience frustrations with the content or exposition of a textbook.

Previous research on high school geometry textbooks has surfaced important criticisms.

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First, textbooks are criticized for the lack of rigorous proof-based tasks that will help

students develop their mathematical acumen. According to Nasser and Aguilar Júnior (2017),

“Argumentation and mathematical proof [are] . . . goals of education and learning, [but the

textbooks do not come] . . . constructed with activities that stimulate this type of work” (p. 112).

Battista (2009) supported this viewpoint, stating that “most U.S. geometry [textbooks] consist of

a hodgepodge of superficially covered concepts with no systematic support for students’

progression to higher levels of geometric thinking” (p. 98). Meinerz and Doering (2017) also

agreed and commented on a lack of support in teachers’ manuals, stating,

There are no proposals to encourage students in the classroom to develop their own justifications and deductions. . . . We need textbooks that present proposals that involve students and encourage them to conjecture, to justify their conjectures, and to deduce them. (p. 104)

Second, textbooks often imply the “common, but false conception that only one correct

definition exists for each defined object in mathematics. . . . For example, . . . too often textbooks

give the impression that a rectangle can and must be defined only one way” (de Villiers et al.,

2009, pp. 190–191). This can lead students and teachers alike to overgeneralize and make

erroneous conclusions, while negatively impacting their concept images. Cannon and Krajcevski

(2017) supported this sentiment about definitions in their analysis of standard and nonstandard

examples. They cited Cunningham and Roberts’s (2010) idea that “geometry textbooks often

present only the most prototypical shapes and lack a sufficient number of non-standard

examples” (Cannon & Krajcevski, 2017, p. 181). They also provided the example of how

textbooks usually draw altitudes of acute triangles, after which students will be perplexed with

how to draw the altitude of an obtuse triangle, asserting that “if a visual representation of a

certain object is always presented in the same way, . . . this may lead students to form inadequate

concept images of [the] object” (Cannon & Krajcevski, 2017, p. 181). Moreover, after analyzing

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a textbook, Cannon and Krajcevski found that “most of the images of polygons were drawn in

such a way that at least one side of the polygon was aligned with the text, with no content-based

reason for this alignment” (2017, p. 183). This finding is concerning because it can affect the

concept image students have and lead students to believe that polygons without a vertical or

horizontal side cannot be drawn. On a similar note, just as the aforementioned researchers found

that textbooks did not present multiple definitions of a term and more than prototypical

examples, Truelove (2004) found that high school geometry textbooks did not sufficiently

address different types of proofs. Most secondary level geometry textbooks pay

little attention . . . to indirect proofs. At most, indirect proofs are briefly mentioned at the end. . . . With limited opportunity to . . . explore indirect proof skills, students may not develop the art of logical argumentation. (Truelove, 2004, p. 21)

Third, the “unique problems which occur in the teaching and learning of mathematics are

exemplified in mathematics textbooks” (O’Halloran, 2017, p. 241). The mathematical writing

and symbolism are too difficult to understand and seem convoluted in the eyes of students.

Meinerz and Doering (2017) similarly found that many books present formulas without any

justification for those formulas, and those that do “prove” formulas often do so through

“incomplete arguments . . . and language beyond the reach of the students” (p. 103).

Fourth, the proof tasks in geometry textbooks can add to students’ misunderstandings

about the nature of proof. Otten et al. (2011) conducted a textbook analysis and found that “the

majority of reasoning-and-proving exercises in geometry textbooks are around particular, not

general, mathematical statements. In exposition sections, on the other hand, the majority of

mathematical statements are general in nature” (p. 347). The researchers explained that this

discrepancy can cause geometry students to “believe that proof is merely an application of

recently learned theorems” because they are “proving things about contrived, particular

situations” (Otten et al., 2011, p. 353). Dreyfus et al. (2012) commented on the directions

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presented for different tasks: “Sometimes students do not know what they should produce when

asked to ‘explain’, ‘demonstrate’, ‘show’, ‘justify’ or ‘prove.’ . . . Textbooks [do] not make this

clear and . . . [leave] students confused” (p. 201). This can also add to students not understanding

that a proof would be a sufficient response to a task in which they are prompted to “explain,”

“demonstrate,” “show,” or “justify.” This sentiment is far from new. In Proofs and Refutations,

Lakatos

noted that mathematics grows not in the order presented in textbooks but rather through a recurring cycle in which relationships are discovered, proofs are proposed and refuted, and then relationships are ultimately proved by refining appropriate arguments and definitions. (Clements, 2003, as cited in Blair & Canada, 2009, p. 288)

The exposition of proofs in geometry textbooks, therefore, do little to help students come

to an organic understanding of what proof is and why it is needed. An additional criticism related

to the way textbooks communicate the nature of proof, highlighted by Reid and Knipping (2010),

is the ineffectiveness of the foundational approach in some textbooks:

One approach sometimes taken is to inform the students that they must forget everything they already know and start from the axioms and definitions given in the textbook. But experience tells us that the students do not forget everything they know, and the axioms and definitions given in the textbook are rarely (never) complete themselves. The result is that the class pretends to base its arguments on the given axioms and definitions, while being guided by their prior knowledge and . . . observing what is omitted in the teacher’s and textbook’s proofs. (p. 214)

These collective criticisms reveal that little has been done to sufficiently improve the

exposition of proof in secondary geometry textbooks since this issue was first raised by Fawcett

(1938) in the 13th NCTM Yearbook, when he cited Young’s 1925 work:

Our texts in this subject are still patterned more or less closely after the model of Euclid, who wrote over two thousand years ago, and whose text, moreover, was not intended for the use of boys and girls, but for mature men. (p. 2)

Fawcett (1938) pointed out that “teachers of mathematics recognized the advisability of

so modifying the subject matter as to make it more palatable to less mature pupils in the

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secondary schools” (p. 2); however, Fawcett lamented that the “solution” to this problem became

to remove many theorems from texts. This sentiment was more recently supported by Davis

(2009), who asserted “still in use in classrooms across the United States are many textbooks that

depict geometry as a collection of theorems, postulates, and axioms developed by unknown

persons for unknown reasons to be passively consumed by the student” (p. 335). It is evident that

the issues with proof in geometry textbooks of the 20th century are still present in those of the

21st.

The Frequency of Proof Tasks in Geometry Textbooks

Studies have been conducted with the intention of quantifying the proof tasks in

geometry textbooks. Otten et al. (2011) examined “six geometry textbooks” for “the

justifications given and the reasoning-and-proving activities expected of students [along

with] . . . the nature of the mathematical statements around which . . . proving takes place” (p.

347). They found very few activities which made reasoning and proving “an explicit object of

reflection” (Otten et al., 2011, p. 347).

Of the 12,468 coded exercises, only 67 asked students about the reasoning-and-proving process (as opposed to asking them to engage in that process). . . . Opportunities are rare . . . [to] reflect on the core mathematical practice of reasoning-and-proving. . . . Furthermore, even in geometry, the traditional home of reasoning-and-proving, students were asked to develop a mathematical proof in less than 7% of the textbook exercises. (Otten et al., 2011, pp. 352–252)

Supporting Otten et al. (2011), Sears (2012) concluded that

more proof tasks should be included in geometry textbooks to facilitate students learning to prove. Of . . . three chapters examined, [only] 7.41% of the tasks in Prentice Hall Geometry and 13.1% of the tasks in McDougal Littell Geometry were proof tasks. (pp. 199–200)

Nasser and Aguilar Júnior’s (2017) later findings supported the work of these

researchers, reporting that previous research found an overall “absence of activities exploring the

inquiry process, essential and basic in the construction of . . . mathematical knowledge” (p. 112).

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It is clear that mathematics education reform has not sufficiently addressed the content,

sequencing, or exposition of proof and proving in secondary geometry textbooks. Considering

the major role that textbooks play in the teaching and learning of proof, this is of grave concern

to mathematics educators and has provided the impetus for numerous studies to date.

Concluding Remarks

Case studies, descriptive research, dissertations, literature reviews, and conference

proceedings regarding the teaching and learning of geometric proof support the conclusions that

• teachers will be better equipped to teach proof if their own understandings of the

material improve,

• teachers’ beliefs about the importance and usefulness of proof impact their teaching

methods,

• a teacher’s perception of his or her own ability to teach geometry impacts his or her

classroom presence and teaching style, and

• student learning and achievement in geometry in the United States have the potential

to improve, but further research is needed.

Mathematics educators should develop a clearer understanding of the views of practicing geometry teachers [so that] . . . strategies may be developed and implemented to improve current instructional practices and educational opportunities that meet the needs of in-service geometry teachers. Mathematics educators may gain insight into the areas that need improvement in the area of teacher preparation, which should be beneficial to future teachers and students. (Truelove, 2004, p. 70)

Selden (2012) suggested studying “how secondary pre-service teachers can acquire the

abilities necessary to effectively teach . . . proof and proving to their future pupils” (p. 414).

Sears (2012) addressed this point, asserting that “[providing] opportunities for teachers to write

proofs and to examine examples of [proof] . . . may help [them] to reflect on students’

conceptions and misconceptions” (p. 209).

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After an extensive review of the related literature, it seems that although a large amount

of research has been conducted regarding elementary mathematics teachers’ and veteran

secondary mathematics teachers’ preparation to teach geometry, there is a scarcity of research

regarding the same issues for novice secondary mathematics teachers. According to Hodge and

Frick (2009), “There is little research and information about what preparation helps a teacher

succeed at effectively teaching students in the geometry classroom” (p. 34). The study of teacher

learning and understanding of teaching geometry . . . needs to be studied in much greater

depth . . . [through] comparative studies, qualitative studies, and quantitative studies (Hodge &

Frick, 2009, p. 35). Additionally, much research seems to exist which “focusses on students’

understandings of proof and ability to prove” (Reid & Knipping, 2010, p. 223), but there is a lack

of research on teachers’ ability to prove. Preservice and novice mathematics teachers’ conceptual

understanding of proof, their ability to write proofs, their beliefs about proofs, and the textbooks

they use should be examined further in order to identify next steps in improving students’

success in proof and proving.

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Chapter III: Methodology

Research Design

A two-part qualitative study was conducted to answer the research questions, What kinds

of proof tasks and proofs are offered in a typical high school geometry textbook? and How

prepared and how confident are preservice and novice secondary mathematics teachers to teach

proofs? First, a qualitative analysis of a common core textbook ([Textbook Author], 2016) was

conducted. I recorded each instance of proof in the textbook and wrote commentary regarding

what was required of students in each proof exercise and each proof-related exercise. Second, I

interviewed 29 preservice or novice secondary mathematics teachers while administering to them

a self-created five-question content assessment. The questions on the content assessment were

chosen and adapted from New York State Regents Exams (The University of the State of New

York, 2015, 2016) as well as eMATHinstruction (2017, 2018a, 2018b) online curriculum

resources.

I asked interview questions while administering the content assessment in order to gain

an understanding of the participants’ knowledge and beliefs about geometry, and, more

specifically, the teaching and learning of geometric proofs. I audio-recorded the administration

of the content assessment for each participant and asked the participants to think out loud while

they were working on the problems. I recorded notes during the administration of each

participant’s content assessment regarding the participant’s knowledge, beliefs, and

mathematical (mis)understandings. The data were then collectively analyzed and the findings

were related back to the literature. This chapter describes the participants, research context, data

sources, instruments and procedures, and data collection and analysis methods.

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Role of the Researcher

I am a White female in my early 30s with eight years of high school and college

mathematics teaching experience. When I was a novice high school teacher, I began teaching a

graduate-level course in Euclidean geometry, mainly for preservice and novice secondary

mathematics teachers. I realized, from the beginning of my experience teaching this course, that

the students in the course overall demonstrated negative feelings about geometry, often citing

their difficulties with writing geometric proofs as high school students as the reason for these

feelings. Many of the students transparently stated their fear of having to teach geometry in the

future because they dislike proofs and believe they do not know how to teach students how to

prove. I developed teaching strategies specific to helping students learn how to write a proof by

identifying, and working backwards from, a goal.

Selection of Participants

At the time of the study, the participants were all initially certified, professionally

certified, or soon-to-be certified preservice or novice secondary mathematics teachers in New

York State. I identified potential participants by reaching out to my personal network of

mathematics education colleagues and their students, inviting them to participate in the study.

Participants were invited to participate using the Institutional Review Board approved consent

form (see Appendix A). The most inexperienced teachers in the study were preservice teachers

who had completed, or were completing, their initial certification in secondary mathematics

education at the time of the study. The most experienced teacher had years of teaching

experience but was newly certified in mathematics and was in her second year of teaching

secondary mathematics. Some of the novice teachers had limited teaching experience but were in

their first or second year of teaching high school geometry.

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All participants had completed, or were in the process of completing, a traditional teacher

preparation program at a college or university in New York State. The majority of the preservice

and novice teacher participants were employed or student teaching in public schools in New

York City, although some worked on Long Island and/or in private schools. A small handful of

participants was student teaching at the time of data collection, but had completed all other

requirements leading to their teaching certifications.

I reached out to potential participants via email and administered consent forms to the

first 30 persons that indicated they would participate in the study. However, upon trying to

schedule an appointment with the 30th participant, the schedule conflicts far exceeded the data

collection time. I therefore settled on 29 participants.

Data Collection and Analysis

Confidentiality

To ensure participants’ confidentiality, the participants were each given a number rather

than being referred to by name. The list of numbers and corresponding names was only viewable

to me. The participants’ undergraduate and/or graduate universities as well as the schools where

they teach or student taught were not disclosed or included. I reminded all participants to not

provide any identifying information about themselves during the audio-recorded administrations

of the content assessments. Furthermore, the audio-recordings and scanned copies of assessments

were kept on a password protected computer that only I had access to. To ensure that the ethical

standards were maintained, I stored the signed consent forms and content assessments in a

locked cabinet in my apartment to which only I had access.

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Data Collection

The data collection took place in January and February of 2019. First, I conducted a

qualitative analysis of a New York State common core-aligned geometry textbook ([Textbook

Author], 2016). The textbook was chosen based on availability and popularity of the publisher.

Then, I used the data from the textbook analysis to inform the creation of the content assessment

that was administered to participants. Each participant was given a number that was kept on a

list, visible only to me, for the duration of the study and throughout all data collection and

analysis.

Initially, I planned to administer the content assessment to each participant and then

separately interview each participant about their thoughts and feelings on teaching geometry (the

initially planned interview questions are included in Appendix C). Then, it came to my attention

that I might need to interview only a subset of participants whose responses to the content

assessment were particularly illuminating for any reason. Finally, upon beginning the

administration of the content assessment to the participants, it became apparent that the

conversations were providing valuable information about the participants’ thoughts, attitudes,

and beliefs that were not otherwise evident on their content assessment papers (they also wrote

down their answers). Therefore, I merged the content assessment and interviews, audiotaped the

administration of every participant’s content assessment, and coded for beliefs and attitudes in

addition to the mathematical accuracy, misconceptions, and mathematical language instances in

their responses.

Due to this evolving methodology, the first four participants’ interviews were recorded in

separate audio-recordings from their content assessments, but only one recording was kept for

each of Participants 5 to 29, which included both their mathematical content thinking as well as

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questions from me about their thoughts, beliefs, and attitudes. Content assessment data are

provided in Appendix D. In conducting the content assessment interviews, I asked questions

stemming from the four categories of Schoenfeld’s (1985) approach to interviewing and thinking

about mathematical behavior.

Data Collection Instruments

There were three data collection instruments. The first was a New York State common

core-aligned geometry textbook ([Textbook Author], 2016), chosen by me. The second was a

content assessment, designed by me with tasks chosen and adapted from New York State

common core-aligned geometry resources (eMATHinstruction, 2017, 2018a, 2018b; The

University of the State of New York, 2015, 2016). The third were the interviews (audiotapes of

the discussions that I had with the participants as they took the content assessment).

The Textbook. The first data collection instrument was a textbook I chose. Because all

the participants were New York State teachers, I chose a New York State textbook for common

core geometry. I have anonymized the title and author of the textbook to avoid the possibility of

the analysis being considered an indictment on it. The author asserted that the textbook was

designed so that “students will use inductive reasoning to make conjectures and learn the

meaning and nature of mathematical proofs . . . [and that] the eight Mathematical Practice

Standards are embedded throughout [the text]” ([Textbook Author], 2016, p. 1). The New York

State Regents Examination, which is the typical culminating exam for students enrolled in

common core geometry in New York State, is addressed in this book (several New York State

Regents Exams are printed in the back of the book for practice). Considering the textbook’s

stated goals and stated alignment to the Regents Exams, I believed that it was a good choice for

this study. The methods used for analyzing the textbook are discussed in the next section.

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After a thorough examination of the chosen textbook, I used the data to inform the design

of the content assessments. I surveyed all available problems from two common core aligned

geometry curricular resources and chose (and adapted) tasks that spanned a variety of difficulty

levels and invoked topics of concern, issues of language, and common misconceptions as found

from my experiences and as highlighted in the literature.

The Content Assessments. When choosing questions for the content assessments, I

relied on Schoenfeld’s (1985) framework for the analysis of mathematical behavior to inform my

choices. Schoenfeld’s (1985) framework examines “what people know, and what people do, as

they work on problems with substantial mathematical content” (p. 11). Schoenfeld (1985) broke

mathematical behavior into four categories—resources, heuristics, control, and belief systems—

and asserted that “the four categories . . . must be dealt with, if one wishes to ‘explain’ human

problem-solving behavior” (p. 12). Because I aimed to determine information about how the

participants’ content knowledge, proof strategies, and beliefs and attitudes all impacted their

performance and feelings, and because proof-writing is certainly a subset of problem solving,

Schoenfeld’s ideas provided the impetus to choose problems that would bring these factors to

light, and ask accompanying questions which would illuminate the focus areas of the research

questions (to be discussed in the next section). The items for the content assessment were

selected after I conducted the analysis of the common core geometry textbook.

I wanted the assessment to take no longer than 45 minutes for participants to complete (in

order to make the administration feasible and secure participants), so I decided on five problems.

The problems were chosen from two common core-aligned resources: the New York State

geometry Regents Exams (The University of the State of New York, 2015, 2016), as well as the

eMATHinstruction common core geometry student workbook (eMATHinstruction, 2017, 2018a,

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2018b). Initially, I was going to choose five proofs. However, after the textbook analysis I

decided on four proofs and one introductory problem, the development of which is explained

below.

Question 1. Question 1 (“Introductory Problem”) comes from eMATHinstruction (2017)

and appears in Figure 1.

Figure 1

Question 1: Introductory Problem

After performing the textbook analysis and seeing the number of exercises in the

common core textbook that asked students to “explain” or “justify” without asking them to

“prove,” I decided to include one such question (Question 1) on the content assessment. I was

curious to see whether the participants would inquire whether the wording of the question meant

that they had to write a formal proof, since I believed that the directions for “explain” and

“justify” were ambiguous when presented as such in the textbook. Based on the related literature,

I knew that this ambiguity of language had been found to cause issues in previous studies about

proof and proving (e.g., Otten et al., 2011), and was the topic of many well-known articles and

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studies about justification and reasoning in the secondary mathematics classroom. I chose an

explain task about transformations (specifically, rotations) for two reasons. First, congruence

mappings are an important topic in common core geometry. Second, the connection between

rotations and the triangle in the diagram being isosceles is not obvious.

Question 2. For Question 2 (Proof 1), students were given two options and could choose

which one to complete. Option 1 came from a common core geometry Regents Examination

(The University of the State of New York, 2016), and Option 2 was from eMATHinstruction

(2018b). Options 1 and 2 are shown in Figures 2 and 3, respectively.

Figure 2

Question 2: Proof 1, Option 1

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Figure 3

Question 2: Proof 1, Option 2

In designing Question 2, I decided to give a choice to participants. Option 1 was a fill-in-

the-blank proof task, which was one type of proof task that frequently appeared in the textbook

that was analyzed. A fill-in-the-blank proof task refers to a proof task in which a proof is

partially presented in statement/reason form and several of the statements and/or reasons are

omitted. The directions for Option 2 did not specify which format participants were supposed to

follow. I decided to give a choice so that I could analyze the number of participants who chose

each task, and ask them their reason(s) for their choice. Moreover, I planned to record instances

of participants changing their minds after beginning one of the options, or those who wanted to

do both. The goal of this was to determine whether the presentation of the task had an impact on

its psychology and impacted the participants’ choices (and therefore shed light on teachers’

reasons for providing fill-in-the-blank proof tasks or students’ preferences/ aversions to them).

These two options were paired in the same question because they are both interlinked with the

same mathematical fact. In Option 1, participants are tasked with proving that the sum of the

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measures of a triangle’s interior angles is 180 degrees, whereas in Option 2 the participants

needed to use that fact in order to utilize the exterior angle theorem for triangles.

The fill-in-the-blank proof in Option 1 was chosen for several reasons. First, the fact that

is being proven through this exercise is entry-level, and I was confident that all participants

would already know the sum of the measures of the interior angles of a triangle. Moreover,

although the task presents as easy because some of it is done for the student, and it is referencing

an already-known fact, I hypothesized that the participants might have trouble filling in reasons

(2) and (5) because their corresponding statements seem obvious. I supposed that the participants

might overcomplicate the problem. Last, I was interested in seeing how many of the mathematics

teacher participants were not already familiar with a proof of the triangle sum theorem, despite

having learned it in their own studies and having relied on it since it was first learned (how many

of them accepted it without proof, or did not remember its proof, or had not considered how to

prove it as teachers).

Another reason for my choice of Option 2, and for pairing it with Option 1 in the same

question, was to determine participants’ feelings about proof types (statement/reason versus

paragraph proofs). I was interested to see whether any of the participants who chose Option 2

would ask whether they had to use a statement/reason chart, and whether they would default to

writing a paragraph proof (since the chart was not provided, as it was for Option 1), or if they

would simply draw in the statement/reason chart for Option 2 because that is what they are most

used to or comfortable with. These decisions were not shared with the participants.

Question 3. Question 3 (Proof 2) comes from eMATHinstruction (2018a) and appears in

Figure 4.

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Figure 4

Question 3: Proof 2

I chose this proof task for several reasons. First, participants could connect it to the

introductory problem and/or Option 2 of Proof 1 (which both involved isosceles triangles).

Second, I hypothesized that participants would perhaps struggle with the overlapping triangles,

which are commonly included in proof tasks for high school geometry students. I was curious to

see how many participants drew the overlapping triangles separately, and how many mistakenly

assumed that angles AEC, BED, and BEC were all the same angle when drawing triangles AEC

and BED separately, without considering the different rays making up each of these angles.

I intentionally used a statement/reason chart with a short vertical bar down the middle for

two reasons. First, participants could feel free to omit the chart and write any type of proof they

wanted underneath the chart. Second, I did not want to give any hints about the length of the

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proof based on the length of the chart, which was to be completely filled in by the participant. I

did not choose a task in which the givens were explicitly stated as such, but planned to pay close

attention to how the participants utilized the givens and whether they realized the need to invoke

the fact that corresponding parts of congruent triangles are congruent (CPCTC) right away, or

only after analyzing the given information and its consequences. Last, I was curious to see how

many participants knew how to correctly state CPCTC, and this made way for discussion with

participants about their knowledge of congruence criteria, another major topic in high school

common core geometry.

Question 4. Question 4 (Proof 3) comes from a common core geometry Regents Exam

(The University of the State of New York, 2015) and appears in Figure 5.

Figure 5

Question 4: Proof 3

Question 4 (Proof 3) was chosen for several reasons. First, the wording of the question

was appealing to me because the “given” that ABCD is a parallelogram is not clearly stated as

such, and I hypothesized that participants might miss this piece of information. Second, the

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distinguishing characteristics among the quadrilaterals in the quadrilateral family are largely

unknown and stressful to both teachers and students of geometry, and I supposed that

participants might feel stuck because they would not remember the distinguishing features of a

rhombus (or know that they needed to identify said features). Last, the statement of the problem

included notation that I wanted to assess whether the participants could interpret. This problem

was strategically placed after Question 3 (Proof 2) because successful completion of this task

also necessitates participants’ ability to work with overlapping triangles (in this case, triangles

BCE and DCF). I thought that Proof 2 provided a nice transition to Proof 3, since the subset of

skills needed for Proof 2 is contained within those of Proof 3. I planned to ask participants who

got stuck on this proof task to try to verbalize what would help them in completing it (to put

themselves in the mind of a student, and determine a tool, scaffolding, or intervention that might

help move them forward).

Question 5. Question 5 (Proof 4) was adapted from an eMATHinstruction (2018b) task,

and appears in Figure 6.

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Figure 6

Question 5: Proof 4

I chose Question 5 (Proof 4) for several reasons. First, overlapping triangles are still

involved which interweaves the content of the preceding proofs; however, in this proof, the

participants need to utilize similarity of triangles versus congruence. Second, this is an important

theorem in Euclidean geometry which is often memorized without meaning, although the proof

is rather uncomplicated and approachable to most levels of students. Third, I supposed that the

participants might struggle with the vocabulary of the theorem statement, even though it was

demonstrated in the line below using the letters on the diagram. Last, I knew from experience

that the many facts and theorems associated with circles and segment lengths inside and outside

circles caused stress for the majority of preservice teachers in my charge, and was curious to gain

insight as to why. To make the problem as accessible as possible, I did not dictate that the

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participants had to use a statement/reason chart, or write a paragraph proof—they had the

autonomy to choose their approach. I ended with this question because I thought it would be

perceived as the hardest problem on the content assessment for the majority of participants.

By designing a content assessment which included not only different types of proofs but

also different standards from common core high school geometry, I hoped to acquire a wealth of

information about the participants’ understanding, mathematical issues, feelings, beliefs, and

goals related to proof and proving.

The interviews. I met with each participant individually and audio-recorded the

administration of the content assessments. Participants were asked to read the problems aloud

and think out loud as much as possible. I first gave participants the chance to solve each problem

on their own first, but asked them to clarify their thinking if it was not made visible to me. When

they were stuck, participants were asked to identify what was troubling them about specific

problems. Often, participants expressed their feelings candidly about the problems and shared

their reflections of their own learning as high school and/or college geometry students. At these

instances, I paused the administration of the content assessment to probe the participants for

more information regarding to Schoenfeld’s (1985) suggested areas for analyzing mathematical

behavior, as this information is directly related to the participants’ beliefs and attitudes and

would provide helpful data for answering research Question 2.

Two of the 29 participants were audio-recorded together because they shared with me

that they had mutually discovered their roles in the study as they happened to be married. I gave

all participants the choice of where to conduct the study, and the two participants under

discussion preferred that the interview take place in their home. The plan was to interview them

separately; however, when I arrived for the audiotaping, one of these two participants disclosed

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the nervousness they were feeling, and the other participant offered to work together, asking if

that was an option. I believed that this could be an interesting opportunity to provide other

analysis and suggestions for future research, if the audiotape of the pair revealed new insights

related to working together that did not arise from other subjects. Therefore, I allowed this

accommodation. The audio-recorded pair was of Participants 16 and 17.

One of the participants (Participants 25) did not consent to be audio-recorded (did not

feel comfortable with her English language speaking skills). I asked her to write down all her

thoughts and feelings. I did not interfere with her taking the content assessment at all, and did not

question her, but I did transcribe her notes on her thoughts and feelings into the data collection

tables where appropriate, as if she had voiced them out loud.

The audio-recording for Participant 19 was partially lost (I made an error with the

recording device), so that participant’s recording begins at the conclusion of the second proof. I

relied on my notes to summarize that participant’s thinking up to that point.

Upon the conclusion of each interview, I submitted the tapes for transcribing to an online

transcription service with no identifying information about any of the participants. Upon

receiving the transcriptions back, I listened to the duration of each tape while simultaneously

reading the transcriptions and corrected any transcription errors that were made by the service

(these were usually minor mathematical notation or mathematical vocabulary adjustments). I

made sure that no identifying information was present in any of the transcripts for any of the

participants. Then, I determined a plan for analyzing the interview data, which is detailed in the

next section.

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Data Analysis

Textbook Data Analysis

I read the selected textbook ([Textbook Author], 2016) in its entirety and noted every

instance of a proof or proof-related task. To determine what exactly was expected of students

learning proofs via this textbook (and teachers teaching proofs via this textbook), I kept a list, by

chapter and section, of each of the proof or proof-related tasks, and commented on any missed

opportunities for proof tasks in the textbook (see Appendix B).

As I collected the data, I recorded commentary about the tasks and how they could

potentially help or impede a student’s learning of geometric proof. Some examples of this

commentary include comments about the presentation of proofs (for instance, the textbook

presents statement/reason chart proofs with an unlabeled third column that was puzzling to me)

and the fact that the order of the “given” and the “prove” did not reflect the standard order of the

geometry Regents Examination, which is what students in New York State largely take at the end

of the year and which this book is geared towards preparing them for. I also noted when the

proofs of presented theorems were omitted, as well as when all or part of the proof task was

completed for the student (e.g., filling in certain statements/reasons, providing the statement of

the theorem with a diagram, the given, the goal).

To determine what is expected of students learning to prove using this textbook, I tallied

the instances of tasks that explicitly instructed students to prove. Such tasks were regarded as

genuine proof tasks (GPTs), compared to those that prompted students to “explain,” “show,”

“demonstrate,” or “justify,” which were regarded as proof-related tasks (PRTs). I tallied the

frequency of PRTs and GPTs in end-of-section exercises, the frequency of PRTs and GPTs in

end-of-chapter problem sets, and the frequency of PRTs and GPTs in cumulative review

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problems at the end of each chapter. Further, I computed the percentages of the PRTs and GPTS

in the end-of-section exercises, end-of-chapter problem sets, and cumulative review problems at

the end of each chapter in order to determine the magnitude of representation of proof and

proving across the textbook. To gain a closer look at what, specifically, is required of students

when tasked with a GPT, I selected a subset of GPTs from the textbook, and analyzed what is

required for students to successfully complete each of these GPTs. The results are presented in

the results section, and discussed thereafter.

Interview and Content Assessment Data

Development of codes. For each of the participant’s audiotapes and corresponding

content assessment papers, mathematical content occurrences as well as beliefs and attitudes

needed to be analyzed. Due to the volume of the data collected, I knew that codes would be

useful in analyzing the interview data. I did not determine the codes before data collection,

because it was impossible to predict what would come out of the interviews. Upon doing an

initial readthrough of the data and listening to all of the tapes, I came up with this initial list of

potential codes:

• Mathematical error;

• Positive or negative feeling about geometry;

• Value judgment;

• Dislike/discomfort or pleasure for/with specific geometry topic;

• Discomfort with mathematical language or notation;

• Expresses need for standard for mathematical language in geometry;

• Goal-oriented approach to proving;

• States importance of goal;

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• Correctly/incorrectly reads geometric notation;

• Can explain how to prove, but not write the proof;

• Explains what is difficult about writing a proof;

• Expresses self-confidence/belief in own ability or self-doubt/lack of confidence;

• Stuck, but can progress with prompting;

• Demonstrates knowledge, or lack thereof, of geometry curriculum;

• Correctly or incorrectly states meaning of CPCTC;

• Demonstrates correct knowledge or erroneous thinking of congruence criteria;

• Explains belief as to why it is or is not important to know something as a geometry

teacher;

• Makes assumption based on what a diagram looks like visually;

• Verbalizes correct or erroneous proof plan;

• Assumes what is supposed to be proven; and,

• Struggles to remember a mathematical term.

It was clear that coding so many categories was not feasible and would not be fruitful for

making general conclusions from the data. Considering the research questions and the goals of

the research, the codes were refined and streamlined as follows to the following five categories:

• Pure mathematical issue;

• Beliefs and attitudes;

• Issues of correspondence between substance and notation (e.g., segment vs. length);

• Mathematical language (including notation mistake, or CPCTC mistake); and,

• Understanding/self-doubt with mathematical idea (not knowing where to stop, asking

if response is correct, expressing doubt or understanding of a mathematical idea).

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The codes were abbreviated as follows, respectively:

• PMI: pure mathematical issue;

• BAA: beliefs and attitudes;

• ICSN: issue of correspondence between substance and notation;

• ML: mathematical language; and

• EUSD: expressing understanding or self-doubt.

Analyzing the content assessment and interview data. For the introductory problem,

starting with Participant 1, I completed the following steps. First, I took out the participant’s

content assessment and organized it so that the participant’s work and my notes were side by

side. Then, I loaded the interview transcript and the audio-recording on my computer and played

the portion of the tape for the introductory problem while reading along with the transcript and

checking the participant’s mathematical work. I kept a cumulative list of each instance that

reflected an instance of each of the five codes; for example, if a participant said something that

expressed a belief, the quotation was copied and pasted into a list of BAA instances. I used a

spreadsheet for each of the five codes to collect the aggregate data with the following columns:

Participant #, Question #, Quotation/Instance.

For each of the five codes, I made a separate sheet. At first, I just was copying and

pasting the instances into each of the sheets, but soon into the process for the introductory

problem realized that the codes needed to sometimes be commented on or split, according to a

positive or negative instance of the code (for example, did a participant express a positive

attitude, or a negative attitude? Did they demonstrate use of correct or incorrect mathematical

language?) So, the following adjustments were made when needed:

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• PMI: “Comments” column was added to note any explanation of the error the

participant was having with the problem.

• BAA: “Comments” column was added to either clarify how what the participant said

was demonstrative of a belief or attitude, or to shed light on implications of a

quotation/ instance.

• ICSN: “ICSN+?” column was added to denote whether the participants made clear

their knowledge of the difference between notation and substance, and a “Comments”

section was added to describe the instance.

• ML: “ML+?” column was added to denote whether it was a positive (ML+) instance

or negative/incorrect (ML-) instance of mathematical language, and a “Comments”

column was added to state whether there was commentary that would clarify my

intention in citing the instance.

• EUSD: Added “Positive (EU) or Negative (ESD)” column to clarify whether the

participants were expressing understanding or self-doubt for the instance at hand (or

both within the same instance).

Then, I looked for trends within each subcode. For example, did the majority of

participants express self-doubt with a specific part of a specific question? The results are

presented and discussed in the results section. After collecting and analyzing the data for

Research Questions 1 and 2, I related the findings back to the literature in order to form robust

conclusions about the roles of the textbook and teachers’ knowledge and beliefs in the teaching

and learning of geometric proof.

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Chapter IV: Results and Discussion for Research Question 1

To answer Research Question 1 (What kinds of proof tasks and proofs are offered in a

typical high school geometry textbook?), I conducted a qualitative analysis of a New York State

common core-aligned geometry textbook ([Textbook Author], 2016). I began by reading the

book in its entirety. Due to the ambiguity of “explain/ justify” and “proof” tasks as described in

the literature, I recorded each instance of an exercise in which justification/ reasoning skills were

required (either explicitly or inherently), and those which explicitly tasked the reader/student

with “proving” something. The former types of exercises were regarded as Proof-Related Tasks

(PRTs) and the latter types of exercises were regarded as Genuine Proof Tasks (GPTs). I

recorded each instance of a PRT and GPT in the textbook, and categorized the list by the

textbook’s chapters and sections. I made additional commentary on some of the PRTs and GPTs,

on missed opportunities for GPTs, and on theorems that were presented in the textbook without

proof. Then, I selected a subset of GPTs from the textbook, and analyzed what is required for

students to successfully complete each of these GPTs. The raw data, which contain my

commentary and classifications, appear in Appendix B. The results regarding several aspects of

the textbook analysis data are presented and discussed in this chapter.

General Remarks

Because this textbook ([Textbook Author], 2016) is not foundational (it does not build

solely on the axioms and definitions given in the textbook, as they are presented), many

theorems are presented with their proofs omitted. For example, on page 241, the “Perpendicular

Bisector Theorem Converse” is stated: “If a point is the same distance from a segment’s

endpoints, then it is on the segment’s perpendicular bisector.” No proof appears. Other instances

of theorems with omitted proofs are presented in the raw data in Appendix B. This presentation

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of material makes for a piecemeal trajectory of learning and implies that it is not necessary to

prove theorems, which is problematic for students who are learning to understand the nature and

purpose of proof. A foundational text would alleviate this issue because students would be tasked

only with proving theorems using the results of what they have already learned.

In considering what the textbook communicates explicitly or implicitly regarding the

nature of proof, I made note of PRTs and GPTs that could confuse students about the nature of

proof. For example, exercises 24–27 in the cumulative review for Chapter 4 appear below:

Write the biconditional for each definition. 24. The real numbers are composed of the rational and irrational numbers. 25. An integer prime is a number with exactly two positive distinct factors. 26. Two angles that sum to 180° are called supplementary angles. 27. Parallel lines are lines on a plane that do not intersect. ([Textbook Author], 2016, p. 193)

Although it is valuable for students to practice turning statements into biconditionals,

these are definitions and there is therefore nothing to be “proven” after they rewrite the

statements using “if and only if” language. Students may struggle to understand the connection

between biconditional statements and proof writing, if this is all they are exposed to. They

should alternatively or additionally be tasked with rewriting biconditional statements of theorems

as a pair of equivalent statements (a conditional statement and its converse). This would help

students to realize that an “if and only if” theorem requires proof in two directions, thereby

providing more opportunities for students to engage in GPTs.

In many of the GPTs, rigor was lacking because students were still given a lot of leading

information (marked up diagram, goal stated for them, hint laying out the entire proof plan,

translating questions for them by providing a diagram that shows a case of the theorem

statement, etc.). For example, Question 28 reads, “Using the diagram provided and the

knowledge that ∡3 ≅ ∡10 and ∡1 ≅ ∡6, prove that ∡15 ≅ ∡13. Hint: First prove line 𝑎 is

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parallel to line 𝑏. Then prove line 𝑐 is parallel to line 𝑑” ([Textbook Author], 2016, p. 166). In

the hint, students are given the entire proof plan. Rigor would increase if they were prompted to

do a proof of something nonroutine and not told how to do it. Other instances of this are included

in the raw data in Appendix B.

Ambiguity in PRTs and GPTs Due to Verbiage

As previously mentioned, and as expressed in the literature, unease regarding proofs in

geometry often comes from not knowing whether a formal proof is required when the

instructions include the words show, justify, or explain versus being explicitly told to prove. I

decided to separate the former from the latter and call the former PRTs and the latter GPTs. For

the most part, distinguishing between PRTs and GPTs was straightforward; however, some

instances in the textbook illustrate the need for clarification regarding the verbiage used in proof

tasks at large, and why a proof is or is not explicitly asked for. The following examples are

highlighted with accompanying commentary:

Section 7.2, practice problems 22 and 23:

22. Given that 𝐴𝐵0000 ∥ 𝐸𝐷0000, determine if the triangles in the diagram below are similar. If the triangles are similar, write a short proof. If the triangles are not similar, write a short explanation. Diagram is provided. 23. Given that 𝐴𝐵0000 ∥ 𝐸𝐷0000, determine if the triangles in the diagram below are similar. If the triangles are similar, write a short proof. If the triangles are not similar, write a short explanation. Diagram (different from question 22) is provided. ([Textbook Author], 2016, p. 295)

The directions in each of these exercises can confuse students about the nature of proof,

and when proof is needed. Why would declaring similarity require proof but saying the triangles

are not similar not require proof? Restating these exercises as “prove or disprove” tasks would

not only communicate that both directions are valid exercises, but also increase the number of

GPTs in the textbook. Another example comes from the cumulative review for Chapters 1–11:

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12. In the triangle below, show that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. Prove: 𝑀𝐴 = 𝑀𝐵 = 𝑀𝐶, Given: 𝐴𝐵𝐶 is a right triangle with vertices 𝐴(0, 0), 𝐵(0, 𝑏), and 𝐶(𝑐, 0).” Diagram is provided. ([Textbook Author], 2016, p. 572)

This question is regarded as a GPT because it explicitly tasks students with writing a

proof. However, the use of the word show in the first part of the task compared with prove in the

second part of the task confuse students. The first part of the prompt states that this is only for

the triangle in the provided diagram, but the second part of the same sentence communicates that

it is a general statement. Then, what the students are tasked with “proving” refers only to the

triangle in the given diagram. This question is an example of how providing diagrams can

communicate an incorrect takeaway that students are proving something only for a specific case,

and also how the words prove and show are sometimes used interchangeably, and sometimes not

(if the second part of the prompt was not included, it is unclear whether students would know

they are supposed to write a proof).

A particularly interesting breakdown between PRTs and GPTs was found in Section 9.5

([Textbook Author], 2016, pp. 442–445). The problems appear below, with PRT or GPT

indicated in parentheses and commentary following the list of tasks.

o (PRT) 25: “The vertices of a quadrilateral are 𝐴(−3, 2), 𝐵(3, 4), 𝐶(5,−2) and

𝐷(−4,−5). Make a sketch and show that 𝐴𝐵𝐶𝐷 is a trapezoid.”

o (PRT) 15: “Show that in an isosceles trapezoid 𝐴𝐵𝐶𝐷 where 𝐴𝐷0000 and 𝐵𝐶0000 are the

bases, ∆𝐴𝐵𝐷 ≅ ∆𝐷𝐶𝐴. Use the theorem of base angles of an isosceles trapezoid.”

o (PRT) 26: “Consider a trapezoid with vertices at the points

𝐴(0, 0), 𝐵(2𝑎, 2𝑏), 𝐶(2𝑐, 2𝑏) and 𝐷(2𝑑, 0). Show that the midsegment 𝑀𝑁00000 of the

trapezoid is parallel to its bases.” Diagram is provided.

o (GPT) 30: “Prove that the base angles of an isosceles trapezoid are congruent.”

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o (GPT) 31: “Prove that the diagonals of a kite are perpendicular to each other. Hint:

Consider perpendicular bisectors.”

o (GPT) 32: “In kite 𝐴𝐵𝐶𝐷, 𝐴𝐵 = 𝐵𝐶 and 𝐶𝐷 = 𝐷𝐴. Prove that the diagonal 𝐵𝐷0000 of the

kite is an angle bisector of ∡𝐵 and ∡𝐷.” Diagram is provided.

o (PRT) 33: “A quadrilateral is obtained by connecting the midpoints of all sides of an

isosceles trapezoid. Determine the type of quadrilateral, being as specific as possible.

Justify your reasoning.” Diagram is provided.

o (PRT) 34: “Show that the part of a midsegment of a trapezoid between its diagonals is

equal to half the difference of the trapezoid’s bases. Make a sketch and refer to it in

your proof.”

o (PRT) 36: “The diagonals of a trapezoid divide its midsegment into three congruent

segments. Show that one base of the trapezoid is twice as long as the other.” Diagram

is provided.

Through examination of this collection of problems, it is evident that certain tasks in this

section explicitly prompt students to prove, and others do not—this could unintentionally

communicate that some of the properties to be shown or proven are less or more important than

others. Question 34 is especially confusing because it directs students to show and then later in

the problem tells them to refer to their proof ([Textbook Author], 2016, p. 444). The ambiguity

between show, explain, justify, and prove tasks referred to in the literature (Otten et al., 2011) is

evident here. If a textbook uses the words show and prove interchangeably, this usage should be

explained—and, if not, the expectations for each directive should be clearly outlined for teachers

and students alike.

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Frequency of PRTs and GPTs in End-of-Section Exercises

For each section of practice exercises for each chapter in the textbook, I recorded the

PRTs and GPTs and then computed the percentages of the exercises which were PRTs and

GPTs. The frequency and percentage tables are listed in Appendix E, Table E1. Of 75 total

sections, 39 of them had 0% GPTs and 36 of them had 0% PRTs. Many of those sections

containing 0% of PRTs also had 0% GPTs, with exceptions in Section 5.1, 7.1, 7.3, 7.6, 7.8, 8.1,

8.2, 8.3, 9.1, 9.2, 9.3, 9.4, 9.6, 9.7, and 10.5. Those sections’ frequencies appear in Table 1 and

frame the commentary that follows.

In nine of the rows in Table 1, the sections contained zero PRTs and only one GPT,

whereas in Sections 9.2 and 9.3, the sections contained zero PRTs yet seven and 12 GPTs,

respectively. This raises the questions as to why, and whether, the content of these sections led to

the inclusion of more genuine-proof tasks than the content of other sections. Section 9.2 is

entitled Deciding if a Parallelogram Is Also a Rectangle, Square, or Rhombus, and Section 9.3 is

entitled Deciding if a Quadrilateral is a Parallelogram. Section 9.3 has the highest percentage of

GPTs of the entire book, followed by Section 3.3, Deductive Reasoning.

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Table 1

Sections With 0% PRTs and Non 0% GPTs

Section

PRTs GPTs

n % n %

5.1 (Part 2) 0/20 0% 2/20 10%

7.1 0/33 0% 3/33 9%

7.3 (Part 1) 0/26 0% 1/26 4%

7.6 0/53 0% 1/53 2%

7.8 0/39 0% 1/39 3%

8.1 0/32 0% 1/32 3%

8.2 0/37 0% 4/37 11%

8.3 0/34 0% 1/34 3%

9.1 0/52 0% 6/52 12%

9.2 0/37 0% 7/37 19%

9.3 0/27 0% 12/27 44%

9.4 (Optional) 0/41 0% 1/41 2%

9.6 0/39 0% 1/39 3%

9.7 0/29 0% 1/29 3%

10.5 0/28 0% 1/28 4%

The percentage of GPTs exceeded 10% in 13 out of 75 sections, whereas the percentage

of PRTs exceeded 10% in 14 out of 75 sections. This points to the fact that the book placed much

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stronger emphasis on computing and procedural problems than on proof tasks in general,

including both proof-related and genuine proof tasks.

Frequency of PRTs and GPTs in End-of-Chapter Problem Sets

Each chapter of the textbook concluded with a set of practice problems, specific to the

skills uncovered in the chapter. I recorded the PRTs and GPTs within the end-of-chapter

exercises for each chapter, and then computed the percentages of these exercises which were

PRTs and GPTs. The frequency and percentage tables are listed in Appendix F, Tables F1 and F2.

Of 12 chapters in the textbook, seven contained zero PRTs in their end-of-chapter exercises (with

the highest percentage of the 12 chapters being 13%), and seven contained zero GPTs in their

end-of-chapter exercises (with the highest percentage of the 12 chapters being 17%). This

finding further demonstrates the lack of emphasis on proof in the textbook. As teachers plan units

of lessons using a textbook, they often start at the end of the chapter and deduce what the

learning goals for students will be based on the problems students will be tasked with solving at

the chapter’s conclusion. Given the lack of PRTs and GPTs in the end-of-chapter practice sets, it

is not surprising that proof is often underemphasized and avoided by teachers and students alike,

whereas computation and procedural problems are heavily emphasized.

Frequency of PRTs and GPTs in End-of-Chapter Cumulative Review Problems

Each chapter of the textbook (beginning with Chapter 2) concluded with a set of

cumulative practice problems from the preceding chapters ([Textbook Author], 2016). For

example, after the conclusion of Chapter 4, there is a set of practice problems on the material

from Chapters 1, 2, 3, and 4. I recorded the PRTs and GPTs within these cumulative problem

sets at the end of each chapter, and then computed the percentages of these exercises which were

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PRTs and GPTs. For the 11 sets of cumulative practice problems, the frequencies and

percentages appear in Table 2.

The data in the Table 2 support the conclusion that proof is massively underemphasized

in the textbook chosen for the study. The low percentages in the later chapters are especially

concerning, since students would presumably complete these problems under the assumption that

they represent important skills they were to have acquired throughout their course in geometry.

Table 2

Cumulative Review Problems at the Conclusion of Each Chapter

Chapters

PRTs GPTs

n % n %

1-2 1/29 3% 0/29 0%

1-3 2/29 7% 2/29 7%

1-4 7/34 21% 2/34 6%

1-5 3/38 8% 0/38 0%

1-6 2/29 7% 7/29 24%

1-7 1/29 3% 2/29 7%

1-8 0/31 0% 2/31 6%

1-9 1/35 3% 1/35 3%

1-10 3/32 9% 0/32 0%

1-11 1/36 3% 2/36 6%

1-12 0/36 0% 3/36 8%

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Upon first glance, the GPTs in the cumulative review for Chapters 1–6 (which are highest

in percentage of all the cumulative reviews, at 24%) seem to contain a variety of proof tasks,

reproduced here ([Textbook Author], 2016, pp. 279–281):

14. Prove: ∆𝐴𝐶𝐵 ≅ ∆𝐸𝐶𝐹,Given: The midpoint of 𝐴𝐸0000 is 𝐶; 𝐵𝐶 = 𝐶𝐹. [Diagram is provided.]

15. Prove the concurrency and perpendicular bisectors theorem. Prove: 𝐷𝐴0000 ≅ 𝐷𝐵0000 ≅ 𝐷𝐶0000, Given: 𝐷𝐸0000, 𝐷𝐹0000,and 𝐷𝐺0000 are ⊥ bisectors. [Diagram is provided.]

16. Two non-congruent line segments 𝐽𝐾 and 𝐿𝑀 bisect each other at point 𝑁. Connecting points 𝐽 and 𝐿 with line segment 𝐽𝐿, and points 𝐾 and 𝑀 with line segment 𝐾𝑀, forms triangles 𝐽𝑁𝐿 and 𝐾𝑁𝑀. (a) Complete a sketch of the situation. (b) Prove that ∆𝐽𝑁𝐿 ≅ ∆𝐾𝑁𝑀.

17. Prove that the median of a triangle cannot form two obtuse angles with the side that it bisects.

18. Sketch an isosceles triangle in the coordinate plane. Then, prove that your triangle is not scalene using indirect reasoning.”

19. Quadrilateral 𝐺𝑂𝐴𝑇 has vertices located at 𝐺(−3, 3), 𝑂(4, 4), 𝐴(6,−1) and 𝑇(−2,−2). Use indirect reasoning to prove that 𝐺𝑂𝐴𝑇 is not a parallelogram.

20. Given: ∆𝐹𝐸𝐷 and ∆𝐹𝐴𝐷, which share side 𝐹𝐷0000;𝐸𝐷0000 ≅ 𝐴𝐷0000, and 𝐹𝐷0000 bisects ∡𝐷. Prove: ∆𝐹𝐸𝐷 ≅ ∆𝐹𝐴𝐷.

Reading these problems without solving them gives the impression that they vary in their

content, levels of rigor, approaches, and visual cues: proofs are included about congruence,

angles, types of triangles, and bisectors. I solved and analyzed each of these problems to

determine what specifically is required of students in completing each of these tasks. The

analysis appears later in this chapter (Analysis of a Subset of GPTs section) and reveals that this

collection of tasks lacks variability and rigor.

Interpreting the Lowest and Highest Percentages of GPTs

Throughout the end-of-section practice exercises, end-of-chapter review exercises, and

cumulative review exercises, there were 39 out of 75, 7 out of 12, and 3 out of 11, respectively,

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that contained zero GPTs ([Textbook Author], 2016). These ratios can imply to teachers and

students alike that proof need not be the focus of a typical high school geometry course, and

support the sentiments from the existing literature that the need for proof is not effectively

communicated or understood by students (e.g., Zaslavsky et al., 2012). It should be noted that an

additional 22 out of 75, 2 out of 12, and 7 out of 11 sections, respectively, contained less than

10% GPTs ([Textbook Author], 2016). In these cases, although GPTs were present, they were

not abundant.

On the other end of the spectrum, there were some instances of a seemingly high

percentage of GPTs: the end-of-section review exercises for Section 3.3 contained 39%, and in

Section 9.3, 44% ([Textbook Author], 2016). A closer look at these sections, however, reveals a

lack of diversity of GPTs. Section 3.3, Deductive Reasoning, contains the following GPTs

([Textbook Author], 2016, p. 133–134):

9–12. Use a two-column proof to solve the equation. Refer to the properties on page 5 as needed to answer these questions.

§ 9. 2𝑥 = 6(2 + 𝑥)

§ 10. 4𝑥 = 2(9 − 𝑥)

§ 11. −7𝑥 = 2(1 − 3𝑥)

§ 12 −2(2 − 3𝑥) = 10 − 𝑥

13–15. Use the diagram below and the theorems you’ve learned thus far to create a two-column proof to find the value of 𝑥.” A diagram is provided of three angles (∡𝐴𝑂𝐵, ∡𝐵𝑂𝐶, ∡𝐶𝑂𝐷) lying on straight line 𝐴𝑂𝐷O⃖OOOOOO⃗ .

§ 13. 𝑚∡𝐴𝑂𝐵 = 𝑥 + 45°; 𝑚∡𝐵𝑂𝐶 = 2𝑥 − 17°; 𝑚∡𝐶𝑂𝐷 = 𝑥

§ 14. 𝑚∡𝐴𝑂𝐵 = 5𝑥; 𝑚∡𝐵𝑂𝐶 = 3𝑥; 𝑚∡𝐶𝑂𝐷 = 2(𝑥 + 20°)

§ 15. 𝑚∡𝐴𝑂𝐵 = 3𝑥 − 41°; 𝑚∡𝐵𝑂𝐶 = 𝑥; 𝑚∡𝐶𝑂𝐷 = 2𝑥 − 19

Although there are seven GPTs in this section, there is limited variation in the questions.

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The only difference in process is in algebraic steps, rather than reasoning. These are the first

GPTs in the textbook, and are all related to algebraic reasoning. Although this placement could

be helpful in reinforcing the important and overlooked idea that proof is applicable to all

mathematics content, rather than just geometric theorems (Pruitt, 1969), the textbook does a

limited job at reinforcing this notion beyond Chapter 3. For example, the large numbers of zeros

and ones (frequencies) in the data for Chapters 10, 11, and 12 reinforce the misconception that

proof is applicable only to certain topics.

For Section 9.3, Deciding if a Quadrilateral Is a Parallelogram ([Textbook Author], 2016,

pp. 429–432), the GPTs appear below:

10. In quadrilateral 𝐴𝐵𝐶𝐷, ∡𝐵𝐴𝐶 ≅ ∡𝐴𝐶𝐷 and ∡𝐵𝐶𝐴 ≅ ∡𝐷𝐴𝐶. Prove that 𝐴𝐵𝐶𝐷 is a parallelogram.

15. Michael has a parallelogram-shaped wooden board. He measures out the same distance on each side of his board, starting from each vertex, and nails in the pegs. After he stretches a rubber band on the pegs, he gets the shape shown below. Prove that the shape Michael obtained is a parallelogram. [Diagram is provided.]

16. In a parallelogram 𝐵𝐸𝑆𝑇, the point 𝐴 is the midpoint of 𝐸𝑆0000 and point 𝑀 is the midpoint of 𝐵𝑇0000. Make a sketch and prove that the quadrilateral 𝐸𝐴𝑇𝑀 is a parallelogram.

17. In the diagram below, 𝐴𝐵𝐶𝐷 and 𝐴𝑀𝑁𝐷 are parallelograms. Prove that 𝑀𝐵𝐶𝑁 is also a parallelogram. [Diagram is provided.]

18. 𝑅𝑂𝑆𝐸 is a parallelogram. The points 𝐴, 𝐵, 𝐶, and 𝐷 are midpoints of segments 𝑋𝑅0000, 𝑋𝑂0000, 𝑋𝑆0000, and 𝑋𝐸0000, respectively. Prove that 𝐴𝐵𝐶𝐷 is also a parallelogram. [Diagram is provided.]

19. In the parallelogram 𝐸𝐹𝐺𝐻, the diagonals intersect at point 𝑂. A line passes through the point 𝑂 and intersects the sides of parallelogram at points 𝑀 and 𝑁. Prove that 𝑂𝑀 =𝑂𝑁. [Diagram is provided.]

20. A midsegment of a triangle is a segment connecting the midpoints of two of the triangle’s sides. In the triangle 𝐴𝐵𝐶,𝑀𝑁00000 is a midsegment connecting the sides 𝐴𝐵0000 and 𝐵𝐶0000. Extend the line 𝑀𝑁00000 outside the triangle and mark the point 𝑃 such that 𝑀𝑁 = 𝑁𝑃. Prove that 𝐴𝑀𝑃𝐶 is a parallelogram. Refer to your sketch in your proof.

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23. Use the diagram below to prove the parallelogram opposite angles theorem converse. Hint: Begin by using the fact that 2𝑎 + 2𝑏 = 360°. Given: ∡𝐴 ≅ ∡𝐵𝐶𝐷and ∡𝐷 ≅∡𝐴𝐵𝐶, Prove: 𝐴𝐵𝐶𝐷 is a parallelogram. [Diagram is provided.]

24. Use the diagram below to prove the parallelogram supplementary angles theorem converse. Hint: First prove that 𝐴𝐵0000 ∥ 𝐶𝐷0000 by showing 𝐴𝐸OOOOO⃗ ∥ 𝐷𝐹OOOOO⃗ . Then, write a similar series of steps to prove 𝐴𝐷0000 ∥ 𝐵𝐶0000 by showing 𝐴𝐺OOOOO⃗ ∥ 𝐵𝐻OOOOOO⃗ . Given: Consecutive interior angles are supplementary, Prove: 𝐴𝐵𝐶𝐷 is a parallelogram. [Diagram is provided.]

25. Prove that, in a parallelogram, the sum of the squares of the diagonals is equal to the sum of the squares of all of its sides. In other words, in parallelogram 𝐴𝐵𝐶𝐷, 𝐴𝐶W +𝐵𝐷W = 𝐴𝐵W + 𝐵𝐶W + 𝐶𝐷W + 𝐴𝐷W.

26. Prove the parallelogram diagonals theorem converse. Hint: Prove that the opposite sides of the quadrilateral are congruent, then use the parallelogram opposite sides theorem converse. Parallelogram diagonals theorem converse: If the diagonals of a quadrilateral bisect one another, then it is a parallelogram. [Diagram is provided with statement of theorem.]

27. Prove the parallelogram opposite sides theorem converse. Hint: Construct 𝐵𝐷0000 and work backward through the proof for the parallelogram opposite sides theorem (non-converse). Parallelogram opposite sides theorem converse: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. [Diagram is provided with statement of theorem.]

Section 9.3 clearly contains a wider variety of GPTs than Section 3.3, which utilizes the

different properties of parallelograms. A question is raised as to why this section has the highest

percentage of GPTs out of the entire book. The GPTs in Section 9.3 are a starting point, but they

could be improved in the following ways to increase rigor and better assess students’ reasoning

skills:

• Regarding Question 15, this is not a typical proof task, and would be ambiguous

without the diagram since it does not state that Michael measures the same distance

from each vertex going in the same order (clockwise or counterclockwise around the

figure).

• More focus should be placed on when (and why) students are given hints in GPTs.

For instance, in Question 25, the exercise is difficult and provides no hint; however,

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in Questions 24, 26, and 27, the hints are more than just that—all are leading and tell

the students exactly what to do.

• Question 23 also begins with a hint, although it does not tell students exactly what to

do. Can students just divide both sides of the given equation by two, and say that

ABCD is a parallelogram since consecutive angles are supplementary? This was

another theorem given in the section that was not proved (and, in fact, students are

tasked with its proof in the next question).

• Regarding Question 26, the “parallelogram opposite sides theorem converse” which

is referenced in the GPT was presented but never proved, and students were never

tasked with its proof to this point; however, the following question tells them to prove

it. Order is an issue.

This analysis revealed that it is imperative to study the content of the GPTs rather than

simply count the number of exercises that direct students to “prove.” A closer look at the GPTs

reveals that some of the high percentages do not necessarily make for varied and rigorous

exercises.

Analysis of a Subset of GPTs

To obtain an understanding of what, specifically, is required of students in completing the

GPTs in the textbook, I chose a subset of seven GPTs to solve and analyze. To represent as much

of the content in the book as possible, I chose the section of cumulative review problems with the

highest percentage of GPTs in the entire book. This was the cumulative review for Chapters 1–6,

with 7/29 (or 24%) of problems classified as GPTs. Each of the GPTs, its solution, and the

subsequent analysis appears below. The first ([Textbook Author], 2016, p. 280) is shown in

Figure 7.

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Figure 7

Cumulative Review GPT: Question 14

To complete this proof, students are required to complete the following steps:

• Apply the definition of a segment’s midpoint;

• Recognize vertical angles formed by intersecting lines; and

• Apply the side-angle-side congruence postulate.

It should be noted that for this solution and those that follow, I solved these problems

using the models and standards set forth by the textbook (for example, in the textbook, the side-

angle-side congruence criterion is treated as a postulate, rather than stated as a theorem and then

proved as a valid means of proving two triangles are congruent). The textbook does not

communicate that students should express that “intersecting lines form vertical angles, and all

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vertical angles are congruent” or identify the corresponding parts of the two triangles that justify

the side-angle-side congruence.

The next cumulative review GPT ([Textbook Author], 2016, p. 280) is shown in Figure 8.

This GPT is very misleading. The concurrency and perpendicular bisectors theorem states, “A

triangle’s perpendicular bisectors intersect at a point of concurrency that is equidistant from the

triangle’s vertices” ([Textbook Author], 2016, p. 249). The theorem consists of two parts

(concurrency and equidistance), but the cited task assumes the first and contains only the second.

Moreover, the provided diagram also assumes the concurrency of the perpendicular bisectors. In

order for the text to accurately task students with proving this theorem, it should just say to

“Prove the concurrency and perpendicular bisectors theorem.” The approach, then, would be to

first prove the existence of a triangle’s circumcenter using the definition of a perpendicular

bisector, and then prove its equidistance from the triangle’s sides.

In this exercise as it is stated, students are tasked with proving part of the theorem that

was initially stated without proof ([Textbook Author], 2016, p. 249). This specific task could be

made more rigorous by removing the provided diagram (and, consequently, not telling students

what they have to prove about it). On the page where the theorem is stated, the auxiliary lines are

drawn into the diagram already, so any student who looks back to the statement of the theorem

(which is not given in the exercise) would have this step done for them ([Textbook Author],

2016, p. 249). If the task were adjusted by removing the diagram and the auxiliary lines drawn in

on page 249, as well as including both parts of the theorem, teachers would be able to assess

whether students understand what is required to begin and complete the proof on their own.

Below the proof is given as it can be expected from a student in these circumstances.

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Figure 8



• Apply the definition of perpendicular bisector;

• Draw auxiliary lines;

• Apply the side-angle-side congruence postulate;

• Apply CPCTC; and,

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• Apply the substitution property.

The third cumulative review GPT question, Question 16 ([Textbook Author], 2016, p.

281), is shown in Figure 9.

Figure 9



• Draw a diagram of the given information;

• Apply the definition of “bisect each other” (the wording used in the solution above is

reflective of the models set forth in the textbook);

• Recognize vertical angles formed by intersecting lines; and


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With the exception of the diagram not being provided, this is essentially the same

exercise as Question 14, and requires the same application of the same geometric knowledge.

Figure 10 shows the fourth GPT question presented to participants ([Textbook Author],

2016, p. 281).

Figure 10



• Apply the definitions of “median” and “obtuse angle”;

• Recognize that drawing a median of a triangle creates a linear pair of angles;

• Know that the sum of the measures of the angles in a linear pair is 180°.

It should be noted that there is a slight mathematical language error in the statement of

the task. It refers to “the” median of a triangle, but there are three medians in any triangle, and

this property applies to all of them.

The fifth GPT question I chose was Question 18 ([Textbook Author], 2016, p. 281),

shown in Figure 11.

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Figure 11


Two solutions are possible. The first solution uses the distance formula:

Assume ∆𝐴𝐵𝐶 is scalene. Then none of the sides of ∆𝐴𝐵𝐶 have the same length. But, by the distance formula, ∆𝐴𝐵𝐶 is isosceles, because two of its sides have the same length (𝐴𝐵 = 𝐴𝐶 = 2√5𝑢𝑛𝑖𝑡𝑠) are congruent. We have reached a contradiction: ∆𝐴𝐵𝐶 cannot be both scalene and isosceles. So, our assumption was false. ∆𝐴𝐵𝐶 is not scalene.

The second solution is to complete this proof without a coordinate plane sketch at all:

Given an isosceles triangle, suppose the triangle is scalene. Then all three sides have different lengths. But two of the sides of the triangle have the same length, because it is isosceles. Therefore, the triangle is not scalene.


• Sketch an isosceles triangle in the coordinate plane; and

• Apply the definitions of isosceles and scalene.

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Although the students are exposed to practice with indirect reasoning via this task, the

task is trivial and, when accompanied by a coordinate plane sketch, visually obvious. If students

are instructed to draw an isosceles triangle, there is no genuine need to prove via indirect

reasoning (or via any method) that the triangle is not scalene. Moreover, since the terms isosceles

and scalene are not new for to the average high school geometry student, students may feel that

the simplicity of the prompt is a trick, as the nature of the question poses questions regarding its

purpose (they might struggle to understand why they need proof to demonstrate that a purposely

drawn isosceles triangle is not scalene).

The sixth GPT question ([Textbook Author], 2016, p. 281) is shown in Figure 12.

Figure 12



• Apply the property that opposite sides of a parallelogram are parallel; and

• Calculate slopes of line segments.

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Similar to Question 18, this task should not necessarily require students to use indirect

reasoning, and is better suited as a PRT. It forces the method of proof by contradiction and can

disrupt students’ learning of the genuine need for indirect reasoning in some cases. The question

could be rephrased as, “Quadrilateral 𝐺𝑂𝐴𝑇 has vertices located at 𝐺(−3, 3), 𝑂(4, 4), 𝐴(6,−1)

and 𝑇(−2,−2). Determine whether𝐺𝑂𝐴𝑇 is a parallelogram.”

The seventh and final GPT question for analysis ([Textbook Author], 2016, p. 281) is

shown in Figure 13.

Figure 13



• Apply the definition of angle bisector;

• Apply the reflexive postulate; and

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The knowledge required to complete this problem, and its proof process, is similar to two

of the other GPTs in the section (Questions 14 and 16), both of which also utilize and conclude

with side-angle-side congruence.

Overall, this subset of GPTs illustrates the fact that the textbook does the majority of the

work for the students, and lacks diversity and variability in proof types. The level of rigor of the

proofs contained in the section does not vary much from one problem to another. Rewriting some

of the book’s exercises, and turning some of the non-proof tasks into proof-tasks, would make

proof more pronounced and rigorous in the textbook, and could foster impactful takeaways for

teachers and students.

Missed Opportunities for Proof Tasks

I counted the number of instances in which the textbook ([Textbook Author], 2016)

missed an opportunity for tasking students with a GPT. I recorded 62 instances of missed

opportunities. Some of these instances took place within the section content, and others within

the end-of-section, end-of-chapter, or cumulative-chapter-review exercises. The missed

opportunities are each presented in the raw textbook data (see Appendix B) and several instances

are highlighted below:

A Missed Opportunity for an Entry-Level Proof Task

Section 4.4, practice exercises, Question 20: “The line 𝑎 goes through the points (6,−4)

and (−5, 0). The line 𝑏 passes through the points (2,−7) and (−9,−3). Are these lines parallel?

Why or why not?” ([Textbook Author], 2016, p. 181). Although this question can be answered

purely computationally, it is a missed entry-level opportunity for a proof. Suppose the exercise

were worded as follows: “The line 𝑎 goes through the points (6,−4) and (−5, 0). The line 𝑏

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passes through the points (2,−7) and (−9,−3). Prove that line 𝑎 is parallel to line 𝑏.” This

adjustment would not only provide an entry-level proof opportunity, but also give students

practice applying a definition within proof-writing and would support the text’s effort to

demonstrate that proof is used in algebra and other branches of mathematics aside from

geometry.

A Missed Opportunity for a Construction-Based Proof Task

In the Chapter 4 review problems, Question 21 reads, “Point 𝑀 does not lie on line 𝑚.

Explain how to construct a line 𝑝 through point 𝑀 and perpendicular to line 𝑚 using a compass

and a straightedge. Perform the constructions” ([Textbook Author], 2016, p. 189). This task

could be rewritten to increase its rigor and demonstrate the applicability of proof to geometric

constructions. Students often memorize the steps to constructions and rely on procedural

repetition rather than conceptual understanding of why the steps result in the desired

construction. If students were required to prove that their construction satisfies the desired

conditions, they would be more deeply engaging in the content and simultaneously improving

their proof-writing skills.

A Missed Opportunity for a GPT

The textbook often presents theorems without proof. An example of these missed

opportunities is the presentation of the isosceles triangle theorems, as follows:

If two sides of a triangle are congruent, then the angles opposite them are congruent. If two angles of a triangle are congruent, then the sides opposite them are congruent. ([Textbook Author], 2016, pp. 198–199)

A diagram appears which shows an isosceles triangle with congruent base angles and

sides marked, and two statements underneath which employ geometric notation. A blurb appears

to the right of the diagram: “The first statement is the base angles theorem. The second statement

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is the converse of the base angles theorem. We will prove the base angles theorem later”

([Textbook Author], 2016, p. 198). The next page begins with “Isosceles Bisector Theorem: If a

line bisects an isosceles triangle’s vertex angle, then it is a perpendicular bisector of the base”

([Textbook Author], 2016, p. 199). Then, “Isosceles Bisector Theorem Converse: If a line is the

perpendicular bisector of an isosceles triangle’s base, then it is also the angle bisector of the

vertex angle” ([Textbook Author], 2016, p. 199).

All of this information is given to the students without direction for them to investigate or

prove these theorems. These are missed opportunities since they are less visually obvious and

could motivate the need for proof.

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Chapter V: Results and Discussion for Research Question 2

To answer Research Question 2 (How prepared and how confident are preservice and

novice secondary mathematics teachers to teach proofs?), I administered a content assessment to

29 preservice and novice teachers in New York State. I audio-recorded and transcribed the

administration of the content assessment, and asked participants to make their thoughts and

feelings clear as they progressed through the assessment. Five common core aligned tasks were

chosen: one “explain” task (PRT) and four GPTs based on their relevance to the related

literature, their adherence to the common core, the geometry skills embedded within them, and

my knowledge of trouble areas for preservice secondary mathematics teachers

(eMATHinstruction, 2017, 2018a, 2018b; The University of the State of New York, 2015, 2016).

The content assessment and interview data were analyzed qualitatively using the following

coding system, the evolution of which was described in the Methodology chapter (Chapter III) in

detail:

• BAA (Beliefs and Attitudes)

• ICSN (Issues of Correspondence Between Substance and Notation)

• Expressing Understanding or Self-Doubt (EUSD)

• Mathematical Language (ML)

• Pure Mathematical Issue (PMI)

The results for the interviews and content assessment items are described in this section,

in the order of the content assessment questions. For each of the content assessment items,

general remarks appear first, followed by specific findings based on the coding system I created.

For brevity’s sake, each instance of each code is not explained in this section, although they are

all present in Appendix D.

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Content Assessment Item 1

General Remarks

Content Assessment Item 1 was the introductory problem (eMATHinstruction, 2017),

presented in Figure 1. I hypothesized that participants would ask for clarification regarding what

was meant or required by the word explain in the question. It turns out that 0 out of 29 of the

participants asked what was meant by the instructions to the introductory problem. This does not

necessarily mean that the participants did not question the difference between explain and prove

prompts, but it suggests a variety of implications:

1. The participants do not think about writing proofs unless they are specifically

prompted to do so;

2. The nature of proof and proving does not raise these questions within the participants;

and,

3. The participants were unsure of how to approach the problem anyway, so they would

not think to prove their results (23 of 29 participants struggled with the problem and

were unable to move forward in solving it without scaffolding questions from me).

Specific Findings Based on Codes

Beliefs and Attitudes. I tallied 25 instances of BAA coding for the introductory problem

(Content Assessment Item 1). Sixteen of 29 participants’ interviews resulted in at least one BAA

code. I found that two of the participants (Participants 1 and 22) transparently stated that my

questioning helped them change their beliefs and attitudes about the problem. Participant 1, who

initially said “this is hard,” followed by “this is so hard,” eventually said, “Now I’m confident”

after a discussion about the problem. After an initial struggle, I asked scaffolding questions to

Participant 22, which resulted in the following exchange:

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Participant 22: Okay, it’s the Socratic method. Investigator: The questioning worked, right? Participant 22: It works.

Two participants expressed distaste for transformations. Immediately upon looking at the

problem, Participant 14 said, “I hate transformations.” Participant 28, after expressing her dislike

for the word ‘image’, said,

Image bothers me because I don’t like. . . image is in math and I remember it from like multi-variable calculus or I just remember it from some other class and I just didn’t like it because I didn’t understand it; however, if I was looking at this from the high. . . if I was in high school where I didn’t learn all that, I would be like, “Oh? What does that mean?” Now, I just. . . I get bad juju from that.

Participant 28, despite not understanding the language in the problem, did persevere in

trying to solve it and was encouraged by her (perceived) understanding, later on exclaiming,

“And the ‘counterclockwise’, I guess, doesn’t matter ‘cause if you go in any direction, it’s. . . oh!

Oh my God! Okay, I’m getting this now. Okay, I got it.” I made a note that the participant was

encouraged by her perception of her understanding although she still did not demonstrate

mathematical understanding. This speaks to the role of one’s beliefs and attitudes in persevering

through a geometry problem that might at first present as difficult.

Issue of Correspondence Between Substance and Notation. For the data for the

introductory problem, I tallied only one instance of ISCN code. It was a positive instance of the

code (ICSN+), meaning that the participant clearly communicated understanding of the

distinction between substance and notation. Participant 12 correctly distinguished between

congruence of two angles and the equivalence of their measures:

So, we can say that the measurement of. . . we can say that Angle A is congruent to Angle B, which means that the measure of angle A is equal to the measure of Angle B.

Expressing Understanding or Self-Doubt. There were 107 instances of the EUSD code

for the first problem. Expressing understanding (EU) was then separated from expressing self-

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doubt (ESD) and the instances of those codes were 66 and 33, respectively, with 8 instances in

which participants expressed both understanding and self-doubt. The overwhelming majority of

the EU code compared to the ESD code should not be interpreted to mean that the participants

did not express self-doubt overall; the EU code was more frequent because it tallied the instances

of when the participants explained correct mathematical thinking. For example, Participant 4

asserted,

We know that this is 30 degrees, because it tells us that. . . . And you know that these are congruent so you would do 30 plus 2x is equal to 180 because that’s the measure of two angles in a triangle, and then you’re given 2x is equal to 150, then x is equal to. . .

An example of a mixed instance (coded ESD/EU) was when Participant 5 said,

I don’t remember rotations. Explain why triangle ABC must be isosceles. Okay, well, an isosceles triangle has two congruent sides. So, A and B, . . . they’re rotated. . . . What about the rotation tells us that these two have to be congruent?

The participant demonstrated understanding of the definition of an isosceles triangle, but

then expressed doubt regarding how that knowledge connected to what the problem was asking.

A Closer Look at the ESD Code. This code was examined more closely for its potential

to reveal consistent areas of trouble for preservice and novice teachers regarding the mathematics

and mathematical language contained within the introductory problem. Of the 33 instances of the

ESD code, many expressed difficulties with the problem, stating “I don’t know” what it means

(mostly about the given information that “B is the image of A after a 30-degree counterclockwise

rotation about C”). An illuminating moment transpired between Participants 16 and 17 (who

were audio-recorded together, as previously explained in the methodology chapter). During the

interaction (transcribed below), the participants tried to reach understanding by relating the

problem to parts of a circle, but then were unable to arrive at a correct conclusion due to their

misconceptions about circles.

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Investigator: What motion were you making with your pen? Participant 16: I was making a, like an arc motion? Investigator: Between? Participant 16: Between A and B. Because originally, I labeled that arc as 30 degrees. I was thinking the value of that, as an arc of 30 degrees, what does that mean for the angles that are left inside? Investigator: Does this ‘X’ mean that you no longer think this arc is 30 degrees? Participant 16: Yes, but now remembering that, that could be true, so . . . Participant 17: If this was a circular piece, then the measure of the arc is related to the central angle. Participant 16: Right. So, this could be the radius and that could be this . . . Investigator: What could be the radius? Participant 16: AC could be the radius. BC could be the radius and there could be . . . Was that inscribed inside the circle? Is that what it is? Participant 17: An inscribed angle is when it touches on the circle. Participant 16: Angle A is an inscribed angle? No, you’re right. Never mind. I’m wrong. Investigator: That’s a different type of angle. Participant 17: Yeah, it could be, if CB was the diameter. Participant 16: But we don’t know that. Participant 17: We don’t know.

As evidenced by the above dialogue, the participants were on the right track to

connecting the parts of a circle with the explanation for why triangle ABC must be isosceles, but

were unable to reach the correct conclusion.

Participant 22, who had the right idea, was not confident enough in their reasoning. When

I asked, “So, how would you write down your reasoning for A?” the participant responded,

I guess I would say triangle ABC must be isosceles because line segment BC is produced from pretty much rotating. . . . Does it make sense to say rotating line segment AC? Is that a thing, or no?

This interaction with Participant 22 further demonstrates the need for the audiotapes to

reveal the participants’ thought processes, since this doubt would not have been realized by

looking at the participant’s written work alone.

Mathematical Language. Mathematical language was coded as either ML+ or ML-,

with my added commentary as needed. I tallied 20 instances of ML+ and 18 instances of ML-,

along with 1 mixed instance. These codes occurred over a total of 23 participants (with the ML-

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codes occurring over a total of 13 participants). ML errors included referring to the “pre-image”

as the “image” or as the “original,” referring to base angles as the “bottom” angles in the

triangle, stating that “rotations preserve shape,” referring to the center of a circle as the “origin,”

and referring to an angle of rotation as a “swivel” or a “swing.” Participants struggled with much

of the language around rigid motions, including the following examples:

Participant 9: “So since C is here and A is here, then the line CA moves 30 degrees like that.” Participant 10: “So the figure tends to be congruent to its original.” Participant 23: “When something is rotated nothing happens to the thing, it stays rigid. Or it stays. . . the pieces stay congruent.” Participant 24: “Because it says below ‘B is the image of A after a counterclockwise rotation about C.’ So I’m assuming we’re starting here, and then we’d go over there.” Participant 25: “Since B is the image of A, these two are the same points.” Participant 27: “If A was B . . .” Participant 28: “Okay, so image, I would think of like an exact replica, just moved a different way and so I’m trying to relate B to A now.”

Pure Mathematical Issue. For the introductory problem, there were 68 instances of the

PMI code, which were collected over 26 participants (only Participants 4, 15, and 23 did not

demonstrate a PMI with the introductory problem). This result was surprising to me, as I

anticipated difficulty with the problem due to the ambiguity of the task. I believed that many

participants would question whether the words “explain why triangle ABC must be isosceles”

meant they had to write a formal proof for their answers.

The PMI encountered by the participants had a lot in common. I commented on instances

of PMI codes to determine the common areas of difficulty that the participants experienced.

Below, trends in the findings are discussed.

• Thinking that an angle, rather than a point, is being rotated: 12 of the PMI instances

for the introductory problem shed light on the fact that the participants incorrectly

assumed that the angle A was being rotated when the problem never said that. In fact,

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several participants even read the problem aloud incorrectly, stating, “Angle B is the

image of Angle A” when the problem read, “B is the image of A.”

• Thinking that the pre-image was not drawn on the diagram and that it was a triangle:

Several participants drew in what they believed to be the pre-image onto the picture.

This demonstrates a complete lack of understanding of what is described in the

problem statement. For example, Participant 6 said, “I know that the pre-image is

over here” and drew a triangle that they believed represented the pre-image.

• Assuming that the triangle is isosceles based on a visual assumption: Several

participants said that the triangle “looks” isosceles because it looks like AC = BC;

however, it was not given that the picture is drawn to scale, and this assumption

completely disregards the given information.

• Assuming that the described rotation means that two angles are congruent, when in

actuality, it means that two distances are equal. In essence, several participants tried

to use properties of isosceles triangles to prove that the triangle is isosceles (circular

reasoning), rather than apply the given information about the rotation and its

properties in order to prove something about the triangle.

• Participants largely saw transformations in the plane as applying to figures rather than

sets of points. For example, Participant 10 said, “Well, it’s like isometric, so nothing

changes . . . angle measure and side lengths, none of that changes. So, the figure tends

to be congruent to its original.”

• Participants thought that an isosceles triangle has exactly two congruent sides, rather

than at least two congruent sides. For example, the following ensued between

Participants 16 and 17:

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Participant 16: How do you know all three angles aren’t congruent? Participant 17: I don’t! Participant 16: I don’t either. I made that assumption. Participant 17: But I think it’s safe to say that an isosceles triangle can’t be equilateral, but an equilateral triangle can be considered isosceles? Participant 16: I think it has to do with the fact that you’re rotating only 30 degrees. Participant 17: But I was just talking about the—. . . I don’t . . . it’s not equilateral.

• One participant (Participant 24) believed that there was not enough information given

in the problem to determine that the triangle must be isosceles. The participant did not

consider the properties of rotations and said, “The other angles, it could be like both

have to add up to 150 so it could be like one could be 30, it could be isosceles, but

then there’s not enough given information.”

The issues in both mathematical language and mathematical content brought to light by

participants’ responses to the introductory problem demonstrate that most participants were

unable to bridge two concepts that are fundamentally related (properties of rigid motions

(specifically rotations), and properties of isosceles triangles) without support from my

scaffolding questioning during the administration of the content assessment.


General Remarks

Content Assessment Item 2 (Proof 1), Option 1 (The University of the State of New York,

2016) and Option 2 (eMATHinstruction, 2018b), are presented in Figures 2 and 3, respectively.

Seventeen out of 29 participants chose to attempt Option 1, 8 chose to attempt Option 2, and 4

participants chose to attempt both. I reviewed each of the audiotapes to determine participants’

reasoning for selecting one option over the other, in order to help answer the part of Research

Question 2 about teachers’ beliefs and knowledge. The data (see Appendix D) were analyzed for

trends in participants’ knowledge and beliefs.

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Reasons for Participants’ Choices. Several participants communicated that they chose

Option 1 because they believed that it was easier to complete a proof task that was already

partially filled in. For example, Participant 1 stated that they would need to do both to determine

which his easier, but was tempted to start with the first one because “it’s filled in.” Participant 2

provided the reason that “things are written out, and I could make sense of the problem by the

statements given.” Participant 5 chose Option 1, stating,

It’s been a while since I’ve done proofs. So, I like that it tells you what the steps are to proving what they want at the end. Whereas in Option 2. . . I’m not sure what steps I’d have to take.

Participant 6 expressed similar feelings, stating that Option 1 “is kind of giving you the

answer in a way because this tells you exactly what to prove. I wouldn’t have thought to draw

the parallel line.” Participant 7 also agreed that Option 1 had prompts that would “lead me

through the correct steps.” Participant 8 stated that they would choose Option 1, although there

are “pros and cons to both options,” because the set-up of Option 1 looked more familiar from

high school geometry. Participant 17 chose Option 1 because it looked easier.

Participant 4 chose Option 2 due to the freedom it affords the proof-writer, and stated, “I

don’t like when it’s set up like [Option 1] just because I get nervous that I’m not writing the right

thing. Where if it’s in a paragraph form . . . it looks less intimidating to me.” This comment

speaks to the fact that the visual presentation of a problem impacts one’s perception of its

difficulty, and demonstrates the importance of incorporating student choice whenever possible.

Doing so provides assessment data regarding students’ preferences, comfort levels with certain

formats of questions, and confidence. Instead of not answering a question at all, students can

express why they prefer one problem over another.

Participants 9 and 11 both chose Option 2 because they were “looking for a challenge,”

whereas Participant 12 chose Option 1 for the same reason (“I’m more interested in Option 1,

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[because] I never saw this particular proof”). On the contrary, Participant 15 chose the first

option because although they thought both proofs were “easy,” Option 1 required less work since

the statements were already provided. This comparison sheds light on the fact that teachers’

attitudes and beliefs probably have much to do with the proof tasks that they choose to assign to

their students.

Participant 10 described the advantages of both options, but ended up choosing Option 1.

They stated that they “wished [only] the first and the last statement were given” because that

would be reminiscent of high school, when they were given the first and the last and “had to

make up [the rest] ourselves.” This comment implies that some teachers do the important work

of beginning and concluding a proof for their students as standard practice, which puts less onus

on the students to set up and formulate a rigorous proof, as well as realize when their proof is

complete on their own.


Beliefs and Attitudes. There were 17 instances of the beliefs and attitudes code for

Content Assessment Item 2. The participants expressed a range of emotions (for instance,

Participant 1 called part of the problem “so annoying” and expressed that it was “not fair”) and

communicated personal values through their responses to the prompt. Participants 4, 14, 15, and

28 gave the most insight into how a teacher’s beliefs and attitudes impact their performance on

the task and/or their teaching. Below, the exchanges with these participants are highlighted, and

my commentary accompanies each instance.

Participant 4. The exchange proceeded as follows:

Investigator: Beautiful. How do you feel about that? Participant 4: It was good, I just get nervous that I’m taking too long but I don’t know. Investigator: What’s too long? Participant 4: I don’t know.

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Investigator: You’re not taking too long at all. I don’t think there is a too long with this stuff. Sometimes stuff doesn’t come to me and then I come back hours later and I’m like okay I get it. So, this I just want to make sure I’m interpreting what you’re saying right. This proof you chose because you kind of had the freedom to try out whatever you wanted versus here you felt pigeonholed into explaining things you might not be able to explain. Participant 4: Yeah. Like I always think my biggest problem with proofs and just math in general is that sometimes the terminology I use isn’t always exactly correct so then these make me more intimidated just because I feel like even the students that I tutor they’ll be like well my teacher is very particular about wording and I like stuff like this because it’s a little . . . I feel like I get more of an opportunity to explain what I’m doing a little bit better.

Participant 4’s explanation of her choice to complete Option 2 is rooted in her beliefs

about herself as a mathematician. Participant 4 expressed doubt about the completeness and

correctness of her mathematical language and feels that she took “too long” to complete the

proof. This brings to light an expectation felt by teachers to be able to perform proofs

immediately and within a certain amount of time. This can unintentionally be passed on to

students and make them feel incapable of mastering a concept. Participant 4 also commented

about how she believed that different teachers have different standards and expectations for

proof-writing, which is mentioned in the literature as a common sentiment (Mariotti, 2006;

Weiss, 2017).

Participant 14. Three separate exchanges with Participant 14 during the administration of

Content Assessment Item 2 demonstrate how Participant 14’s attitudes about geometry and

proof-writing impeded her ability to make progress on the question. For example, this was

Exchange 1:

Investigator: Okay. You also said when you saw statement two, it reminded you of constructions, you said you don’t like constructions. Participant 14: I don’t like constructions. Investigator: Can you speak a little bit more about that? Participant 14: Anything that’s very spatial, that’s another reason why geometry, I’ve always been like, “Ew,” because I feel like my spatial awareness is not really up there.

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In this exchange, Participant 14 transparently communicated her dislike for constructions,

although the problem in this content assessment item had nothing to do with a construction. The

participant made the incorrect assumption that “drawing a line” into a proof diagram meant that

constructions were being used, and projected negative feelings about constructions onto this

problem.

In Exchange 2, Participant 14 stated,

And it’s something I’ve never enjoyed. Which is something that I like about algebra, that there’s less having to visualize and it’s more like, I like the math where it’s not so much visualization, because that’s something I struggle with. I think I prefer it more when it’s just, this is what you’re given, go from there.

In this exchange, Participant 14 stated that geometry is something she has never enjoyed

and her preference for algebra. It is interesting that in the last line, the participant expressed a

preference for a “this is what you’re given; go from there” style task, but associated that type of

task with her prior experiences in algebra rather than geometry, when this specific language is

explicitly and commonly used in geometry.

Finally, in Exchange 3, I asked, “And now how are you going to word that?” Participant

14 answered, “This is something also I never liked about geometry. It’s like so much writing.”

The participant expresses distaste for geometry again, and generalizes her feelings about

geometry, stating that it requires “too much writing.” It should be noted that the desired response

to my question in this instance was the phrase “supplementary angles.”

Shortly after these interactions, Participant 14 gave up on proving the interior angle sum

theorem for triangles. Through each of the aforementioned exchanges, it is clear that her beliefs

and attitudes towards the subject matter influenced her perceived inability to complete the proof

task. This has implications for students in geometry who are easily and powerfully influenced by

the opinions expressed by their teachers.

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Participant 15. Participant 15 made one comment that communicated her beliefs about

an important topic in the high school geometry curriculum. Regarding the use of auxiliary lines,

the following transpired:

Investigator: Have you ever done something like this with your students? Participant 15: No. Investigator: Okay. Participant 15: Yeah, they don’t need to know that.

This exchange is particularly illuminating because it shows how a teacher’s lack of

knowledge of the curriculum can directly impede student learning and achievement. Not only are

auxiliary lines a fundamental and important tool for completing proofs, but also routinely needed

for completing common core aligned tasks, such as the one in Content Assessment Item 2. This

task is the proof of a fundamental theorem in geometry (the triangle interior angle sum) that is

often explored via hands-on manipulatives in middle school grades, and later formalized via

rigorous proof. Participant 15’s erroneous statement that students “don’t need to know” about

auxiliary lines leaves room to question what other topics are overlooked by geometry teachers.

Participant 28. Participant 28 expressed joy after completing Content Assessment Item 2.

Participant 28: That was cool. So, I would actually . . . take this one over Option 1. Investigator: Because you feel more confident about your reasoning. Okay. Awesome. Participant 28: That was— Investigator: That was— Participant 28: That was a fun one. I liked that one.

Participant 28, despite having significant trouble completing the proof task and using an

unconventional method, enjoyed the task and called it fun. I had encouraged Participant 28 to

keep going by calling her response creative and complimenting her for thinking of a starting

point, despite not initially knowing how to start. This demonstrates the power of positivity in

completing mathematical problems that initially pose as difficult. The participant was

encouraged by her own creativity and persevered through a problem that she first doubted her

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own ability to complete. This is a powerful demonstration of how attitudes towards proof-writing

impact successful completion, and how teachers’ attitudes and beliefs can play a direct role in the

learning going on in their classrooms.

Issue of Correspondence Between Substance and Notation. There was only one

instance of the ICSN code for Content Assessment Item 2, and it was a negative instance.

Participant 20, who chose proof Option 2, said, “Since A and B are congruent, that’s really

saying that twice the measure of angle A is congruent to angle BCD.” Participant 20

demonstrated a lack of understanding of the seemingly subtle language differences in stating that

two angles are congruent, and that the measures of those angles are equal. This distinction is

important because it can directly impact the precision of students’ proof writing and

mathematical language.


for Content Assessment Item 2. Eight of those instances were of participants expressing

understanding, 11 of participants expressing self-doubt, and four mixed instances (see Appendix

D for a complete list of these instances). This section discusses some of the ESD instances in

conjunction with the entire collection of mixed instances.

Three of the 11 ESD codes were instances of participants not knowing whether their

reason for a geometric assertion was good enough. Participant 1 stated:

Participant 1: It’s saying, we have to draw that they’re parallel. Investigator: Okay. Is there a reason that you know you can do that? Participant 1: Yes, because they never intersect. Investigator: Okay. So why do you seem hesitant in that answer? Participant 1: Because it’s not a good reason.

The exchange with Participant 2 went as follows:

Investigator: For two . . . do you know why you are able to draw a line parallel to AB? Participant 2: Do I know why? Because I could just extend the line from point C.

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Investigator: Okay. Participant 2: I could write that?

Finally, Participant 6 stated,

If I had to write something, if I was forced to, that’s what I would write ‘cause that’s what I think makes the most sense. I don’t know if there’s a better, answer though.” In all four instances of the mixed (EU/ESD) code, the participants expressed understanding of a mathematical idea followed by self-doubt in the expression of the justification for the idea.

The following excerpts are similar in nature to those highlighted from the ESD codes,

because they all express similar doubts regarding the completeness/ correctness of a written

justification. For example, Participant 1 said:

Participant 1: Yes. Oh. I got it. I got it. Investigator: What do you got? Participant 1: ‘Cause a straight line has 180 degrees. Investigator: Oh. Participant 1: But how do I say it?

Participant 5 said:

Oh okay. So, okay. And then angle one plus angle two plus angle three is 180 degrees because well, if they’re congruent and we already said that all of them add to 180 degrees, then when they’re inside the triangle, they’ll also add up to 180 degrees. Investigator: Okay. Participant 5: I don’t know how to write that.

Participant 7 expressed doubts in this way:

They are on alternate . . . well first they’re in between two parallel lines, that was given in two. So now you can form angle relationships and AC acts as the transverse. So these are I guess definition of alternate interior angles. Alright, so then four is the measure of AC, ACD plus the measure of angle two plus the measure of angle BCE. BCE, is 180. I mean, they’re supplementary because they’re on a line, right, so this whole line is 180. I’m not sure what I would write here, though.

Finally, Participant 12 stated, “I’m just trying to remember, is there like a name for that

postulate?”

The above excerpts from the data demonstrate participants’ overall uncertainty with

writing justifications for geometric propositions. Of course, this brings to light the question of

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how impactfully their students can be learning to write proofs with well-developed justifications,

but another question also comes to light about the preparation of the teachers. In the literature, it

is expressed that teachers feel unease with writing justifications because there do not exist

standards which would help to determine whether a response is “good enough.” This provides

the impetus for a suggestion of a specific set of standards for geometric proof writing which can

accompany the common core state standards for geometry.

Mathematical Language. There were 19 instances of the mathematical language code

for Content Assessment Item 2. Five were positive instances and 14 were negative instances

(meaning the participants expressed erroneous or imprecise mathematical language). Although

all of the instances are presented in the data (see Appendix D), this section focuses on a subset of

the negative instances of the mathematical language code within this content assessment item, in

order to identify gaps in participants’ knowledge and later form suggestions for improvement.

Two participants (Participants 2 and 5) struggled to come up with the term “substitution”

and instead called it “congruence.” Participant 5 later corrected the language, after first switching

to “equivalence.” This brings back into focus the previously mentioned ICSN code once more,

because it shows that another participant was unaware of when it is appropriate to use the term

congruence versus equivalence. Participant 15 had a similar issue: “So measure of angle one is

congruent to measure of angle ACD. Measure of angle three congruent to measure of angle

BCE.”

Other examples of imprecise or incorrect mathematical language are highlighted below:

Participant 6: So the measurement of angle one is equal to the measurement of angle ACD and the measurement of angle three is equal to BCE. That’s alternate interior angles. Investigator: How do you know that? Participant 6: I remember in middle school someone taught me to look for the Z.

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Participant 21 also related alternate interior angles to the nonmathematical explanation of

“looking for a Z”:

Participant 21: Okay. So Angle 1 is equal to ACD and Angle 3 is congruent to BCE, so I’ll pick up a highlighter to show my parallel lines. Investigator: Okay. Participant 21: And I’m going to show my Z’s, so my Z’s are representing my alternate interior angles. And by definition, my alternate interior angles are congruent.

Participant 12 provided a more mathematical explanation of alternate interior angles than

Participants 6 and 21, but still was unable to fully explain where alternate interior angles are

located in relation to a transversal:

Investigator: Okay. How do you know that those are alternate interior? Participant 12: By definition, if you have a pair of parallel lines and you have a transversal cut through them, interior means it’ll be inside the two parallel lines. It’d have to be between them. Investigator: Okay. Participant 12: And alternate means . . . I guess it’s kind of synonymous to opposite, so if one’s on the left inside of the top line, it’d be on the right inside of the bottom line.

Continuing on the topic of alternate interior angles, Participant 14 used an incorrect and

nonexistent term (alterior): “From there you can see different angle relationships, like alterior

and interior angles, vertical angles.” Although not about alternate interior angles, Participant 18

also used a nonexistent term (for expressing a group of three angles which lie on a line):

Participant 18: What do they call the supplementary? What’s the word? Investigator: If two angles are on the line? Participant 18: Yeah, there was a word for this. Investigator: Linear pair? Participant 18: Linear pair, yeah. I used to say linear triple.

Participant 20 was unable to correctly express the statement of the exterior angle theorem

for triangles:

Participant 20: Oh. Then I know that angle BCD, is equal to the sum of A and B. Investigator: How do you know that? Participant 20: Because the exterior angle of a triangle is congruent to the opposite interior angles.

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Whether the participants were expressing the statements of theorems in incorrect ways,

inappropriately using one term in the place of another, or making up their own terms, the use of

imprecise and inaccurate mathematical language is concerning because it raises the question of

whether students are taking incorrect notions away from class. Mathematics teachers are

traditionally the highest form of authority within mathematics classrooms, and therefore need to

be role models for precise and correct mathematical language.

Pure Mathematical Issue. There were 24 instances of the PMI code for Content

Assessment Item 2; however, the 24 instances were not distinct. The instances of PMI codes can

all be found in Appendix D, and a subset of them (those that surfaced for more than one

participant) are grouped accordingly and discussed in this section.

Incorrect Marking of the Proof Diagram. Participants 13 and 19 both chose Option 1

and marked their diagrams incorrectly (they indicated that two not-necessarily-congruent angles

were congruent by placing one arc on each of the angles). Although Participant 13 did fix the

markings of the diagram as a result of my questioning about the error, Participant 19 did not. The

questioning which resulted in Participant 13’s successful correction is reproduced below:

Investigator: Okay. Can you explain to me the markings that you just made on the diagram? Participant 13: Oh. I have to fix that. Investigator: What did you have, and what did you change it to? Participant 13: I had two angle marks on angles ACD and angles BCE, but after you asked me about the markings, I noticed that if I left it that way, that would mean that angle ACD and the measure of angle BCE are congruent.

Circular Reasoning. Participants 1, 2, and 3 all ran into an issue of assuming what they

were trying to prove (circular reasoning). Participant 2, who realized her own error via thinking

out loud, was still unable to move forward without my help:

Participant 2: Measure of angle 1 plus measure of angle 2, plus measure of angle 3 is 180 degrees. It’s because all of the angles add up to 180, but I have to prove it from using the reasoning before. I can’t write that as proving it because that’s...

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Investigator: That’s what you’re trying to prove. Participant 2: Correct. Okay, one, two, three. D, C, E. Okay, measure of angle 1 equals measure of angle 2, equals measure of angle 3 is 180. Now we have measure of angle 1 equals ACD. Measure of angle 3 equals BCE. Measure of ACD plus the measure of angle 2 plus the measure of angle 3 equals 180. I need help.

Not Knowing Euclid’s Parallel Postulate. Participants 3, 5, 6, 10, 12, 15, 17, 19, and 29

were unaware of the existence of Euclid’s parallel postulate and unable to reason in the moment

about its existence or purpose in proving the theorem. This knowledge was necessary for

successful completion of the “reason” for Statement 2 in proof Option 1. The participants’

remarks which support the aforementioned claim appear below:

Participant 3: I don’t know what the reason would be, I guess I would just write . . . I don’t know what I would write, actually.

Participant 5: Right. So, isn’t it just the definition of parallel lines? Wouldn’t I just write that?

Participant 6: Through point C draw DCE parallel to AB. I wanna say this is just a definition of a parallel line.

Participant 10: I don’t know the reason for point 2. I don’t know how to explain it.

Participant 12: I think it is possible because, since it’s a triangle, I just don’t know how to state it as a reason. I’d have to think about that, but I know one of it is that because side AC and side CB are not parallel. So, they intersect out of point C. Since those two are not parallel, you can perhaps . . . Yeah. So in any triangle, acute, obtuse, or right, you can always make a parallel line to the opposite side of that vertex.

Participant 15: I mean we don’t have a specific reason we can use for statement number two right?

Participant 17: The statement given was, “Through C, draw DCE parallel to AB” and I didn’t have an adequate reason as to why I would just draw the line. Now, I know why the line is drawn, but, in terms of this format, I’m not comfortable with say . . . It’s not a given and it’s not anything mathematically that I would say to draw this line.

Participant 19: We can draw this line because of creativity. The line makes it easier to prove it. I’ll come back to that.

Participant 29: Sure. So, the only thing that was confusing me was for number two, statement number two. Through point C, draw DCE parallel to AB. I don’t understand how to write a reason for that. It just seems like we’re adding something on that we couldn’t use, so I don’t know what I would write as a reason.

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Approximately one-third of the participants were unaware of a fundamental and

foundational concept in Euclidean geometry, raising questions about not only their confidence

and preparedness to teach the subject, but also their students’ subsequent learning.

It should be noted that Participants 13, 21, and 23 knew the concept of Euclid’s parallel

postulate (they were able to state that there is only one line through point 𝐶 that is parallel to

𝐴𝐵0000), but were unable to name it.

Participant 13: The reason would be that there is only one line that goes through point C that would make a line that is parallel to AB.

Participant 21: That is a rule that you could draw any auxiliary line through a point.

Participant 23: I guess . . . I don’t think this would be . . . Okay . . . I think the reason I might give is with geometry, through any point not on a line you can construct one parallel line. I’m just gonna say that and call it a day.

One of these three participants (Participant 13) stated that she did not learn about the

parallel postulate until college, and asserted that she did not learn it in a mathematics course, but

rather in a mathematics education course. This sentiment was supported by Participant 10, who

recalled “Euclid’s postulates” after my interference and questioning later on in the interview, and

also stated that she had first been exposed to them in a mathematics education course in college.


General Remarks

Content Assessment Item 3 (Proof 2), taken from eMATHinstruction (2018a), is

presented in Figure 4. For this content assessment item, the participants were told that the use of

the statement-reason chart was optional. Of 29 participants, 18 attempted to write a statement-

reason style proof (although only 10 of the 18 were able to complete a proof, and not all

“complete” solutions were fully correct), four attempted to write a paragraph proof, one

attempted to write both, and six left their papers blank (or nearly blank). This means that 62% of

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participants felt more comfortable attempting to prove that △ 𝐴𝐸𝐷 is isosceles using a statement-

reason format (although only 34% were able to use a statement-reason chart to successfully

complete the task), and 21% of participants were unable to make any progress towards writing a

proof, despite the help of my scaffolding questions.

One of the key realizations necessary for successful completion of this proof is that △

𝐴𝐸𝐶 and △ 𝐵𝐸𝐷 are overlapping triangles. Drawing these two triangles separately from the

original diagram is a useful tool in creating a proof plan. Of 29 participants, five used this

strategy, and four of those five wrote fully correct or almost fully correct proofs. The remaining

one of five participants who drew separate triangles (Participant 14) was one of the six

participants who left the paper nearly blank (despite having correct congruence marks on her

paper on the newly formed separate diagrams, the participant was unable to write any parts of the

proof). Some participants did not realize that “∡𝐸” is not the same angle in △ 𝐴𝐸𝐶 as it is in △

𝐵𝐸𝐶. This issue was exacerbated in the cases of the participants who did not draw separate

triangles to assist them with the completion of the proof.


Although the different codes for Content Assessment Item 3 shed light on difference

facets of the participants’ confidence and preparation, there was a common theme of comments

about congruence criteria within each of the five collections of coded instances. This section

delves into the findings for Content Assessment Item 3 and specifically mention the prevalence

of congruence criteria within the coded responses of the participants.

Beliefs and Attitudes. Content Assessment Item 3 contained 27 instances of the Beliefs

and Attitudes code. All instances are recorded in Appendix D, and some excerpts are recorded

and discussed in this section.

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Participant 14, who was one of six participants that left their papers blank or nearly

blank, made negative comments about the task immediately upon seeing it. She sarcastically

said, “Why do they do this to children? I don’t remember ever seeing something like this in high

school. . . . Oh, yuck.” This raises the question of whether Participant 14 could have progressed

through the task, given a change in attitude.

Participants 1, 7, and 11 all expressed the belief that they were ill-prepared to become

mathematics teachers because of their perceived inability to complete the proof task. Participant

1, when asked how she felt about her response to the task, said, “Horrible.” When asked to

explain that feeling, she added, “I can’t write down what I think. I’m becoming a math teacher.”

Participant 7 said, “I feel like I’m skipping stuff. I feel like I can’t just state this stuff. If my

students did this I, would say that there’s something that’s missing, but I don’t know what it is.”

Participant 11 expressed similar sentiments, saying, “The reason I didn’t like geometry in high

school, was because I didn’t know what to write.”

Several participants expressed the belief that teachers have different sets of standards for

geometric proof writing, and that this lack of standardization leads to not knowing if a proof is

complete, or not feeling confident in their completed proofs. For instance, Participant 21 (who is

an in-service geometry teacher) said,

Honestly, I think I did all right. But to be honest with you, sometimes with my reasoning, I don’t know if it’s sufficient enough based on the Regents rubric or based on what other teachers write, or based on the definitions, because I’ve learned so many different ways. I remember when I was in middle school, my teacher would just tell us write “by the definition of a midpoint” or “definition of a right triangle.” Whereas once I started tutoring, I saw teachers would be like, “No, you have to write it all out.”

Regarding congruence criteria, there were several illuminating instances wherein

participants expressed beliefs and attitudes about the teaching and learning of the different

criteria. For example, the following exchange occurred with Participant 10:

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Investigator: Interesting. But you’ve never written out a proof of why side-angle-side works, and why angle-angle-side works, but that angle-side-side doesn’t? Participant 10: I feel like we talked about it, but I can’t remember writing it. I mean, maybe I did. Investigator: As a future secondary teacher, do you think that that is something that you would want your students to know? Participant 10: Yeah. Because we just memorized everything. We never really understood why something was the reason it was.

Similarly, Participant 11 stated,

I want to say I was just told, that these were the methods. It was on our graphic organizer, and you just kind of . . . And they’re like, okay, these are the methods. And then, to never use angle, side, side.

A final example occurred with Participant 15:

Investigator: Okay, do you know why . . . so for example you wrote here, side angle side, do you know why side-angle-side is sufficient or angle-side-angle, or side-side-side. Has it ever been shown to you why those are good cases? Participant 15: No. Investigator: Okay, do you think that you should know that as a teacher? Or do you think it not necessary? Participant 15: I mean, it’s not necessary. Investigator: Okay, why not? Participant 15: To be honest, I don’t know why we need to prove two triangles are congruent, so why do we need to know, the reason behind the reasons? Investigator: Okay, so you’re saying you don’t really see the benefit of students doing this? Participant 15: I mean they can do proof, but like the part of proof two triangles are congruent . . . Investigator: You don’t see much value in that? Participant 15: Yes.

All three of the aforementioned participants mentioned that they were told to memorize

the permissible congruence criteria in proof-writing; however, as demonstrated by the above

excerpts, their beliefs about whether it is necessary to teach this in a more effective way to

students varied.

Issue of Correspondence Between Substance and Notation. There were three instances

of the ICSN code for Content Assessment Item 3 (one positive, and two negative). All three of

these instances were related to substance versus notation—in the negative instances, the two

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participants were unsure of whether to say “congruent” versus “equals” and “measures” as

mentioned within the context of congruence. On the other hand, in the positive instance,

Participant 23 demonstrated knowledge of the difference when the participant self-corrected

mid-sentence: “I’m subtracting AC minus BC is equal to, congruent. . . . Oh no, no. I’ll say

congruent, no, equal to BD minus BC.”


for Content Assessment Item 3, broken down into five instances of EU, 11 of ESD, and two

mixed instances. The excerpts highlighted in this section focus specifically on the permissible

congruence criteria, and the other instances are presented in Appendix D.

Supporting what will be mentioned in the coming section on the PMI code, only two

participants demonstrated understanding of why “angle-side-side” is not a valid congruence

criterion. The first was Participant 15:

Investigator: Okay, what can’t you use? What are you not allowed to use? Participant 15: A-S-S. Investigator: Okay, do you know why? Participant 15: For A-S-S . . . I mean if you have two sides and a non-included angle, you can draw more than one triangle. The triangle is not unique. Investigator: Okay, and when did learn that? Do you think you knew that as a high school student? Like the reason that you can’t use A-S-S? Participant 15: No, when I taught precalculus, as a student teacher. Investigator: When you taught precalculus as a student teacher? Participant 15: Yeah. Investigator: Okay, so that came up with the law of sines? Participant 15: Law of sines ambiguous case- Investigator: And the use case. Oh interesting, so now you’re using that knowledge as a geometry teacher? Participant 15: Yeah.

Participant 23 also demonstrated understanding of this concept:

Investigator: Okay, and what are other methods that you know of to prove triangles congruent Participant 23: You could use side, side, side . . . Angle, angle, side . . . There’s some other ones. Hypotenuse, leg if you had that’s a right triangle. Investigator: Do you know what case you cannot use?

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Participant 23: You can’t use A-S-S. Investigator: Do you know why? Participant 23: Because when you have two sides and a non-included angle there could be the ambiguous case and so you could have two triangles.

The excerpts above support the findings of the BAA code for this content assessment

item, in which participants expressed that they were told to memorize the permissible

congruence criteria and had not built their own meaning of these concepts prior to becoming

mathematics teachers.

Mathematical Language. There were 20 instances of the mathematical language code

for Content Assessment Item 3 (six were positive, and 14 negative). Selected excerpts are

highlighted and discussed below, and the contents in their entirety are expressed in Appendix D.

Four participants (4, 7, 15, and 17), when asked to state the meaning of CPCTC, stated,

“Congruent parts of congruent triangles are congruent.” Participants 13 and 19, originally also

having made this same mistake, later corrected it. Participant 13 stated,

And then through congruent parts of congruent triangles are congruent, we can say that DE is congruent to AE. Investigator: Okay. Participant 13: And therefore, triangle AED is isosceles. Investigator: Okay. Participant 13: Oh, corresponding parts. Investigator: Where does the corresponding go? Participant 13: In the first. Investigator: Okay. What do you think made you catch that? Participant 13: Because I was reading it as I was writing it, and I was like, “How do I already know that they’re congruent parts?” Investigator: Okay. Participant 13: That’s what I was trying to prove, and then I was like, “Oh, corresponding.”

Participant 19 stated,

Oh, hey, no no. No. It’s something else, not congruent. Investigator: Which one is not congruent? Participant 19: The first one. Investigator: The first C?

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Participant 19: The first C is not congruent. Oh, “corresponding parts of congruent triangle are congruent.”

Not all participants knew the accurate meaning of this acronym, even though they used it

as a part of their proofs. In addition to numerous issues with expressing the meaning of CPCTC,

I found that a subset of participants (Participants 1, 7, 9, 13, and 16) did not utilize correct

mathematical language for expressing what happens when using the reflexive and subtraction

postulates together to subtract the lengths of overlapping segments. My interaction with

Participant 9 appears below:

Participant 9: Okay. So they told us that AC and BD are congruent, and both these lines share BC, so if you subtract BC from both of these, then we get that AB is congruent to BD. Investigator: Okay. Participant 9: I think there’s a name for it, but I don’t remember. Investigator: A name for what? Participant 9: This . . . That if they share part of the line segment you can subtract it or something.

Although Participant 9 demonstrated a correct understanding of the concept, there are

issues within the mathematical language pertaining to substance versus notation (specifically

related to congruence), the reflexive postulate, and the subtraction postulate. Participant 9

expressed an awareness of the incorrectness of the language, but was unable to think of the

correct terms.


Assessment Item 3. Three categorizations of pure mathematical issues are discussed in this

section, and the raw data are presented in Appendix D.

Inclusivity of Isosceles Triangles Within the Definition of Equilateral. Three

participants (Participants 1, 2, and 13) struggled with Content Assessment Item 3 in part because

they were unsure of whether an equilateral triangle is also an isosceles triangle. That is, they

were unsure whether an isosceles triangle has at least two, or exactly two, congruent sides and/or

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angles. This led to a perceived roadblock in their proof processes because they struggled to move

forward after questioning whether an equilateral triangle would classify as isosceles. For

example, Participant 1 said,

I think angle A and angle D should be congruent but then I also have to . . . I think I also have to prove that angle E is not the same as angle A and angle D. Because if angle E is the same then it’s gonna be equilateral.

Making Assumptions Based on the Diagram. Participants 28 and 29 both made visual

assumptions based on how the diagram in the content assessment item looked, despite both

acknowledging that they know this approach is not permissible or mathematically justified. The

exchange with Participant 28 went as follows:

Participant 28: Well, if anything, I would say BEC is isosceles. Investigator: How would you know that? Participant 28: Well, the reason why I would know that . . . I mean, visually, it looks like an isosceles. It doesn’t look like a . . . AED looks like an equilateral. Just visually it looks like it, and I know you can’t use visually in math; however, that’s just I what I do, and the BEC looks like an isosceles just because the fact that it’s thinner than the equilateral triangle, than AED, and it just . . . all sides are equal. BC is smaller than BE and CE. So visually, that’s just how I look at it. And then what I would say is . . . well this one’s a little tricky because if I had . . . if it said like . . . if it said AD is congruent to AE, then I would know those two sides are exactly the same so that means . . . and that would prove as isosceles, not an equilateral but then I would also feel that . . . well, I’m stuck.

Participant 29 explained the visual assumptions this way:

Investigator: Okay. Is there any reason why you’re leaning towards those angles? Participant 29: Honestly? Investigator: Mm-hmm (affirmative). Participant 29: I know it shouldn’t be, but it’s the picture. Investigator: Just visually Participant 29: Yeah, visually, it’s making me lean that way. And even though I know I shouldn’t, now I’m thinking of . . . It’s . . . I think if E weren’t where it’s placed, then it wouldn’t be isosceles, so maybe that is a good way of thinking. I don’t know. I am lost.

It is important to note that neither participant, following the making of visual

assumptions, was able to complete the proof task.

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Congruence Criteria. Once again, the issue of congruence criteria arose as a pure

mathematical issue which impeded participants’ understanding of the task and, consequently,

their proof writing. My interactions with Participants 5, 10, 11, and 17 support the earlier

findings of the other codes related to this issue and add several new considerations: Participants

were unaware of the different criteria; unsure of the difference between congruence criteria and

similarity criteria; unsure of why the congruence criteria were sufficient; and unsure of why

angle-side-side was insufficient. Participant 5 said,

Investigator: So, do you remember how to prove triangles are congruent? Participant 5: Angle side angle. Investigator: Ah. Is that the only one? Participant 5: Or side angle side.

The exchange with Participant 10 went as follows:

Investigator: Another thing you mentioned is side-angle-side. So you can use that to prove that two triangles are congruent. Do you know why you can use that? Why side-angle-side is sufficient to say that the triangles are congruent? Participant 10: Honestly, no. Investigator: Okay. So that’s just something that you’ve kind of— Participant 10: It’s just ingrained. Investigator: Okay. Do you know the other congruence methods? Participant 10: Side-side-side, side-angle-side, and then angle-angle-side. Hypotenuse-leg, but that’s only specific cases. Not angle-angle-angle. Investigator: So angle-angle . . . what can that give you? Because that’s something else. Participant 10: I don’t remember.

On this topic, Participant 11 said,

Participant 11: Side, side, side, angle, angle, angle. I know I’m not supposed to use angle, side, side. Investigator: Okay, do you know why? Participant 11: I’m sure I’ve done it before, I just don’t remember it right now.

Finally, I had the following exchange with Participant 17:

Investigator: Do you know, or have you ever proved why something like side angle side does work? Participant 17: Have I proved it? Yes. Do I remember it? I could probably figure it out, but I wouldn’t be able to say it right now.

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General Remarks

For Content Assessment Item 4, taken from a common core geometry Regents Exam

(The University of the State of New York, 2015; see Figure 5), 11 out of 29 (or 38%) of

participants left their papers blank or nearly blank. Moreover, only 10 out of 29 (or 34%) of the

participants were able to complete the proof with completely correct (or almost completely

correct) responses. Given the number of participants who struggled with the problem, I asked

many of the participants what they would need in order to be more successful and confident in

approaching Content Assessment Item 4. Several of the participants expressed that they would

need to study the distinguishing features of quadrilaterals prior to teaching a unit on this topic.

For instance, Participant 20 said they would have to do “research” before teaching quadrilateral

proofs, about “being able to prove them and actually know all the properties of the different

quadrilaterals.” Participant 14 expressed a desire for a graphic organizer that laid out the

relationships between the quadrilaterals:

Participant 14: Like kind of like the Regents graphic organizer, they love to give. Investigator: Okay, what does that organizer have on it? Participant 14: Like, ‘to prove something is a parallelogram you need to check these boxes. . . to prove something’s a square it has to be this.’ You know what I mean? List of all parallelograms and their relationships, like a square is a rectangle. Stuff like that.

Participant 16 stated that they wanted to be provided a definition: “I would want the

definition of a rhombus. I would want the parts of a rhombus that make it uniquely a rhombus.”

The collection of comments that express the participants’ unease regarding the relationships

within the quadrilateral family is contained within Appendix D.


Beliefs and Attitudes. There were 25 instances of the BAA code for Content Assessment

Item 4, spread out over 15/29 participants. The BAA-coded instances for this content assessment

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item were mostly negative, with participants expressing a lack of confidence or a general dislike

for quadrilaterals. This section specifically highlights the attitudes expressed by various

participants regarding this proof task.

Participant 1: This is a waste of time. Participant 3: It looks scary at first. Participant 4: Oh gosh. . . . I just get nervous. Participant 10: I’m just worried that I don’t really know the material inside-out. Participant 13: I’m not confident with the properties of quadrilaterals in general. Participant 16: I have no freaking idea how to do this. Participant 19: But, I feel like this question is going to take me a long time to think about it because I have no clue what’s going on. Participant 27: I don’t like rhombuses. . . . I hate rhombuses. Participant 28: I don’t have enough confidence.

There was widespread unease regarding the task, not just regarding the proof-writing

(which participants also expressed unease about), but additionally the specific content knowledge

required to distinguish one type of quadrilateral from another.

Issue of Correspondence Between Substance and Notation. There were no instances

of the ICSN code for Content Assessment Item 4.

Expressing Understanding or Self-Doubt. There were seven instances of the EUSD

code for Content Assessment Item 4. Four participants expressed understanding, two expressed

self-doubt, and one expressed a combination of understanding and self-doubt. Although all of the

data are contained in Appendix D, this section focuses on Participant 21, who expressed a

mixture of understanding and self-doubt as she progressed through this proof task.

Participant 21 expressed understanding when she stated that she knew the properties of a

rhombus. When I questioned how she knew those properties, she stated that it was because she

“just taught this unit.” Although Participant 21 was confident regarding her knowledge of the

properties of a rhombus, she expressed self-doubt in two instances regarding the writing of her

proof. Instance 1 went as follows:

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Investigator: So, you’re wondering would it be sufficient for reason for you to write “definition of parallelogram.” Participant 21: Yeah. . . . I never know.

In Instance 2, Participant 21 stated,

Participant 21: And now I think I’m done. Investigator: Why are you done? Participant 21: Because I’ve now shown that two consecutive sides of a parallelogram are congruent. Investigator: Okay. And does that make it a rhombus? Participant 21: I think so. . . . But I don’t think it’s sufficient.

Although Participant 21 stated earlier in the interview that she had just taught a unit about

quadrilaterals, she was unable to judge whether her proof about one such quadrilateral was

sufficient and complete.

Mathematical Language. There were five instances of the ML code for Content

Assessment Item 4, all which were “ML” instances. None of the instances were particularly

illuminating regarding the specific content of this assessment item, but it is important to note that

two participants (6 and 17) both referred to the term “reflexive” as “reflective.”


Assessment Item 4. Two main pure mathematical issues surfaced amongst the participants that

are discussed in this section; all coded instances are presented in Appendix D.

The first pure mathematical issue encountered was that since the piece of given

information that 𝐴𝐵𝐶𝐷 is a parallelogram was not explicitly stated as “given,” participants

skipped over this during the reading of the problem and thought they had to prove that 𝐴𝐵𝐶𝐷

was a parallelogram. For example:

Participant 29: I mean, then we have that those two sides are congruent. I don’t feel like it’s enough . . . to prove that it’s a rhombus. Just because those two sides are congruent, we leave out the fact that they need to be parallel. Not those sides, but the opposite sides are parallel, and that the other two are also congruent. Investigator: Can you list all the givens in this problem? Participant 29: Yeah. So, the ones that are stated in the problem?

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Investigator: Mm-hmm (affirmative). Yeah, anything you’re given. Participant 29: BE is perpendicular to CED. DF is perpendicular to BFC. CE is congruent to CF. Then, also, what I can deduce from that . . . Okay. Investigator: Are those the only givens? Or is there anything else that’s given? Participant 29: Oh my God. Parallelogram. ABCD.

Participants 3, 8, and 14 also made the same mistake and got stuck because they thought

they had to prove that 𝐴𝐵𝐶𝐷 is a parallelogram before moving on to prove that it is a rhombus.

The second pure mathematical issue that many participants ran into was making

erroneous statements about distinguishing features of quadrilaterals and relationships within the

quadrilateral family. Participants 1, 2, 5, 8, 9, 11, 14, 18, and 21 all struggled in this area. I have

included some excerpts, beginning with Participant 5:

Participant 5: I don’t know what a rhombus is. Investigator: What do you mean when you say you don’t know what a rhombus is? Participant 5: Because I . . . Is it a square? Is it a rectangle? I feel like this is something we’ve talked about in grad school and I still don’t know. So I picture a diamond in my head, but that’s an over-generalization, right?

Participant 9 stated,

And why . . . Now I’m just saying this just because we had this conversation, but why a square is a square, why it may or may not be considered a rectangle, because I don’t know. Investigator: Okay. Participant 9: Or why a rhombus isn’t a parallelogram. Because when I was a student, I always felt like a rhombus and a parallelogram are like they’re the same thing. Investigator: Why? Participant 9: Because they’re slanted quadrilaterals.

The exchange with Participant 11 was as follows:

Investigator: And do you know properties of a rhombus? Participant 11: I think that their diagonals are not congruent . . . Oh no, wait. No, no, no, no, I have to back out. Wait, wait, hold on. No. In a rhombus, all four sides are congruent. But, in a parallelogram, diagonals are not congruent . . . Investigator: Okay. Participant 11: Because not all four sides are congruent, which means, if it’s stretched then the diagonals are not congruent. Yeah. Okay. For a rhombus, since all their sides . . . Oh no. Can that be stretched? I’m not sure. Now, I’m just confusing myself. I think the diagonal part is, I don’t know if they’re congruent. I think, I’m thinking I can’t just prove that the sides are congruent, because that’s just a square.

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Investigator: Okay. Participant 11: And, I think, I have to prove that opposite angles are congruent. And then I’m also, it’s not . . . I already know it’s a parallelogram. I already know that. . . Since it is a parallelogram, I feel like I already know that their opposite angles are congruent. Investigator: Okay. Do you think easier or harder to make a plan for this proof than the last one? Participant 11: Definitely [harder]. I think it’s because I don’t know enough. I don’t know what I need to prove that it’s a rhombus.

Participant 14 commented,

Or do I have to prove, I always confuse is a rhombus a square. I always forget that. Like I think of my geometry teacher in my head, but I always forget. And then it’s also if it is a square, do I have to prove that it has exactly four right angles, which I don’t think I have to.

Finally, Participant 21 said,

Investigator: Okay. So you wrote down all these properties of a rhombus. How many of them do you have to prove? Participant 21: One. Investigator: Any one? Participant 21: No, because all sides are congruent. That could prove a square. Investigator: If something’s a square, is it a rhombus? Participant 21: If something is a square is it a rhombus? No.

It is clear from the excerpts presented that the preservice and novice teacher participants

struggled to state the relationships between quadrilaterals as well as their distinguishing features.

This added challenge impeded their proof-writing in these instances.


General Remarks

Content Assessment Item 5 (Proof 4) is shown in Figure 6 and came from

eMATHinstruction (2018b). Most participants (23/29) left this question blank or nearly blank;

only Participants 3, 6, 12, 15, 18, 23 attempted it. Of the six participants who did nearly or fully

complete it, the process was not necessarily smooth (for example, Participant 3 persevered

through the proof task but needed my help every step of the way). Although some participants

simply ran out of time, others expressed discomfort with the problem to the point that they chose

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not to attempt it. All the data are contained in Appendix D, and this section details some specific

findings based on the coding system used to analyze the data.


Beliefs and Attitudes. There were 10 instances of the BAA code for Content Assessment

Item 5. These instances revealed a general dislike for circles. For example, Participant 1 said she

“hates” circles; Participant 2 said she was “finished” upon seeing the word “secants”; Participant

18 said, “I frankly don’t care”; and Participant 20 said, “Please don’t make me do this.” This

negative display of attitudes amongst preservice and novice teacher participants raises the

question of why circles in particular pose such problems and discomfort as an area of geometry

requiring proof.

Issue of Correspondence Between Substance and Notation. There were no instances

of the ICSN code for Content Assessment Item 5.

Expressing Understanding or Self-Doubt. There were seven instances of the EUSD

code for Content Assessment Item 5; six instances of EU and one of ESD. There was a common

trend found in the instances of the EU code for this Content Assessment Item. All six of the EU-

coded instances contained participants expressing that they knew they needed to use the concept

of similar triangles (see Appendix D). Although most of these instances also included the

participants’ sentiments that they only knew this because they realized they needed to end up

with equal products of means and extremes, they still were able to either discuss their proof path

or actually write a solution due to the entry point that their understanding provided.

Mathematical Language. There was one negative instance (ML-) for Content

Assessment Item 5, within Participant 28’s explanation of her thinking. What is interesting is that

the language expressed contained many errors, but Participant 28 was one of only a few

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participants who was able to think along the correct lines about the problem. An excerpt from

Participant 28’s interview is below:

Oh, that’s a good question. What would be similar? Oh! Well, okay, this would sound, I don’t know if it sounds stupid but the distance from H to C, I think, would be similar to the distance from B to D if that makes sense or, I mean, from H to D. So, it’s almost as if this, the HC, is almost in ratio with the HD so that’s what I would assume ‘cause I don’t really know how to solve this. So because if they’re in, I would say, similar ratio, I would also assume that HB is in ratio with HE and that’s what I would assume and that’s why I would say that they’re equal but I wouldn’t know how to . . . if there was any other reason, I wouldn’t know how. I don’t . . . see. I mean, I’m sure that there’s triangles involved in angles because the fact that they’re inscribed in the same angle, that means that their angle measure is the same and then . . . oh yeah.

The above excerpt contains a variety of errors in mathematical language, but

demonstrates that Participant 28 is thinking along the correct lines of the necessary steps to

proving the theorem in Content Assessment Item 5.

Pure Mathematical Issue. There were five instances of the PMI code for Content

Assessment Item 5. This small number does not mean that the participants did not, overall,

struggle with this question; the number was impacted by the 23 of 29 participants who left the

problem blank or nearly blank. The biggest mathematical issue for this content assessment item

was a lack of the necessary content knowledge to approach or sustain the writing of the proof.

For instance, Participants 5, 14, and 24 all expressed being unable to remember rules about

secants and circles. The participants’ focus on not being able to remember the rules distracted

them from the task at hand; in fact, nothing about secants needed to be remembered in order to

complete this proof task. All that was necessary was the recognition that the two triangles in the

picture were similar. So, although there was a circle present in the diagram of the proof, this was

really another proof task about triangles (specifically, the similarity of two triangles).

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Chapter VI: Summary, Conclusions, and Recommendations

Summary

Despite the widespread teaching of geometry, a great majority of students lack an

adequate understanding of geometric concepts, and the ability to write proofs. In order to

improve the teaching and learning of geometric proofs, the struggles faced by teachers and

students need to be analyzed. The content of a course textbook and a teacher’s confidence and

preparedness are critical factors which impact student learning. Although many studies have

been conducted which examine either curriculum materials or teachers’ preparedness, there was

a need for research examining these factors in conjunction. This study therefore examined a

typical high school geometry textbook for its proof-related contents, as well as the content

knowledge and beliefs of preservice and novice secondary mathematics teachers in order to

determine their preparedness for, and feelings about, the teaching and learning of geometric

proof.

This study contributed to the pre-existing research on Shulman’s (1986) categories of

teachers’ knowledge. Although many studies distinctly examined subject matter content

knowledge, pedagogical content knowledge, or curricular knowledge, the present study began to

examine these factors in conjunction.

The first purpose of the study was to investigate and describe the types of proof tasks

offered in a typical high school geometry textbook. The second purpose was to explore

preservice and novice secondary mathematics teachers’ proof-writing abilities and beliefs about

geometry (particularly proofs), in order to propose ideas about how to improve the teaching and

learning of geometric proofs.

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I conducted a two-part qualitative study. Part one was intended to achieve the first

purpose of the study, and involved an analysis of the selected textbook ([Textbook Author],

2016) in which I read the textbook in its entirety and then composed a collection of different

proof tasks within the book’s end-of-chapter, end-of-section, and cumulative review exercise

sets. I split these tasks into two categories: Genuine Proof Tasks (GPTs), which explicitly

instructed students to write a proof (“prove”), and Proof-Related Tasks (PRTs), which prompted

students to “explain,” “justify,” “demonstrate,” or “show.” I tallied the frequencies of each type

of task, and then computed the percentages of these tasks in each of the sections of the textbook

in order to determine the presence of proof at large in the text. I also noted specific instances of

missed opportunities for proof and proving throughout the selected text.

To gain a deeper understanding of what, specifically, the selected text required of

students in regard to proof and proving, I chose a subset of GPTs to analyze. In this analysis, I

listed the specific prior knowledge that students would have to invoke to successfully solve the

problems, and posed potential pathways towards the writing of each proof.

After acquiring an understanding of what is offered regarding proof in a typical high

school geometry textbook, I created a five-item content assessment (eMATHinstruction, 2017,

2018a, 2018b; The University of the State of New York, 2015, 2016) to administer to preservice

and novice secondary mathematics teachers in order to achieve the second purpose of the study. I

secured 29 participants and met with them separately to administer the assessment while asking

questions regarding their feelings, beliefs, and confidence in regard to teaching proof in high

school geometry. The administration of these content assessments and simultaneous interviews

was audio-recorded and transcribed, and subsequently analyzed for trends in the participants’

responses using five qualitative codes: Beliefs and Attitudes (BAA), Issues of Correspondence

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Between Substance and Notation (ICSN), Expressing Understanding and Self-Doubt (EUSD),

Mathematical Language (ML), and Pure Mathematical Issues (PMI). Analyzing the responses of

the 29 participants, in conjunction with the consideration of what is required by the textbook,

enabled me to draw conclusions about the teaching and learning of proof and proving in

secondary mathematics education.

Conclusions

The first research question was: What kinds of proof tasks and proofs are offered in a

typical high school geometry textbook? The results of the textbook analysis show that proof is

largely underemphasized in the selected textbook, although the textbook asserts that its tasks will

help students to develop an evolving intuition about proof and its nature ([Textbook Author],

2016). In the majority of the end-of-section, end-of-chapter, and cumulative review exercise sets,

the percentage of proof-related tasks and genuine proof tasks was less than 10%, and in many

instances was 0%. Although there were some opportunities for students to write rigorous

geometric proofs, there were many instances of missed opportunities for proof tasks and

instances of tasks in which some, or the majority, of the work was done for the students. The

data support the conclusion that proof is often avoided or underemphasized in secondary

mathematics classrooms, and support the findings of Sears (2012) that quality proof tasks in

geometry textbooks are scarce. Sears asserted that teachers primarily complete proofs for

students during whole class instruction. Reid and Knipping (2010) cited Herbst’s (2002)

summary of such a situation as a “division of labor that could be described by saying the teacher

proves and the students write down the proof” (pp. 216–217). The findings of the present study

suggest that this practice can be learned from textbooks that include proof tasks which require

students to do very little, and then transferred to a teacher’s classroom practice.

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The data for Research Question 1 also support Davis’s (2009) idea that geometry

textbooks too often “depict geometry as a collection of theorems, postulates, and axioms

developed by unknown persons for unknown reasons to be passively consumed by the student”

(p. 335). I highlighted many instances of theorems that were presented without proof, to

presumably be read and accepted without justification. This undermines both the need for proof

as well as the nature of proof, as well as the stated purposes of the text ([Textbook Author],

2016), and supports the idea mentioned in Chapter I that “mathematics grows not in the order

presented in textbooks but rather through a recurring cycle in which . . . proofs are proposed and

refuted” (Clements, 2003, as cited in Blair & Canada, 2009, p. 288). This quotation speaks to the

importance of a teacher being prepared to develop a balance between the careful and intentional

use of curricular materials and the facilitation of discovery. A natural and successful endeavor in

the teaching and learning of proof cannot be achieved through a textbook alone, yet the majority

of the teacher participants in the study felt unprepared to facilitate proof activities outside of

what would be provided by their assigned textbook. This concern is directly connected to the

second phase of the study.

The second research question was: How prepared and how confident are preservice and

novice secondary mathematics teachers to teach proofs? The results of the study showed that the

majority of the preservice and novice secondary mathematics teacher participants are not

currently prepared to teach high school geometry, from standpoints of both mathematical content

knowledge and confidence. Overall, the participants struggled with the proof tasks on the content

assessment, and demonstrated negative feelings regarding their own abilities to write proofs. In

some instances, for select content assessment items, although the participants were able to get

started, they found themselves stuck at a certain point and unable to move forward.

139

The participants struggled with the use of mathematical language, and expressed

discomfort regarding the fact that there is no published and uniformly accepted standard for

proper proof writing, including mathematical language requirements. This finding supports the

sentiment that Mariotti (2006) expressed when summarizing the role of organization when one

seeks to determine the truth of a conjecture: “Organization becomes functional to understanding,

which consequently becomes strictly tied to the constraints of acceptability and validation shared

within a given community” (p. 176). This idea relates to the psychology of proof and proving for

the participants, because the “constraints of acceptability” cause much of the performance

anxiety that teachers and students have when it comes to constructing and sharing their

mathematical proofs.

In other instances, the participants wrote proofs but doubted that they were complete and

expressed frustration regarding their responses. The participants who had no geometry teaching

experience mainly drew on their own experiences from high school, and struggled to remember

the specific rules and standards that their particular teachers tried to instill. This finding reflects

the ideas of Schoenfeld (1985), who said:

As a result of instruction that focuses heavily on writing results in specific ways, and grading procedures that penalize students for not expressing otherwise correct answers in those ways, students can come to believe that ‘being mathematical’ means no less—and no more—than expressing oneself via the prescribed forms. (p. 369)

As noted in Chapter I, Battista (2009) also raised this issue in the 71st NCTM Yearbook,

citing his 1995 work with Clements, which established that most students see proof-writing as

nothing more than following a set of prescribed rules. Other participants who were geometry

tutors expressed a lack of consistency of standards and expectations for proof writing across the

students that they tutor, leading to their own feelings of uncertainty when tasked with teaching

140

about proof writing or writing proofs on their own, confirming the 1999 work of Netz and the

2010 work of Reid and Knipping.

There were many instances of pure mathematical content issues across the five content

assessment items, which came to light via the analysis of the participants’ work and their

interviews with me. The first content assessment item was a proof-related task, tasking the

participants with explaining the connection between a rotation and congruence. Although I

hypothesized that the participants would ask if the word “explain” meant that they had to write a

proof, zero of the participants did so, and none wrote a proof without asking. This relates to the

sentiment expressed by Otten et al. (2011), who questioned whether students realize that a proof

would satisfactorily fulfill the requirements of an exercise in which they are prompted to

“explain.” Clearly, in order for students to understand this component of the nature of proof,

teachers need to understand it first.

Another mathematical issue that arose was the participants’ weakness with definitions

and special cases of definitions. On Content Assessment Items 1 and 2, both which dealt with

isosceles triangles, many participants asked questions about whether an equilateral triangle is an

isosceles triangle, too. Several were unable to move forward in their paths of reasoning because

they erroneously believed that an equilateral triangle is not isosceles. On Content Assessment

Item 4, many participants struggled to complete the proof task because they were unsure of the

distinguishing features of a rhombus. These participants expressed frustration because they felt

they would be able to be successful had they been equipped with the prior knowledge. This

points to a need for preservice teacher educators to incorporate assessment of candidates’

knowledge of definitions, which is of critical importance for teachers and students alike.

141

In response to the interview questions that addressed participants’ beliefs and feelings

about proof and proving, I found that the participants overall voiced preferences to teach other

courses over geometry. They stated reasons ranging from negative experiences in high school, to

not having a formal geometry course in college, to disliking proofs, to the seeming

disconnectedness of geometry from the other subjects (for instance, several participants stated

that although Algebra 1 and Algebra 2 are closely related, that geometry content seems far

outside of these other courses). Since teachers’ attitudes are so easily transferrable to the students

in their charge, it is important to continue to conduct research in this area so that the state of the

teaching and learning of geometric proof can be improved.

Recommendations

I now offer recommendations based on the literature reviewed and the study’s findings,

analysis, and conclusions. The limitations of this study highlighted some areas that might have

been conducted differently. First, I studied only one common core textbook ([Textbook Author],

2016) because it was not the primary purpose of the study to compare different textbooks, but

rather to determine how preservice and novice teachers interpret and feel about proof tasks.

Second, I was unable to schedule interviews with all 30 participants who initially signed consent

forms; I was able to schedule meetings with 29 participants and settled on that number. Third,

the analysis of the data for the content assessment showed that the codes overlapped one another

quite frequently. For instance, expressing beliefs and attitudes is often inclusive of expressing

understanding or expressing self-doubt. Similarly, expressing a negative attitude is often

indicative of a pure mathematical issue, and expressing understanding often includes correct and

precise mathematical language. On the other hand, issues with mathematical language arose

within the context of participants expression of pure mathematical issues, while they were

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expressing self-doubt (which, in many instances, was about the mathematical language itself). I

decided to choose only one code per selected quotation based on its prevalence to the content

assessment item and the most common sentiments and areas of struggle amongst the participants.

Fourth, to maintain consistency in the analysis of the data, the same codes were used for each of

the content assessment items; however, due to the limited number of participants who actually

engaged in Content Assessment Item 5, the sample sizes of quotations were smaller than that of

any other item. The analysis was based on what was available to me.

A recommendation for future research is to conduct a similar study which considers the

aforementioned limitations and develops a coding system with more distinct categories, to more

specifically target the participants’ issues. The present study aimed at an overall assessment of

teachers’ confidence, knowledge, and preparedness; a future study could serve the purpose of

identifying specific trouble areas within a smaller group of participants’ levels of confidence,

command of content knowledge, and/or beliefs and attitudes, and measure the impact of an

intervention on any of these factors.

Moreover, it would be interesting to study more than one textbook using the same data

collection and analysis procedures, and determine whether the issues found in the textbook

analyzed for this study are common or standalone. This could lead to the improvement of

curricular materials and better alignment of textbooks to the goals set forth for students by the

NCTM.

For mathematics educators, this study emphasized several factors to consider promoting

and implementing in their schools and classrooms, at both the secondary and university levels.

The issues encountered by participants on the content assessment reflected deficiencies within

Schoenfeld’s (1985) four categories of beliefs, heuristics, resources, and control, and therefore

143

raise concerns about the participants’ preparedness, confidence, and readiness to teach high

school geometry. In order to for future teachers to be better prepared, professional educators

should create college-level materials aimed at improving candidates’ mathematical knowledge

for teaching, specifically in geometry.

The issues in both mathematical language and mathematical content brought to light by

participants’ responses raise concerns about preservice and novice teachers’ knowledge of

transformations and their properties (participants struggled, overall, to connect the idea of a

rotation with the idea of congruence). Due to the fact that proving congruence using rigid

transformations is a new topic in high school geometry (since the implementation of the

Common Core Standards for Mathematics), pre-service teacher candidates and novice teachers

should engage in curricular activities which will foster their own deep understandings of these

concepts. Moreover, a stronger understanding of the notion of the congruence of segments versus

the equivalence of their lengths may provide the necessary foundation for stronger understanding

of congruence criteria, which could enable them to facilitate investigation and discovery learning

of this important concept in their classrooms.

The importance of justification in geometry teaching and learning need be stressed. Many

participants revealed that they employ the use of congruence criteria as “justification” within a

proof, without knowing why the justification holds. This raises the issue of the legitimacy and

validity of their argument, and teachers’ comprehension of how justification links axioms,

definitions, theorems, and proofs.

It is concerning that preservice and novice secondary mathematics teachers did not all

know the accurate meaning of the CPCTC acronym, especially considering the fact that they

used it as a part of their proofs. This finding not only raises questions about whether incorrect

144

mathematics is being taught in classrooms, but further supports the sentiments expressed within

the commentary about the EUSD code regarding genuine justifications of geometric

propositions. The question also arises as to how current curricular materials are re-engaging

teachers with topics that they all too often have not revisited since their own high school

experiences. This inquiry provides the impetus for more research on how curricular materials and

teacher knowledge interact within a teacher’s preparedness and teaching practice, as the present

study aimed to explore.

The findings suggest the need for increased focus on mathematical language in preservice

teacher education courses as well as mathematics content courses, as a teacher’s ability to

communicate effectively mathematically directly impacts students’ understanding of the content.

The discomfort that was expressed by many participants regarding the first proof on the

content assessment that did not focus on triangles points to the need for novice teachers’

increased preparation (and confidence) to teach proof-writing on a variety of topics, beyond

triangle congruence.

Teacher educators should build in methods within their teacher preparation programs to

determine gaps within candidates’ mathematical knowledge, perhaps by inquiring about their

mathematical backgrounds. Often, pre-service and novice teachers are confident in their abilities

to teach secondary mathematics because they were university mathematics majors. However, this

study showed that even those equipped with degrees in mathematics can have a shallow

understanding of more basic secondary mathematics concepts, and not necessarily be prepared to

teach high-school level content meaningfully.

Courses on teacher identity and teacher development could help with the issues of

confidence that many of the participants expressed or demonstrated. Participants’ perceptions of

145

their own knowledge (or the lack thereof) play a major role in impeding their proof-writing, and

potentially their students’ confidence levels, as well.

146

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[PowerPoint presentation]. Teachers College, Columbia University. Weiss, Y. (2017, May 7–11). Textbook analysis in university teacher education. In G. Schubring,

L. Fan, & V. Geraldo (Eds.), Proceedings of the Second International Conference on Mathematics Textbook Research and Development (pp. 226–230). Universidade Federal do Rio de Janeiro: Instituto de Matemática.

Wohlhuter, K. A. (1996). Decision making in mathematics reform context: Factors influencing

geometry teachers’ planning and interactive decisions [Unpublished doctoral dissertation]. Oregon State University.

Zaslavsky, O., Nickerson, S. D., Stylianides, I. K., & Winicki-Landman, G. (2012). The need for

proof and proving: Mathematical and pedagogical perspectives. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: The 19th ICMI study (pp. 215–230). Springer.

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Appendix A: Informed Consent

Protocol Title: A Study of Pre-Service and Novice Teachers’ Preparedness to Teach Geometry Proofs

Principal Investigator: Mara Markinson, PhD Candidate, Teachers College, [telephone number] [email address]

INTRODUCTION: You are being invited to participate in this research study called “A Study of Pre-Service and Novice Teachers’ Preparedness to Teach Geometry Proofs.” You may qualify to take part in this research study because you are a pre-service or novice secondary mathematics teacher. Approximately thirty people will participate in this study and it will take at most 1.5 hours of your time to complete. WHY IS THIS STUDY BEING DONE? This study is being done to determine what is expected of high school students in regard to writing geometry proofs, and how their teachers’ preparation impacts their confidence about teaching geometry proof writing. WHAT WILL I BE ASKED TO DO IF I AGREE TO TAKE PART IN THIS STUDY? If you decide to participate, you will take a content assessment about geometry proofs for about 30 minutes. The content assessment is paper-based and will consist of four geometry proofs. The principal investigator will observe you take the content assessment. This is not a test with a grade, but being used as a way to understand your approaches to solving geometry proofs. After taking the content assessment, if you are willing to participate in an interview, the principal investigator will interview you. This interview can take place at the same time as the content assessment, or at a later time that is convenient for you. During the interview, you will be asked to discuss your preparation as a secondary mathematics teacher and your thoughts about writing and teaching geometry proofs. The interview will take approximately one hour. You will be given a de-identified code in order to keep your identity confidential. All of these procedures will be done at locations mutually agreed upon by you and the principal investigator, at times that are convenient to you. WHAT POSSIBLE RISKS OR DISCOMFORTS CAN I EXPECT FROM TAKING PART IN THIS STUDY? This is a minimal risk study, which means the harms or discomforts that you may experience are not greater than you would ordinarily encounter in daily life while taking routine physical or psychological examinations or tests. However, there are some risks to consider. You might feel uncomfortable to discuss your preparation and readiness to teach high school geometry. However, you do not have to answer any questions or divulge anything you don’t want to talk about. You can stop participating in the study at any time without penalty.

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The principal investigator is taking precautions to keep your information confidential and prevent anyone from discovering or guessing your identity, such as using a pseudonym instead of your name and keeping all information on a password protected computer and locked in a drawer. WHAT POSSIBLE BENEFITS CAN I EXPECT FROM TAKING PART IN THIS STUDY? There is no direct benefit to you for participating in this study. Participation may benefit the field of teacher education to better understand the best way to prepare secondary mathematics teachers. WILL I BE PAID FOR BEING IN THIS STUDY? You will not be paid to participate; however, you will be entered into a lottery to win a $100 cash gift card. Your name will be entered into the lottery at the conclusion of the content assessment. Your chances of winning are approximately 1/30. At the conclusion of the study, the lottery winner will be determined. If you are the winner, you will be notified via phone. There are no costs to you for taking part in this study. WHEN IS THE STUDY OVER? CAN I LEAVE THE STUDY BEFORE IT ENDS? The study is over when you have completed the content assessment, and, if applicable, the interview. However, you can leave the study at any time even if you haven’t finished. PROTECTION OF YOUR CONFIDENTIALITY: The investigator will keep all written materials locked in a desk drawer in a locked office. The master list matching participants with de-identified codes will be kept locked and separate from the list of codes. For quality assurance, the study team, the study sponsor (grant agency), and/or members of the Teachers College Institutional Review Board (IRB) may review the data collected from you as part of this study. Otherwise, all information obtained from your participation in this study will be held strictly confidential and will be disclosed only with your permission or as required by U.S. or State law. HOW WILL THE RESULTS BE USED? The results of this study will be published in journals and presented at academic conferences. Your identity will be removed from any data you provide before publication or use for educational purposes. This study is being conducted as part of the dissertation of the principal investigator.

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WHO MAY VIEW MY PARTICIPATION IN THIS STUDY? I consent to allow written materials viewed at an educational setting or at a conference outside of Teachers College Columbia University.

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OPTIONAL CONSENT FOR FUTURE CONTACT: The investigator may wish to contact you in the future. Please initial the appropriate statements to indicate whether or not you give permission for future contact. I give permission to be contacted in the future for research purposes:

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WHO CAN ANSWER MY QUESTIONS ABOUT THIS STUDY? If you have any questions about taking part in this research study, you should contact the principal investigator, Mara Markinson, at [telephone number] or at [email address].

If you have questions or concerns about your rights as a research subject, you should contact the Institutional Review Board (IRB) (the human research ethics committee) at 212-678-4105 or email [email protected]. Or you can write to the IRB at Teachers College, Columbia University, 525 W. 120th Street, New York, NY 1002. The IRB is the committee that oversees human research protection for Teachers College, Columbia University.

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PARTICIPANT’S RIGHTS: • I have read and discussed the informed consent with the researcher. I have had

ample opportunity to ask questions about the purposes, procedures, risks and benefits regarding this research study.

• I understand that my participation is voluntary. I may refuse to participate or withdraw participation at any time without penalty.

• The researcher may withdraw me from the research at his or her professional discretion.

• If, during the course of the study, significant new information that has been developed becomes available which may relate to my willingness to continue my participation, the investigator will provide this information to me.

• Any information derived from the research study that personally identifies me will not be voluntarily released or disclosed without my separate consent, except as specifically required by law.

• De-identifiable data may be used for future research studies, or distributed to another investigator for future research without additional informed consent from the subject or the representative.

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Appendix B: Textbook Data

All notes and page numbers are based on the analyzed textbook ([Textbook Author],

2016). GPTs, which explicitly direct students to prove, are italicized.

Chapter 1 Notes

Chapter 1, Geometry Fundamentals, begins on page 36. Section 1.1, Geometry

Essentials, begins on page 37. Practice problems begin on page 38. Of 10 problems, the

following two are related to proof and proving:

Question 9: What is the minimum number of points needed to define two distinct planes? Explain your reasoning.

Question 10: Use the postulates in the text to explain why every plane must contain at least one line.

A second part of the section called Construction: Copying Segments and Angles begins

on page 40, and it has its own set of practice problems beginning on page 41. Of 24 problems,

one is related to proof and proving:

Question 24: Identify 3 angles in the figure below. Which angle is the largest? Explain how you know.

Section 1.2, Measuring Distances, begins on page 43. The ruler postulate and the segment

addition postulate are defined. Practice problems begin on page 45. Of 13 problems, the

following two relate to proof and proving:

Question 12: The endpoints of 𝑀𝑁00000are at (7, 5) and (7,−2), and the endpoints of 𝐺𝐻0000are (6,−11) and (3,−11). Is 𝑀𝑁00000 ≅ 𝐺𝐻0000? Explain.

Question 13: The endpoints of 𝐴𝐵0000are at (1, 432) and (−1, 432), and the endpoints of 𝑌𝑍0000are (0, −455) and (2,−455). Is 𝐴𝐵0000 ≅ 𝑌𝑍0000? Explain.

A second part of the section called The Distance and Midpoint Formulas begins on page

46, and it has its own set of practice problems beginning on page 48. Of 18 problems, the

following one relates to proof and proving:

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Question 18: Alan states, ‘The midpoint of any line segment can be found by multiplying the difference between the two 𝑥 −values by c

W and multiplying the difference between

the two 𝑦 −values by cW.’ Is he correct? Why or why not?

A third part of the section called Ratios and Line Segments begins on page 49, with its

own set of practice problems beginning on page 50. Of 11 problems, none are related to proof

and proving.

Section 1.3, Transformations and Congruent Figures, begins on page 50. Practice

problems begin on page 53. Of 24 problems, the following two relate to proof and proving:

Question 7: Did the transformation produce a congruent figure? Explain.

Question 22: A figure is flipped, moved up, and then rotated, producing a new figure. Is the new figure congruent to the original figure? Explain your reasoning.

Section 1.4, Translation in a Coordinate Plane, begins on page 56. No proof is mentioned

at all until page 59, where a paragraph proof appears (and is done) to pose the solution to

“Determine if translation is an isometry for all figures. Explain your reasoning.” Practice

problems begin on page 60. Of 34 problems, only Question 30 relates to proof and proving:

“Explain in words why a figure and its translated image are congruent.”

Section 1.5, Rotation, begins on page 62. Practice problems begin on page 67. Of 45

problems, the following two relate to proof and proving:

Question 43: Using the point (𝑎, 𝑏), show algebraically that two 90° clockwise rotations about the origin are equivalent to a 180° rotation about the origin.

Question 44: Using the point (𝑎, 𝑏), show algebraically that two 90° counterclockwise rotations about the origin are equivalent to a 180° rotation about the origin.

Section 1.6, Reflection, Rotation, and Symmetry, begins on page 70. Practice problems

begin on page 75. Of 32 problems, the following two relate to proof and proving:

Question 26 and Question 27: Determine whether each statement is always true, sometimes true, or always false. Justify your answer.

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Question 26: Reflections of figures preserve distances between pairs of corresponding vertices. That is, if figure 𝐴 is reflected across a reflection axis to form figure 𝐴′, then the distance between two vertices on 𝐴 is equal to the distance between the corresponding vertices on 𝐴′.

Question 27: Reflections of polygons preserve area. That is, if figure 𝐴 is reflected across a reflection axis to form figure 𝐴′, then the area of 𝐴 is equal to the area of 𝐴′.

Other parts of the section, called Rotational Symmetry and Three-Dimensional

Symmetry, begin on page 77. Their own set of practice problems begins on page 78. Of nine

problems, the following two relate to proof and proving:

Question 1: Does a cube have rotational symmetry? Justify your answer.

Question 8: John states that the number of lines of symmetry of a two-dimensional figure is equal to the number of rotational symmetries of the figure. Is his statement true? Explain.

Section 1.7, Composition of Transformations, begins on page 79. On page 81, the first

theorem of the text, reflections in parallel lines theorem, is introduced and a definition is given

for theorem: “A theorem is a mathematical statement established by means by [sic] a proof. Here

we apply the reflections in parallel lines theorem to calculate the distance of a double reflection

across two parallel lines.” Practice problems begin on page 82. Of 31 problems, the following

two relate to proof and proving:

Question 9: Does the order in which you perform multiple translations matter? Explain.

Question 26: Suppose you have two lines, 𝑙 and 𝑚, that are parallel. If you reflect a figure first across 𝑙 and then across 𝑚, will the distance the figure travels be the same as if you reflected it first across 𝑚 and then across 𝑙? How far does the figure travel in each case? How do you know? Sketch a labeled diagram to support your explanation.

The Chapter 1 review begins on page 85. Of 38 problems, the following five have to do

with proof and proving:

Question 23: Can the composition of translations be written as a single translation? Explain.

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Question 25: Using the point (𝑎, 𝑏), show algebraically that two 180° rotations about the origin return you to the original point (𝑎, 𝑏).

Question 26: Using the point (𝑎, 𝑏), show algebraically that four 90° rotations about the origin return you to the original point (𝑎, 𝑏).

Question 36: In addition to representing a 180° rotation, the transformation (𝑥, 𝑦) →(−𝑥, −𝑦) results from the composition of which two reflections? Justify your response.

Question 37: Jose explains that since the reflections across two parallel lines that are 𝑥 units apart make a point move a distance of 2𝑥 units, a single reflection across the first line would make the point move half the total distance, which is 𝑥units. Is his explanation correct? Why or why not?

Chapter 2 Notes

Chapter 2, Similar Figures and Dilations, begins on page 88. Section 2.1, Similar Figures,

begins on page 89. Practice problems begin on page 91. Of 25 problems, none relate to proof and

proving.

Section 2.2, Dilation and Similar Figures, begins on page 93. Practice problems begin on

page 100. Of 24 problems, the following one relates to proof and proving:

Question 24: Dilate the figure below by a factor of 2. The center of dilation is the origin. Do the equations of the lines containing the sides of the triangle change? If so, describe in general how they change, using the concepts of slope and intercept.

Section 2.3, Similarity, Polygons, and Circles, begins on page 102. Practice problems

begin on page 105. Of 27 problems, the following two relate to proof and proving:

Question 26: Laney states, “All rectangles are similar because they all have four right angles.” Is she correct? Why or why not?

Question 27: Triangle 𝑀𝐴𝑃 has vertices located at 𝑀(𝑎, 𝑏), 𝐴(0, 0)and 𝑃(0, 𝑐). Triangle 𝑄𝑆𝑇 has vertices located at 𝑄(3𝑎, 3𝑏), 𝑆(0, 0)and 𝑇(0, 3𝑐). Show that triangle 𝑀𝐴𝑃 is similar to triangle 𝑄𝑆𝑇.

A second part of the section called Similar Polygons and Area begins on page 107:

“Similar polygons area theorem: If two polygons are similar, then the ratio of their areas is equal

to the ratio of the squared lengths of any pair of corresponding sides.” No proof appears. Model

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problems follow. One practice problem (Question 4) on page 108 says, “Shawna states, ‘The

ratio of the areas of two similar polygons is equal to the ratio of their corresponding sides.’ Is she

correct? Explain.” The students were just given the theorem on the page before.

Section 2.4 is called Similarity and Transformations and begins on page 109. Practice

problems begin on page 110. Of 20 problems, the following are related to proof and proving:

Question 17–Question 20: Sketch the resulting triangle after the transformations. State whether the resulting figure is similar, congruent, or both. Explain your reasoning.

Question 17: Rotated 90° clockwise; translated 5 units vertically.

Question 18: Rotated 90° counterclockwise; translated 3 units horizontally; dilated by a factor of 2.

Question 19: Dilated by a factor of 2; reflected across the 𝑦 −axis; translated 4 units vertically.

Question 20: Translated -2 units vertically; dilated by a factor of cW; reflected across

the 𝑦 −axis.

The Chapter 2 review begins on page 112. Of 28 problems, one relates to proof and

proving:

Question 18: Triangle 𝐴𝐵𝐶 is transformed to triangle 𝐴′𝐵′𝐶′. Josh calculates the ratio ijkj

ik and finds that it is equal to 6. He then states that triangle 𝐴′𝐵′𝐶′ is a

dilation of triangle 𝐴𝐵𝐶 with a scale factor of 6. Is his reasoning sufficient? Explain.

Page 115 begins a cumulative review for Chapters 1–2. Of 29 questions in the cumulative

review, one is related to proof and proving:

Question 28: Triangle 𝐴𝐵𝐶 is transformed to triangle 𝐴′𝐵′𝐶′. Sophia calculates the ratio ik

ijkj to be 5 and states that the scale factor for this dilation is 5. Is Sophia

correct? Explain.

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Chapter 3 Notes

Chapter 3 is called Reasoning and begins on page 118. Section 3.1, Inductive Reasoning,

begins on page 119. Practice problems begin on page 122. Of 23 problems, the following are

related to proof and proving:

Question 4: Meng skims a table of prime numbers and concludes that all prime numbers are odd. Which of the following would be a valid counterexample to prove that his conjecture is false?

Question 5: Tabetha says, “All parallelograms have 2 acute angles and 2 obtuse angles.” Which of the following would be a valid counterexample to prove that her conjecture is false?

Question 6: Sebastian says, “All rhombuses have 2 acute angles and 2 obtuse angles.” Which of the following would be a valid counterexample to prove that his conjecture is false?

Question 17–Question 20: Find a counterexample for the following statements.

Question 17: 𝑎 − 𝑏 ≤ 𝑎.

Question 18: mn+ o

n= mpo

Wn for any 𝑧 ≠ 0.

Question 19: 𝑎W > 𝑎.

Question 20: All numbers are greater than 0 and less than or equal to 1.

Section 3.2, Conditional Statements, begins on page 124. Page 125 includes some

examples of how to transform statements with words such as “all” and “never” into conditional

statements. Page 126 explains the terms conditional, converse, inverse, contrapositive, and

biconditional.

Page 127 has three model problems about writing the converse, inverse, and

contrapositive of conditional statements. All the problems are done for the reader, and the truth

values of each statement are indicated. Practice problems begin on page 128. Of 35 problems, the

following are related to proof and proving:

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Question 1–Question 2: Use the following statement to answer the problems. “If you live on the east side of Bank Street, then your house number is even.”

Question 1: If Erica lives on the east side of Bank Street, then what can you say about her address?

Question 2: If Candace lives on Bank Street, then what can you say about her address?

Question 9–Question 12: Write an equivalent conditional statement in the form “If 𝐴, then 𝐵” for the following statements:

Question 9: All squares are rectangles.

Question 10: All dogs chase cats.

Question 11: All liquids are consumable.

Question 12: No vegans eat meat.

Question 35: Let’s consider the conditional statement “If it is raining, then it is cloudy” to be a true statement. (a) In your own words, explain how the converse statement is different from the conditional statement. (b) In your own words, explain how the contrapositive statement is equivalent to the conditional statement. (c) If we switched the hypothesis and conclusion of “All squares have four right angles” as a conditional statement, would the statement be true or false? Explain.

Section 3.3, Deductive Reasoning, begins on page 130. The book defines deductive

reasoning as when “we start with a true claim and then systemically build a valid argument,

based on previously known true statements, that ends with a logical conclusion” (p. 130). It later

says, “In this lesson and the next, we formalize the language of these arguments as ‘proofs’ and

provide practice with simple proofs to familiarize you with the process” (p. 130). Next, a

definition of proof is given:

A sequence of statements, such as a given, an axiom, a postulate, a theorem, or is somehow deduced from previously established statements, which establishes the truth of a mathematical statement. As we learn to write proofs, we must give a reason, or justification, for each step of the proof. (p. 130)

The last paragraph of text, which is intended to explain why one needs to prove that two

right angles are congruent, says,

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Now, you might be thinking, “Well, of course they have the same measure! They look exactly alike.” Yes, that is true, but in mathematics, it is not enough that something appears to be true on a diagram. You must prove it. (p. 130)

The first example of a two-column proof appears. The proof’s purpose is to show that

right angles are congruent. Model problems then appear. The first model problem uses the

segment addition postulate. The second model problem uses a “proof” to solve a problem which

requires substituting into a given formula (this is not an example of what students are ever

required to do in common core geometry). The third model problem also is a nonroutine

problem, which students could easily solve but is made much more convoluted by the presence

of the proof prompt. The question reads, “𝑚∡𝐶𝐴𝐵 is three times 𝑚∡𝐷𝐴𝐶. If 𝑚∡𝐷𝐴𝐵 is 52°,

what is 𝑚∡𝐶𝐴𝐵?” (p. 130).

On page 132, the activity prompts students to place statements and reasons from a bank

in a two-column proof format. This activity uses transitivity, but the book does not explain the

difference between transitivity and substitution. Practice problems begin on page 133. Of 18

problems, the following are related to proof and proving:

Question 3: Which is the first mistake made in the following proof?

Question 4: Sarah attends high school with Carolyn. Sarah also attends high school with Beth. If Sarah attends only one high school, what conclusion can be drawn?

Question 7: Keara states that if the ratios tu and v

w are equal, then it must be true that 𝑎 =

𝑚 and 𝑏 = 𝑛. Is her statement correct? Explain.

Question 8: Is the expression (5𝑎 ∙ 𝑏) + (4𝑎 ∙ 𝑐) equivalent to the expression (𝑐 ∙ 4𝑎) +(𝑏 ∙ 5𝑎)? Explain your reasoning.

Question 9–Question 12: Use a two-column proof to solve the equation. Refer to the properties on page 5 as needed to answer these questions.

Question 9: 2𝑥 = 6(2 + 𝑥).

Question 10: 4𝑥 = 2(9 − 𝑥).

Question 11: −7𝑥 = 2(1 − 3𝑥).

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Question 12: −2(2 − 3𝑥) = 10 − 𝑥.

Question 13–Question 15: Use the diagram below and the theorems you’ve learned thus far to create a two-column proof to find the value of 𝑥. [A diagram is provided of three angles (∡𝐴𝑂𝐵, ∡𝐵𝑂𝐶, ∡𝐶𝑂𝐷) lying on straight line 𝐴𝑂𝐷O⃖OOOOOO⃗ .]

Question 13: 𝑚∡𝐴𝑂𝐵 = 𝑥 + 45°; 𝑚∡𝐵𝑂𝐶 = 2𝑥 − 17°; 𝑚∡𝐶𝑂𝐷 = 𝑥.

Question 14: 𝑚∡𝐴𝑂𝐵 = 5𝑥; 𝑚∡𝐵𝑂𝐶 = 3𝑥; 𝑚∡𝐶𝑂𝐷 = 2(𝑥 + 20°).

Question 15: 𝑚∡𝐴𝑂𝐵 = 3𝑥 − 41°; 𝑚∡𝐵𝑂𝐶 = 𝑥; 𝑚∡𝐶𝑂𝐷 = 2𝑥 − 19°.

Question 18: Suppose we have the true biconditionals “𝐴 if and only if 𝐵” and “𝐵 if and only if 𝐶.” Is the biconditional “𝐴 if and only if 𝐶” also true? Explain.

Section 3.4, Reasoning in Geometry, begins on page 134. Definitions of complementary,

supplementary, adjacent, and vertical angles are given. It is given that vertical angles are

congruent without proof. The model problems are solved both algebraically and with

explanation, but it is impossible to tell if they are supposed to be examples of statement-reason

proofs.

Transitive, symmetric, and reflective properties of congruence are introduced on pages

136–137 and related to the corresponding properties of equality from algebra. (Of note,

transitivity was listed as a “reason” for students to place into a proof on page 132, which

precedes this introduction.) It is not detailed why the students need to say, for example, that

𝐴𝐵0000 ≅ 𝐴𝐵0000 rather than 𝐴𝐵 = 𝐴𝐵.

On page 137, students are tasked to “prove the congruent complements theorem” without

a statement of what the theorem is. The textbook simply gives a diagram and says, “Prove A =

C.” Again, there is a statement-reason chart provided, and students are to place the statements

and reasons in the correct boxes from a word bank. The following page (p. 138) then states the

congruent complements theorem.

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Several relationships about angles are stated on page 138: (a) congruent supplements

theorem, (b) congruent complements theorem, (c) linear pair postulate, and (d) vertical angles

theorem. Proofs of these relationships are presented in statement-reason form over the following

two pages of model problems. The proofs vary in how they present the theorem statements. For

example, the first one just says “Prove 𝑚∡𝐴 = 𝑚∡𝐶,” but the second one says, “Prove the

vertical angles theorem by showing that angles A and C are congruent.” In the model proofs that

are given, some of the listed “reasons” simply state the word “diagram.”

Practice problems begin on page 141. Of 36 exercises, few qualify as proof exercises.

Those somewhat aligned with proof and proving are as follows:

Question 3: What can be concluded from 𝑚∡𝐴 = 𝑚∡𝐵 = 90°?

Question 24: If ∡𝑅 is supplementary to a 47° angle and ∡𝑆 is supplementary to a 56° angle, could they be vertical angles? Explain.

Question 25: 𝑋 is supplementary to a 135° angle and 𝑌 is supplementary to a 125° angle. Could they be vertical angles? Why or why not?

Question 26: 𝑀 is supplementary to a 25° angle and 𝑚∡𝑁 = 155°. Could they be vertical angles? Why or why not?

Question 33: “Two parallel lines never intersect” is accepted without proof. Is this a postulate or a theorem? Explain.

Question 34: Scientists once accepted without proof that the speed of light is not constant. For these scientists, was the belief that the speed of light could change a postulate or a theorem? Explain.

Question 35: Prove the vertical angles theorem. Given: Lines 𝑘 and 𝑚 intersect. Prove: ∡𝐴 ≅ ∡𝐶.” [Diagram is given of lines k and m which intersect. The four angles created by the intersecting lines are labeled A, B, C, and D, with vertical pairs A/C and B/D. On the given diagram, Angles A and C are marked as congruent and B and D are marked as congruent.]

Question 36: Given: 𝑚∡1 ≅ 𝑚∡3; 𝑚∡2 ≅ 𝑚∡4. Prove 𝑚∡𝑌𝑋𝑍 ≅ 𝑚∡𝐵𝐴𝐶. [This is an angle addition postulate question. A diagram is given of two angles, ∡𝑌𝑋𝑍 and ∡𝐵𝐴𝐶. Each are split into two parts by a ray (∡1 and ∡2, and ∡3, and ∡4, respectively).]

168

The Chapter 3 review begins on page 143. There are multiple choice questions about

classifying types of reasoning as deductive or inductive, applying students’ knowledge of

conditional statements and their converses, contrapositives, and selecting equations that must be

true based on given information about pairs of angles. Then, Question 13–19 prompt the students

to state whether given situations are examples of inductive or deductive reasoning. Question 20,

a GPT, says, “Explain why the following two-column proof is incorrect, and then write the

correct algebraic proof.” The “proof” presented is justifying the steps of solving the equation

4(5 − 𝑥) + 3 = 8, and it calls the statement of the equation the “given.” The first (incorrect)

statement after that is 5 − 𝑥 + 3 = 2, with the corresponding reason, multiplication property of

equality.

There are no other proof questions in the entire chapter review. Questions 31–36 tell

students to determine whether statements are true or false, and to justify their answers. However,

in this section of problems, two of the questions are not statements, so there is no way for

students to follow the directions given. These two questions say, “How many linear pairs are

formed when two non-parallel lines intersect?” and “How many linear pairs are formed when

three non-parallel lines intersect at one point?” (p. 145).


review, the following four are related to proof and proving:

Question 6: If 𝐴𝐵0000 ≅ 𝐵𝐶0000 and 𝐵𝐶0000 ≅ 𝐶𝐷0000, then which of these statements must be true? The points do not necessarily lie on the same line. Choose all that apply.

Question 13: Two rectangles are similar. The length of a side in the larger rectangle is 𝑎 times the length of a corresponding side in the smaller rectangle. Algebraically prove the ratio of the perimeter of the larger rectangle to the perimeter of the smaller rectangle is 𝑎: 1.

Question 17: Using a two-column proof, show that if two congruent angles 𝐴 and 𝐵 form a linear pair, then 𝐴 and 𝐵 are right angles.

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Question 20: Amy states, “Vertical angles are adjacent angles.” Is she correct? Explain.

Chapter 4 Notes

Page 148 begins Chapter 4, Parallel and Perpendicular Lines. On page 150, the

corresponding angles postulate is given: “If a transversal intersects parallel lines, then all the

pairs of corresponding angles are congruent.” The reader is told that “the converse of the

corresponding angles postulate is also true.” Then, model problems with their solutions are

presented.

On page 151, the solution to the second model problem resembles a proof in that it has

writing in columns with one of the columns appearing to be a list of justifications. However, it is

not stated whether this is intended to be a proof, and one of the explanations has questionable

language: “𝐴 and 𝐷 are linear angles, which means they are supplementary and sum to 180°” (p.

151). Alternate interior and exterior angles are then introduced, and the alternate interior angles

theorem is stated: “If a transversal intersects parallel lines, then all of the pairs of alternate

interior angles are congruent” (p. 151). Page 152 states the alternate exterior angles theorem as,

“If a transversal intersects parallel lines, then all of the pairs of alternate exterior angles are

congruent.”

After the statement of the alternate exterior angles theorem, the book says:

Instead of stating that corresponding angles are congruent (when a transversal intersects parallel lines) as a postulate, we could state it as a theorem. To do this, assume that the alternate exterior angles theorem is a postulate. We use a paragraph proof to prove the theorem. Paragraph proof: With alternate exterior angles stated as a postulate, we have one set of angles shown to be congruent. Alternate interior angles would then be congruent too, since they are vertical angles to the alternate exterior angles, and vertical angles are congruent. The other pairs of corresponding angles would have to be congruent since they are supplementary to congruent angles. (p. 152)

Immediately after this long block of text, the book says, “Prove the alternate exterior

angles theorem converse by proving that lines 𝑚 and 𝑛 are parallel” (p. 152). A diagram is given

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with fill-in-the-blank statements and reasons for a two-column proof. The “prove” and the

“given” are also stated for the students. Two model problems are then given about finding the

measures of angles formed by parallel lines cut by a transversal.

The consecutive interior angles theorem is then stated as, “If a transversal intersects

parallel lines, then all pairs of consecutive interior angles are supplementary” (p. 154). No proof

is provided. The second model problem on that page tells students to “Prove the converse of the

corresponding angles postulate: If two lines and a transversal form corresponding angles that are

congruent, then the lines are parallel.” The proof of this problem, which is shown as the solution

in two-column format, requires the use of the consecutive interior angles theorem. In the problem

statement, the converse is given to the students, so they do not need to figure out what to prove.

The section of the chapter ends with 32 practice problems starting on page 155. The

following problems are the only ones that relate to proof and proving:

Question 18: In the diagram below, line 𝑚 intersects the sides of angle 𝑀at the points 𝐴 and 𝐵. Can both 𝑀𝐴OOOOOO⃗ and 𝑀𝐵OOOOOO⃗ be perpendicular to line 𝑚? Use the perpendicular postulate to explain your answer.

Question 25: The lines 𝑎 and 𝑏 intersect. Selena says she can draw a third line, 𝑐, that is parallel to both 𝑎 and 𝑏. Is Selena right? Explain why or why not.

Question 27: Lines 𝑎, 𝑏, and 𝑐 are coplanar. The lines 𝑎 and 𝑏 are both perpendicular to line 𝑐. Show that 𝑎 is parallel to 𝑏. Make a sketch to explain your answer. Hint: Suppose that line 𝑎 is not parallel to line 𝑏, and use the perpendicular postulate to show that this leads to a contradiction.

Questions 29–32 are proof questions. A diagram of two lines cut by a transversal is

given, with the eight angles created labeled 1–8.

Question 29: Prove: ∡3 ≅ ∡6; ∡4 ≅ ∡5. Given: 𝑚 ∥ 𝑛.

Question 30: Prove: 𝑚 ∥ 𝑛. Given: ∡3 ≅ ∡6.

Question 31: Prove: 𝑚∡3+ 𝑚∡5 = 180°;𝑚∡4 + 𝑚∡6 = 180°. Given: 𝑚 ∥ 𝑛.

Question 32: Prove: 𝑚 ∥ 𝑛. Given: 𝑚∡3 +𝑚∡5 = 180°.

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On page 158, Section 4.2 (More on Parallel Lines and Angles) begins with a statement-

reason proof of the alternate exterior angles theorem. After this proof is done for the students,

they are given a fill-in-the-blank activity of a statement-reason proof of the alternate interior

angles theorem. Here, the number of statements given is one more than the number of reasons,

but it is not indicated in the directions that reasons can be used more than once. Model problems

follow these proofs, but they are related to computation rather than proving (unless the format of

the presented solutions is supposed to be a proof).

Earlier, we stated a postulate: If two parallel lines are intersected by a transversal, then the corresponding angles are congruent. The converse is true: If the corresponding angles are congruent, then the lines are parallel. Since converses are not always true, we must carefully determine their validity. (p. 160)

On page 160, the corresponding angles postulate converse is stated as, “If the lines

intersected by a transversal form congruent corresponding angles, then the lines intersected by

the transversal are parallel.” The text continues to say, “This postulate lets us write a series of

theorem converses” (p. 160) and the following statements are provided:

Alternate interior angles converse: If alternate interior angles are congruent, then the lines intersected by the transversal are parallel.

Alternate exterior angles converse: If alternate exterior angles are congruent, then the lines intersected by the transversal are parallel.

Consecutive interior angles converse: If consecutive interior angles are supplementary, then the lines intersected by the transversal are parallel.

Model problems about parallel lines and angles are presented with their solutions on

pages 161–162. Practice problems begin on page 163. There are 28 practice problems. An

analysis of the proof tasks is below:

Question 21: Using the diagram, prove the alternate interior angles theorem converse. [A diagram has two lines cut by a transversal, with eight angles marked 1–8, and a given of two congruent angles.] Prove: 𝑚 ∥ 𝑛.

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Question 22: Explain how you would prove that if the alternate interior angles formed by two lines and a transversal are not congruent, then the two lines intersect.

Question 23: Are lines 𝑚 and 𝑛 parallel? Justify your answer. [A diagram is given which has two lines cut by a transversal and two angles labeled with their degree measures.]

Question 26: Two parallel lines are intersected by a transversal. Prove that angle bisectors (lines dividing an angle into two adjacent congruent angles) of the alternate interior angles are parallel to each other. Make a sketch and label the angles, referring to them in your proof. [A diagram is given which shows the students exactly how to set up the diagram they are requested to make in the task.]

Question 27: Prove the alternate exterior angles postulate converse: If alternate exterior angles are congruent, then the lines intersected by the transversal are parallel. [A diagram is given with two lines cut by a transversal and eight angles labeled 1–8.] Prove: 𝑚 ∥ 𝑛. Given: ∡1 ≅ ∡8.

Question 28: Use the diagram provided and the knowledge that ∡3 ≅ ∡10 and ∡1 ≅ ∡6, prove that ∡15 ≅ ∡13. Hint: First prove line 𝑎 is parallel to line 𝑏. Then prove line 𝑐 is parallel to line 𝑑.

Section 4.3, Perpendicular Lines, begins on page 166. The perpendicular lines and right

angles theorem is stated as, “If two lines are perpendicular, then they intersect to form four right

angles.” Lines perpendicular to a transversal theorem is defined next: “If two lines in a plane are

perpendicular to the same line, then they are parallel to one another” (p. 166). Finally, the

perpendicular transversal theorem is given as, “If a transversal is perpendicular to one of a pair

of parallel lines, then it is perpendicular to the other parallel line” (p. 167).

Then, a proof appears. It says “Prove 𝑚 ⊥ 𝑦” and has a picture of two parallel lines, 𝑥

and 𝑦, with a third line 𝑚intersecting line 𝑥 at a right angle. A side note on the page says, “We

prove the perpendicular transversal theorem.” In the exposition of this proof, there is a statement-

reason chart, with a third column that has no heading. It seems that the third column contains

explanations of the reasons in the second column. Also, the proof presented is not easy to follow.

The statement and reason for why angle 𝐵 is 90° is rather murky; in fact, the statement is also

listed in the reason column.

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On page 168, two model problems appear. One of them is to prove that two lines are

parallel. The solution is given in the form of a statement-reason proof. The third column

referenced in the previous bullet is also present here.

Page 169 introduces a construction of a perpendicular bisector by outlining the steps

students should perform with their compass and straightedge to construct the perpendicular

bisector of a segment. On page 170, after the conclusion of the construction, appears the

question, “Why did the construction above create perpendicular lines?” A statement-reason proof

follows (it is filled in for students). Again, the third mysterious column is there which re-explains

all the reasons. Of note here is the fact that “SSS congruence” is used as a reason, but the book

has not yet introduced this topic. It says in the third column, “The triangles have three pairs of

congruent sides. (We discuss SSS in depth in the next chapter.)” (p. 170).

Pages 171–172 detail the steps of two more constructions—constructing two parallel

lines and a line parallel to a given line through a point not on the line—and then show the proof

of the latter. Again, the third explanatory column is there, and again, “SSS congruence” is used

as a reason although the book has not yet introduced this topic. On page 173, students are told

that the distance between a point and a line refers to the shortest distance and that it will be the

perpendicular segment connecting the point and the line.

Practice problems begin on page 173. Of 36 problems, three relate to proof and proving.

Question 10 reads, “Given that 𝑚 ∥ 𝑛 and 𝑝 ⊥ 𝑚, prove that 𝑝 ⊥ 𝑛.” A diagram is given of two

parallel lines cut by a transversal. Although I have marked it as GPT based on the instructions to

“prove”, the students do not have anything to genuinely prove here because the set-up is the

same as the perpendicular transversal theorem (all they have to do is invoke the theorem). Also,

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it should be noted that the eight angles measured on this diagram have no need to be labeled (the

labeled points serve no function).

The other two questions are as follows:

Question 11: Prove that the angle bisectors of adjacent supplementary angles are perpendicular to each other. [A diagram illustrates the situation and provides the given.]

Question 12: On a plane, 𝑎 is perpendicular to 𝑏, 𝑏 is parallel to 𝑐, and 𝑛 is perpendicular to 𝑎. Is line 𝑛 perpendicular to 𝑐? Explain your answer.

Section 4.4 (Parallel Lines, Perpendicular Lines, and Slope) begins on page 176. On page

177, students are told that “slopes of perpendicular lines are negative reciprocals.” Then, the

textbook says, “We show how this is true with a 90° clockwise rotation in the graph to the right”

and a nice exposition is provided to use the rotation to justify that the slopes are negative

reciprocals. This explanation is not labeled as a proof, and it is true that it is for a special case

(both lines intersect the origin).

Two model problems on page 178 have nothing to do with proof or proving. Page 179

presents a model problem on how to find the distance between a point and a line. The process is

clearly modeled using an example; nothing is proved. Practice problems begin on page 180. Of

34 problems, the following (potentially) relate to proof and proving:

Question 12: Are the lines described by the equations 𝑦 = 5𝑥 + 7 and 𝑦 = 2𝑥 + 3 parallel? Justify your answer.

Question 20 The line 𝑎 goes through the points (6,−4) and (−5, 0). The line 𝑏 passes through the points (2,−7) and (−9, −3). Are these lines parallel? Why or why not?

Question 23: On the diagram, the triangles 𝐴𝐵𝐶 and 𝐷𝐵𝐾 are similar, that is, their corresponding angles are congruent. Prove that the lines have the same slope.

Question 25: Is the triangle in the graph below a right triangle? Justify your answer.

Question 27: Suppose you have a positively sloped line. If you rotate that line 90° counterclockwise about the origin, will the rotation form a line that is perpendicular to the first? Explain your answer.

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Question 34: A teacher says: ‘Triangle 𝐴𝐵𝐶 has vertex 𝐴 at (−5,−2), vertex 𝐵 at (2,−10), and vertex 𝐶 at (0, 0). 𝐴𝐵𝐶 is a right triangle, with 𝐶 its right angle.’ Is he correct? Justify your answer.

Page 182 contains a multipart problem practice in which students are given four

coordinates that are the vertices of a quadrilateral. They must graph it, explain what type of

quadrilateral it is, justify how to move a vertex to make it a parallelogram but not a rectangle,

justify how to move a vertex to make it a rectangle, and then explain whether the move from the

previous part makes the quadrilateral a square.

Section 4.5 (Parallel Lines and Triangles) begins on page 182. First, interior and exterior

angles of a polygon are defined. Then, the triangle sum theorem is stated as “the sum of the

interior angle measures in a triangle is 180 degrees” (p. 183). The theorem is proved for the

students using an auxiliary line. The proof appears in statement-reason form, but again with the

third unlabeled column.

Page 184 begins with the statement of the exterior angle theorem: “The measure of a

triangle’s exterior angle equals the sum of the measures of the two non-adjacent interior angles.”

There is a labeled diagram. No proof is presented, and the students are not tasked with proving

the theorem. Two computational model problems follow, which both show students how to use

the exterior angle sum to find the measure of a missing angle (one gives numeric values, the

other involves algebraic expressions).

Page185 begins with the statement of a corollary, the triangle sum corollary: “If a triangle

is a right triangle, then the measures of its two other angles sum to 90 degrees.” Above the

statement of the corollary, it says, “The corollary below follows from the theorem that a

triangle’s angles sum to 180°. A corollary is a statement that follows from an already proven

theorem or postulate” (p. 185).

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A model problem follows that shows how to use the triangle sum corollary to find the

measure of an angle in a right triangle. In this example, the two non-right angles are labeled 𝑧 +

15° and 2(𝑧 − 12°). Practice problems begin on page 185. Of 35 problems, the following

(potentially) relate to proof and proving:

Question 16: Can there be both an obtuse and a right angle in a triangle? If so, give an example. If not, explain why.

Question 30: In the diagram, 𝐴𝐵𝐶 is a right triangle and 𝐶𝐷0000 ⊥ 𝐴𝐵0000. Prove that 𝑚∡𝐴 =𝑚∡𝐵𝐶𝐷.

Question 33: Identify one exterior angle for each vertex of a triangle. What is the sum of these angles? Justify your response.

Question 34: The sum of the measures of any two interior angles of a particular triangle is greater than 90°. Determine the type of triangle: obtuse, right, or acute. In acute triangles, the measures of all angles are less than 90°, and obtuse triangles have an angle whose measure is greater than 90°. Explain your answer.

Question 35: Prove the sum of the measures of the exterior angles of a triangle is 360°.

Page 187 begins the Chapter 4 review. Question 7 is a multiple-choice question that

makes students reason about a given diagram: “If 𝑚 ∥ 𝑛, how do we know that 𝑝 ⊥ 𝑛?” Seven

other questions involve proof and proving:

Question 9: Using the diagram, prove the consecutive interior angles theorem converse.

Question 13: Prove that the opposite angles of quadrilateral 𝐴𝐵𝐶𝐷 are congruent. That is, prove that ∡1 ≅ ∡2 and ∡3 ≅ ∡4. [Diagram is given with two pairs of parallel lines forming a quadrilateral.]

Question 15: When two parallel lines 𝑝 and 𝑞 are intersected by a transversal 𝑛, the interior consecutive angles 𝐴 and 𝐵 are congruent. Prove that the transversal is perpendicular to both parallel lines.

Question 21: Point 𝑀 does not lie on line 𝑚. Explain how to construct a line 𝑝 through point 𝑀 and perpendicular to line 𝑚 using a compass and a straightedge. Perform the constructions.

Question 22: Using the diagram, prove the perpendicular lines and right angles theorem. Prove: Angles 𝐴, 𝐵, 𝐶, and 𝐷 are right angles. Given: 𝑚 ⊥ 𝑛.

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Question 25: Explain why a triangle cannot have two right angles.

Question 29: Using the diagram, prove the triangle sum corollary.

Page 191 begins with a cumulative review for Chapters 1–4. Of 34 questions, the

following have to do with proof and proving:

Question 13: Suppose 3 straight lines 𝑎, 𝑏, and 𝑐, all lie in the same plane. If lines 𝑎 and 𝑏 are parallel and lines 𝑏 and 𝑐 are parallel, what can you conclude about the relationship between lines 𝑎 and 𝑐? How do you know?

Question 14: In the diagram below, line 𝑎 is parallel to line 𝑏, ∡1 ≅ ∡3, and the measure of ∡9 = 120°. Prove the shaded triangle is equilateral.

Question 16: Two parallel lines are intersected by a transversal. Prove that the angle bisectors of two consecutive interior angles are perpendicular. Make a sketch, label the angles, and refer to them in your proof.

Question 17: A teacher says: “Triangle 𝐴𝐵𝐶 has vertex 𝐴 at (−4, −1), vertex 𝐵 at (2,−8), and vertex 𝐶 at (0, 0). 𝐴𝐵𝐶 is a right triangle, with 𝐶 its right angle.” Is she correct? Justify your answer.

Question 22: Using the point (𝑎, 𝑏), show algebraically that four 90° counterclockwise rotations about the origin return you to the original point (𝑎, 𝑏).

Question 24–Question 27: Write the biconditional for each definition.

Question 24: The real numbers are composed of the rational and irrational numbers.

Question 25: An integer prime is a number with exactly two positive distinct factors.

Question 26: Two angles that sum to 180° are called supplementary angles.

Question 27: Parallel lines are lines on a plane that do not intersect.

Chapter 5 Notes

Page 194 begins Chapter 5, Congruent Triangles. To begin Section 5.1, Isosceles and

Equilateral Triangles, types of triangles are classified based on their angles and sides (p. 195).

Perhaps there is a missed opportunity for students to prove that isosceles triangles have two

congruent sides (but this might come up later in the chapter). Two model problems are presented

that use the interior angle sum and the distance formula to classify triangles based on their angles

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and/or sides. Starting on page 197, there are 14 practice problems, none of which relate to proof

or proving.

The isosceles triangle theorems are presented as follows: “If two sides of a triangle are

congruent, then the angles opposite them are congruent. If two angles of a triangle are congruent,

then the sides opposite them are congruent” (p. 198). A diagram appears that shows an isosceles

triangle with congruent base angles and sides marked, and two statements underneath that

employ geometric notation. A blurb appears to the right of the diagram which says, “The first

statement is the base angles theorem. The second statement is the converse of the base angles

theorem. We will prove the base angles theorem later” (p. 198).

Page 199 begins with the isosceles bisector theorem: “If a line bisects an isosceles

triangle’s vertex angle, then it is a perpendicular bisector of the base.” Then it introduces the

isosceles bisector theorem converse: “If a line is the perpendicular bisector of an isosceles

triangle’s base, then it is also the angle bisector of the vertex angle” (p. 199). A model problem

for how to find the measure of a base angle in an isosceles triangle immediately follows.

Equilateral triangle theorems are presented next: “If three sides of a triangle are

congruent, then the angles opposite them are congruent. If three angles of a triangle are

congruent, then the sides opposite them are congruent” (p. 200). Students are then told, “The

second statement is the converse of the first” (p. 200).

At the bottom of the page, construction of an equilateral triangle begins, and two methods

are presented. The process of the first method is outlined for students. This is a missed

opportunity for a proof—students could justify the steps of the construction. The first method is

the traditional compass-straightedge method, and the second uses a reflective device. The second

method is proven using a statement–reason chart (p. 202). The proof is not easily readable or

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comprehensible given that it invokes new terminology such as “axis of reflection symmetry” and

the “reflective device.”

Page 203 has a model problem which says, “Prove the first equilateral triangle theorem:

If three sides of a triangle are congruent, then the angles opposite them are congruent.” Then the

proof is given for students in statement-reason form based on a given diagram. Here, there is no

third column. Practice problems begin on page 203. Of 20 problems, the following two relate to

proof and proving:

Question 18: 𝑚∡𝑊𝑍𝑋 = 60°. Prove that ∆𝑊𝑍𝑋 is equilateral.

Question 20: Prove the second equilateral triangle theorem: If three angles of a triangle are congruent, then the sides opposite them are congruent.

Section 5.2, Congruent Figures, begins with a statement of the third angles theorem: “If

two angles in a triangle are congruent to those of another triangle, then the triangles’ third angles

are congruent” (p. 205). A paragraph proof is presented for this theorem (p. 206).

Four model problems follow. The first is finding the measure of a missing angle, but the

solution is presented in statement-reason third column format. The second is a multiple-choice

question about classifying triangles as similar/congruent. The third and fourth questions are both

proof exercises that target proving two triangles are congruent. The proofs are done for the

students, and the text uses the reasoning, “Corresponding angles, corresponding sides all

congruent” (p. 207) as the reason for saying two triangles are congruent.

Practice problems begin on page 208. Of 14 practice problems, the following three relate

to proof and proving:

Question 3: Which of the following could be used in a proof to justify the statement ∡𝐴𝐸𝐵 ≅ ∡𝐶𝐸𝐷? Select all that apply. [A diagram is shown with side-side-side congruence marked, two congruent angles marked, and the third angle congruent via vertical angles or third angles theorem. The choices are “third angles theorem,” “reflexive property,” “adjacency theorem,” and “vertical angles theorem.”]

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Question 13: Given 𝐶𝐹0000 ≅ 𝐺𝐸0000, prove ∆𝐵𝐶𝐺 ≅ ∆𝐷𝐸𝐹 by showing that all pairs of corresponding sides and angles are congruent. Be sure to justify each step in your proof. [Diagram is provided.]

Question 14: In isosceles triangle 𝐴𝐵𝐶, 𝐵𝐷0000 bisects ∡𝐵 and intersects base 𝐴𝐶0000 at point 𝐷, as shown in the diagram. Prove ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷. [Diagram is provided.]

Page 209 begins Section 5.3, Proving Triangles Congruent with SSS and SAS:

We discussed previously how rigid motion transformations create congruent figures. Another way in geometry to prove that triangles are congruent is a series of postulates and theorems. We start with the side-side-side (SSS) postulate. . . . If three sides of one triangle are congruent to those of another triangle, then the triangles are congruent. (p. 209)

Justification of “copy an angle” construction is then given (p. 210) using SSS postulate

(statement-reason proof). Model Problem 1 reads, “Are the triangles congruent? Apply the SSS

postulate to prove that they are” (p. 210). Model Problem 2 is a good exercise (“Use SSS to

prove the base angles theorem”), but it is leading (tells students to use SSS) and is done for them.

Model Problem 3 is another SSS congruence proof, appropriately stated because it does not tell

the students the route to go on (although the problem is solved for them given that it is a model).

Three out of five model problems are proof problems (p. 210).

The hypotenuse-leg congruence theorem is stated on page 211: “If the hypotenuse and a

leg of a right triangle are congruent to those of another right triangle, then the triangles are

congruent.” This theorem is immediately followed with text that reads,

The Pythagorean theorem and the SSS postulate can be used to prove the hypotenuse-leg theorem. If one leg and the hypotenuse are congruent to those of another triangle, then the other legs must be congruent as well due to the Pythagorean theorem, and then the triangles are congruent according to the SSS postulate. (p. 212)

On page 212 there is a small blurb-type mention of why there is no SSA postulate, with a

diagram demonstrating an example of how two sides and a non-included angle do not generate a

unique triangle. Two model problems appear; both are proofs. The first is to prove that two

triangles are congruent given a condition about the diagram. The solution is presented in a

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statement-reason third column chart. The second is stated as a “show that” question, but the

response is written as a proof with a statement-reason chart (no third column).

Side-angle-side (SAS) postulate is introduced next: “If two sides and the included angle

of a triangle are congruent to those of another triangle, then the triangles are congruent” (p. 213).

There is a demonstration of how to use SAS to prove two triangles are congruent (statement–

reason chart), and three model problems are presented (p. 214), the first two of which are proofs.

The first one involves properties of parallelograms, the reflexive property, and SAS; the second

is more leading (tells students to use SAS). Both are done for the students because they are

model problems.

Practice problems begin on page 215. Of 32 problems, the following relate to proof and

proving:

Question 4: Based on the diagram, by which of the following methods can the triangles be proved congruent?

Question 7: ∆𝐴𝐵𝐶 ≅ ∆𝑌𝑍𝑋 and 𝐴 and 𝑌 are right angles. 𝐴𝐵0000 ≅ 𝑌𝑍0000 and 𝐵𝐶0000 ≅ 𝑍𝑋0000. Can we use the hypotenuse-leg theorem to prove that these two triangles are congruent?

Question 16–Question 22: Using the diagram, provide a justification for each statement. The statements are part of a proof that triangle BAT is congruent to triangle ENR.

Question 24: What additional piece of information is needed to prove ∆𝑃𝑈𝐸 ≅ ∆𝑅𝐸𝑈 by the hypotenuse-leg theorem? [Diagram is provided.]

Question 25: What additional piece of information is needed to prove ∆𝑆𝑌𝑅 ≅ ∆𝐵𝑈𝐹 by the hypotenuse-leg theorem? [Diagram is provided.]

Question 26: Prove ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 by SSS. [Diagram is provided.]

Question 27: Prove ∆𝐽𝐾𝐿 ≅ ∆𝑅𝑆𝑇 using the side-side-side postulate. [Diagram is provided that has two pairs of congruent sides and two pairs of congruent base angles.]

Question 28: Tricia needs to prove the two right triangles below are congruent, but she says she can’t do it because she doesn’t know the length of each triangle’s hypotenuse. Is Tricia correct in saying there is no way to prove the triangles congruent? Explain your reasoning. [Diagram is provided with SAS marks.]

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Question 30: Luis says ∆𝑆𝐸𝑇 ≅ ∆𝐵𝐴𝑅 by the side-angle-side postulate. Do you agree? Explain your reasoning.

Question 31: Given 𝐽𝐸O⃖O⃗ ∥ 𝐴𝑃O⃖OOO⃗ , prove ∆𝐽𝐸𝑇 ≅ ∆𝑃𝐴𝐿. [Diagram is provided.]

Question 32: Using the diagram, prove the Isosceles Bisector Theorem Converse: If a line through the vertex of an isosceles triangle is the perpendicular bisector of the base, then it is also the angle bisector of the vertex angle. [Diagram is provided.]

Section 5.4, Proving Triangles Congruent with ASA and AAS, begins on page 218.

Angle-side-angle (ASA) postulate is defined: “If two angles of a triangle and the side between

them are congruent to those of another triangle, then the triangles are congruent” (p. 218). Two

model problems follow, the second of which is a proof. There is a diagram given of an isosceles

triangle (two sides marked congruent) and the perpendicular bisector of the third side. The model

problem shows how to prove that the two triangles formed are congruent. A paragraph proof is

used to show how ASA can be used, and commentary afterwards explains that SAS could have

been used as well.

Page 219 begins with a five-paragraph explanation of how the criteria ASA, SAS, and

SSS follow from rigid motion transformations. Some examples are explained using a diagram.

The bottom half of the page is called Introduction to Coordinate Proofs. “Now we begin doing

coordinate proofs. We prove that a translation creates a congruent figure and is an example of

rigid motion” (p. 220). A diagram shows a translation and a statement-reason third column proof

is given. The text then reads,

Rigid motion provides a definition of congruent figures, including triangles. In a rigid motion transformation, all corresponding angles and all corresponding sides remain congruent. In other words, a rigid motion transformation does not change the measure of an angle or the length of a triangle side. This means that two triangles are congruent if and only if one can be transformed into the other by rigid motion transformations. (p. 221)

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A model problem with a contrived scenario is presented. On page 222, angle-angle-side

(AAS) theorem is given: “If two angles and a side (not between them) of a triangle are congruent

to those of another triangle, then the triangles are congruent.” Then it says,

Instead of a two-column proof, we do a flow proof. These proofs are intended to let you see the various “paths” of a proof—the sets of steps that could be done in parallel. For instance, showing that we have congruent angles and sides occurs at the same level in the proof. (p. 222)

The flow proof is presented with an explanation of its structure and what is occurring at

each step underneath the proof. Practice problems begin on page 223. Of 33 problems, the

following have to do with proof and proving:

Question 1: Which of these are valid postulates to prove that two triangles are congruent? Select all that apply.

Question 4: Two triangles have one pair of corresponding congruent sides. Which additional information would be needed to prove that the triangles are congruent?

Question 5: Which of the following justifications cannot be used to prove two triangles are congruent?

Question 6: Which set of statements can be used to prove ∆𝑀𝐴𝑁 ≅ ∆𝐷𝑂𝐺?

Question 7: How could you prove ∆𝐷𝐸𝐺 ≅ ∆𝐹𝐸𝐺? Select all postulates that apply.

Question 8–Question 14: [Each present two triangles with given information and ask students to “explain” whether the triangles are congruent.]

Question 15: Which additional piece of information is needed to prove ∆𝐻𝐼𝐾 ≅ ∆𝐽𝐼𝐾 by the 𝐴𝑆𝐴 postulate? [Diagram is provided.]

Question 18: Prove ∆𝐷𝐸𝐺 ≅ ∆𝐹𝐸𝐺 using the ASA postulate.

Question 19: Prove that ∆𝐴𝐵𝐶 is congruent to ∆𝐸𝐷𝐶 using the ASA postulate.

Question 20: Prove that ∆𝑆𝑅𝐾 ≅ ∆𝑃𝑅𝐴 using the ASA postulate.

Question 21: Taylor proved ∆𝐵𝑊𝐼 ≅ ∆𝐿𝐺𝐴 by the AAS theorem using three given pieces of information about corresponding congruent parts of the two triangles. Two of the given pieces of information were that ∡𝑊 ≅ ∡𝐺 and 𝑊𝐼0000 ≅ 𝐺𝐴0000. (a) Sketch the two triangles and indicate the known congruent parts based on the given information. (b) What is the third piece of information Taylor must have been given in order to complete her proof?

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Question 23: Prove ∆𝑃𝐴𝑆 ≅ ∆𝐴𝑃𝐿 using the angle-angle-side theorem.

Question 24: Two triangles each contain two angles which both measure 50° and a non-included side which measures 10 centimeters. What is the most efficient method for proving the triangles congruent?

Question 25: What method(s) can be used to prove the two triangles are congruent? [Diagram is given with markings of congruence.]

Question 26–Question 27: State which method could be used to prove ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹, based on the given information. If it is not possible to prove the triangles congruent, state “Not possible.”

Question 28: The three angles of triangle 𝐾𝐽𝑅 are congruent to the three angles of triangle 𝑀𝐴𝑅. It is known that 𝐾𝑅0000 ≅ 𝑀𝑅00000. Are the two triangles necessarily congruent? Explain.

Question 29: ∆𝑇𝑅𝐴 is equilateral. What method could be used to prove ∆𝐺𝑅𝑇 ≅ ∆𝐸𝑅𝐴? [Diagram is given of overlapping triangles.]

Question 30: Regular pentagon 𝑃𝑄𝑅𝑆𝑇 has one diagonal connecting vertex 𝑇 and vertex 𝑄, and another diagonal connecting vertex 𝑇 and vertex 𝑅. By what method can ∆𝑇𝑃𝑄 be proved congruent to ∆𝑇𝑆𝑅?

Question 33: Prove the isosceles bisector theorem: If a line bisects an isosceles triangle’s vertex angle, then it is a perpendicular bisector of the base. [Diagram is provided.]

Page 226 begins the Chapter 5 Review, and four of the questions relate to proof or

proving:

Question 16: Prove: ∆𝑆𝐴𝑅 ≅ ∆𝑇𝑅𝐴. [Diagram is provided.]

Question 19: Triangle 𝐵𝐶𝐸 is isosceles. Prove that ∆𝐴𝐶𝐹 is congruent to ∆𝐷𝐵𝐺. [Diagram is given with two overlapping triangles.]

Question 21: Is the triangle below an isosceles triangle? Explain your reasoning. [Diagram is provided of a triangle on a coordinate plane.]

Question 30: Parallelogram 𝑃𝐸𝐴𝑇 contains diagonal 𝑃𝐴0000. Prove ∆𝐴𝑃𝐸 ≅ ∆𝑃𝐴𝑇.


review, the following three are the proof-related questions:

Question 24: What is the difference between postulates and theorems?

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Question 35: Any angle of a particular triangle is smaller than the sum of the two other angles. Determine the type of the triangle: obtuse, right, or acute. Explain your answer.

Question 36: A teacher says: “Triangle 𝐴𝐵𝐶 has vertex 𝐴 at (5,−4), vertex 𝐵 at (8, 15), and vertex 𝐶 at (0, 0). 𝐴𝐵𝐶 is a right triangle, with 𝐶 its right angle.” Is she correct? Justify your answer.

Chapter 6 Notes

Chapter 6, Relationships Within Triangles, begins with Section 6.1, Midsegments.

Midsegment is defined and then the midsegment theorem—parallel to the third side is presented

as, “If a segment joins the midpoints of two triangle sides, then the segment is parallel to the

third triangle side” (p. 233). Following, midsegment theorem—length is half of third side is

presented as, “If a segment joins the midpoints of two triangle sides, then the segment’s length is

one-half the third side’s length” (p. 233).

Midsegment theorem—parallel to the third side is proved using a statement-reason third

column proof. The proof employs “SAS similarity theorem,” (p. 234), and the third column box

for this reason says,

Because two corresponding sides of the triangles are proportional, and the angle between them is congruent, the triangles are similar by SAS for similar triangles. We discuss the similarity theorem in depth in the next chapter. (p. 234)

Midsegment theorem—length is half of third side is proved using a coordinate geometry

proof along with a statement-reason third column chart (p. 235). The method used involves

choosing coordinate points for “convenience” and is hard to understand. Model problems begin

on page 236. Of four problems, none are proofs. Practice problems begin on page 237. Of 27

problems, the following have to do with proof and proving:

Question 14: Prove: ∡𝑂𝑅𝐶 ≅ ∡𝑂𝐸𝑆. [Diagram is provided of a triangle 𝑆𝑂𝐸 with midsegment 𝐶𝑅drawn in (𝐶 ∈ 𝑂𝑆, 𝑅 ∈ 𝑂𝐸).]

Question 22: Refer to the diagram of ∆𝐴𝑇𝑅. Without using the slope formula, prove 𝑇𝑅0000is parallel to 𝐿𝐼� . [Diagram is provided of the triangle graphed on a coordinate plane (vertices have integer coordinates).]

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Question 27: Triangle 𝐴𝐵𝐶 includes point 𝐸 located on line segment 𝐴𝐵 and point 𝐹 located on line segment 𝐴𝐶. [Diagram is provided of triangle graphed on coordinate plane.] Derrick says that all he needs to do to prove 𝐸𝐹0000 is a midsegment of ∆𝐴𝐵𝐶 is to show that 𝐸𝐹0000 is parallel to 𝐵𝐶0000. Do you agree with Derrick? Why or why not?

Section 6.2, Perpendicular and Angle Bisectors, begins on page 240. Perpendicular

bisector theorem is defined: “If a point is on a line segment’s perpendicular bisector, then it is the

same distance from each of the line segment’s endpoints” (p. 240). Page 241 has a statement-

reason third column proof of the theorem, which uses SAS. The concluding statement, 𝑃𝐴 = 𝑃𝐵,

has the reason “definition of congruent triangles” and the third column says, “Corresponding

parts of congruent triangles are congruent” (p. 241).

Perpendicular bisector theorem converse is defined next: “If a point is the same distance

from a segment’s endpoints, then it is on the segment’s perpendicular bisector” (p. 241). Here,

the proof is omitted. Model problem 2 is a proof of the fact that a triangle for which an altitude is

also a median is isosceles. The proof is presented in statement-reason third column format.

The text defines angle bisector theorem as follows: “If a point is on the bisector of an

angle, then that point is the same distance from the sides that form that angle” (p. 243). No proof

is presented; rather, students are tasked with proving this later in the chapter. Similarly, bisector

theorem converse is defined but no proof is presented: “If a point in the interior of an angle is the

same distance from that angle’s sides, then it is on the bisector” (p. 244).

Page 245 presents construction: angle bisector in a four-step process. Afterwards, the

question, “Why does this construction create two congruent angles?” precedes a paragraph proof.

The proof uses the diagram to explain how SSS is employed via the construction. This

information is nicely presented.

Practice begins on page 246. Of 25 problems, the following two task students with proof

and proving:

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Question 24: Will the angle bisectors of the interior angles of a polygon all intersect at the same point? Explain your reasoning.

Question 25: Prove the Angle Bisector Theorem: If a point is on the bisector of an angle, then that point is the same distance from the sides that form that angle. [Diagram is provided which essentially shows students the given, the “prove,” and the geometric markings.]

Section 6.3, Circumcenters, begins on page 248. Students are first told that the

perpendicular bisectors of the sides of a polygon meet at a single point. The exact wording reads,

In lesson 6.2 we learned that we can create a perpendicular bisector to any line or line segment. Similarly, we can create the perpendicular bisectors to each side of a polygon, such as a triangle. When we do so, those bisectors meet at a single point, which is called a point of concurrency, and the bisectors are known as concurrent segments, concurrent rays, or concurrent lines. (p. 248)

Page 249 presents the concurrency and perpendicular bisectors theorem: “A triangle’s

perpendicular bisectors intersect at a point of concurrency that is equidistant from the triangle’s

vertices.” Circumcenter is defined as “a point of concurrency” (p. 250), and then a model

problem appears which shows how to calculate the circumcenter of a right triangle. The result is

not generalized to indicate that the circumcenter of a right triangle will always lie at the midpoint

of the triangle’s hypotenuse.

Practice problems begin on page 253. Of 28 problems, two have to do with proof and

proving:

Question 22: Explain why connecting any three distinct points on the circumference of a circle will result in a triangle whose circumcenter is located at the center of the circle.

Question 28: Why are only two perpendicular bisectors needed to find the circumcenter? How do you know the third will intersect? Explain your answer.

Section 6.4, Centroids and Orthocenters, begins on page 254. Centroid is defined and

then the medians intersection theorem is presented as “the medians of a triangle meet at a single

point” (p. 255). The theorem is not proved. Concurrency and medians theorem is next defined:

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“The distance from a vertex to the centroid is two-thirds the length of the median” (p. 255). The

text next states,

We want to prove the theorem that states the distance from a vertex to the centroid is two-thirds the length of the median. Before we can prove the theorem, we need to show how three medians divide a triangle into six smaller triangles of equal area. (p. 255)

A diagram is drawn, and the statement of that “theorem” (it is not regarded as one) is

provided specific to the diagram. It is proved using a statement-reason third column proof. The

proof is not broken down enough for students to understand that the equal areas argument is

being applied to the pairs of not only the smallest triangles, but also the larger triangles formed

by one median.

Page 256 states: “We now prove the concurrency and medians theorem.” A statement-

reason third column proof is presented that employs the six triangles of equal area property of the

centroid. Within the context of a model problem, the proof of the concurrency of the three

medians is presented for students in statement-reason third column format (p. 257).

Altitude is defined, and it is stated that “the altitudes of a triangle intersect at a point

called the orthocenter” (p. 258). This statement is not proved. The location of the orthocenter

depends on the type of triangle. It is then explained, via words and diagrams, where the

orthocenter lies for acute, obtuse, and right triangles.

Practice problems begin on page 260. Of 31 problems, the following have to do with

proof and proving:

Question 13: Use the diagram to answer the questions. . . . (c) What additional information is needed to prove that 𝐹𝐵0000 is a median of ∆𝐴𝐷𝐹?

Question 31: If 𝐾𝑇 = 9, 𝐴𝐶 = 36, 𝐴𝑃 = 33, and 𝐴𝑇 = 22, prove ∆𝐴𝑅𝐾 ≅∆𝐴𝑇𝐾.[Diagram is provided.]

Section 6.5, Incenters, begins with readers being told that

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the angle bisectors of a triangle also have a point of concurrency. . . . This point of concurrency is called the incenter. A triangle’s incenter is equidistant from each side of the triangle. (p. 262)

Then, the concurrency and angle bisectors theorem is stated: “The angle bisectors of a

triangle intersect at a point that is the same distance away from each side of the triangle” (p.

262). No proof is provided.

Practice problems begin on page 264. Of 21 problems, the following two have to do with

proof and proving:

Question 18: When two angle bisector segments are constructed to stop at the incenter of a triangle, is it possible for the angle formed between them to measure less than 90°? Explain your reasoning.

Question 19: Using the diagram below as a reference, prove that the incenter, 𝐷, is equidistant from each side of the triangle. Hint: Use the angle bisector theorem. Prove: 𝐷𝐸 = 𝐷𝐹 = 𝐷𝐺,Given: 𝐷 is the incenter of ∆𝐴𝐵𝐶. [Diagram is provided.]

Section 6.6, Inequalities in One Triangle, is labeled “optional” and begins on page 265.

Although labeled optional, the section contains five important theorems (none of which are

proved in the section):

Triangle Inequality Theorem: Each side of a triangle is shorter than the sum of its other two sides.

Unequal Sides Theorem: If, in a triangle, two sides are unequal in length, then the angle opposite the longer side is greater than the angle opposite the shorter side.

Unequal Sides Theorem Converse: If, in a triangle, two angles are unequal in measure, then the side opposite the greater angle is longer than the side opposite the smaller angle.

Hinge Theorem: If two sides of a triangle are congruent to those of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

Converse of Hinge Theorem: If the two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

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Practice problems begin on page 269. Of 20 problems, the following five relate to proof

and proving:

Question 16: Triangle 𝑄𝑈𝐴 has vertices located at 𝑄(−2, 2), 𝑈(3,−3), and 𝐴(2, 5). Graph the triangle, and then prove ∡𝑄 is the largest angle. Hint: Consider using the distance formula.

Question 17: Use the hinge theorem to explain why it is impossible to prove two non-right triangles congruent using SSA. You may wish to include an illustration to support your answer.

Question 18: Given point 𝐷 is the centroid of ∆𝐴𝐵𝐶, 𝐵𝑀00000 ≅ 𝐵𝐶0000, and 𝑚∡𝑀𝐵𝐶 = 40°, prove 𝐴𝐵 > 𝐵𝐶. [Diagram is provided.]

Question 19: Given triangle 𝐹𝐴𝐶 with vertices at 𝐹(4, 6), 𝐴(5, 4), and 𝐶(1,2) and triangle 𝑇𝑂𝑅 with vertices at 𝑇(5, 1), 𝑂(7, 0), and 𝑅(1,−2), use the hinge theorem to show that 𝑚∡𝑇 > 𝑚∡𝐹.

Question 20: Use the diagram below to prove the hinge theorem for triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹. Hint: First find ways to prove that 𝐴𝑃0000 ≅ 𝐷𝐹0000 and 𝑃𝑄0000 ≅ 𝐶𝑄0000. Then prove that 𝐴𝑄 +𝑄𝐶 > 𝐴𝑃 and 𝐴𝐶 > 𝐴𝑃. Given 𝑚∡𝐴𝐵𝐶 > 𝑚∡𝐷𝐸𝐹, prove 𝐴𝐶 > 𝐴𝑃. [Diagram is provided.]

Section 6.7, Indirect Reasoning, is labeled “optional” and begins on page 271. General

comments are made about the process for proving by contradiction, and two model proofs

appear. The solutions are not written in statement-reason form but rather columns and rows that

lay out “what we want to prove,” “assume the opposite conclusion,” and “a contradiction” (p. .

Neither model is reader friendly.

Practice problems begin on page 272. Of 25 problems, the following are proof tasks:

Question 14: A triangle has vertices located at 𝐶(0, 5), 𝐴(−2, 1), and 𝑇(4, 3). Prove ∆𝐶𝐴𝑇 is not equilateral using indirect reasoning.

Question 20: Given triangle 𝐽𝐾𝐿 and triangle 𝐴𝐵𝐶 where 𝐽𝐾 is congruent to 𝐴𝐵, 𝐾𝐿 is congruent to 𝐵𝐶, and 𝐽𝐿 is not congruent to 𝐴𝐶. (b) You need to prove ∆𝐽𝐾𝐿 is not congruent to ∆𝐴𝐵𝐶 using indirect reasoning. What statement should you assume to be true to begin your proof? (c) Based on the given information and an assumption of the opposite conclusion, what theorem or postulate could you use to attempt to prove the triangles congruent? (d) Use the information and your answers to parts (a)–(c) to write a proof that ∆𝐽𝐾𝐿 is not congruent to ∆𝐴𝐵𝐶.

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Question 21: Given the sides of a triangle measure 5 units, 11 units, and 14 units, answer the following questions. (b) You are asked to prove the triangle is not right by indirect reasoning. What initial assumption should you make to begin your proof? (c) Use the given information and your answers to parts (a) and (b) to prove that the triangle is not right using indirect reasoning.

Question 22: Triangle 𝐿𝐺𝐴 has vertices 𝐿(3, 5), 𝐺(0, 3), and 𝐴(6, 1). . . . (e) Use your graph and information from parts (a)–(d) to prove that ∆𝐿𝐺𝐴 is a [sic] not a right triangle by indirect reasoning.

Question 24: Use indirect reasoning to show that (−3, 11) is not the midpoint of the line segment with endpoints at (−5, 13) and (−1, 15).

Question 25: Quentin wanted to prove that the sum of two consecutive odd integers is not an odd integer. He did so in the proof shown. Refer to his proof to answer the questions. [Quentin’s statement–reason chart is shown. Students are asked to fill in one missing reason, identify where the contradiction is, and provide a counterexample to one of Quentin’s statements.]

Page 275 begins the Chapter 6 review, which has four questions related to proof and

proving:

Question 26: The measures of the interior angles of a triangle can be represented by (2𝑥 + 7)°, (3𝑥 − 9)°, and (5𝑥 − 18)°. Prove that the triangle is not isosceles using indirect reasoning.

Question 28: Use the diagram to prove 𝐵𝐷OOOOOO⃗ is the angle bisector of ∡𝐴𝐵𝐶. [Diagram is provided.]

Question 35: Two fishing trawlers left an island at the same time. One headed east for 8 miles, then turned southeast and went an additional 7 miles, and then returned to the island. The other boat headed north for seven miles, then turned northwest and went another 8 miles, and then returned to the island. Based on the diagram, prove that the two boats did not travel the same total distance. Use indirect reasoning in your proof. [Diagram is provided.]

Question 36 (Optional): Prove the Triangle Inequality Theorem: Each side of a triangle is shorter than the sum of its other two sides.


review, the following nine questions are the proof-related questions:

Question 9: The phrase, “All triangles are polygons” can be written as which conditional statement?

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Question 14: Prove: ∆𝐴𝐶𝐵 ≅ ∆𝐸𝐶𝐹,Given: The midpoint of 𝐴𝐸0000 is 𝐶; 𝐵𝐶 = 𝐶𝐹. [Diagram is provided.]

Question 15: Prove the concurrency and perpendicular bisectors theorem. Prove: 𝐷𝐴0000 ≅𝐷𝐵0000 ≅ 𝐷𝐶0000, Given: 𝐷𝐸0000, 𝐷𝐹0000,and 𝐷𝐺0000 are ⊥ bisectors. [Diagram is provided.]

Question 16: Two non-congruent line segments 𝐽𝐾 and 𝐿𝑀 bisect each other at point 𝑁. Connecting points 𝐽 and 𝐿 with line segment 𝐽𝐿, and points 𝐾 and 𝑀 with line segment 𝐾𝑀, forms triangles 𝐽𝑁𝐿 and 𝐾𝑁𝑀. (a) Complete a sketch of the situation. (b) Prove that ∆𝐽𝑁𝐿 ≅ ∆𝐾𝑁𝑀.

Question 17: Prove that the median of a triangle cannot form two obtuse angles with the side that it bisects.

Question 18: Sketch an isosceles triangle in the coordinate plane. Then, prove that your triangle is not scalene using indirect reasoning.

Question 19: Quadrilateral 𝐺𝑂𝐴𝑇 has vertices located at 𝐺(−3, 3), 𝑂(4, 4), 𝐴(6,−1) and 𝑇(−2, −2). Use indirect reasoning to prove that 𝐺𝑂𝐴𝑇 is not a parallelogram.

Question 20: Given: ∆𝐹𝐸𝐷 and ∆𝐹𝐴𝐷, which share side 𝐹𝐷0000; 𝐸𝐷0000 ≅ 𝐴𝐷0000, and 𝐹𝐷0000 bisects ∡𝐷. Prove: ∆𝐹𝐸𝐷 ≅ ∆𝐹𝐴𝐷.

Question 24: What additional piece of information is needed to prove ∆𝑇𝑈𝑉 ≅ ∆𝑊𝑋𝑌 by SSS? [Diagram is provided with side lengths of one triangle all labeled, and two side lengths of the other triangle labeled.]

Chapter 7 Notes

Chapter 7, Similarity and Trigonometry, begins on page 282. Section 7.1, Similarity:

Angle-Angle and Side-Side-Side, begins on page 283 with the angle-angle (AA) similarity

postulate: “If two angles of one triangle are congruent to two angles of another triangle, then the

triangles are similar.” Page 284 contains a numeric example and a paragraph proof of the fact

that dilations preserve angle measure. The proof is not readily understandable, and the center of

dilation is not stated. The text explains that it is arbitrary, but that explanation does not help

students understand the diagram. The bottom of page 284 contains a mistake; it says that under a

dilation that creates similar figures, “side lengths do not change.”

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Five model problems appear on page 285, the third of which is a proof. Provided a

marked-up diagram, the problem is to “Prove: ∆𝐶𝐴𝐵~∆𝑍𝑋𝑌.” The proof is presented in

statement-reason third column format.

On page 286, model problem 4b is stated as, “Prove the slope of 𝑚c is equal to the slope

of 𝑚W by showing that ∆𝐴𝐵𝐶~∆𝑌𝑋𝐶.” A diagram is provided. The solution is presented, but not

in a proof format that has been explained up to this point. The last part of the solution can be

misleading here; the expressions presented as slopes will both work out to positive numbers

(they use segments), but the slopes of both lines are negative.

Page 287 introduces the side-side-side (SSS) similarity theorem: “If the corresponding

sides of two triangles are proportional, then the triangles are similar.” No proof is presented, and

model problems begin immediately on page 288. Neither solution is presented as a proof,

although Question 2 says to “Show ∆𝐹𝐸𝐷~∆𝐺𝐼𝐻~∆𝐴𝐵𝐶.”

Practice problems begin on page 289. Of 33 problems, the following three are (arguably)

proof tasks:

Question 27: Given: 𝑚∡𝐴 = 𝑚∡𝐸, Prove: ∆𝐴𝐵𝐶~∆𝐸𝐵𝐷. [Diagram is provided.]

Question 31: Given the diagram below and the statement that 𝐵𝐷0000 is not parallel to 𝐴𝐸0000, prove ∆𝐶𝐵𝐷 is not similar to ∆𝐶𝐴𝐸. [Diagram is provided.]

Question 33: In the diagram below, 𝐴𝐵 = 10, 𝐵𝐸 = 8, 𝐴𝐸 = 7, 𝐷𝐶 = 15, 𝐷𝐸 = 12, and 𝐶𝐸 = 10.5. Prove 𝐴𝐵0000 ∥ 𝐷𝐶0000. [Diagram is provided.]

Section 7.2, Similar Triangles: Side-Angle-Side Theorem, begins on page 291. Side-

angle-side (SAS) similarity theorem is defined: “If the lengths of two pairs of corresponding

sides of two triangles are proportional and the angles the sides form are congruent, then the

triangles are similar.” Of three model problems, one is a proof. There is a side-splitter type

diagram presented, and the problem says, “Prove: ∆𝐴𝐵𝐶~∆𝐴𝐷𝐸.” The proof is presented in


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Practice problems begin on page 293. Of 38 problems, the following six are proof tasks:

Question 11: Show that the two triangles are similar. [Diagram is provided.]

Question 22: Given that 𝐴𝐵0000 ∥ 𝐸𝐷0000, determine if the triangles in the diagram below are similar. If the triangles are similar, write a short proof. If the triangles are not similar, write a short explanation.

Question 23: Given that 𝐴𝐵0000 ∥ 𝐸𝐷0000, determine if the triangles in the diagram below are similar. If the triangles are similar, write a short proof. If the triangles are not similar, write a short explanation.

Question 36: Given: ik��= k�

��, Prove: ∆𝑍𝐶𝑃 is isosceles. [Diagram is provided with

overlapping triangles.]

Question 37: Given: ∡𝐻𝐺𝐼 = ∡𝐺𝐹𝐽, Prove: ∆𝐴𝐵𝐶~∆𝐹𝐸𝐷.

Question 38: Prove the side-angle-side similarity theorem. Hint: Place a point, 𝑋, on 𝐴𝐶0000 so that 𝐴𝑋0000 ≅ 𝐷𝐹0000. Then draw a line segment 𝑋𝑌0000 where point 𝑌 is on 𝐴𝐵0000 and 𝑋𝑌0000 ∥ 𝐵𝐶0000. Side-angle-side similarity theorem: If the lengths of two pairs of corresponding sides of two triangles are proportional and the angles the sides form are congruent, then the triangles are similar. Use the diagram below to begin your proof.

Section 7.3, Pythagorean Theorem, begins on page 298. A proof of the Pythagorean

theorem is presented on page 298, although it does not state what form it is intending to be in

(paragraph vs. statement-reason is not clear). Two model problems follow, neither of which is a

proof. Practice problems begin on page 300. Of 26 problems, only Question 26 is a proof task:

“Prove: ∡𝐸𝐻𝐺 is a right angle.” A diagram is provided.

The section has another part, which is not numbered, called, Pythagorean Triples. After

defining what a triple is, the formula for generating Pythagorean triples is presented as

“optional.” The formula is not proved, but an example is presented and a model problem follows.

Practice problems for this unnumbered section begin on page 303. Of nine problems, none are

proof tasks, although there are several good critical-thinking problems.

Section 7.4, Similar Right Triangles, begins with the words, “an altitude in a right

triangle creates similar triangles, as the theorem describes” (p. 304). Next appears the statement

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of the altitude-hypotenuse theorem: “If a triangle is a right triangle, then the altitude from the

right angle to the hypotenuse creates two triangles that are similar to the original right triangle

and to each other.” A diagram is provided that demonstrates the similarity of the three triangles.

Proof of the altitude-hypotenuse theorem is presented on page 305 in statement-reason

third column format. “We use the altitude-hypotenuse theorem for another proof of the

Pythagorean Theorem” (p. 305). Proof is presented in statement-reason third column format.

Students are next presented the converse of Pythagorean theorem: “If the sides of a triangle have

lengths 𝑎, 𝑏, 𝑐 that satisfy 𝑎W + 𝑏W = 𝑐W, then the triangle is a right triangle” (p. 307). No proof is

provided.

The Pythagorean acute inequality theorem is also presented: “If the square of the

triangle’s longest side is less than the sum of the squares of the other two sides, then the triangle

is acute” (p. 308). The diagram is misleading because it looks like a right triangle. It should be

specified that these theorems are for any triangle. A misleading diagram is also given for the

Pythagorean obtuse inequality theorem: “If the square of the triangle’s longest side is greater

than the sum of the squares of the other two sides, then the triangle is obtuse” (p. 308). The

diagram is misleading because it looks like a right triangle, and it should be specified that these

theorems are for any triangle. There is a blurb on the side of the page that says, “The converses

of both of these theorems are also true” (p. 308). No proofs are provided for the Pythagorean

inequality theorems or their converses.

Practice problems begin on page 309. Of 34 problems, the following are (arguably) proof

tasks:

Question 25: Show that ∆𝑊𝑋𝑌 and ∆𝑊𝑌𝑍 are similar. [Diagram is provided, and numeric values are given for selected segments.]

Question 26: Use the converse of the Pythagorean theorem to show that a triangle with side lengths of 9, 12, and 15 is a right triangle.

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Question 27: Use the contrapositive of the Pythagorean theorem to show that a triangle with side lengths of √13, 9, and 11 is not a right triangle.

Question 30: Use the diagram below to prove the converse of the Pythagorean theorem. Hint: You are allowed to use the Pythagorean theorem to prove its converse. Given: 𝑎W +𝑏W = 𝑐W, Prove: 𝑚∡𝐶 = 90°. [Diagram is provided. However, the diagram contains two triangles, and the hint/directions do not indicate what students are supposed to do with the second.]

Question 33: Show that the triangle with vertices located at (3, 4), (6, 5), and (2,−1) is an obtuse triangle.

Section 7.5, Special Right Triangles, begins on page 312, and 45-45-90 and 30-60-90

triangles are explained. Practice problems begin on page 315. Of 35 problems, only Question 32

is a proof task: “Assuming that the legs of an isosceles right triangle each measure 𝑥, show that

the length of the hypotenuse will be 𝑥√2.”

Section 7.6, Trigonometric Ratios, begins on page 317. First, the definitions of

trigonometry, trigonometric ratios, adjacent leg, opposite leg, sine, cosine, and tangent are

given. “If two right triangles are similar, then the values of their trigonometric ratios will be

equal, since the ratios only depend on the acute angle measures” (p. 318).

Sine, Cosine, and Complementary Angles: In right triangles, the two acute angles are always complementary. Because of this, the sine and cosine ratios have a special relationship. See if you can figure it out by looking at the table without reading ahead. All numbers are rounded to the nearest thousandth. (p. 320)

After the table, a general case is shown where 𝐴 and 𝐶 are acute angles and 𝑠𝑖𝑛𝐴 =

𝑐𝑜𝑠𝐶. On page 324, sine, cosine, and tangent for special triangles are calculated and summarized

in a table. Practice problems begin on page 326. Of 53 problems, the following is the only proof

task:

Question 53: Prove that the tangent of an angle is equal to the angle’s sine divided by its cosine, or 𝑡𝑎𝑛𝐴 = ��wi

��i. [Diagram of a right triangle is provided, with angles and sides

labeled.]

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Section 7.7, Inverses of Trigonometric Functions, is labeled “optional” and begins on

page 330. There is an incorrect mathematical statement: “This means if we know the sine of an

angle is cW, then the measure of the angle must be 30°” (p. 330). The text has no mention of

restricted domain before claiming this statement. In the following sentence, it is stated that

discussion will be limited to angles with measures 0° to 90°, without explaining why. Practice

problems begin on page 332. Of 28 problems, none are proof tasks.

Section 7.8, Law of Cosines and Law of Sines, begins on page 334. First, the Law of

Cosines is stated, and its proof is presented in statement-reason form on page 335. Trigonometry

and triangle area is then explained: “The area of a triangle can be calculated as one-half of the

product of the lengths of the triangle’s base and height. Using trigonometry, we can state another

formula for the area” (p. 338). The Law of Sines and solving ASA and AAS triangles are also

covered.

On page 339, readers are tasked to “prove the law of sines using ∆𝐴𝐵𝐶.” The proof is

presented in an unknown format (two columns labeled “diagram” and “explanation”). The proof

relies on knowledge of central and inscribed angles, which does not come until a further chapter.

There is also an error in the proof. It says, “Using the sine ratio, ∡𝐷𝑃𝐶 =��= t

W�” but it should

say “𝑠𝑖𝑛∡𝐷𝑃𝐶 =��= t

W�.” It is nice that the proof mentions that “the proof is similar if the center

of the circle is outside the inscribed triangle” (p. 339).

Practice problems begin on page 342. Of 39 problems, only Question 39 is a proof task:

Use the law of cosines to prove the Pythagorean acute inequality theorem. Hint: What are the values of 𝑐𝑜𝑠𝐶 if 𝑚∡𝐶 is between 0° and 90°? Between 90° and 180°? Pythagorean acute inequality theorem: If the square of the triangle’s longest side is less than the sum of the squares of the other two sides, then the triangle is acute.

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Chapter 7 review begins on page 346. Of 30 problems, none are proof tasks. The

cumulative review for Chapters 1–7 begins on page 349. Of 29 problems, the following three are

proof tasks:

Question 23: A teacher says: “Triangle 𝐴𝐵𝐶 has vertex 𝐴 at (8, 4), vertex 𝐵 at (0, 0), and vertex 𝐶 at (4,−2). 𝐴𝐵𝐶 is a right triangle, with 𝐶 its right angle.” Is he correct? Justify your answer.

Question 25: Using the diagram, prove the exterior angle theorem. [Diagram is provided, along with the “prove” statement.]

Question 29: Use the law of cosines to prove the Pythagorean obtuse inequality theorem. Hint: What are the values of 𝑐𝑜𝑠𝐶 if 𝑚∡𝐶 is between 0° and 90°? Between 90° and 180°? Pythagorean obtuse inequality theorem: If the square of the triangle’s longest side is greater than the sum of the squares of the other two sides, then the triangle is obtuse.

Chapter 8 Notes

Chapter 8, Circles, begins on page 352. Lesson 8.1, Circles, Tangents, and Secants,

begins with explaining the parts of a circle (center, radius, diameter, and circumference). A short

history of 𝜋 is given and the area formula for a circle is presented. Chord, secant, tangent, and

common tangent are defined on the top of page 355, and the process for constructing inscribed

and circumscribed circles (of triangles) is presented. Both constructions are proved via paragraph

proofs (p. 356). The proof of the inscribed circle construction relies on the tangent to a circle

theorem: “A line is tangent to a circle if and only if that line is perpendicular to the radius drawn

at the point of tangency” (p. 356). The theorem is not proved.

“How to construct a tangent line to a circle through a given point” (p. 357) is presented,

with a paragraph proof of the process immediately following. The proof relies on the tangent to a

circle theorem that was presented on the previous page but was not proven. Next, “how to

construct tangent lines through a point not on the circle” (p. 357) is presented, with a statement-

reason third column proof of this construction (p. 358). Finally, the tangent endpoint theorem is

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defined: “If two tangent segments share a common endpoint, then they are congruent” (p. 360).

No proof is provided.

Practice problems begin on page 360. Of 32 problems, only Question 32 is a proof task:

Prove the tangent endpoint theorem. Hint: Construct auxiliary segments 𝐴𝐶0000, 𝐵𝐶0000, and 𝐸𝐶0000. Tangent Endpoint Theorem: If two tangent segments share a common endpoint, then they are congruent. [Diagram is provided with theorem statement.]

Section 8.2, Chords and Arcs, begins on page 363. Page 364 defines the arc addition

postulate: “The measure of an arc formed by two adjacent arcs in a circle is the sum of the

measures of each arc.” Congruent minor arcs theorem is also defined: “In the same or congruent

circles, two minor arcs are congruent if and only if their chords are congruent” (p. 364). No proof

is provided. Similarly, no proof is provided for the perpendicular chords theorem: “If a radius is

perpendicular to a chord, then it bisects the chord” (p. 364). For the perpendicular chords

theorem converse, defined as “a radius that bisects a chord is perpendicular to the chord” (p.

365), the theorem is proved using a paragraph proof. However, again, no proof is provided for

the congruent chords theorem: “Two chords are congruent if and only if they are the same

distance from the circle’s center” (p. 365).

Practice problems begin on page 366. Of 37 problems, the following four are proof tasks:

Question 34: Given a circle with a center at (0, 0), a radius 𝑟, and the point (𝑥, 𝑦) located on the circle, prove 𝑥W + 𝑦W = 𝑟W. Hint: Use the distance formula.

Question 35: Prove the congruent minor arcs theorem. Remember, since it is an “if and only if” theorem, there are two if/then statements to prove. Congruent minor arcs theorem: In the same or congruent circles, two minor arcs are congruent if and only if their chords are congruent.

Question 36: Prove the congruent chords theorem. Remember, since it is an “if and only if” theorem, there are two if/then statements to prove. Hint: Construct auxiliary segments 𝐶𝐴0000, 𝐶𝐷0000, 𝐶𝐵0000,and 𝐶𝐸0000. Congruent chords theorem: Two chords are congruent if and only if they are the same distance from the circle’s center.

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Question 37:Prove the perpendicular chords theorem. Hint: Let 𝐷 be the intersection point of 𝐴𝐵0000 and 𝐸𝐹0000, and use an indirect proof. Perpendicular chords theorem: If a chord is a perpendicular bisector of another chord, then the first chord is a diameter.

Section 8.3, Inscribed Figures, begins on page 370. Inscribed Angle Case 1 is defined:

“In a circle, the measure of an inscribed angle is half the measure of the central angle with the

same intercepted arc” (p. 370). No proof is provided. Inscribed Angle Case 2, defined as “in a

circle, two inscribed angles with the same intercepted arc are congruent” (p. 371), likewise has

no proof provided.

On page 372, the process for “Construction: Regular Hexagon in a Circle” is presented. It

says, “The construction relies on the fact that the length of the side of a hexagon equals the

distance from the circle’s center to a vertex, which is the circle’s radius” (p. 372). No proof of

the construction is provided, although there is a blurb that explains why the construction works.

Inscribed quadrilateral theorem is defined as, “A quadrilateral can be inscribed in a circle if and

only if its opposite angles are supplementary” (p. 373). A blurb states that because the theorem is

an “if and only if” theorem, two if–then statements are needed. On page 374, a statement-reason

third column proof is given of the condition, “If a quadrilateral can be inscribed in a circle, then

its opposite angles are supplementary.” The statement of the conditional is given to students, as

is the proof. Here, there is a blurb that states the students will have to prove the other conditional

in Practice Problem 34.

Page 375 describes the cyclic quadrilateral inscribed angle theorem: “The inscribed angle

formed by one diagonal and side of a cyclic quadrilateral is congruent to the inscribed angle

formed by the other diagonal and opposite side.” The theorem is proved using a statement-reason

third column proof on page 376, but it is reliant on the inscribed angle case 2 theorem, which

was not proved.

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Last in this section, students learn the inscribed right triangle theorem: “A side of an

inscribed triangle is a diameter, if and only if the triangle is a right triangle with its right angle

opposite the diameter” (p. 376). No proof is provided.

Practice problems begin on page 378. Of 34 problems, only one is a proof task:

Question 34: Prove the other half of the inscribed quadrilateral theorem using the diagram below: If 𝑚∡𝐴 +𝑚∡𝐷 = 180° and 𝑚∡𝐵 +𝑚∡𝐸 = 180°, then 𝐴𝐵𝐷𝐸 can be inscribed in a circle. [Diagram is provided.]

Section 8.4, More on Chords and Angles, begins on page 380. Tangent and chord

theorem is defined: “If a tangent and chord intersect on a circle, then the measure of each angle

formed is one-half the measure of its intercepted arc” (p. 380). No proof is provided. The next

theorem, chord angles inside circle theorem, does not have a proof provided either. It is defined

as, “If two chords intersect inside a circle to form an angle, then the measure of the angle equals

one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle” (p.

381).

Page 382 defines angles formed outside circle theorems, but again no proofs are

provided:

Angle formed by a secant line and a tangent line: When a secant and a tangent intersect outside a circle, the measure of the angle they form is equal to one-half the difference of the intercepted arcs.

Angle formed by two secant lines: When two secants intersect outside a circle, the measure of the angle they form equals one-half the difference of their intercepted arcs.

Angle formed by two tangent lines: When two tangent lines intersect outside a circle, the measure of the angle they form equals one-half the difference of their intercepted arcs.

Page 383 informs readers that “the theorem below is an extension of one of the ‘angle

formed outside circle’ theorems presented on the previous page.” It is the secants and tangents

theorem: “If secant and tangent segments intersect outside a circle, then the product of the

lengths of the entire secant segment and the secant segment outside of the circle equals the

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square of the tangent segment’s length” (p. 383). No proof is provided, nor is there one for the

intersecting chords theorem: “The products of the segment lengths of each of two chords that

intersect inside a circle equal one another” (p. 385). Following is a discovery activity to

“discover a relationship concerning the lengths of chord segments formed” (p. 385)—but the

students were just told the relationship. Practice problems begin on page 386. Of 25 problems,

none are proof tasks.

Section 8.5, Arc Lengths and Area, begins on page 388. No theorems are stated;

however, concepts of arc length, sector area, radian measure of angles, and conversion between

degrees and radians are presented.

Using similarity, we derive the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality, or scale factor, between the two similar triangles within the circles shown in the diagram. (p. 393)

A statement-reason third column “proof” appears here, even though it is not labeled as a

proof, and it was not the result of a theorem statement. The “proof” is easily readable until the

last two statements and their justifications.

Practice problems begin on page 395. Of 42 problems, none are proof tasks. Chapter 8

review begins on page 399. Of 28 problems, none are proof tasks. Cumulative review for

Chapters 1–8 begins on page 402. Of 31 problems, the following two are proof tasks:

Question 22: Prove the consecutive interior angles theorem: Prove: 𝑚∡𝐷 +𝑚∡𝐸 =180°, Given: 𝑚 ∥ 𝑛, 𝑡 is a transversal. [Diagram is provided.]

Question 31: Prove the transversal and parallel lines theorem for the case that 𝐴𝑁O⃖OOO⃗ and 𝑆𝑀O⃖OOO⃗ are not parallel. Hint: Use the triangle proportionality theorem. A transversal and parallel lines theorem: When parallel lines pass through transversals, they divide the transversals into proportional line segments. [Here, a mostly-filled-in statement and reason chart is provided. Students are tasked with filling in two statements and three reasons.]

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Chapter 9 Notes

Chapter 9, Polygons, begins on page 406. Section 9.1, Parallelograms and Their

Diagonals, begins on page 407. The parallelogram opposite sides theorem is introduced: “If a

quadrilateral is a parallelogram, then its opposite sides are congruent” (p. 408). Proof is

presented in statement-reason third column format. Proof is also presented of the fact that

“parallel lines have the same slope” (p. 409). It is not presented as a theorem statement, but the

solution is presented as a proof in statement-reason third column format. Page 409 presents the

parallelogram angle theorems:

If a quadrilateral is a parallelogram, then its opposite angles are congruent. If a quadrilateral is a parallelogram, then its adjacent angles are supplementary. [No proof of this second theorem is provided.] The proof of the first theorem is provided in statement-reason third column format.

Page 412 defines the parallelogram diagonals theorem: “If the quadrilateral is a

parallelogram, then its diagonals bisect one another.” Proof is provided on page 413 in

statement-reason third column format. Practice problems begin on page 414. Of 52 problems, the

following six are proof tasks:

Question 27: In parallelogram 𝐴𝐵𝐶𝐷, the diagonals intersect at point 𝑂. Prove that ∡𝐴𝐵𝑂 ≅ ∡𝐶𝐷𝑂 without using the alternate interior angles theorem.

Question 36: The coordinates of quadrilateral 𝑀𝑁𝑃𝑄 are 𝑀(−5, 1), 𝑁(−4, 4), 𝑃(−1, 5) and 𝑄(−2, 2). Prove that 𝑀𝑁𝑃𝑄 is a parallelogram.

Question 40: 𝐴𝐵𝐷𝐶 and 𝐷𝐸𝐺𝐹 are parallelograms, and 𝐵𝐹0000 and 𝐶𝐸0000 are continuous line segments. Prove that 𝐴𝐶 ∥ 𝐸𝐺. [Diagram is provided.]

Question 41: The diagonals of a parallelogram 𝐴𝐵𝐶𝐷 intersect at point 𝑂. The line 𝑀𝑁O⃖OOOO⃗ passes through point 𝑂 and intersects the sides 𝐵𝐶0000 and 𝐴𝐷0000 at points 𝑀 and 𝑁. Make a sketch and then write a proof that 𝑂𝑀 = 𝑂𝑁.

Question 46: Consider a quadrilateral 𝐴𝐵𝐶𝐷 whose interior angles all measure less than 180°. Prove that if ∡𝐵𝐴𝐶 ≅ ∡𝐴𝐶𝐷 and ∡𝐵𝐶𝐴 ≅ ∡𝐷𝐴𝐶, then 𝐴𝐵𝐶𝐷 is a parallelogram. Make a sketch and use it in your proof.

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Question 47: Prove the second parallelogram angle theorem using the diagram below. Given: 𝐴𝐵𝐶𝐷 is a parallelogram, Prove: Adjacent angles in the parallelogram are supplementary. [Diagram is provided.]

Section 9.2, Deciding If a Parallelogram Is Also a Rectangle, Square, or Rhombus, begins

on page 417 with the rectangle diagonals theorem: “A parallelogram is a rectangle if and only if

its diagonals are congruent.” One of the conditionals (“If a parallelogram has congruent

diagonals, then it is a rectangle”) is proved using statement-reason format (p. 418). No third

column is present. No proof of the other conditional is present.

Parallelogram and rhombus theorems are defined next: “A parallelogram is a rhombus if

and only if its diagonals are perpendicular. A parallelogram is a rhombus if and only if its

diagonals bisect the figure’s angles” (p. 418). No proofs are provided. Practice problems begin

on page 422. Of 37 problems, the following seven are proof tasks:

Question 11: The diagonals of a rectangle 𝐸𝐹𝐺𝐻 below intersect at point 𝐴. Prove that the triangles 𝐸𝐴𝐻 and 𝐸𝐴𝐹 are isosceles. [Diagram is provided.]

Question 12: Prove that in a rhombus the diagonals bisect the rhombus’s angles and are perpendicular to each other.

Question 22: Prove that the midpoints of the sides of a rectangle are the vertices of a rhombus. Make a sketch and refer to it in your proof. [Diagram is provided.]

Question 25: Prove the distances from the point of intersection of the diagonals of a rhombus to all its sides are equal. Prove: 𝑂𝐴 = 𝑂𝐵 = 𝑂𝐶 = 𝑂𝐷, Given: 𝐾𝐿𝑀𝑁 is a rhombus, 𝐾𝑀00000 and 𝐿𝑁0000 are diagonals, and the diagonals intersect at 𝑂. [Diagram is provided.]

Question 29: 𝐴𝐵𝐶𝐷 is a square. 𝐴𝐸0000, 𝐵𝐹0000, 𝐶𝐺0000,and 𝐷𝐻0000 are all congruent. Prove that 𝐸𝐹𝐺𝐻 is a square. [Diagram is provided.]

Question 30: Prove that the midpoints of the sides of a rhombus are the vertices of a rectangle. In other words, if 𝑀𝑂𝑅𝐸 is a rhombus and 𝐷, 𝑈, 𝑆, and 𝑇 are the midpoints of its sides, prove that 𝐷𝑈𝑆𝑇is a rectangle. [Diagram is provided.]

Question 33: Prove that a parallelogram is a rhombus if and only if its diagonals bisect its angles.

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Section 9.3, Deciding If a Quadrilateral Is a Parallelogram, begins on page 425. Of five

theorems presented, the first four do not have any proof provided. The first is parallelogram

opposite sides theorem converse: “If both pairs of opposite sides of a quadrilateral are congruent,

then the quadrilateral is a parallelogram” (p. 425). The second theorem provided without proof is

the parallelogram opposite angles theorem converse: “If both pairs of opposite angles of a

quadrilateral are congruent, then the quadrilateral is a parallelogram” (p. 425). The third is

parallelogram supplementary angles theorem converse: “If all angles in a quadrilateral are

supplementary to their consecutive angles, then the quadrilateral is a parallelogram” (p. 425).

Fourth, no proof is provided for the parallelogram diagonals theorem converse: “If the diagonals

of a quadrilateral bisect one another, then it is a parallelogram” (p. 426). The only theorem to

show proof is the parallelogram and quadrilateral theorem: “If one pair of sides of a quadrilateral

is congruent and parallel, then the quadrilateral is a parallelogram” (p. 426). Proof is provided in

statement-reason third column form.

Practice problems begin on page 429. Of 27 problems, the following 12 are proof tasks:

Question 10: In quadrilateral 𝐴𝐵𝐶𝐷, ∡𝐵𝐴𝐶 ≅ ∡𝐴𝐶𝐷 and ∡𝐵𝐶𝐴 ≅ ∡𝐷𝐴𝐶. Prove that 𝐴𝐵𝐶𝐷 is a parallelogram.

Question 15: Michael has a parallelogram-shaped wooden board. He measures out the same distance on each side of his board, starting from each vertex, and nails in the pegs. After he stretches a rubber band on the pegs, he gets the shape shown below. Prove that the shape Michael obtained is a parallelogram. [Diagram is provided.]

Question 16: In a parallelogram 𝐵𝐸𝑆𝑇, the point 𝐴 is the midpoint of 𝐸𝑆0000 and point 𝑀 is the midpoint of 𝐵𝑇0000. Make a sketch and prove that the quadrilateral 𝐸𝐴𝑇𝑀 is a parallelogram.

Question 17: In the diagram below, 𝐴𝐵𝐶𝐷 and 𝐴𝑀𝑁𝐷 are parallelograms. Prove that 𝑀𝐵𝐶𝑁 is also a parallelogram. [Diagram is provided.]

Question 18: 𝑅𝑂𝑆𝐸 is a parallelogram. The points 𝐴,𝐵, 𝐶, and 𝐷 are midpoints of segments 𝑋𝑅0000, 𝑋𝑂0000, 𝑋𝑆0000, and 𝑋𝐸0000, respectively. Prove that 𝐴𝐵𝐶𝐷 is also a parallelogram. [Diagram is provided.]

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Question 19: In the parallelogram 𝐸𝐹𝐺𝐻, the diagonals intersect at point 𝑂. A line passes through the point 𝑂 and intersects the sides of parallelogram at points 𝑀 and 𝑁. Prove that 𝑂𝑀 = 𝑂𝑁. [Diagram is provided.]

Question 20: A midsegment of a triangle is a segment connecting the midpoints of two of the triangle’s sides. In the triangle 𝐴𝐵𝐶, 𝑀𝑁00000 is a midsegment connecting the sides 𝐴𝐵0000 and 𝐵𝐶0000. Extend the line 𝑀𝑁00000 outside the triangle and mark the point 𝑃 such that 𝑀𝑁 =𝑁𝑃. Prove that 𝐴𝑀𝑃𝐶 is a parallelogram. Refer to your sketch in your proof.

Question 23: Use the diagram below to prove the parallelogram opposite angles theorem converse. Hint: Begin by using the fact that 2𝑎 + 2𝑏 = 360°. Given: ∡𝐴 ≅ ∡𝐵𝐶𝐷and ∡𝐷 ≅ ∡𝐴𝐵𝐶, Prove: 𝐴𝐵𝐶𝐷 is a parallelogram. [Diagram is provided.]

Question 24: Use the diagram below to prove the parallelogram supplementary angles theorem converse. Hint: First prove that 𝐴𝐵0000 ∥ 𝐶𝐷0000 by showing 𝐴𝐸OOOOO⃗ ∥ 𝐷𝐹OOOOO⃗ . Then, write a similar series of steps to prove 𝐴𝐷0000 ∥ 𝐵𝐶0000 by showing 𝐴𝐺OOOOO⃗ ∥ 𝐵𝐻OOOOOO⃗ . Given: Consecutive interior angles are supplementary, Prove: 𝐴𝐵𝐶𝐷 is a parallelogram. [Diagram is provided.]

Question 25: Prove that, in a parallelogram, the sum of the squares of the diagonals is equal to the sum of the squares of all of its sides. In other words, in parallelogram 𝐴𝐵𝐶𝐷, 𝐴𝐶W + 𝐵𝐷W = 𝐴𝐵W + 𝐵𝐶W + 𝐶𝐷W + 𝐴𝐷W.

Question 26: Prove the parallelogram diagonals theorem converse. Hint: Prove that the opposite sides of the quadrilateral are congruent, then use the parallelogram opposite sides theorem converse. Parallelogram diagonals theorem converse: If the diagonals of a quadrilateral bisect one another, then it is a parallelogram. [Diagram is provided with statement of theorem.]

Question 27: Prove the parallelogram opposite sides theorem converse. Hint: Construct 𝐵𝐷0000 and work backward through the proof for the parallelogram opposite sides theorem (non-converse). Parallelogram opposite sides theorem converse: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. [Diagram is provided with statement of theorem.]

Section 9.4, Polygons and Their Angles, is labeled “optional” and begins on page 432.

No theorems are presented. Formulas are given, and some have “informal derivations,” but none

are proved. Practice problems begin on page 435. Of 41 problems, only Question 39 is a proof

task: “Prove that the sum of the interior angles of a convex polygon with 𝑛 sides is (𝑛 − 2) ∙

180°.”

207

Section 9.5, Trapezoids and Kites, begins on page 437. The Common Core definition of

trapezoid is presented on page 437, followed by the isosceles trapezoid theorem: “If the

quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent” (p. 438). No

proof is provided. On page 438, isosceles trapezoid diagonals theorem is presented: “If a

quadrilateral is an isosceles trapezoid, then its diagonals are congruent.” Proof is provided in


Trapezoid midsegment theorems is defined as, “A trapezoid’s midsegment is parallel to

each base. Midsegment’s length is one half the sum of the bases’ lengths” (p. 439). No proof is

provided for the first theorem; however, the second theorem is proved on page 440 in statement-

reason third column format.

Practice problems begin on page 442. Of 38 problems, the following nine are proof tasks:

Question 15: Show that in an isosceles trapezoid 𝐴𝐵𝐶𝐷 where 𝐴𝐷0000 and 𝐵𝐶0000 are the bases, ∆𝐴𝐵𝐷 ≅ ∆𝐷𝐶𝐴. Use the theorem of base angles of an isosceles trapezoid.

Question 25: The vertices of a quadrilateral are 𝐴(−3, 2), 𝐵(3, 4), 𝐶(5,−2) and 𝐷(−4,−5). Make a sketch and show that 𝐴𝐵𝐶𝐷 is a trapezoid.

Question 26: Consider a trapezoid with vertices at the points 𝐴(0, 0), 𝐵(2𝑎, 2𝑏), 𝐶(2𝑐, 2𝑏) and 𝐷(2𝑑, 0). Show that the midsegment 𝑀𝑁00000 of the trapezoid is parallel to its bases. [Diagram is provided.]

Question 30: Prove that the base angles of an isosceles trapezoid are congruent.

Question 31: Prove that the diagonals of a kite are perpendicular to each other. Hint: Consider perpendicular bisectors.

Question 32: In kite 𝐴𝐵𝐶𝐷, 𝐴𝐵 = 𝐵𝐶 and 𝐶𝐷 = 𝐷𝐴. Prove that the diagonal 𝐵𝐷0000 of the kite is an angle bisector of ∡𝐵 and ∡𝐷. [Diagram is provided.]

Question 33: A quadrilateral is obtained by connecting the midpoints of all sides of an isosceles trapezoid. Determine the type of quadrilateral, being as specific as possible. Justify your reasoning. [Diagram is provided.]

Question 34: Show that the part of a midsegment of a trapezoid between its diagonals is equal to half the difference of the trapezoid’s bases. Make a sketch and refer to it in your proof.

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Question 36: The diagonals of a trapezoid divide its midsegment into three congruent segments. Show that one base of the trapezoid is twice as long as the other. [Diagram is provided.]

Section 9.6, Areas and the Coordinate Plane, begins on page 445. The area formula for

parallelogram and rhombus is given without proof. It is unclear why it is given for a rhombus

because a rhombus is a parallelogram. Two methods for constructing a square are presented on

page 447. The first method is not proved, but the second is proved via paragraph proof on page

448.

Page 450 shows how the area formula for a trapezoid is derived from the area formula for

a parallelogram. The area formula is not proved. Page 452 shows how the area formula for a kite

and rhombus is derived using triangle area and diagonals. Here again, the area formula is not

proved. Practice problems begin on page 454. Of 39 problems, only Question 37 is a proof task:

“Prove the diagonals of a parallelogram divide the parallelogram into four triangles with equal

area.”

Section 9.7, Area of Regular Polygons, begins on page 457. In the first paragraph of the

section, the “center” of a regular polygon is referred to. The formula for the area of a regular

polygon is presented and its derivation is explained in terms of triangles. The formula is not

proved. Practice problems begin on page 459. Of 29 problems, only Question 25 is a proof task:

“Prove the area of a regular polygon is cW𝑎 ∙ 𝑠 ∙ 𝑛.”

Section 9.8, Area and Trigonometry, begins on page 461. There are no theorems stated in

the section. There is a small blurb explaining that trigonometry can be used to find the areas of

regular polygons given that all regular polygons are composed of triangles, followed by two

model problems. Practice problems begin on page 463. Of 28 problems, none are proof tasks.

209

The Chapter 9 review begins on page 465. Of 35 problems, the following four are proof

tasks:

Question 31: In kite 𝐴𝐵𝐶𝐷, 𝐴𝐵 = 𝐵𝐶 and 𝐶𝐷 = 𝐷𝐴. Prove that diagonal 𝐴𝐶0000 of the kite is bisected by diagonal 𝐵𝐷0000.

Question 32: Prove that a trapezoid is isosceles if and only if its diagonals are congruent. That is, prove both the theorem and its converse.

Question 33: A plastic triangle needs to be cut into exactly three pieces, each having a trapezoid shape. Show and explain how that can be done. The pieces don’t have to be equal.

Question 34: The points 𝐸, 𝐹, 𝐺, and 𝐻 are the midpoints of the sides of a parallelogram 𝐴𝐵𝐶𝐷. Prove that 𝐸𝐹𝐺𝐻 is also a parallelogram. [Diagram is provided.]

The cumulative review for Chapters 1–9 begins on page 468. Of 35 problems, the

following two are proof tasks:

Question 14: Prove the similar polygons area theorem for rectangles given similar rectangles 𝐴𝐵𝐶𝐷 and 𝐸𝐹𝐺𝐻. Let 𝐴𝐵𝐶𝐷 have length 𝑝 and width 𝑞. Let 𝐸𝐹𝐺𝐻 have length 𝑟 and width 𝑠. Also, the ratio of �

� and �

� is 𝑘. Hint: Use substitution.

Question 18: In a parallelogram, the adjacent angles formed by the diagonals and the sides are congruent. Show that the diagonals of the parallelogram are perpendicular to each other.

Chapter 10 Notes

Chapter 10, Solids, begins on page 472. Section 10.1, Three-Dimensional Figures, Cross-

Sections, and Drawings, begins on page 473. On page 476, “paragraph derivation” appears to

explain the extension of the Pythagorean theorem to three dimensions. Practice problems begin

on page 481. Of 27 problems, none are proof tasks.

Section 10.2, Surface Area, begins on page 483. Surface area formulas are given for

cubes, prisms, cylinders, pyramids, cones, and spheres. None of the formulas are proved, but

some are derived. Practice problems begin on page 492. Of 40 problems, none are proof tasks.

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Section 10.3, Volume, begins on page 495. Volume formulas are given for prisms, cubes,

cylinders, pyramids, cones, and spheres. None of the formulas are proved, but some are derived

informally. Practice problems begin on page 505. Of 41 problems, none are proof tasks.

Section 10.4, Cavalieri’s Principle, begins on page 508. After stating Cavalieri’s

principle, the text says, “Derive the volume for two figures: a hemisphere and a sphere” (p. 508).

The derivation is shown in three parts starting on page 509, each using a statement-reason third

column format, although none are presented as a “proof.” The reading is difficult, and it is not

clear what is intended to be shown, or why Cavalieri’s principle is needed.

Practice problems begin on page 511. Of 33 problems, one is a proof task:

Question 33: It is possible to divide a cube into congruent rectangular pyramids that share a vertex at the exact center of the cube and whose bases form the faces of the cube. If an edge of the cube is called 𝑏 and the height of each pyramid is called ℎ, use the volume of the cube to show that the formula for the volume of each of the six congruent pyramids is 𝑉 = c

�𝑏Wℎ.

Section 10.5, Similar Solids, begins on page 513. The similar solids theorem is given: “If

two figures are similar with a scale factor 𝑥: 𝑦, then their surface areas have the ratio 𝑥W: 𝑦W, and

their volumes have the ratio 𝑥�: 𝑦�” (p. 514). No proof is provided. Practice problems begin on

page 516. Of 28 problems, only one is a proof task:

Question 27: Prove the similar solids theorem for cylinders given similar cylinders 𝐶c and 𝐶W with heights ℎc and ℎW, and radii 𝑟c and 𝑟W,respectively. The ratio of ��

�� and ��

�� is 𝑘.

Hint: Use substitution.

Chapter 10 review begins on page 519. Of 30 problems, none are proof tasks. The

cumulative review for Chapters 1–10 begins on page 522. Of 32 problems, the following three

are proof tasks:

Question 23: In the parallelogram 𝐴𝐵𝐶𝐷, the points 𝑄 and 𝑀 are the midpoints of the sides 𝐴𝐷0000 and 𝐵𝐶0000, respectively. Also, 𝐿𝐵 = 𝑃𝐷 = c

�𝐷𝐶.Is quadrilateral 𝐿𝑀𝑃𝑄 a

parallelogram? Justify your answer. [Diagram is provided.]

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Question 31: Sketch an acute triangle. Then, sketch all three midsegments, which divide the original triangle into four smaller triangles. Explain how you know the four smaller triangles are all congruent to one another.

Question 32: Lines 𝑎, 𝑏, and 𝑐 are coplanar. Two lines 𝑎 and 𝑏 are parallel. The line 𝑐 intersects the line 𝑎. Show that the line 𝑐 also intersects the line 𝑏. Make a sketch, label 𝐷 the point of intersection of the lines 𝑐 and 𝑎, and use the parallel postulate.

Chapter 11 Notes

Chapter 11, Conics, begins on page 526. Section 11.1, Circles at the Origin, begins on

page 527. Although not stated as a theorem, the equation for a circle centered at the origin with

radius 𝑟 is given on page 528 as 𝑥W + 𝑦W = 𝑟W; and a “paragraph proof” is presented for the

equation. Practice problems begin on page 530. Of 33 problems, none are proof tasks.

Section 11.2, Parabolas at the Origin, begins on page 531. Page 532 shows a derivation of

the equation of a concave up parabola with its vertex at the origin. On page 533, the changes in

the equation that will determine whether the parabola opens up, down, right, or left are

described. These results are summarized in a table on page 534.

A model problem on page 536 tasks the reader with “showing” that the length of the latus

rectum for any parabola is 4𝑝. The solution is presented as a paragraph proof. Practice problems

begin on page 536. Of 31 problems, only Question 31 is a proof task: “Prove that the equation

for a horizontal parabola can be written 𝑥 = c��𝑦W, where (𝑝, 0) is the focus and 𝑥 = −𝑝 is the

directrix.”

Section 11.3, Circles Translated From the Origin, begins on page 538. The “standard

form equation for a circle” is given; No proof is provided (p. 538). Page 541 demonstrates, using

one example, how to complete the square to transform a circle equation into standard form. Proof

is not given, and a “general case” is not mentioned.

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Page 542 reviews the idea that all circles are similar and shows an example of a similarity

transformation. Practice problems begin on page 542. Of 29 problems, only Question 29 is a

proof task: “Prove that 𝑥W − 𝑎𝑥 + 𝑦W − 𝑏𝑦 = 𝑐 is the equation for a circle whose center has

been translated from the origin to the point �tW, uW�.”

Section 11.4, Parabolas Translated From the Origin, is labeled “optional” and begins on

page 544. Page 544 explains the standard form equations for a parabola. The formulas are not

derived or proved. Practice problems begin on page 548. Of 43 problems, none are proof tasks.

Section 11.5, Ellipses at the Origin, is labeled “optional” and begins on page 550. Page

551 gives the formula for an ellipse centered at the origin and explains the difference between a

horizontal and vertical ellipse. On pages 554–556, derivations of ellipse equations are presented.

Practice problems begin on page 557. Of 26 problems, none are proof tasks.

Section 11.6, Hyperbolas at the Origin, is labeled as “optional” and begins on page 558.

A geometric definition of a hyperbola is given, as are equations for hyperbolas centered at the

origin, without proof (pp. 558–559). Parts of a hyperbola are named and described, and a

summary table is shown of equations of hyperbolas at the origin. Derivation of hyperbola

equations is presented but is not labeled as a proof (p. 564). Practice problems begin on page

565. Of 26 problems, none are proof tasks.

Chapter 11 review begins on page 567. Of 43 problems, just one, Question 42, is a proof

task: “For which values of 𝑘 will the hyperbola m�

t�− o�

u�= 1 intersect the line 𝑦 = 𝑘𝑥? Justify

your answer.”

The cumulative review for Chapters 1–11 begins on page 571. Of 36 problems, the

following three are proof tasks:

213

Question 12: In the triangle below, show that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. Prove: 𝑀𝐴 = 𝑀𝐵 = 𝑀𝐶, Given: 𝐴𝐵𝐶 is a right triangle with vertices 𝐴(0, 0), 𝐵(0, 𝑏), and 𝐶(𝑐, 0). [Diagram is provided.]

Question 14: In the parallelogram 𝐵𝑂𝐷𝑌 below, the segments 𝐵𝐻0000 and 𝐷𝐾0000 are the heights. Is the quadrilateral 𝐵𝐾𝐷𝐻 a parallelogram? Justify your reasoning. [Diagram is provided.]

Question 36: Prove the angle bisector theorem converse. Angle Bisector Theorem Converse: If a point of an angle is the same distance from that angle’s sides, then it is on the bisector. [Diagram is provided with theorem statement.]

Chapter 12 Notes

Chapter 12, Probability, begins on page 574. Section 12.1, Introduction to Probability,

begins on page 575. No theorems appear in the section. Practice problems begin on page 581. Of

32 problems, none are proof tasks. Practice problems begin again on page 585. Of eight

problems, none are proof tasks.

Section 12.2, Permutations and Combinations, begins on page 586. No theorems appear

in the section. Practice problems begin on page 596. Of 51 problems, none are proof tasks.

Section 12.3, Independent Events and the Multiplication Rule, begins on page 599. No

theorems appear in the section. Practice problems begin on page 603. Of 22 problems, none are

proof tasks.

Section 12.4, Addition and Subtraction Rules, begin on page 605. No theorems appear in

the section. Practice problems begin on page 611. Of 20 problems, only Question 18 is a proof

task: “Show algebraically that the probability of the complement of not 𝐴 equals the probability

of 𝐴. That is, 𝑃�𝑛𝑜𝑡(𝑛𝑜𝑡𝐴)� = 𝑃(𝐴).”

Section 12.5, Conditional Probability, starts on page 613.

In most sections of this book, we show a formula and then provide examples of how to apply it. In this section, we will reverse the order by first discussing a situation that requires Bayes’ theorem and then explaining the theorem. The theorem is used to model conditional probabilities. (p. 616)

214

Bayes’ theorem is presented on page 616, as “optional.” Before presenting the theorem

on page 618, the text says,

We state the formula: Bayes’ Theorem: 𝑃(𝐵|𝐴) = (i|k)∙ (k)

�𝐴¡𝐵�∙ (k)p (i|w�¢k)∙ (w�¢k).

Practice problems begin on page 619. Of 32 problems, none are proof tasks.

The Chapter 12 review begins on page 623. Of 30 problems, none are proof tasks. The

cumulative review for Chapters 1–12 begins on page 626. Of 36 problems, the following three

are proof tasks:

Question 25: 𝐴𝐵𝐶𝐷 is a parallelogram. Two equilateral triangles 𝐴𝐵𝑋 and 𝐵𝐶𝑌 are constructed outside of the parallelogram on its sides. Make a sketch and prove that triangle 𝑋𝑌𝐷 is equilateral. [Diagram is provided.]

Question 26: The midpoints 𝐾 and 𝐿 of the opposite sides of the parallelogram 𝐴𝐵𝐶𝐷 below are connected with the vertices 𝐵 and 𝐷. Prove that the segments 𝐵𝐿0000 and 𝐷𝐾0000 divide the diagonal 𝐴𝐶0000 of the parallelogram into three equal parts. In other words, prove that 𝐴𝑋 = 𝑋𝑌 = 𝑌𝐶. [Diagram is provided.]

Question 31: Prove that the perpendicular bisectors of a triangle intersect at a concurrency point (the circumcenter). Hint: Use a coordinate proof.

215

Appendix C: Interview Questions

Interview: Background Information for Participants

You have been invited to participate in a quantitative study which will examine

mathematics teachers’ preparation in, and beliefs about, geometric proof writing, and their

abilities to teach how to write geometric proofs.

This interview was designed to gather your background information and thoughts and

opinions about writing geometry proofs and teaching students how to write geometry proofs.

Participation in this interview is voluntary. The interview will take approximately one

hour. Most of the questions require you to reflect on your own preparation and skills, as well as

discuss your confidence and perceived readiness to teach high school students how to write

geometric proofs.

Please be assured that your responses will be kept absolutely confidential and will not be

shared with any parties. To ensure confidentiality, your identity will be coded. At any time

during the study, you have the right to withdraw your participation. Please respond to the

questions as openly, as thoroughly, and as honestly as you can.

1. Please explain your preparation as a secondary mathematics teacher (what kind of

teacher preparation program, if any, did you take part in?).

2. Did any of your teacher preparation include a course (or part of a course) in

geometry?

3. What grade level/subject do you prefer to teach within secondary math?

4. Please reflect on your experiences as a high school geometry student.

5. Please reflect on your experiences writing proofs as a high school student.

6. What do you remember about writing geometry proofs as a student?

216

7. What types of proofs do you have experience with, in both content (topics) and form

(two-column; paragraph)?

8. Do you have any experience teaching proofs to students? If so, reflect on it.

9. How do you feel about teaching geometry proofs to students of your own in the near

future?

10. Please rate your confidence on a scale of 1-10 (1: low, 10: high) about teaching

geometry in general. Explain your rating.

11. Please rate your confidence on a scale of 1-10 (1: low, 10: high) about teaching

geometry proofs. Explain your rating.

12. While preparing to become a mathematics teacher, have you ever witnessed a

particularly powerful lesson on proofs (be it in a classroom during fieldwork, at a

conference, at a workshop, etc.)? If so, please explain what made the lesson

impactful.

13. What do you think are the most important takeaways for students when learning to

write geometry proofs? Why?

14. Please rate your confidence on a scale of 1-10 (1: low, 10: high) about your

knowledge of the common core state standards about geometry proofs. Explain your

rating.

15. What, if anything, do you think could be done to improve your confidence about

teaching geometry proofs?

16. How would you feel if you were assigned to teach high school geometry? Why?

217

17. If you already teach high school geometry, how did you feel when you found out

about your assignment? Why? How, if at all, have your feelings changed since that

time?

18. Do you believe it is important for you to be an expert at high school geometry, even if

you are not planning on teaching the course?

19. How do you plan on improving your geometry content knowledge should you be

assigned to teach the course in the future?

218

Appendix D: Content Assessment Data

Table D1

Instances of Beliefs and Attitudes Code for Content Assessment Item 1

P# Quotation/instance 1 This is hard.

1 This is so hard.

1 Now I’m confident.

2 Investigator: There is four proofs after this, but you can stop at any time. Participant 2: No problem. Investigator: . . . [crosstalk 00:06:18]. Participant 2: No, I want to do them all.

4 Participant 4: B is the image of A, a clockwise rota—oh, because I didn’t read the question, this is what happens. Investigator: So, you just looked at the diagram and then . . . Participant 4: I’m the worst.

5 Participant 5: So, 50 plus 30 . . . Whoops. Why did you let me make that mistake? Investigator: Well, what happened? Participant 5: 100 degrees. No, shit. Okay. This is embarrassing. Can we delete that off of the recording?

5 Investigator: There you go. How’d you get that? Participant 5: Because of my retarded arithmetic. Let’s work back. 180 degrees minus 30 degrees is 150 degrees. And 150 degrees divided by two is 75.

11 Investigator: Okay. And the interior angles of a triangle adding up to 180, is that something you’ve known for a long time? Participant 11: Yes, since, geez, I wanna say elementary school, but that might be silly. Investigator: Okay. Participant 11: Maybe, middle school. Investigator: Do you remember, was it that ever proven to you? Participant 11: I’m not sure. Investigator: Or were you just supposed to accept that? Participant 11: I was just supposed to accept that.

12 Investigator: Great. At the beginning of Part B, you said that you know the angle sum in a triangle is 180 degrees. Do you remember when you learned that? Participant 12: Probably ninth grade geometry. Investigator: Okay. Do you think that that’s something you proved? Participant 12: Not until college.

14 Investigator: Okay so if you can look at the first problem and let me know what you think about it. Participant 14: I hate transformations.

219

P# Quotation/instance Investigator: Okay why? Participant 14: Because I taught them for a math education class. Investigator: Okay. Participant 14: And that week gave me so much anxiety. Investigator: This is in your methods class? Participant 14: Yeah. Investigator: You did a series of mini lessons? Participant 14: Yes. Investigator: On transformations. Okay, in a group. Participant 14: In a group. Investigator: Okay. And you had anxiety because of the content, or because of the students? Participant 14: I don’t know. Before then I actually really liked geometry. I had it with this great teacher in high school. Investigator: Okay. Participant 14: He was amazing, he’s still there. And it was just so good, but I never liked transformations. I don’t know why. Investigator: Okay, when you took geometry you were in 10th grade or 9th grade? Participant 14: I was in 9th grade. Investigator: Okay. And were you in a common core class or was it before common core? Participant 14: I took the Regents. . . Investigator: Okay. Participant 14: I’m not sure if it was common core or not. Investigator: Okay. Participant 14: Because this was 2011. Investigator: Okay so a few years before the common core. So, you had a good experience in high school geometry? Participant 14: Yes but—everything before that, like pre-, like middle school and all that, when they start introducing transformations, it was very, very terrible. Investigator: Okay. Is this the only area of math that you remember struggling in? Participant 14: I did terribly in algebra, to the point where I had extreme math anxiety and I had tutoring for years. That was my worst Regents score. But after the geometry class, which now, since it’s been so long, like I knew I wanted to teach Algebra and Algebra II first. Investigator: Okay. Participant 14: Yeah. Investigator: So, it’s kind of like you flipped. Participant 14: Yes. Investigator: You’re now more confident in Algebra and Algebra II. Participant 14: I think mostly because I tutored in Algebra and I tutored in Algebra II during undergrad. And geometry, since I haven’t seen it for so long, I know it’s something I would have to study up before I would be able to teach it.

15 Investigator: Okay, so then how would you explain this? Participant 15: To my students?

220

P# Quotation/instance Investigator: On the paper, or to your students. Participant 15: They will fail.

18 Participant 18: AC congruent to BC. That one [inaudible 00:01:45]. Alright, let’s see. Do you want it to be a formal explanation, or like . . .? Investigator: Yeah, like what you would hand in. Participant 18: What I would hand in? Investigator: Yes. Participant 18: Alright, that’s a little bit more effort then.

18 This is simpler than I remember. What is this called, it feels like there’s a phrase for it. Congruence by . . . was it mapping? This is like new stuff right, because I don’t remember learning it like this until recently.

20 Participant 20: Can I come back to a question, or does it just need to just be skipped completely? Investigator: We can come back to it later. Can you tell me why you want to come back to it, or what’s making you uncomfortable about it? Participant 20: I’m almost certain that A and B are the same distance away from C. Investigator: Okay. So why do you want to come back to it? Participant 20: Because I don’t know why.

20 Investigator: Okay. So, if Angle A is 30 degrees, can you mark that on the picture? What angle do you also know? Participant 20: Angle B. Investigator: Angle B, and what would that mean about the measure of Angle C? Participant 20: I don’t like this.

21 Participant 21: A rigid motion that preserves distance and angle measure, which I learned more about this definition as I taught this year. Investigator: Okay. Participant 21: Preserves angle measure. . . Investigator: This is your first year teaching geometry? Participant 21: Yes.

22 Participant 22: Okay. It’s the Socratic method. Investigator: The questioning worked, right? Participant 22: It works.

24 Participant 24: If it was isosceles, it would have to be BC and AC. Investigator: And why do you think that? Participant 24: Because they’re opposite from the . . . Oh, oh no. I’m doing it to scale. I’m looking at it if it was scaled, but it doesn’t say that.

25 First of all, at first read I couldn’t visualize what was going on until I had to do it by myself following the steps. B is the image of A, which means A’=B. So how does this happen? By rotating 30 degrees about C, and then connect the 3 points. Something that helped me is knowing the definition of isosceles about two segments are the same and two angles are the same.

221

P# Quotation/instance 26 Investigator: You were talking about your experience in high school geometry and . . .

Participant 26: Yeah, so . . . Investigator: Yeah. Participant 26: Yeah. Basically, I didn’t take the initiative to study, and . . . Well, the teacher, I believe that she was okay. I just felt like she was going way too much in-depth in geometry. Investigator: Too much. Participant 26: Too much in-depth, and she . . . Well, I don’t know if in the curriculum, they’re supposed to talk axioms and certain postulates. Well, actually, my memory’s pretty vague on that point, because I don’t remember much, but I don’t know. I think it was just the way that she was teaching. She was just throwing these things at you, and then the students weren’t comprehending, in a way. We were just nodding, like, “Yes. We understand.” Investigator: You wouldn’t say anything if you didn’t understand? Participant 26: Right. Wouldn’t.

26 Participant 26: I see now, like in eighth grade, they’re actually introducing geometry now. Investigator: Yes. Participant 26: Right? Investigator: Is that not something you remember as a student? Participant 26: No. All I remember was some basics of algebra, and that’s it.

28 Participant 28: Okay. All right. So, in the measure below, Angle B is the image . . . okay. I don’t like that. Investigator: Image? Why not? Participant 28: Image bothers me because I don’t like . . . image is in math and I remember it from like multi-variable calculus or I just remember it from some other class and I just didn’t like it because I didn’t understand it; however, if I was looking at this from the high— . . . If I was in high school where I didn’t learn all that, I would be like, “Oh? What does that mean?” Now, I just . . . I get bad juju from that.

28 Participant 28: So, in the measure below, be is the image of A after counterclockwise rotation of 30 degrees about C. That’s crazy.

28 Participant 28: So, I’ve never heard of this. So, I actually don’t . . . I don’t know how to solve this. I— Investigator: Because the language is not comprehensible? Participant 28: The language . . . I mean, if someone explained to me, “Okay, image means this.” When they say counterclockwise rotation of 30 degrees, if they say this means that, then I would but because I have no knowledge of this, I can’t. I mean, the 30 degrees I’m trying to see what that means. The only thing . . . okay, so I’m going to see if I can figure this out even though I don’t have any knowledge.

28 And the counterclockwise, I guess, doesn’t matter ‘cause if you go in any direction, it’s . . . oh! Oh my God! Okay, I’m getting this now. Okay, I got it.

Note. P# = participant number.

222

Table D2

Instances of Issue of Correspondence Between Substance and Notation Code for Content

Assessment Item 1

P# Quotation/instance ICSN

+? 12 So, we can say that the measurement of . . . Angle A is congruent to Angle B,

which means that the measure of Angle A is equal to the measure of Angle B. ICSN+


223

Table D3

Instances of Mathematical Language Code for Content Assessment Item 1

P# Quotation/instance ML 2 And a rotation around a point doesn’t change the angle. [Trying to say that

rotation preserves distance.] ML-

3 Rotations . . . like if it was an image and you’re rotating it, the angle measures are the same. [Calls pre-image an image.]

ML-

4 Because I mean . . . I don’t really know. I would just think . . . I would just think that if this is the triangle that you’re given after it rotated, these are the two angles, this is horrible, but these are the two angles at the bottom so they would be the two angles at the base. [Knows “it’s horrible” but still refers to base angles as “bottom.”]

ML-

4 Because you know based on the definition of a triangle or triangle angle theorem that the angles add up to . . . angle sum theory, right? I don’t know.

ML-

6 In the diagram below, B is the image of A after a counterclockwise rotation of 30 degrees about C.

ML+

6 Rotations only affect the angles of things. ML-

7 Okay, so B is the image after A. ML-

7 So, A is your original, and B is your new image. ML-

7 When you rotate something, it holds congruence. ML-

7 Because the bases of an isosceles triangle are congruent. ML-

8 It means that if I were to make a counterclockwise rotation of 30 degrees having C as the center of rotation, then the image of A would be B.

ML+

8 Well, first I would think about the properties of isosceles triangles to see how I can apply that to the triangle. So, some properties I might consider would be the base angles would be congruent and opposite sides of those base angles would also be congruent.

ML+

8 Well, the base angles of a triangle are congruent. ML-

9 Okay. So, A is the pre-image, and B is the image, and when you do a rotation the angle measures are preserved. So, A and B are still the same angle.

ML+

9 Participant 9: . . . 30 degrees, then Angle C is 30 degrees. Investigator: Okay. And can you just try to explain how you know that. Participant 9: So, since C is here and A is here, then the line CA moves 30 degrees like that.

ML-

10 So, the figure tends to be congruent to its original. ML-

10 In an isosceles triangle, at least two angles must be congruent. ML+

224

P# Quotation/instance ML 12 And then, in rotation, distance is preserved. So, I know that the distance from C

to A has to be the same as the distance from C to B, so I could say that CA is congruent to CB. Right away, that’s already two lines in a triangle, so that’s already enough to prove that Triangle ABC is isosceles.

ML+

12 After a counterclockwise rotation of 30 degrees about C. Since it says about, I know that’s the center of rotation.

ML+

12 So, we can say that the measurement of . . . Angle A is congruent to Angle B, which means that the measure of Angle A is equal to the measure of Angle B.

ML+

13 Investigator: So, does the word rotation make you think of anything? Participant 13: It preserves distance.

ML+

15 Investigator: Okay. Do you know what this means? B is the image of A? Participant 15: Yeah. Investigator: What does that mean? Participant 15: That means . . . I mean B would be the . . . No. A is the original point, so after rotation, counterclockwise rotation of 30 degrees, you will get B as the image, pre-image.

ML+ML-

15 Determine the degree measure of Angle A. So, Angle C equal to 30 degrees. For each triangle, the sum of interior angles is 180, so it’ll be 180 minus Angle C: 30, equals 150 and based on the base angle theorems, Angle A will be congruent to Angle B, so Angle A will be the half of 150, equal to 75.

ML-

15 Since CA is congruent to CB, that means Angle C is a vertex angle. So, Angle A and Angle B are base angles.

ML+

16 Yeah, I finished first. Angle A and Angle B are congruent, rotations preserve shape—rigid motion? Therefore, the angle measure stays the same.

ML-

17 Since B and A are images of each other—I should really say, since B is an image of A.

ML+

17 Because, when you rotate, we’re talking about. . . . When I think of rotation, it’s like around a center point and then a bunch of points outside, and that’s like a locus. And so you have this circle that’s created.

ML+

18 Oh, B is the image of A. Okay. So, B is the image of A. That helps, ‘cause that means we have a line mapped to another line. Yeah, so if a line maps another line, then they must be congruent.

ML+

18 I’m mostly thinking about this in terms of a circle. It’s a swivel. ML-

18 Participant 18: You have the origin, you have the radius, you have another radius. You can draw an arc if you wanted to, which is why I went for this first. Investigator: Okay, what do you mean when you say origin? Participant 18: Center of a circle.

ML-

18 A triangle . . . any triangle with center of a circle and two points on the circle will be isosceles.

ML-

225

P# Quotation/instance ML 20 Participant 20: I don’t know how to explain why, but I know it has to be,

because I’m rotating about Point C. Investigator: So, what does that mean about Point C? Participant 20: That C is my center of rotation.

ML+

20 Investigator: Okay, so how can you relate this to a circle? Participant 20: Maybe that I’m swinging the radius.

ML-

21 So, if it’s being rotated from the Point C . . . ML-

21 Investigator: Okay. So, you said ABC is isosceles, because AC is congruent to BC. How would you explain why those are congruent? Participant 21: Well, I mean I kind of assumed it at first just by turning the page and seeing that AC and BC looks the same length. Investigator: Okay. Participant 21: But after figuring out that Angle A and Angle B are congruent based on the fact that Angle C is 30 degrees, B is the image of A. So, if it’s being rotated, a rotation is a rigid motion. Investigator: What does that mean? Participant 21: A rigid motion that preserves distance and angle measure, which I learned more about this definition as I taught this year.

ML+

21 Therefore, if Angle A is congruent to Angle B, those are the base angles of Triangle ABC. Therefore, stating that AC is congruent to BC and by the converse isosceles theorem, ABC must be an isosceles triangle.

ML+

22 It means when you rotate A counterclockwise about C 30 degrees, you’ll get B. ML+

22 Because it’s forward. Because CA is a line. Therefore, when you rotate the Point C about A, this is kind of like the terminal side of the angle, and then you form a 30-degree angle, which is C.

ML+

23 And also, rotations preserve length. ML+

23 Determine the degree measure of Angle A. I know A and B are congruent, so I know that the angle sum of a triangle is 180, and taking away the sum of the vertex angle I have that the base angles must have a sum of 150, dividing by two I get . . . What is it? 75?

ML+

23 . . .When something is rotated nothing happens to the thing, it stays rigid. Or it stays . . . The pieces stay congruent.

ML-

24 Because it says below B is the image of A after a counterclockwise rotation about C. So, I’m assuming we’re starting here and then we’d go over there.

ML-

25 In order for Triangle ABC to be isosceles, two segments are congruent, and the base angles of them must have the same measurements.

ML-

25 Since B is an image of A, these two are the same points. ML-

226

P# Quotation/instance ML 25 B was created by rotating A 30 degrees from that Point C, so first the distance

from C to A must be the same distance from C to B which means CA is congruent to CB.

ML+

25 We know that if they are congruent their bases must have the same angles. ML-

26 Investigator: Okay, and then what . . . If this is C, where would A and B be in relation to the circle? Participant 26: Don’t know if I’m using the word right. I think it’s the . . . It’s inscribed . . . No? . . . On the circle? Investigator: Okay, so on the circumference. Participant 26: Right. On the circumference.

ML-

27 Investigator: Okay, and what do you mean when you say base angles? Participant 27: The bottom angles of an isosceles.

ML-

27 Because if A was B, then, and it’s rotated about C, then being rotated up like this, so that would make this 30.

ML-

28 Because it has to be 180. So, I was saying, okay, so image, I would think of like an exact replica, just moved a different way, and so I’m trying to relate B to A now.

28 Investigator: Let’s say I could say to you, “Okay, I’ll answer any question you have about it,” what would you start by asking? Participant 28: What does “image” mean? Investigator: Okay. Is there anything else you would ask after that? Participant 28: I would say, “Okay, I want to know, so what?” I’d be like, “What does image mean?” And then actually, in a way, “What is about C mean?” Because that’s . . . I’ve never heard of that phrasing. Counterclockwise rotation of 30 degrees, I think I could figure that out on my own. I kind of did it, but still. Yeah, no. I . . . would guess it would only be those two, and if I still couldn’t figure it out, then I would say, “Okay, what do you mean by counterclockwise rotation? Is there something in here that I’m missing?”

Note. P# = participant number; ML+ = positive instance of mathematical language; ML- = negative or incorrect instance of mathematical language.

227

Table D4

Instances of the Expressing Understanding or Self-Doubt Code for Content Assessment Item 1

P# Quotation/instance EU or ESD

1 I don’t know. ESD

1 Because this angle cannot be 120 degrees. [This realization fixed initial misconception that Angle A = 30 degrees, since it led to an impossibility. This determination was made based on a visual assumption, though.]

EU

1 For sure. [Expressed understanding that initial idea about Angle A was incorrect.]

EU

1 This is the center of rotation technically, and then you’re rotating A 30 degrees this way, correct?

EU

1 So counterclockwise, you’re going this way. EU

1 Because this is the angle, and if I’m going this way, I’m rotating Angle C. So, Angle C would have to be 30.

EU

1 Because if these two angles are 75, right? Then these two sides are the same, and for it to be an isosceles triangle, you need two sides.

EU

1 Investigator: Okay. So, do you have any questions about this problem? Participant 1: Yeah, I just don’t like it. Investigator: Do you feel confident in your answer? Participant 1: No. Investigator: So, you think it could be wrong? Participant 1: Yeah. For sure.

ESD

1 At first, I thought this whole triangle, something was happening with the triangle, not just the point.

EU

1 Investigator: I see. Is there a pre-image on this picture? Participant 1: Technically. Because the . . . it’s a point.

EU

1 So, this would be A and this would be A prime. So, the pre image is A and then it’s B.

EU

1 That the angle measure between these two are the same, no? See, but that’s my issue because now I’m getting mixed up with rigid motions because I keep on thinking that this is an image. I’m not thinking of it as a point.

ESD

1 Investigator: What is a rigid motion? Participant 1: When angle measure, distance, and something else is preserved. And orientation is preserved.

EU

1 Investigator: So, in this picture, what would be preserved? What is being preserved? Participant 1: The distance between these two. Investigator: Between which two?

EU

228


Participant 1: A and . . . oh. The distance from A to C and B to C, so that’s why AC and BC must be the same length. Investigator: Ah. Does that make it a little more clear? Participant 1: So AC and BC must be the same length, then this is correct.

2 This is a transformation problem. EU

2 I’m not sure. ESD

2 And a rotation around a point doesn’t change the angle. EU

2 Investigator: Where do you think the 30-degree rotation is present in the picture? What would have to be 30 degrees? Participant 2: C.

EU

2 Investigator: Can you just explain that thought process? Participant 2: Yes. Since I know the degree measure of one of the angles, I subtracted that from the total measure of all of the angles in the triangle which are 180, giving me 150. Since they are congruent, then I just divided the 150 by 2 to get 75 degrees each.

EU

3 I don’t know. ESD

3 Rotations . . . like if it was an image and you’re rotating it, the angle measures are the same.

EU

3 Investigator: Mm-hmm [affirmative]. And then what does that tell you about isosceles? Is that triangle isosceles? Participant 3: If all that’s true, then yeah. [Is not confident that what was stated previously is true.]

ESD

3 Oh, wait, no. It is isosceles because the rotation preserves distance, so these have to be congruent.

EU

3 Investigator: What does that mean, rotation preserves distance? Participant 3: The distance from A to C . . . so A prime to C, or B to C is also the same distance.

EU

3 Investigator: 75? Okay, and do you know what the measure of B would be, also? Or A prime? Participant 3: 75.

EU

4 I don’t know. Do we have enough information to figure that out? ESD

4 Investigator: Okay. So, in this picture, where is this 30-degree angle? Participant 4: This . . . rotate. See this is what I’m bad about with rotations, but I think it’s . . . is it outside or is it here? I don’t know.

ESD

4 Because I mean . . . I don’t really know. I would just think . . . I would just think that if this is the triangle that you’re given after it rotated, these are the two angles, this is horrible, but these are the two angles at the bottom, so they would be the two angles at the base.

ESD

229


4 Investigator: Thank you. Okay and how would you determine the degree measure of Angle A? Participant 4: Well, we know that this is 30 degrees, because it tells us that. So then and you know that these are congruent, so you would do 30 plus 2x is equal to 180 because that’s the measure of two angles in a triangle and then you’re given 2x is equal to 150, then x is equal to . . . can I use my calculator?

EU

4 Because you know based on the definition of a triangle or triangle angle theorem that the angles add up to . . . angle sum theory, right? I don’t know.

ESD

5 I don’t remember rotations. Explain why Triangle ABC must be isosceles. Okay, well, isosceles triangle has two congruent sides. So, A and B . . . They’re rotated . . . What about the rotation tells us that these two have to be congruent?

ESD/ EU

5 Because C is the center, I guess. So, you’re imagining yourself rotating from C. So that . . . I mean . . . Oh, wait a minute. So, if this were a circle, the radii would be congruent. Not radii . . . Diameter. No, radius would be congruent. So, AC and BC would have to be congruent. Is that correct?

ESD/ EU

5 Participant 5: Okay. So, explain why Triangle ABC must be isosceles . . . Investigator: Go ahead. Participant 5: Because if you are rotating about C, I imagine . . . Well, I can’t write that as the explanation, can I?

ESD

5 Investigator: What part of this picture is the radius? Participant 5: B is the image if A after a counterclockwise . . . So, AC is a radius and BC is a radius.

EU

5 So, 50 plus 30 . . . Whoops. Why did you let me make that mistake? EU

5 Investigator: Okay, awesome. How would you feel about your— . . . If I asked you to now explain this problem to somebody else who was struggling, do you think you could do it? Participant 5: Yes.

EU

6 I’m not sure I know how to do this one. ESD

6 There’s no information given. I’m not given any of the side lengths or the angles. The only thing that’s given is a 30-degree rotation around C.

ESD

6 Investigator: Okay. What does it mean when it says that B is the image of A? Participant 6: Oh, oh, oh. Wait! In the area B is the image of A after a countwards . . . Oh, wait, wait, wait. Okay, if B is the image of A, then this is 30 degrees.

EU

6 Participant 6: Because A . . . Okay, I misunderstood the question. I thought the entire triangle was being rotated, but . . . I mean it is, but okay. Because A moved to Point B, so that means that the angle that’s created ACB must be the 30-degree angle. Investigator: Okay. So is the entire triangle being rotated?

EU

230


Participant 6: Yes, but it’s actually over here. This would have been the old side BC, and then it was rotated down.

6 Investigator: Okay, and then what can you say about the distance between E and D and the distance between E prime and D? Participant 6: That they’re equal. Investigator: Why? Participant 6: ‘Cause it’s like creating a circle. It’s like a locus of points. Investigator: Okay, and what- Participant 6: So, each one of these would be the radius of the circle. Investigator: Okay. Does that relate at all to this? Participant 6: Yes!!! Investigator: How? Participant 6: ‘Cause then AC and BC are the radiuses of the circle. Investigator: And what’s the center of the circle? Participant 6: C. Investigator: Okay. So, does that make sense? Participant 6: Yeah. Investigator: And why do two radii of a circle have to be the same? Participant 6: It’s a locus of points and then each radius is the same.

EU

6 Yeah. There’s 180 degrees in a circle so if I minus 30 degrees—I meant, in a triangle. So, if I minus 30, that’s 150 degrees. It’s isosceles, so the two have to be the same, so that’s 75.

EU

7 Investigator: Yeah, which point is A prime? Participant 7: B. Investigator: Good, and how do you know that? Participant 7: Because it’s after the rotation. So, A is your original, and B is your new image.

EU

7 The 30-degree angle would be inside of C. So, Angle A, C, B. EU

7 Yes. A says explain why Triangle ABC must be isosceles. It’s isosceles because the length AC never changes. It’s only rotated. When you rotate something, it holds congruence.

EU

8 It means that if I were to make a counterclockwise rotation of 30 degrees having C as the center of rotation, then the image of A would be B.

EU

8 Investigator: Okay, and so you know which angle on this picture is 30 degrees? Participant 8: In the diagram? Investigator: Yeah. Participant 8: No.

ESD

9 Participant 9: And in an isosceles triangle there are two congruent angles. Investigator: Okay. Participant 9: At least two congruent angles. Investigator: Okay.

EU

231


Participant 9: Yeah. Investigator: And what do you mean when you say at least? Participant 9: Because an equilateral triangle is also an isosceles triangle.

10 Well, if triangle . . . okay, I kind of have no idea. ESD

10 In an isosceles triangle at least two angles must be congruent. EU

10 Participant 10: Determine the degree measure of Angle A. Since to get from A to B we rotated 30 degrees about C, would that make Angle C 30 degrees? Investigator: Why do you think that? Participant 10: I guess the way it’s positioned. We have the line from C to A, and then rotated to B so CB is created. So that’s why Angle C is 30 degrees. And then to find the measure of Angle A, we know that the sum of the interior angles of a triangle is 180. So, subtracting 30 from 180 and then dividing that by 2 would give you the measure of Angle A.

ESD, EU

11 Investigator: Can I ask you how you knew to mark Angle C as 30 and to make those lines on the sides? Participant 11: It was a rotation, so I know the line segments stay the same. I just marked it that way.

EU

11 Investigator: Yes. Okay. And how did you know that . . . I see you marked them Angle A and Angle B, both as X, how did you know that those were the same? Participant 11: In an isosceles triangle, two of the angles are congruent, and so they must be the same. That’s why I labeled both of them X, and then the other one I already knew was 30. Then 30 plus two X’s equal to 180, because the interior angles of a triangle add up to 180.

EU

12 Participant 12: And then, in rotation, distance is preserved. So, I know that the distance from C to A has to be the same as the distance from C to B, so I could say that CA is congruent to CB. Right away, that’s already two lines in a triangle, so that’s already enough to prove that Triangle ABC is isosceles.

EU

12 Investigator: Okay. What’s the name of this angle that you marked off as 30? Participant 12: The central angle.

EU

12 So, we can say that the measurement of . . . Angle A is congruent to Angle B, which means that the measure of Angle A is equal to the measure of Angle B.

EU

13 Investigator: Do you need help? Is that why you’re looking at me? Okay. What do you know about the words in the problem?

ESD

13 Participant 13: A is just A, and B is basically A prime. EU

13 Participant 13: So, A’s the pre-image, and then this is . . . Then, C is the center of rotation. Investigator: Okay, how do you know that? Participant 13: Because it says counterclockwise rotation of 30 degrees about C. Investigator: Okay. Do you know where the 30 degrees would be in the

EU

232


diagram? Participant 13: Oh. 30 degrees is the measure of Angle C.

13 Participant 13: How do I know . . . Investigator: How do you know what? Participant 13: That the length of AC and the length of BC is the same. Investigator: Okay. So, why do you know that you need to know that? Participant 13: Because the question says explain why Triangle ABC must be isosceles.

EU

13 Investigator: Okay, that’s okay. What shape do you think about when you think of rotations? Participant 13: Circle. Investigator: Why? Participant 13: Because of a clock, counterclockwise, clockwise. Investigator: Okay. On a clock, where’s the center of rotation? Participant 13: In the middle, in the center. Investigator: Okay. Okay. So, here you said the center of rotation was C. Participant 13: Mm-hmm [affirmative]. Investigator: Right? Can you kind of relate this to a clock? Participant 13: Yes. Investigator: Okay. How? Cool, so you kind of made 12, 3, 6, and 9. Okay. So, your C is at the center? Participant 13: Mm-hmm [affirmative]. Investigator: Right? Participant 13: And B is my 9. Investigator: Okay. Participant 13: So, I guess after the rotation, AC will lay upon BC. Investigator: Okay. Participant 13: That’s how you know that this triangle is isosceles. Investigator: Excellent. So, you said something about rotations making you think of circles. What part of the circle are we talking about here? Participant 13: The circumference. Investigator: Okay, so what’s on the circumference? Participant 13: A and B. Investigator: Okay. What is C? Participant 13: The center of the circle. Investigator: Okay, so then what would AC be? Participant 13: The radius. Investigator: What would BC be? Participant 13: The radius. Ohhhh! Investigator: Why are you so excited about that? Participant 13: Because all the radii are equal in length. Investigator: Okay, excellent. Do you think that without my prompting, do you think you would’ve come to that in time?

ESD, EU

233


Participant 13: Hm. No. Investigator: Okay. Why not? What do you think you would’ve needed? Would you need more? Participant 13: I would need more information, I think, on the measures of the angle.

13 Participant 13: It says determine the degree measure of Angle A. Since we already know the angle of measure C, which is 30 degrees, I did 180 minus 30, and I get 150. Since we now know that Triangle ABC is isosceles, the measure of Angle A and the measure of Angle B are the same. Investigator: Okay. Participant 13: We have to do 150 divided by two, which would be 75.

EU

14 Investigator: Okay. So when you said ABC is isosceles, what does isosceles mean? Participant 14: Isosceles means that at least two angles must be equal.

EU

15 So in the diagram below, B is the image of A after counterclockwise rotation of 30 degrees about C. Explain why Triangle ABC must be isosceles. . . . Wait, hold on. I don’t know.

ESD

15 Investigator: Okay, so where’s the center of rotation? Participant 15: C.

EU

15 Investigator: Okay. And then do you know which angle is 30 degrees? If any? Participant 15: That means Angle C, Right? Investigator: Yeah, do you know why? Participant 15: I don’t know why.

ESD

15 Investigator: Okay, so what shape do you think of when you think of rotation in general? Participant 15: Circle. Investigator: Okay, so why do you think of a circle? Participant 15: 360. Investigator: Okay, so if you’re at C, what part of the circle would this be? Participant 15: Center. Investigator: Okay, so then what’s CA? Participant 15: Diameter. Investigator: Okay. Participant 15: Ohhhh! Investigator: Oh. You get it now? Participant 15: Yeah. Investigator: Okay. Participant 15: So, BC would also be a diameter, so they are congruent so that’s isosceles triangle, Angle A equals Angle B. Investigator: Okay. When you say diameter, but C is the center? Participant 15: Yeah.

EU

234


Investigator: So, diameter connects . . . Participant 15: Oh wait, hold that, radius.

15 Participant 15: Why Triangle ABC must be isosceles? So right here, that means Triangle ABC is inscribed in the circle, so C is the center of a circle. So, CA and CB both represent the radius of the circle. And for a circle, radius has the same length; that means CB is congruent to CA. According to the definition of isosceles triangle, Triangle ABC is isosceles.

EU

15 Participant 15: Determine the degree measure of Angle A. So, Angle C equal to 30 degrees. For each triangle, the sum of interior angles is 180, so it’ll be 180 minus Angle C: 30, equals 150 and based on the base angle theorems, Angle A will be congruent to Angle B so Angle A will be the half of 150, equal to 75.

EU

15 Participant 15: Since CA is congruent to CB, that means Angle C is a vertex angle. So, Angle A and Angle B are base angles.

EU

16/17

Participant 16: How do you know all three angles aren’t congruent? Participant 17: I don’t! Participant 16: I don’t, either. I made that assumption. Participant 17: But I think it’s safe to say that an isosceles triangle can’t be equilateral, but an equilateral triangle can be considered isosceles? Participant 16: I think it has to do with the fact that you’re rotating only 30 degrees. Participant 17: But I was just talking about the . . . I don’t. . . It’s not equilateral. Participant 16: Yes, so then . . . Participant 17: The rotation of 30 degrees about C. I didn’t read that part. About C. So, it’s actually rotating about C. Therefore, C has to be 30 degrees. Participant 16: Duh! Participant 17: And so the base angles are gonna be 150. Participant 16: 75 degrees. I didn’t read that either. Investigator: In Part A you did not read that it said about C? Participant 17: In the original prompt. Investigator: So, once you read that, what did that help to clarify? Participant 16: Angle C has to be 30 degrees, because that’s the angle measure that’s also giving the rotation to get from A to B.

ESD, EU

16/17

Investigator: What motion were you making with your pen? Participant 16: I was making a, like an arc motion? Investigator: Between? Participant 16: Between A and B. Because originally I labeled that arc as 30 degrees. I was thinking the value of that, as an arc of 30 degrees, what does that mean for the angles that are left inside? Investigator: Does this X mean that you no longer think this arc is 30 degrees? Participant 16: Yes, but now remembering that, that could be true, so . . . Participant 17: If this was a circular piece, then the measure of the arc is related to the central angle.

ESD

235


Participant 16: Right. So, this could be the radius and that could be this . . . Investigator: What could be the radius? Participant 16: AC could be the radius. BC could be the radius and there could be . . . Was that inscribed inside the circle? Is that what it is? Participant 17: An inscribed angle is when it touches on the circle. Participant 16: Angle A is an inscribed angle? No, you’re right. Never mind. I’m wrong. Investigator: That’s a different type of angle. Participant 17: Yeah, it could be, if CB was the diameter. Participant 16: But we don’t know that. Participant 17: We don’t know.

18 Participant 18: Image of A after a counterclockwise rotation 30 degrees about C. Okay, so counterclockwise. Do a little summary here before A, B, C. Explain why it must be isosceles. How can it be isosceles though? Maybe it doesn’t tell you.

ESD

18 Oh, B is the image of A. Okay. So, B is the image of A. That helps, ‘cause that means we have a line mapped to another line. Yeah, so if a line maps another line then they must be congruent.

EU

18 Participant 18: I’m mostly thinking about this in terms of a circle. It’s a swivel. Investigator: Great. What part of the circle is present in this diagram? Participant 18: You have the origin, you have the radius, you have another radius. You can draw an arc if you wanted to, which is why I went for this first. Investigator: Okay, what do you mean when you say origin? Participant 18: Center of a circle. Investigator: Okay, so which point is the center? Participant 18: C. Investigator: Okay, and then what’s the radius? Participant 18: AC, and BC. Investigator: Okay, so can you tie that into the isosceles reasoning? Participant 18: A triangle . . . any triangle with center of a circle and two points on the circle will be isosceles. Is that how you spell that? Investigator: Isosceles? Yes. I-S-O-S-C-E-L-E-S. Nice.

EU

20 Participant 20: Can I come back to a question, or does it just need to just be skipped completely? Investigator: We can come back to it later. Can you tell me why you want to come back to it, or what’s making you uncomfortable about it? Participant 20: I’m almost certain that A and B are the same distance away from C. Investigator: Okay. So why do you want to come back to it? Participant 20: Because I don’t know why. Investigator: Because you don’t know why. Participant 20: I don’t know how to explain why, but I know it has to be, because I’m rotating about Point C.

ESD, EU

236


Investigator: So, what does that mean about Point C? Participant 20: That C is my center of rotation. Investigator: Can you label that? Do you want to lean on this? When you think of rotations, what shape do you think of? Participant 20: Circles. Investigator: Okay, so how can you relate this to a circle? Participant 20: Maybe that I’m swinging the radius. Investigator: Okay, what part is the radius? Participant 20: AB. Investigator: Okay. Then, where is the center of the circle? Participant 20: Well, if AB is the radius, then I don’t know. I don’t think that’s . . . What I just said . . . Here’s what I think. I don’t know how true it is, but because I rotated about Point A . . . I mean I rotated about Point C. Investigator: You’re rotating about Point C, yeah. Participant 20: A and B have to be the same distance away from C. Investigator: Okay. Participant 20: I don’t know what’s giving me the right to say it, but it is. Investigator: Okay, so you’re certain that A and B are the same distance from C? Participant 20: Yes. Investigator: How does that help you with Part A in the problem? Participant 20: Because now AC and BC are equivalent, so I have an isosceles triangle, because two sides are the same. Investigator: Okay, so you don’t know the reason that A and B are the same distance from C, but you’re certain that they are. Participant 20: Yes. Investigator: Okay, can you write that down for Part A? I’m just going to pause this while you’re writing. Okay, can you read your answer for Part A? Participant 20: Segment AC is congruent to BC since A and B are equidistant from Point C. Investigator: Okay, do you have any other thoughts about this? Participant 20: Since I have a triangle with two congruent sides, it has to be isosceles. I don’t think the 30 degrees matters for saying why A and B are equidistant from C. I think had I rotated this point 10 degrees about Point C, A and B would still be equidistant.

20 Investigator: Okay. You’re correct to say that. Alright, can you do Part B? Participant 20: No? Investigator: Why not? Participant 20: I don’t know, it would be 30 or some type of multiple of 30. Investigator: 30 or a multiple of 30? Participant 20: Yeah. Investigator: Okay, can you write that down? Participant 20: If it’s a multiple of 30, it can only be 30 or 60. Investigator: Okay, why can’t it be 90?

ESD

237


Participant 20: It can’t be 90 because we have an isosceles triangle, so two of the sides have to be equivalent, and it can’t be 60 . . . Well, I don’t think it’s 60. Investigator: Why not? Participant 20: I’m going based off the drawing, because it’s not equilateral. Investigator: Based on what the drawing looks like, it’s not equilateral? Participant 20: Yeah. Investigator: Okay, question, you said something interesting. You said, “I don’t think the 30 degrees matters for saying A and B are equidistant from C, even if it was 10 degrees, they would still be equidistant.” Participant 20: Yeah. Investigator: So, what does the 30 degrees mean or do in this picture? Participant 20: I think it creates a 30-degree angle. Investigator: Where? Participant 20: At A. I think Angle A is 30 degrees. Investigator: Okay. Participant 20: If I’m taking this point here and rotating it 30 degrees, yeah, it creates a 30-degree angle. Investigator: Okay. So, if Angle A is 30 degrees, can you mark that on the picture? What angle do you also know? Participant 20: Angle B. Investigator: Angle B, and what would that mean about the measure of Angle C? Participant 20: I don’t like this. Investigator: Why not? Participant 20: Because, then it’s saying that Angle B has to be 120 degrees. Investigator: You mean Angle C? Participant 20: Yes. Investigator: Why don’t you like that Angle C is 120 degrees? Participant 20: Based on the picture, but even if I made my own picture and here was A, and I swing it 30 degrees . . . Yeah, C can’t be bigger than Angle A. Investigator: Okay, so something’s wrong, but we’re not sure what. Participant 20: Yeah.

21 So, B is the image of A, oh, makes sense, after a counterclockwise rotation of 30 degrees about C. If it’s the image, that means that they must have the same angle measure. So, if Angle C is 30 degrees, then 180 - 30 = 150. So that means that the measure of Angle A and the measure of Angle B must be equal 75 degrees. Investigator: Okay. How did you know that C was 30 degrees? Participant 21: Because it’s saying that it’s a counterclockwise rotation of 30 degrees about C. So, if it’s being rotated from the Point C, that means that Angle ACB must be equal to 30 degrees. That’s just, I believe that’s the definition of a rotation about a point, actually that’s the angle measure. Yeah, that’s angle measure that it’s creating.

EU

21 Investigator: Okay. So, you said ABC, is isosceles because AC is congruent to BC. How would you explain why those are congruent?

EU

238


Participant 21: Well, I mean I kind of assumed it at first just by turning the page and seeing that AC and BC looks the same length. Investigator: Okay. Participant 21: But after figuring out that Angle A and Angle B are congruent based on the fact that Angle C is 30 degrees, B is the image of A. So, if it’s being rotated, a rotation is a rigid motion. Investigator: What does that mean? Participant 21: A rigid motion that preserves distance and angle measure, which I learned more about this definition as I taught this year.

22 Participant 22: Why it must be isosceles? B is the image of A after it a counterclockwise rotation of 30 degrees about C. Bro, I don’t know.

ESD

22 Investigator: You don’t know? Okay. Do you know what isosceles means? Participant 22: Yes. I know what isosceles means. Investigator: So, what would that mean about this triangle? Participant 22: That it has two sides that are congruent, and opposite angles are equal. Investigator: Okay. And when you say opposite angles . . . Participant 22: From the sides that are congruent. Investigator: Okay. Do you know what it means when it says B is the image of A? Participant 22: Yes. Investigator: What does that mean? Participant 22: It means when you rotate A counterclockwise about C 30 degrees, you’ll get B.

EU

22 Investigator: Okay. So, do you know where the 30 degrees is in this picture? Participant 22: 30 degrees is . . . rotation of 30 degrees about C. Wait, it’s about C, right? Investigator: Mm-hmm [affirmative]. Participant 22: This one. Is it Angle C? You can’t tell me though, right?

ESD

22 Participant 22: So, I think this is 30 degrees. Investigator: Okay. So, does that help you? Participant 22: Oh, yeah. Because then the line segment AC and BC are the same length. Investigator: Okay. Participant 22: Therefore, Angle A and Angle B . . . Investigator: Okay. Participant 22: . . . which means it’s isosceles.

EU

22 Participant 22: But can it technically be an equilateral? I mean, that’s a special case of isosceles, right? Investigator: So, if it were equilateral, is that still isosceles? Participant 22: Yes. Okay. So, I guess that’s not an issue.

EU

239


22 Investigator: So how would you write down your reasoning for A? Participant 22: I guess I would say Triangle ABC must be isosceles because line segment BC is produced from pretty much rotating. . . . Does it make sense to say rotating line segment AC? Is that a thing, or no?

ESD

23 Investigator: How do you know the angle you marked is 30 degrees? Participant 23: Because it was given that it was a rotation of 30 degrees about C. Investigator: Okay. Participant 23: I know that move, that segment was moved 30 degrees. Investigator: Okay, and what segment? Participant 23: Segment AC. Investigator: Okay. Participant 23: And also, rotations preserve length. Investigator: Okay. Participant 23: I know that AC must be congruent to BC.

EU

23 Participant 23: Determine the degree measure of Angle A, I know A and B are congruent so I know that the angle sum of a triangle is 180 and taking away the sum of the vertex angle I have that the base angles must have a sum of 150, dividing by two I get. . . . What is it? 75?

EU

23 Investigator: Okay. You said something in Part A. . . . You said rotations preserve length. Participant 23: Yeah. Investigator: Can you just say in your own words, you don’t have to write them down, but for the tape, what does that mean? Participant 23: I think it means that when something is rotated nothing happens to the thing, it stays rigid. Or it stays . . . the pieces stay congruent. Investigator: Okay, so that what’s preserve means in that context? Participant 23: Yeah.

EU

24 Participant 24: I feel like it’s not giving enough information. EU

24 Investigator: Okay. What would you— . . . So, in Part A you said if it were isosceles, right? So, let’s say you did know that, would you then be able to say the degree measure of A? Participant 24: Yeah, it would be . . . Oh, if it was, then it would be 75. Investigator: How come? Participant 24: Because it would have to be half of 150. Investigator: Okay. Participant 24: Yeah. Investigator: And can you just say where you got the half of 150 from? Participant 24: Maybe it would be 30. I don’t know.

EU, ESD

24 Investigator: It’s frustrating when you feel like there’s not enough information. Participant 24: Yeah, it is. Right? Investigator: So, you said maybe it would be 30, maybe it would be 75. Participant 24: Mm-hmm [affirmative].

EU

240


Investigator: Okay. Can you just explain why it would be each of those? Participant 24: Okay, it would be 30 if this one was like the same angle as this one. So like, if I’m saying that AC and BC are congruent, then A would Angle BAC or A would have to be 30. But if I got this one wrong and it was over here, if Angle C was 30, then Angle B and Angle A would have to be 75.

25 In order for Triangle ABC to be isosceles, two segments are congruent and the base angles of them must have the same measurements. Since B is an image of A, these two are the same points. B was created by rotating A 30 degrees from that Point C, so first the distance from C to A must be the same distance from C to B which means CA is congruent to CB. We know that if they are congruent their bases must have the same angles.

EU

25 I’m guessing that since B was creating a rotation about C by 30 degrees, the measure of Angle C is 30 degrees. We said it’s isosceles so the measurement of the other two angles must be the same. We know that the sum of the angles of every triangle is 180 degrees.

ESD

25 First of all, at first read I couldn’t visualize what was going on until I had to do it by myself following the steps. B is the image of A, which means A’= B. So how does this happen? By rotating 30 degrees about C, and then connect the 3 points. Something that helped me is knowing the definition of isosceles about two segments are the same and two angles are the same.

EU

26 Investigator: Okay, and then, is that enough to say that the triangle’s isosceles? Participant 26: No, because . . . Investigator: Why not? Participant 26: It’s not enough to say it, because we don’t know if AB, line AB is . . . Don’t know it, to be honest with you.

ESD

26 Participant 26: We have that a triangle is 180, and we have two of these measures right here, which would say that you have 30 degrees plus this measure of Angle A and then plus measure of Angle B. We subtract these right here, all from both sides. We’re going to have 150 right here, and then measure of Angle A plus measure of Angle B. Can I just say that both of these measures would be . . . Say Angle A and Angle B is equal to, say, X, and then we have X plus X, which is equal to 150. This would be just 2X, and then X would be . . . What is it? 75? Investigator: Mm-hmm [affirmative]. Participant 26: Which would make these two angles right here 75. Investigator: Okay, and then, from that, is the triangle isosceles? Participant 26: I believe so. Investigator: Okay. Why do you believe so? Participant 26: Because, well, in an isosceles triangle, we have two angles that have the same equal measure. Investigator: Good. Okay. Were you able to do Part B, which is determine the degree measure of A?

EU

241


Participant 26: Yes, which we just did. Investigator: Okay.

27 Investigator: All right, and Part B? Participant 27: Determine the degree measure of Angle A. Participant 27: Not sure how to go about figuring that out.

ESD

27 Investigator: Okay. Does the 30 degrees in the description have any place in the diagram? Participant 27: I’m thinking this would be the 30? Investigator: The Angle C? Participant 27: Right. Investigator: Okay. Participant 27: Because if A was B, then, and it’s rotated about C, then being rotated up like this, so that would make this 30. Investigator: Okay. Participant 27: Is that correct?

ESD

28 Participant 28: So, in the measure below, be is the image of A after counterclockwise rotation of 30 degrees about C. That’s crazy. That doesn’t . . . that . . . what does that mean? But I would basically . . . okay. So, I’d be like, “Okay, what are they saying?” Okay. I don’t know it. I have never heard of this in— Investigator: Phrasing like that? Participant 28: I’ve never heard of phrasing like this using counterclockwise and rotation about and this was definitely something I didn’t do in geometry in high school.

ESD

28 This is a hard one. ESD

28 It’s supposed to be 180 but it says it’s isosceles and none of the angles are the same so yeah; this is messing me up because basically so I know all the angles have to be 180 but I’m given two angles already that are 60 and 30 so the other one must . . . would be 90; however, that means that all the angles aren’t the same and that means that it’s . . . it wouldn’t be isosceles.

EU

28 Yeah, I don’t know how to solve this. ESD

29 Participant 29: Okay, so we’re saying that B is the image of A after it’s been rotated counterclockwise about C, so I’m picturing taking the Point A and, from C, drawing almost like a circle.

EU

29 Investigator: Okay. Can you say a little more about the circle you’re picturing? Participant 29: Yeah, so it has radius AC, and I’m just confused about the 30 degrees. Like how would I be able . . . So, if it’s getting rotated 30 degrees, would that make the Angle BCA 30 degrees?

ESD

29 Participant 29: Okay. And so why it has to be isosceles . . . We have to prove that side AC and BC are congruent. I’m not sure how I would do that with just the angle. That’s what’s confusing me.

ESD, EU

242


Investigator: Okay. Can you do it by thinking back to the part of that circle that you were referring to earlier? Participant 29: So, then this would also be the radius. Investigator: What would? Participant 29: BC. Investigator: Then what else was a radius? Participant 29: AC. And so that’s why it’s isosceles. Investigator: Why does that make it isosceles? Participant 29: Because then side AC is congruent to side BC.

29 Participant 29: Sure. So . . . I don’t like my explanation. Investigator: Why not? Participant 29: I don’t feel like it’s explaining it enough. Investigator: What did you write? Participant 29: I wrote, “Since we are rotating Point A about Point C, then AC is a radius,” and I said of length x. Thus, BC is also a radius of length x. Investigator: Why don’t you feel like that’s enough of an explanation? Participant 29: I feel like maybe we have to say something about the fact that it’s an image, that it has the same radius. I don’t know if this explains enough of why the length of BC is the same as AC.

ESD

29 Investigator: Are you able to do Part B? Participant 29: Determine the degree measure of Angle A. Yeah. Investigator: And how would you do that? Participant 29: You would take 180° and subtract the 30 degrees, because that’s the angle measure of BCA, and then you would get 150. Then, since it’s isosceles, you would just divide that by 2. You get 75.

EU

Note. P# = participant number; EU = expressing understanding; ESD = expressing self-doubt.

243

Table D5

Instances of Pure Mathematical Issue Code for Content Assessment Item 1

P# Quotation/instance Comments 1 Cause the measure of Angle B should equal the measure of

angle . . . the measure of Angle B should equal the measure of Angle A because if it’s . . . B is the image of A after counterclockwise . . . since it’s about C and it’s 30 degrees, I think A must be equal to B.

Thinks Angle A (and B) both equal 30 degrees based on given information—started incorrectly by labeling A = 30 degrees.

1 One of the other angles must be 30, but which one is it? Incorrectly thinks another angle must be equal in measure to Angle C since the triangle is isosceles.

1 If I’m rotating this angle counterclockwise. . . Incorrectly thinks an ANGLE is being rotated, when it is a POINT.

1 Shouldn’t B also be 30? Gets stuck at same issue as previous.

1 So, let’s call this X and let’s call this X, ‘cause we don’t know those two angle measures.

Without justification that the angles are equal in measure, uses the same unknown to represent them.

2 Investigator: Okay. Is there an angle on this picture that then you know the measure of based on that information? Participant 2: No. Oh, yeah. B. Investigator: Is? Participant 2: The same as A angle. Investigator: Interesting. Why? Participant 2: Because it was just rotated.

Angle is not being rotated.

2 Investigator: Okay. Can you mark that down? And why do you think that that’s 30 degrees? Participant 2: Because everything has to add up to 180.

Reasoning has nothing to do with why Angle C would measure 30 degrees.

2 Participant 2: According to what I know about geometry, if two angles are equal, then their opposite sides are congruent. Investigator: Okay. Great. So, can you tell me which sides those would be? Participant 2: BC, and AC.

Participant is asserting that two angles are equal which means the sides opposite them are congruent. Although

244

P# Quotation/instance Comments this is true, it stemmed from the incorrect assumption that Angle A equals Angle B. The rotation gives equal distances, not equal angles

2 Investigator: What does isosceles mean? Participant 2: Two sides of a triangle are congruent.

At least two sides are congruent.

4 So, Angle B is the image of A after a clockwise rotation 30 degrees about C. Well because if B is the image of A, and it’s a rotation that preserves orientation and angle measurement and distance so they would have to be the same angle. And then if the two base angles are the same, then it’s isosceles based on the definition.

Assumes angle is the pre-image, when it is a point.

4 Because I mean . . . I don’t really know. I would just think . . . I would just think that if this is the triangle that you’re given after it rotated, these are the two angles, this is horrible, but these are the two angles at the bottom so they would be the two angles at the base.

Triangle was not rotated, and bottom is not base.

5 Because these will have the same angle measure, whereas this does not. So, if you do 180 minus 30, that’s 50. And then 50 divided by two. So, each one is 25 degrees.

Incorrect arithmetic.

6 Participant 6: So, I know that the pre-image is over here— Investigator: Okay. Participant 6: —somewhere. If it’s a 30-degree rotation . . . then this is 30 degrees. Yeah, I don’t know how to solve this if I’m not given any of the angle measurements.

Believes pre-image was a triangle that was not drawn in.

6 Participant 6: Because A . . . Okay, I misunderstood the question. I thought the entire triangle was being rotated, but . . . I mean it is, but okay. Because A moved to Point B so that means that the angle that’s created ACB must be the 30-degree angle. Investigator: Okay. So, is the entire triangle being rotated? Participant 6: Yes, but it’s actually over here. This would have been the old side BC and then it was rotated down.

Believes entire triangle was rotated.

6 Participant 6: During a rotation the side lengths are preserved. Investigator: And what does preserved mean? Participant 6: That they’re the same. Investigator: Okay. Participant 6: They’re not getting any bigger or smaller. Investigator: How do you know that?

“Rotations only affect the angles of things”; and the participant is basing her reasoning off of two sides that do not exist.

245

P# Quotation/instance Comments Participant 6: Because it doesn’t . . . Rotations only affect the angles of things, it doesn’t affect the sides. Investigator: Okay. Participant 6: So the side length AC has to be the same as the side length BC. Investigator: Okay. Participant 6: Meaning that there are two sides in this triangle that are congruent- Investigator: Okay. Participant 6: . . . So, the triangle is isosceles.

7 Investigator: Okay. So, can you write that down and what does isosceles mean? Participant 7: Two sides are congruent.

At least two sides are congruent.

8 Participant 8: It means that if I were to make a counterclockwise rotation of 30 degrees having C as the center of rotation then the image of A would be B. Investigator: Okay and when you say center of rotation what do you mean by that? Participant 8: I would have to rotate A, well if I were to use a compass I would use C as the center of rotation in order to rotate A.

Uses proper mathematical language, but then cannot express what a center of rotation is.

8 Investigator: Okay. And can you read Part A and let me know what you think about that? Participant 8: Explain why Triangle ABC must be isosceles. Investigator: Okay, what do you think? Participant 8: Well first I would think about the properties of isosceles triangles to see how I can apply that to the triangle. So, some properties I might consider would be the base angles would be congruent and opposite sides of those base angles would also be congruent.

Circular reasoning, of sorts. Participant is trying to use properties of isosceles triangles to prove that the triangle is isosceles, rather than being able to apply what she just said about the rotation and its properties in order to prove something about the triangle.

8 I was thinking maybe I could use triangle, If I prove that triangle ADE is similar to Triangle ABC then if I could show that the smaller ADE is an isosceles triangle maybe that would help me to see if Triangle ABC would be an isosceles triangle.

Approach, although not mathematically incorrect, has nothing to do with the transformation or the given information. Seems like the participant cannot connect ideas about

246

P# Quotation/instance Comments rigid motions to notions of congruence.

9 Okay. So, A is the pre-image, and B is the image, and when you do a rotation the angle measures are preserved. So, A and B are still the same angle.

Although the participant uses correct mathematical language, she is thinking that the Angle A, rather than the Point A, is rotated.

10 Well, it’s like isometric, so nothing changes . . . angle measure and side lengths, none of that changes. So, the figure tends to be congruent to its original.

Sees transformations in the plane as applying to figures rather than sets of points

10 Oh. In an isosceles triangle, the base angles are congruent. So, if Angle A is the same as Angle B, then the triangle is isosceles.

Participant thinks she can say for certain that the angles are congruent, when only a point was rotated. Bases reasoning on incorrect property of rotation.

10 In a rotation angle measure is preserved, therefore when A is rotated 30 degrees resulting in image B, Angle A is congruent to Angle B. In an isosceles triangle at least two angles must be congruent, therefore Triangle ABC must be isosceles.

Same comment as previous.

10 Participant 10: Determine the degree measure of Angle A. Since to get from A to B we rotated 30 degrees about C, would that make Angle C 30 degrees? Investigator: Why do you think that? Participant 10: I guess the way it’s positioned. We have the line from C to A, and then rotated to B so CB is created. So that’s why Angle C is 30 degrees. And then to find the measure of Angle A, we know that the sum of the interior angles of a triangle is 180. So, subtracting 30 from 180 and then dividing that by 2 would give you the measure of Angle A.

Does not connect this realization (measure of Angle C) with the fact that her reasoning for Part A was based on the wrong property of rotations—does not realize that the identification of Angle C from the beginning was crucial.

11 Participant 11: Yeah, and in an isosceles triangle, two sides must be congruent.

Although this does not impact completion of this problem, what are the implications for the future when participants say two vs. at least two sides are congruent in an isosceles triangle?

247

P# Quotation/instance Comments 12 . . .The 30 degrees is Angle ABC. Identifies incorrect

angle, but later corrects. 13 Investigator: Okay. What does isosceles mean?

Participant 13: That there are two legs within a triangle that have the same length.

13 Investigator: Okay. I see you marked off AC and BC. Why do you think those would be the two with the same length? Participant 13: It was not drawn to scale?

Not connecting knowledge of rotation and preservation of distance to side lengths of the triangle.

13 Investigator: So, does the word rotation make you think of anything? Participant 13: It preserves distance. Investigator: What does that mean? Participant 13: That the distance between all the points will stay the same, they won’t change. Investigator: Okay. Does that help at all? Participant 13: No.

Uses correct mathematical language but cannot apply the concepts to the figure given.

14 Investigator: Okay. All right so we’re going to talk more about that later. So, you hate transformations, how do you know this problem is about transformations? Participant 14: I already see counterclockwise rotation of 30 degrees. Investigator: Okay. Do you know what that means in terms of this picture? Participant 14: Let’s see. In the diagram below B is the image of A after a counterclockwise rotation of 30 degrees about C. So, it means that they took an original isosceles triangle. Investigator: Okay. Participant 14: And then they rotated it about the Point C. Investigator: Okay. Participant 14: For 30 degrees. Investigator: Okay. So why does this triangle have to be isosceles? Participant 14: Because rotation is a type of transformation that doesn’t alter, it’s not like a dilation that alters the angles, the measures of a triangle.

Completely wrong—decides that the pre-image is isosceles, does not use properties of rotations. Also, dilation does Not alter measures of angles.

14 Participant 14: Determine the angle measure of A. Let’s see. . . I’m assuming it would be the same as it was before.

Assumes the Angle A was somehow transformed.

16 Participant 16: Yeah, I finished first. Angle A and Angle B are congruent, rotations preserve shape - rigid motion? Therefore, the angle measure stays the same.

Assumes angle is being transformed, rather than

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P# Quotation/instance Comments the point. Bases reasoning off angles.

17 Participant 17: I said, since B and A are images of each other, I should really say since B is an image of A, they must have the same measure, therefore they are congruent. When two angles in a triangle are congruent, the triangle is isosceles.

Assumes angle is being transformed, rather than the point. Bases reasoning off angles.

16/17

Participant 16: How do you know all three angles aren’t congruent? Participant 17: I don’t! Participant 16: I don’t either. I made that assumption. Participant 17: But I think it’s safe to say that an isosceles triangle can’t be equilateral, but an equilateral triangle can be considered isosceles? Participant 16: I think it has to do with the fact that you’re rotating only 30 degrees. Participant 17: But I was just talking about the . . . I don’t. . . It’s not equilateral.

Isosceles vs. equilateral misconceptions.

17 Investigator: You said a center point. On this diagram, which one would be the center point? Participant 17: C Investigator: And then, is . . . Are there other parts of this diagram that you can relate to a circle, or not? Participant 17: We can talk about circular component, like incircle, circumcircle.

Irrelevant terms.

16/17

Investigator: What motion were you making with your pen? Participant 16: I was making a, like an arc motion? Investigator: Between? Participant 16: Between A and B. Because originally, I labeled that arc as 30 degrees. I was thinking the value of that, as an arc of 30 degrees, what does that mean for the angles that are left inside? Investigator: Does this X mean that you no longer think this arc is 30 degrees? Participant 16: Yes, but now remembering that, that could be true, so . . . Participant 17: If this was a circular piece, then the measure of the arc is related to the central angle. Participant 16: Right. So, this could be the radius and that could be this . . . Investigator: What could be the radius? Participant 16: AC could be the radius. BC could be the radius and there could be . . . Was that inscribed inside the circle? Is that what it is?

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P# Quotation/instance Comments Participant 17: An inscribed angle is when it touches on the circle. Participant 16: Angle A is an inscribed angle? No, you’re right. Never mind. I’m wrong. Investigator: That’s a different type of angle. Participant 17: Yeah, it could be, if CB was the diameter. Participant 16: But we don’t know that. Participant 17: We don’t know.

18 So, that means the original right side is now the new left side, therefore the left and the right are congruent to each other.

No idea what he is talking about with left vs. right sides; however, he drew on the diagram indicating that he is interpreting an incorrect pre-image.

18 Investigator: Yeah, why are you hesitant to use the word maps? Why don’t you like the word maps? Participant 18: No, because they don’t . . . if they gave me distinct ABC and then a new ABC, I’d be like okay, ‘cause you can just name them. I’m having trouble naming exactly what would you call the pre-image? That’s what I’m having issue with.

18 Participant 18: Determine the degree measure of Angle A. Sounds like 30 to me. Let me just think about that for a second. 30 degrees, yeah, it’s obviously 30 degrees. Do I have to explain why? Investigator: You can explain for the tape. You don’t have to write why. Participant 18: If the line is moving, I’m trying to think about this in terms of axiomatic geometry. Investigator: Okay. Participant 18: If you basically . . . because, an angle is formed by two rays. This isn’t two rays obviously but, if the whole line AC is rotated about C, then it logically concludes that this also must be 30 degrees.

Totally incorrect reasoning and answer, but is confident in his incorrect answer.

19 Since B is the image of A, Angle A is congruent to Angle B. And Angle B is the image of Angle A, that shows that these two angles have the same angle measurement.

Assumes angle is being transformed, rather than the point. Bases reasoning off angles.

19 When a triangle has two equal angle measurements, it is an isosceles triangle.

Participant thinks he can say for certain that the angles are congruent, when only a point was

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P# Quotation/instance Comments rotated. Bases reasoning off incorrect property of rotation. Also, does not mention at least two angles for an isosceles triangle, but rather exactly two.

19 2x+c=180. I don’t know the angle of C for now, so I’ll look at the given information again. But I’m stuck. A counterclockwise rotation doesn’t give you any information about the measure of Angle C.

Could not solve for Angle A, although he labeled A and B both as “x,” because he did not utilize the 30-degree given angle and interpreted the problem incorrectly.

20 Investigator: Okay, so how can you relate this to a circle? Participant 20: Maybe that I’m swinging the radius. Investigator: Okay, what part is the radius? Participant 20: AB. Investigator: Okay. Then, where is the center of the circle? Participant 20: Well, if AB is the radius, then I don’t know. I don’t think that’s . . . What I just said . . . Here’s what I think. I don’t know how true it is, but because I rotated about Point A . . . I mean I rotated about Point C. Investigator: You’re rotating about Point C, yeah. Participant 20: A and B have to be the same distance away from C. Investigator: Okay. Participant 20: I don’t know what’s giving me the right to say it, but it is. Investigator: Okay, so you’re certain that A and B are the same distance from C? Participant 20: Yes.

Relates to a circle, and identified C as the center of rotation, but does not realize that must mean C is the center of the circle. Incorrectly identifies AB as a radius.

20 Investigator: Okay. You’re correct to say that. Alright, can you do Part B? Participant 20: No? Investigator: Why not? Participant 20: I don’t know, it would be 30 or some type of multiple of 30. Investigator: 30 or a multiple of 30? Participant 20: Yeah. Investigator: Okay, can you write that down? Participant 20: If it’s a multiple of 30, it can only be 30 or 60. Investigator: Okay, why can’t it be 90?

Knows that a 30-degree counterclockwise rotation must mean that one of the angles is 30 degrees, but incorrectly identifies that angle as Angle A. Continues with that reasoning and arrives at a roadblock, not based on facts but rather on assuming that

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P# Quotation/instance Comments Participant 20: It can’t be 90 because we have an isosceles triangle, so two of the sides have to be equivalent, and it can’t be 60 . . . Well, I don’t think it’s 60. Investigator: Why not? Participant 20: I’m going based off the drawing, because it’s not equilateral. Investigator: Based on what the drawing looks like, it’s not equilateral? Participant 20: Yeah. Investigator: Okay, question, you said something interesting. You said, “I don’t think the 30 degrees matters for saying A and B are equidistant from C, even if it was 10 degrees, they would still be equidistant.” Participant 20: Yeah. Investigator: So, what does the 30 degrees mean or do in this picture? Participant 20: I think it creates a 30-degree angle. Investigator: Where? Participant 20: At A. I think Angle A is 30 degrees. Investigator: Okay. Participant 20: If I’m taking this point here and rotating it 30 degrees, yeah, it creates a 30-degree angle. Investigator: Okay. So, if Angle A is 30 degrees, can you mark that on the picture? What angle do you also know? Participant 20: Angle B. Investigator: Angle B, and what would that mean about the measure of Angle C? Participant 20: I don’t like this. Investigator: Why not? Participant 20: Because, then it’s saying that Angle B has to be 120 degrees. Investigator: You mean Angle C? Participant 20: Yes. Investigator: Why don’t you like that Angle C is 120 degrees? Participant 20: Based on the picture, but even if I made my own picture and here was A, and I swing it 30 degrees . . . Yeah, C can’t be bigger than Angle A. Investigator: Okay, so something’s wrong, but we’re not sure what. Participant 20: Yeah.

the diagram is drawn to scale. Note that based on her assessment paper alone, it would seem that the participant understands the question and uses correct reasoning for Part A, but she has serious misconceptions that would not have come to light without this conversation.

21 I think, why should ABC be isosceles when it . .. okay. Now I see it. If I turned the page, I could see why. I can see that AC is congruent to [crosstalk 00:00:41]. I see that AC is congruent to BC.

Bases reasoning purely on visual interpretation of diagram, without initially considering

252

P# Quotation/instance Comments given information or being told that the diagram is drawn to scale.

21 So, when I rotated, I do see that AC is congruent to BC, but why must it be isosceles?

Rotated paper to “find” congruent sides and considered this to be the described rotation. Note, in this case the initial misconception did not impact the participant arriving at the correct answer. She soon after realized that Angle C is 30 degrees.

21 Investigator: Okay. So, you said ABC is isosceles because AC is congruent to BC. How would you explain why those are congruent? Participant 21: Well, I mean I kind of assumed it at first just by turning the page and seeing that AC and BC looks the same length. Investigator: Okay. Participant 21: But after figuring out that Angle A and Angle B are congruent based on the fact that Angle C is 30 degrees, B is the image of A. So, if it’s being rotated, a rotation is a rigid motion. Investigator: What does that mean? Participant 21: A rigid motion that preserves distance and angle measure, which I learned more about this definition as I taught this year.

First, assumed that the triangle was congruent by visual inspection. Second, says that Angle A is congruent to Angle B “based on the fact that Angle C is 30 degrees.”

21 Angle measure and length. So, if it’s being rotated then Angle A and Angle B must be congruent as well.

Read problem out loud as, “Angle B is the image of Angle A.” Assumes congruence of two angles based on the given rotation, but a point was being rotated, not an angle.

22 Participant 22: But can it technically be an equilateral? I mean, that’s a special case of isosceles, right? Investigator: So, if it were equilateral, is that still isosceles? Participant 22: Yes. Okay. So, I guess that’s not an issue.

Hard to determine whether this would impact participant’s understanding if there was no conversation.

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P# Quotation/instance Comments 22 Okay, so rotate line segment AC, meaning that AC and BC

are same length. Does not link this conclusion to the properties of rotations that necessitate the congruence of AC and BC (although the participant clearly does know that they are congruent).

24 Investigator: Okay. Why do you feel like you don’t have enough information? Participant 24: Because I have 150 degrees left. Investigator: For which two angles? Participant 24: For Angle BAC and BCA. Investigator: Okay, so you labeled Angle B as 30. Participant 24: Yeah. Investigator: And where did you get that from? Participant 24: Because it says below B is the image of A after a counterclockwise rotation about C. So, I’m assuming we’re starting here and then we’d go over there. Investigator: Starting, so you’d go to A? Participant 24: Mm-hmm [affirmative].

Incorrectly labeled Angle B as 30 degrees.

24 Investigator: Okay. And can you say again why you think there’s not enough information? Talk about that 150. Participant 24: Oh, because the other angles, it could be like both have to add up to 150 so it could be like one could be 30, it could be isosceles, but then there’s not enough given information.

Believes there is not enough information given to determine that the triangle MUST be isosceles. Does not consider properties of rotation.

24 Investigator: Okay. Do you have enough information to do Part B? Participant 24: Determines that the degree of measure of Angle A. No, I don’t think so.

Thinks definition of isosceles is exactly two congruent sides and angles.

25 Something that helped me is knowing the definition of isosceles about two segments are the same and two angles are the same.


26 First, I don’t understand when it mentions the image and when . . . I actually found the word “about” confusing.

Does not know the vocabulary necessary for entry into the problem.

26 Investigator: From that description, do you know what would be 30 degrees in the picture?

Investigator explained what the wording means

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P# Quotation/instance Comments Participant 26: Yeah. 30 degrees would be right here. Investigator: Okay. At Angle C. Participant 26: At Angle C. Investigator: Okay, and then, does that help you figure out why this thing is isosceles. Participant 26: Oh. Yeah. We talked about this. Mm. Investigator: What do you know about isosceles triangles? Participant 26: That . . . Isn’t it that two of the angles are congruent? Investigator: Mm-hmm [affirmative], and anything else? Participant 26: Two of the sides have to be . . . Investigator: Have to be . . . Participant 26: Congruent. Investigator: Okay. Great. If this . . . If we go back to the picture and we talk about this rotation, what shape does the rotation make you think of? Participant 26: Like a circle. Investigator: Why does it make you think of a circle? Participant 26: Well, probably because of the direction that it’s going. Investigator: Mm-hmm [affirmative]. In this case, if you have a circle, and you know you have a center point on the circle . . . Right? Where would the center of that circle kind of be in this picture? Participant 26: Point C. Investigator: Okay, and then what . . . If this is C, where would A and B be in relation to the circle? Participant 26: Don’t know if I’m using the word right. I think it’s the . . . It’s inscribed . . . No? . . . on the circle? Investigator: Okay, so on the circumference. Participant 26: Right. On the circumference. Investigator: Okay, and then what part of the circle is CA? Participant 26: It’s the radius. Investigator: Okay, and what about CB? Participant 26: It’s also the radius. Investigator: Okay. What does that tell you about those? Participant 26: Both of them are congruent. Investigator: Okay, and then, is that enough to say that the triangle’s isosceles? Participant 26: No, because . . . Investigator: Why not? Participant 26: It’s not enough to say it, because we don’t know if AB, line AB is . . . Don’t know it, to be honest with you.

in the problem, and scaffolded the participant with questions that he was able to answer; however, he was not able to synthesize (bizarre—was able to say that AC and BC are congruent, and that a triangle has two congruent sides, but not that Triangle ABC is isosceles).

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P# Quotation/instance Comments 26 Investigator: Okay. If it was . . . Let’s say you have a triangle

with three equal sides. Participant 26: Mm-hmm [affirmative]. Investigator: Right? Is that isosceles? Participant 26: No. It’s equilateral.


26 Participant 26: It’s not enough to say it, because we don’t know if AB, line AB is . . . Don’t know it, to be honest with you. Investigator: If . . . What are you thinking about AB? If AB . . . We don’t know . . . You said we don’t know if AB is . . . Do you mean the same as these? Participant 26: Right.

Thinks he needs to know if AB is the same as the other sides because if it is, then he thinks the triangle is NOT isosceles “but rather equilateral.”

26 Participant 26: That the . . . I believe . . . Yeah. This one has to be . . . Well, we say that this . . . This right here is a, little a. This one has to be little b. Investigator: Okay. Participant 26: We could say that it has to be, I think . . . Wouldn’t it just be like the sine, the sine of B? Investigator: What would be the sine of B? Participant 26: And the sine of A? So, B, measure of Angle B will be sine of B. Investigator: Okay, and then the measure of this would be sine of A? Participant 26: Right. Sine of A. Investigator: Okay. That’s using the law of sines? Participant 26: Right. Investigator: Okay, and then, are these going to be the same if the sides are the same? Participant 26: Yes. They would have to.

Law of sines is irrelevant to the problem.

27 I’m thinking it would have to be isosceles because, something having to do with the base angles being the same degree, so rotating it will just swap their positions, but stay the same.

Does not use properties of rotations; basing answer based on property of isosceles triangles; thinks an angle is being rotated rather than a point.

27 Investigator: Okay, and what do you mean when you say base angles? Participant 27: The bottom angles of an isosceles.

Thinks base means bottom.

27 Investigator: Okay. So, since the base angles, then, are congruent, the triangle’s going to be isosceles? Participant 27: Yeah. Writes on paper, “The base angles in an isosceles triangle are

Participant thinks she can conclude that the base angles are

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P# Quotation/instance Comments congruent. If B is the image of A after the rotation, the 2 angles must be congruent.”

congruent based on the given information.

27 Investigator: Okay. Does the 30 degrees in the description have any place in the diagram? Participant 27: I’m thinking this would be the 30? Investigator: The Angle C? Participant 27: Right. Investigator: Okay. Participant 27: Because if A was B, then, and it’s rotated about C, then being rotated up like this, so that would make this 30. Investigator: Okay. Participant 27: Is that correct?

Does not connect realization that Angle C is 30 degrees to the fact that the answer in Part A was incorrect. Also, thinks “A was B.”

28 So, the image of A after counterclockwise rotation of 30 degrees about C. And it also . . . it looks like it’s 30 degrees. It looks like a small enough angle for that, or does it?

Bases reasoning on visual inspection without being told the diagram is drawn to scale.

28 Okay. Just because I’m . . . now I’m using . . . okay. So, what it’s saying is that this is 30 degrees and it’s saying a counterclockwise rotation, so you basically take that 30 degrees and move it up 30 degrees so I’m extending that to another 30 degrees. So now it’s a total of 60 degrees and the image of A . . . so image, if I’m correct, relates to something along like reflection or it’s the same thing just in a different . . . it’s turned a different way. So, what I think this is saying is the image A after a counterclockwise rotation of C . . . so I think that this whole . . . that B is equal to 60 degrees because A . . . wait, hold on. Adding 30-60. Yeah, but that means that A can’t be 60.

28 So, I’m just going to forget about B. So, I’m relating . . . I’m trying to relate C to A now, like how are they the same?

Participant thinks she can “ignore” B.

28 It looks isosceles. Visually it does. Again, bases reasoning on visual assumption.

29 I wrote, “Since we are rotating Point A about Point C, then AC is a radius, and I said of length x. Thus, BC is also a radius of length x.”

Participant uses the concept of a circle correctly, but cannot explain the property of rotations that explains why Triangle ABC must be isosceles.


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Table D6

Participants’ Reason(s) for Option Selection for Content Assessment Item 2

P# Q# Option selected Reason for selection

1 Proof 1

Both Participant 1: Okay. Can I just choose the first one ‘cause it’s filled in for me? Investigator: Why do you want to choose the first one? Participant 1: ‘Cause it’s filled in. Investigator: What’s filled in? Participant 1: The statements. So, I just need to fill out the reasons. Investigator: Okay. Sure. Participant 1: But maybe this one’s easier. Investigator: Well, what do you think about when you’re determining if proof is easy? How do you decide? Participant 1: I would have to do both. Investigator: You would have to do both. Participant 1: Mm-hmm. Investigator: Interesting. Participant 1: Can I do both or no? Investigator: Sure. Which one are you more tempted to start with? Participant 1: This one, obviously. Investigator: Which one? Participant 1: The first. Option 1.

2 Proof 1

Option 1

Participant 2: Yes. I’m going to choose the first problem, Option 1. Investigator: Why? Participant 2: Because I feel that things are written out, and I could make sense out of the problem by the statements given. Investigator: Okay. Participant 2: Option 2 I would have to come up with the reasons, the statements. I don’t know- Investigator: You’d have to come up with the statements? Participant 2: Right, and I don’t know a lot of the theorems offhand. Investigator: Okay, just want to make sure I interpreted what you said correctly, you’re more comfortable with Option 1 because the statements are already given, and you feel more confident with your ability to determine the reasons for those statements rather than to come up with the statements on your own? Participant 2: Correct. Investigator: No matter what the prompt was? Do you feel that you would choose something that looks like Option 1? Participant 2: At this time, yes.

3 Proof 1

Option 1

Investigator: Okay. All right, great. If you can turn to the next page. So, this one gives you two options, so you can just take a minute to look at

258


each and then you’re gonna pick one and explain why you chose that one. And it could be any reason. Participant 3: Like which one I’d wanna do? Investigator: Yes. Or which one you would feel more comfortable attempting. Participant 3: The first one. Investigator: Okay, why? Participant 3: Because I’ve done this one before . . . like in Math 385. Investigator: What is that class? Participant 3: It’s uhh . . . forgot the name for it. Investigator: Okay. What did you— Participant 3: It’s teaching math, like they teach you how to teach math. Investigator: Okay, so have you done this exact proof, or you did an activity that showed—[crosstalk 00:05:03] Participant 3: No, we just did the picture. That was it. Not the actual-[crosstalk 00:05:07] Investigator: Okay. So, you had prior experience with this one? Participant 3: Yeah.

4 Proof 1

Option 2

Investigator: Okay. Thank you. So, if you look at the next page you have two options, so either this one at the top or this one, so you take as much time as you need to just pick one of them and then just tell me why you’re picking the one that you pick. Participant 4: Okay. [Silence.] I think I’ll do Option 2. Investigator: Okay. Why are you more tempted to choose that one? Participant 4: Well, this is just me, but I mean I don’t like it when it’s set up like this just because I get nervous that I’m not writing the right thing. Where I feel like . . . Investigator: For the fill in the blank one. Participant 4: Where if it’s in a paragraph form I feel like I can kind of even if it’s not exactly, I don’t know, it looks less intimidating to me but that might just be me.

5 Proof 1

Option 1

Investigator: Okay. So, on the next page, you’re going to see two options. Participant 5: Okay. Investigator: Option 1 is a proof where you would fill in the blanks. And Option 2 is just totally on your own. So, you if could just read both of them and let me know which one you want to do and why you’re choosing that one. Participant 5: Okay. Given the theorem of sum of the measures . . . is 180. Complete the proof for this theorem. Prove that measure of angle one . . . Add to 180. Through Point C, draw C, BCE parallel to AB.

259


[inaudible 00:06:32] Measure of angle one, record measure of Angle ACD. So, transversal. Angle three, and BCE. ACD plus . . . CD plus angle two is 180. Oh. Okay. Participant 5: And then in an isosceles triangle, Triangle ABC, AC is congruent to BC. The BCD is twice the measure of Angle A. I think I would choose Option 1. Investigator: Okay, why? Participant 5: Because it’s been a while since I’ve done proofs. So, I like that it tells you what the steps to proving what they want at the end. Whereas here in Option 2, I need to be able to prove that BCD is twice A, but I’m not sure what steps I’d have to take, whereas in Option 1 I can remember why these things are true. And I . . . Yeah.

6 Proof 1

Option 1

Investigator: Great, thank you. All right on the next page you have an option to choose between two groups. Participant 6: Okay. Investigator: So just take however long you need to look at each of them and let me know which one you’re choosing and why you’re choosing it. Participant 6: Okay. [inaudible 00:06:11] triangle. Prove that [inaudible 00:06:19]. Through Point C, draw parallel to AB. [inaudible 00:06:30] at one. [inaudible 00:06:35]. Okay. Isosceles triangle below AC [inaudible 00:06:39] BC. Side AC’s extended. Prove that the measure of BCD is twice the measurement of A. So, I think I wanna choose Option 1. Investigator: Why? Participant 6: That proof came a little bit easier to me. Investigator: What do you mean by that? Participant 6: I looked at it and I knew exactly what to write on each of these lines. Investigator: Okay. Participant 6: Option 2 I know how to do, but I had to think for an extra second to figure it out. Investigator: Okay, so do you . . . If what was provided in Option 1 were not provided, would you still know . . . do you think you still would have chosen it? Does the structure that’s given help with your decision? Participant 6: A little bit, yeah, because then I see this . . . It’s kind of giving you the answer in a way because this tells you exactly what to prove. Yeah, because I wouldn’t have thought to draw in the parallel line. Investigator: Okay. Participant 6: I know that’s a thing that you’re supposed to do- Investigator: Okay.

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Participant 6: . . . but without it being given to me, I might not have thought to do that. Investigator: Do you know- Participant 6: I think I would have been stuck.

7 Proof 1

Option 1

Investigator: Okay, awesome. Can you flip to the next page? So, on this page you have two options. So, you’re going to choose one of them, so if you can just look at the two prompts you can take as much time as you need to pick one. Let me know which one you’re choosing and why you’re choosing it. Participant 7: Alright, so given the theorem the sum of the measure of the interior angles of a triangle is 180, complete this proof. Prove that the measure of Angle 1+2+3 is 180, fill in the missing reasons below, okay. Then, an isosceles Triangle ABC below AC is congruent to BC, okay. Side AC has been extended through Point C to D. Prove that the measure of Angle BCD is twice the measure of A. Okay. I guess I’ll do the first one. Investigator: Okay, why are you drawn more to the first one? Participant 7: I guess ‘cause there’s prompts so it’ll lead me through the correct steps.

8 Proof 1

Option 1

Investigator: Okay, great. Okay, if you can turn the page. So, this says proof number one. You have two options here, and if you can just look at both of these and then pick one and just explain why you’re choosing that one, why you’re more drawn to the problem that you chose. Participant 8: Okay. I’m thinking. Well, I think there are pros and cons to both problems. Investigator: Okay. Participant 8: So, the first option I like that there is a chart with statements and reasons because it gives some guidance of how to prove it. With Option 2 I like, without the chart I feel like I have more freedom to do different approaches to prove the problem. Investigator: Okay. Participant 8: But . . . Investigator: What is a disadvantage to each of these. Participant 8: For Option 1, if I wanted to use another reason or something, I couldn’t fit it into this chart. I would have to only rely on these statements. I can’t change it up. While this one there’s absolutely no guidance. I would have to really make sure I’m doing everything correct. Investigator: Yes. Do you have experience with the type of proof in Option 1, a statement–reason chart? Participant 8: Yeah. Investigator: And do you have experience with any other types of proofs like if you weren’t gonna make a statement-reason chart?

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Participant 8: Yes, I have experience with both. Investigator: Okay. And what would it look like if you didn’t make this chart? Participant 8: I would probably maybe write in paragraph format but have different diagrams or take pieces of the diagram to show proof. Investigator: Okay. Participant 8: I think I could still, I think I would just go with Option 1 though since I am familiar from high school geometry with this set up.

9 Proof 1

Option 2

Investigator: Okay. If you could flip the page over, you’ll see proof number one, and there’s two options. So just take a moment to look through the options, and then explain which one you’re choosing, and what draws you to the problem that you’re choosing. Participant 9: Okay, I’ll do Option 2. Investigator: Okay. Why? Participant 9: I’ve taught Option 1 to high school students. But I don’t know, it’s more writing. Investigator: You think Option 1 is more writing? Participant 9: Yes. Investigator: Okay, when did you teach this to high school students? Participant 9: It was during a lesson for my edTPA. Investigator: So, in your preparation to become a teacher. Participant 9: Yes. Investigator: Which you are still completing . . . you’re about to student teach, yes? Participant 9: Yes. Investigator: Okay. So, you were in a field placement in a high school geometry course? Participant 9: Yes. Investigator: And you actually taught them this proof. Participant 9: Yes. Investigator: Did it have this chart with it? Participant 9: Yes. I just copy and pasted the chart, and then I printed it for them. Except when I did this, I didn’t say like “Hey, here’s the proof.” . . . I didn’t walk them through it. Like every five minutes, they would get a hint. Investigator: Okay. Participant 9: So, since this is all just using the relationship between the parallel lines, they learned this in unit one. So, it’s nothing new that they would have to learn to do this. Investigator: Okay. Participant 9: It’s just all that stuff from Unit 1. Investigator: Okay. Participant 9: They would just have to use their vocabulary sheet, which

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most of them had. So that was good. Investigator: Okay. Participant 9: Yeah. Investigator: So, you’re choosing two because you know that you know how to do one? Participant 9: Yes. Investigator: So, it’s kind of like you’re looking for a challenge? Participant 9: Yes, I guess. Investigator: Okay. Participant 9: Yeah, yeah. Investigator: And are you excited to do this? Participant 9: Yes.

10 Proof 1

Option 1

Investigator: So, when you flip this page, you have two options. I just want you to look at each option and let me know which one you would rather do and why. Participant 10: Should I read the options out loud? Investigator: You don’t have to, but if it helps you, you can. Participant 10: I don’t know. I can’t do this one. Investigator: Okay. What do you think about the perceived difficult of each of these? Which one do you think looks harder? Participant 10: I guess Option 2. If you look at Option 1, it’s kind of guided, so your initial reaction would be the Option 1 is easier. But in Option 2 it’s like . . . I guess you have the freedom of writing whatever you want and not restrained. Investigator: Okay. Have you written a proof in this type of chart before? Participant 10: Yeah. Investigator: Have you written a proof without a chart before? Participant 10: Yeah. Investigator: Okay. And do you know what kind of proof that is, if you don’t have statement-reason? Participant 10: I don’t know the name. Investigator: Okay. What does it look like? Participant 10: Just like a paragraph. Investigator: Yeah. It’s called a paragraph proof. Good. Which do you prefer, of the two? Participant 10: I like them equally. It’s okay, I don’t really care about [inaudible 00:08:06]. Investigator: Okay. So, no preference between statement-reason and paragraph. Alright. So, let’s pick either one and then try it out. Participant 10: I guess Option 1. Investigator: Okay. Do you like that the statements are given to you, or do you wish that they weren’t?

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Participant 10: I kind of wish it’s like the first statement given and then the last statement given, because that’s how I always learned it in high school. We were given the first and the last, and then we had to make it up ourselves. Investigator: Okay. Alright, so why don’t you start Option 1, and if you need to stray from what’s there, you can.

11 Proof 1

Option 2

Investigator: Okay. Okay, if you can flip the page over, and you’re gonna choose between Option 1 and Option 2. Participant 11: Okay. Investigator: And, just, you can take a minute to read them and let me know why you’re choosing whichever one you choose. Participant 11: Okay. Some of the measurements [inaudible 00:03:42]. Okay. Prove that, uh huh. Okay, I know how to do Option 1. Investigator: Okay. Participant 11: I’m gonna try Option 2. Investigator: Wow, okay. Participant 11: Yeah. Investigator: Yeah, can you just tell me a little more about that decision? Why are you not sticking with what you know how to do? Participant 11: I don’t know, I want a challenge. Not knowing, not knowing off the top of my head how to do number two is, I feel like I should know how to do it, so I want to try it. Yeah.

12 Proof 1

Option 1

Investigator: Okay, interesting. All right. If you can turn this over, that page. For the first proof, I want you to choose between the top, Option 1, and bottom, Option 2. If you can just take a minute to look at them, and then tell me which one you’re going to choose and why you want to do that one. Participant 12: Okay. I choose to do Option 1. Investigator: Okay. Participant 12: Because I have no idea . . . I mean, proof, I’ll probably think about it while I work on it. Investigator: Okay. Participant 12: I’m more interested because Option 1. I never saw this particular proof, and it’s a more general way of proving that the sum of the measures of interior angle of a triangle is 180 degrees. Investigator: Okay. Participant 12: So, I’m more interested in that. Investigator: Okay.

13 Proof 1

Option 1

Investigator: Okay. All right. When you turn the page, I’m going to present you with two options. Option 1, Option 2. I’d like you to just look at both of them, and you’re going to pick one and just tell me which one you choose and why you chose it. Participant 13: I choose Option A.

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Investigator: Okay. Why do you want to do that one? Participant 13: Because it’s something familiar from when I went to high school. Investigator: Okay. What’s familiar? Participant 13: The format of statement reason. Investigator: Okay. All right, so go ahead.

14 Proof 1

Option 1

Investigator: So, on the back I just want to know which one of these two options you would be most drawn to. Participant 14: Okay. Investigator: If you were given a choice. Like hey you can complete either one of these, which one do you prefer and why? Participant 14: I’d probably choose the first one. Investigator: Okay. Participant 14: Even though I don’t like constructions. Investigator: Okay. Participant 14: And I feel like this might be because it’s saying draw something parallel to AB. Investigator: Okay. Participant 14: It’s just more information, so I feel like— Investigator: More information given? Participant 14: More information given. Investigator: Okay. Participant 14: So even though I don’t feel too comfortable with either one, I feel like I’m more able to break it down. Investigator: Okay. Because you have a starting point? Participant 14: Exactly. Investigator: Okay, when you did proofs in high school, did they more take the form of a statement reason chart like this or just blank page, according to what you remember? Participant 14: They did it, I think the way that teacher did, was he did both. Investigator: Okay. Participant 14: I think we started, for the most part, with something like this and then he would even change it where he would give some statements, some reasons. Investigator: Okay. Participant 14: And you would have to go back and forth. But I think at the end, in order to prepare us for the Regents, we did Option 2. Investigator: Okay. Participant 14: Something like Option 2, where it was kind of like, just prove it. Investigator: And if you want you can use a chart, and if you don’t want

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to you don’t have to. Participant 14: Exactly, and I would always go back to the chart.

15 Proof 1

Option 1

Investigator: Okay, thank you. Okay when you flip the page over, you see two options, so you’re gonna pick one, whichever one you want to do more, and you can just explain to me which one you choose and why. Participant 15: Given the theorem, the sum of the measure of interior angle of a triangle is 180, complete the proof for this theorem. Given ABC, measure of angle one. In ABC . . . prove measure of angle one plus measure of Angle C equals to measure . . . plus measure of angle three plus 180. Draw this parallel to AB. That’s easy. Investigator: This proof is easy? Participant 15: Yeah. Investigator: Why? Participant 15: Right here you have all the statements; there’s nothing you need to do, just fill in the reasons. Investigator: Okay, so you wanna do this one, Option 1? Participant 15: Yeah, I haven’t read Option 2. Investigator: Okay. Participant 15: Let me see. In the isosceles Triangle ABC shown below, AC is congruent to BC [inaudible 00:05:51], AC is extended through Point C to D. Prove the measure of Angle BCD is the twice of . . . twice the measure of Angle A. It’s also easy. Investigator: Why do you think that one’s easy? Participant 15: I mean, I know how to prove that, that’s why. Investigator: Okay, how do you know? Participant 15: For this one I know the angles . . . right here side AC extended through Point C to D, that means Angle BCD is exterior angle of the Triangle ABC. And based on the exterior angle theorem, I think, the measure of Angle BCD is congruent to the sum of two remote interior angles. So, measure of Angle BCD is equal to Angle A plus Angle B, and since Triangle ABC is an isosceles triangle, the two base angles are congruent. Angle A is congruent to Angle B, so by substitution, twice Angle A would equal to the measure of Angle BCD. Investigator: Okay, so which one do you think is easier? Participant 15: First one. Investigator: Still easier, even though you just said the whole proof for Option 2? Participant 15: Yeah, but for the Option 1, you have all the statements. Investigator: So, you just have to write the reasons? Participant 15: Mm-hmm. Investigator: Okay, so can you do Option 1, but I’m glad that I know that you can do both.

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16 Proof 1

Option 2

Investigator: And what would you choose? Participant 16: I would lean more towards Option 2, because I can just do a paragraph proof and I felt I was more comfortable with the actual content, what they wanted me to prove.

17 Proof 1

Option 1

Investigator: Okay. So, when you turn your page for the first proof, you have an option, so just take a minute to look at those and then see which one you prefer and then just explain why you prefer the problem that you chose. Participant 17: So, we’re not doing it. Investigator: No, then you’re gonna do it. But first just, why did you choose it? Participant 17: Just by skimming them, I’m gonna say I prefer Option 1, only because it’s in a format that I’m used to. The statement-reason format? Not saying if Option 1 is more difficult, based on what I read, versus Option 2, but I am more comfortable with that format of proof. Investigator: If the statement reason chart was blank, would you still be tempted to choose Option 1? Participant 17: Let’s see. Participant 17: Looking at the actual proof themselves, neither of them really bother me, but I think Option 1 would be easier to do, so I choose that.

18 Proof 1

Both Investigator: So, here you can choose between Option 1 and Option 2 and I just want you to let me know out loud why you’re choosing what you’re choosing. Participant 18: Okay. Just on first impression, I’ve done this before. Investigator: Option 1? Participant 18: Yeah, I didn’t even read the question yet, but I’m going to assume, yeah, this easy. Investigator: Okay. [LATER] Participant 18: I’m going to look at the other one. Investigator: Sure, you can look at it. Participant 18: AC is over to BC side extended. Prove the measure BCD is twice the measure of . . . congruent. I kind of want to do this one. I’ll do it later, just for fun.

19 Proof 1

Option 1

I’m familiar with this question, so it’s more comfortable.

20 Proof 1

Option 2

Investigator: Can you turn to the next page? There’s two options here. So, I just want you to take a look at them, and let me know which one you want to do more, and why? Okay, which one do you choose? Participant 20: Option 2. Investigator: Why? Participant 20: Because. Option 1 forces me to think on ways that I may

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not want to think. Investigator: Okay, why does it force you? Participant 20: Because it gives me the statements already. Investigator: Okay, if they were both blank, which one would you choose? Participant 20: I would probably choose Option 1 if they were both blank. Investigator: Okay, so you are in the mindset of wanting to approach a geometry proof the way you want to approach it? Participant 20: Yes.

21 Proof 1

Option 1

Investigator: Okay, if you’re going to turn the page, you’re going to choose one of the two options. Option 1 or 2, let me know which one you’re choosing. And why. Participant 21: All right, I’m going to go with— Investigator: You already made a decision? Participant 21: I wanted to go with one cause I’ve given this to my students. Investigator: Okay, that’s a fine reason.

22 Proof 1

Option 2

Investigator: Thank you. Okay. On this page, the next page, you have two choices. Option 1 or Option 2. So, take a minute to look at them, and then let me know which one do you want to do more. Participant 22: And this one is just like fill in the blanks? Investigator: Yeah. The first one you would have to fill in the blanks. Participant 22: I guess shall do Option 2. Investigator: Okay, why are you choosing Option 2? Participant 22: I don’t know. I feel like having the statements written for me tries to make me like . . . like puts me in a mindset that’s not my own sequence of thinking. So, it makes me kind of uncomfortable. Investigator: Okay. So, if it wasn’t blank, you might be more inclined. . . . I mean if it wasn’t filled in, you might be more inclined to choose Option 1? Participant 22: Right. Because like reading it, I’m like, “Okay, now I have to come up with a reason for something someone else wrote.” Investigator: Okay.

23 Proof 1

Option 1

Investigator: Okay, great. If you turn to the second page this is one of the proofs, there’s two options, if you can take a look at each of them and say which one you’re more inclined to do and why you’re more inclined to do it. Participant 23: I’m more inclined to do this one. Investigator: Number one? Participant 23: Number one, because I like to do the thing that is more organized, easier. Investigator: Okay and why do you think number one looks easier?

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Participant 23: Because you have your statements, you have your reasons, and all I have to do is give the reasons for the statements. Investigator: Okay. Participant 23: Less room I guess for error.

24 Proof 1

Option 1

Investigator: Yeah. So, I want you to read each of them and let me know which one you want to do more and why. Participant 24: [inaudible 00:06:45] isosceles. ABC is shown below. AC is congruent to BC. Show on the sides AC has been extended through Point C to Point D. Prove that the measure of BCT is twice the measure of Angle A. . . . I would say the first one because it sounds really familiar. Investigator: Okay, familiar to something you’ve done? Participant 24: I learned in high school, yeah.

25 Proof 1

Both I think that the idea of this relates to interior and exterior angles. After reading Option 1, I think I may use Statement 3 to see what happens [in Option 2]. I don’t know how to address the problem [Option 2], but I think definitely will be related with exterior and interior angles, and I can use some of the statements from Option 1 to prove them.

26 Proof 1

Option 2

Investigator: Great. All right. On the next page, there’s two options that I just want you to look at and choose one of them, and let me know why you want to choose the one that you’re choosing. Participant 26: Okay. I think I’d like to do Option 2. Investigator: How come? Participant 26: First of all . . . I don’t know why. I think I feel a little bit more comfortable by just seeing the diagram. Investigator: Mm-hmm [affirmative]. Rather than this? Participant 26: Yeah. Rather than that one. Investigator: Okay. Why? Participant 26: All I know is because I think it reminds me of supplementary angles, so like— Investigator: Where are the supplementary angles? Participant 26: Would be like AC . . . I mean BCD . . . It’s angles BCA and BCD. Investigator: BCA and BCD. Okay. Participant 26: Mm-hmm [affirmative].

27 Proof 1

Option 2

Investigator: Great, thank you. All right, so, on this page you have two choices, or an Option 1 and an Option 2— Participant 27: Statement, right? Investigator: . . . and I want to know which one you . . . I mean, you can take as long as you need to look at them. Which one would you rather do, and why? And then I’m actually going to ask you to do the one that you pick. Participant 27: [inaudible 00:05:07] measures [inaudible 00:05:08] do

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you want me to read it out loud? Investigator: You don’t have to, you can just read them to yourself. If it will help you to read them out loud, you can. Participant 27: [inaudible 00:05:20] Participant 27: Okay [inaudible 00:06:31] B, C, D. Actually, it kind of does. Okay.

28 Proof 1

Both Investigator: Okay. Well, you have a choice here. Participant 28: Cool. Oh! Investigator: So . . . yeah. I want to know which you would be more inclined to do and why and you can take your time to look at them or you can— Participant 28: So, I’m just . . . I do a weird thing. So I, right off the bat, I don’t want to do one just because of . . . I wouldn’t want to do it just because it looks scary; however, I realize that these statements would be helping me giving reasons and it’s an explaining thing. So sometimes when I have to explain something, like reasons, it feels like I’m not giving enough information or I’m not giving the right information. I’m not confident in that. Investigator: So, if you have a fill in the reasons, you wouldn’t be sure that you’re giving the right ones? Participant 28: I would be upset. Like I have this thing. I have to give the exact reason ‘cause I’m always afraid that teachers are looking for something very specific and if I word it differently than they’re not going to count that. Investigator: Let me ask you a question about that. That’s a very common sentiment. When you say teachers are looking for something specific, do you think all teachers are looking for the same specific thing or do you think it differs teacher to teacher? Participant 28: It would differ teacher to teacher but also if I was taking a test, I would hope that previously they would be giving reasons but the same reasons over and over again and I would use their words because I have- Investigator: Whatever teacher you had? Participant 28: Whatever teacher I had, I would use their words. For example, when I’m writing an essay, I write it how they want it. I might not like it, but I write it how I know they’ll like it. So same with this. I’ll write it the way they want it, not how I want it. So this . . . and it’s funny. If it was actually a test and I was given Option 1 and Option 2, I would do both of them. Investigator: Okay. And then kind of see how you felt about- Participant 28: I would do . . . yes. I would do both of them, and whichever one I felt more . . . I would try to answer both of them fully, and whichever one I didn’t feel comfortable, I would cross the whole

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thing out. I would be like and I would write, “Don’t grade this one. Grade this one.” Investigator: Okay. Participant 28: Because I want to see . . . I want to see number one, if I can do it, but also I don’t like given options because it like, “Is this one better than this one?” Stuff like that. So, I actually, yep. I think I want to try both. Investigator: Okay. Participant 28: I’m going to try the first one though. Investigator: Okay. [LATER] Participant 28: I love . . . geometry was fun. I do like lines and stuff. So then if I felt that if I wasn’t confident in that, I would do this. Investigator: Option 2? Participant 28: I would do both

29 Proof 1

Option 1

Investigator: Great. Okay. On the next page, there’s two options for a proof. One of them, you can see, has a statement-reason chart, and the other one is blank. But you’re free to use any method. If you do Option 1, though, you do have to do a statement-reason chart. So just take a minute to look at them and let me know which one you think seems easier to you, or you would be more drawn to do it, and what your reason is for choosing that one. Participant 29: Okay. I’m more drawn to do this one. Investigator: Option 1? Participant 29: Mm-hmm [affirmative]. Investigator: Why? Participant 29: I mean, I think just because it’s already mapped out. Also, I read the first few, and I do remember the properties with parallel lines and transversals, so I feel like I would remember how to write the reasons for that.

Note. P# = participant number; Q# = question number.

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Table D7


P# BAA instance 1 Participant 1: Honestly, the statement is so annoying.

1 Participant 1: Because I know in a triangle, the sum of the interior angles of a triangle is equal to 180. But I know ACD is the same thing as this. As Angle 1 and measure of Angle 2. See that’s the issue which these proofs because I’m forced to say something that I don’t know what to say. Investigator: So, you don’t . . . I mean you wanted this one because it has been filled in for you. Participant 1: Yeah, but it’s not fair.

1 Investigator: What do you think of it as a whole? Participant 1: It wasn’t that bad. It wasn’t that bad. But I really had to think about it. ‘Cause it forced me to think in a certain way.

2 Participant 2: Right now, I want to cut everything out and maneuver it. Investigator: That’s something that you’re thinking of as a strategy to help you? Participant 2: Yes. Which I think would eventually help the students, ‘cause if I need to see it visually, I think the students would need to prove it by sliding it over to see that they’re all the same, and if I maneuver them, they’re going to form a straight angle.

2 Investigator: Okay. How do you feel about this? Do you think that if you were to hand this in do you think that this would get a full mark? Participant 2: I think it would be close, but I’m not sure it’s written as perfectly as it should be mathematically. Investigator: You would want help with the mathematical language? Participant 2: I would want help with the writing of the reasoning as true mathematics.

4 Investigator: Beautiful. How do you feel about that? Participant 4: It was good, I just get nervous that I’m taking too long but I don’t know. Investigator: What’s too long? Participant 4: I don’t know. Investigator: You’re not taking too long at all. I don’t think there is a too long with this stuff. Sometimes stuff doesn’t come to me and then I come back hours later and I’m like, “Okay, I get it.” So, this I just want to make sure I’m interpreting what you’re saying right. This proof you chose because you kind of had the freedom to try out whatever you wanted versus here you felt pigeonholed into explaining things you might not be able to explain. Participant 4: Yeah. Like I always think my biggest problem with proofs and just math in general is that sometimes the terminology I use isn’t always exactly correct, so then these make me more intimidated just because I feel like even the students that I tutor they’ll be like, “Well, my teacher is very particular about wording.” And I like stuff like this because it’s a little . . . I feel like I get more of an opportunity to explain what I’m doing a little bit better.

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P# BAA instance 8 Participant 8: I’m gonna skip two for now because I don’t remember the reason exactly,

but I’ll come back to it. 9 Investigator: Okay. So, did you end up needing this parallel line that you drew?

Participant 9: No. Investigator: Okay. So, what do you think prompted you to do this? Participant 9: I was looking at Option 1.

10 Investigator: And how confident do you feel about your answers to the rest of these? Participant 10: I guess pretty confident. They’re not detailed, because I know in high school they would always be like, define what alternate interior angles is, and define the whole substitution thing. So, it’s like short form.

14 Investigator: Okay. You also said when you saw Statement 2, it reminded you of constructions, you said you don’t like constructions. Participant 14: I don’t like constructions. Investigator: Can you speak a little bit more about that? Participant 14: Anything that’s very spatial, that’s another reason why geometry, I’ve always been like, “Ew,” because I feel like my spatial awareness is not really up there.

14 Participant 14: And it’s something I’ve never enjoyed. Which is something that I like about algebra, that there’s less having to visualize and it’s more like, I like the math where it’s not so much visualization, because that’s something I struggle with. I think I prefer it more when it’s just, this is what you’re given, go from there.

14 Investigator: And now how are you going to word that? Participant 14: This is something also I never liked about geometry, it’s like so much writing.

15 Investigator: Circle, which is how much? Participant 15: 360. Investigator: Okay, do you know why a circle is 360 degrees? Participant 15: No. Investigator: Okay, have you ever thought about that before? Participant 15: No.

15 [Regarding auxiliary lines] Investigator: No, Okay. Have you ever done something like this with your students? Participant 15: No. Investigator: Okay. Participant 15: Yeah, they don’t need to know that.

20 Investigator: Paragraph. Where do you have experience with paragraph form proofs? Participant 20: In my undergrad college geometry class. Investigator: You took a geometry class, was it required? Participant 20: Yes. Investigator: Okay, was it useful? Participant 20: Yes. Investigator: What about it was useful? Participant 20: I think, honestly, writing the proofs in paragraph form, versus statement

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P# BAA instance reason . . . Because in high school, I did not like proofs, and it wasn’t because I didn’t know or like geometry, the format, it was too structured. Investigator: Too structured, okay. Interesting. I know you’re a first year geometry teacher. Participant 20: Yes. Investigator: Do you teach your students paragraph proofs, or only statement reason? Participant 20: I actually teach one of the classes paragraph, and the rest of them, statement-reason. Investigator: Okay, what’s your reasoning for that? Is it ability levels? Participant 20: Yes, I also wanted to see if it makes a difference, or if the students would like or dislike proofs any more than the other one. Investigator: Interesting, okay. Do you find that there is a difference so far? Participant 20: No. Investigator: Do the students like proofs, or not like proofs? Participant 20: They still dislike proofs.

28 Participant 28: That was cool. So, I would actually . . . I would take this one over Option 1. Investigator: Because you feel more confident about your reasoning. Okay. Awesome. Participant 28: That was— Investigator: That was— Participant 28: That was a fun one. I liked that one.

29 Investigator: Why are you hesitant about using that as a reason? Participant 29: I don’t know. I’m just not used to using that property in geometry. Investigator: In geometry. That’s more of a . . . Participant 29: Algebra. Investigator: . . . algebra thing. Okay.


Table D8


Assessment Item 2

P# ICSN+ ICSN- 20

Since A and B are congruent, that’s really saying that twice the measure of Angle A is congruent to Angle BCD.

Note. P# = participant number; ICSN = issue of correspondence between substance and notation (positive or negative).

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Table D9

Instances of Expressing Understanding or Self-Doubt Code for Content Assessment Item 2

P# EU ESD EUSD 1 Participant 1: So, there’s

definitely something to do with alternate interior angles off the bat.

1

Participant 1: It’s saying, we have to draw that they’re parallel. Investigator: Okay. Is there a reason that you know you can do that? Participant 1: Yes, because they never intersect. Investigator: Okay. So why do you seem hesitant in that answer? Participant 1: Because it’s not a good reason. Investigator: Why isn’t it a good reason? Participant 1: Because I need to prove that they don’t intersect. But I can’t . . . I would say that they have the same slope, but I need to also prove that. But can’t you . . . honestly, the statement is so annoying.

1 Participant 1: So, measure of Angle 1 is equal to the measure of ACD . . . [inaudible 00:13:49] so this. Why? Because alternate interior angles . . . two parallel lines caught by a transversal create congruent alternate interior angles. The measure of Angle 3 is equal to BCE. Yeah, okay. So that’s confident.

1

Participant 1: Yes. Oh. I got it. I got it.

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P# EU ESD EUSD Investigator: What do you got? Participant 1: ‘Cause a straight line has 180 degrees. Investigator: Oh. Participant 1: But how do I say it?

2

Investigator: Okay. Is there a mark you can put on that picture that shows that they’re parallel? Participant 2: Yes. Investigator: What’s that mark? Participant 2: No, that’s congruent. Possibly this. I’m not sure.

2

Investigator: Yep. Those are good, arrows going the same direction. That tells you they’re parallel. Yeah. For two. . . do you know why you are able to draw a line parallel to AB? Participant 2: Do I know why? Because I could just extend the line from Point C. Investigator: Okay. Participant 2: I could write that?

5 Investigator: And how do you know that they’re alternate interior? Participant 5: Because those angles, if I had extended AB, would fall inside of the parallel lines as opposed to outside.

5

Investigator: Okay. And how do you know that they’re supplementary? Participant 5: Because DCE is a line, and when you have . . . So, a line is 180 degrees, I think.

5

Participant 5: Oh okay. So, okay. And

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P# EU ESD EUSD then Angle 1 plus Angle 2 plus Angle 3 is 180 degrees because well, if they’re congruent and we already said that all of them add to 180 degrees, then when they’re inside the triangle, they’ll also add up to 180 degrees. Investigator: Okay. Participant 5: I don’t know how to write that.

6

Participant 6: If I had to write something, if I was forced to, that’s what I would write ‘cause that’s what I think makes the most sense. I don’t know if there’s a better answer though.

6

Investigator: Okay, so would you say the fact that one plus two equals three is an example of the addition postulate? Participant 6: Kind of, but I don’t know if the addition postulate is only supposed to be for geometric entities.

7 Participant 7: Okay. So, through Point C draw DCE parallel to AB. Through Point C, draw DCE parallel to. . . okay, well that’s, oh it’s an auxiliary line, right? Investigator: What does that mean? Participant 7: It’s like a line you can put into the diagram to help you find angle relationships. Alright, so then they’re saying the next step is the measure of angle one is congruent to the measure of

277

P# EU ESD EUSD Angle ACD. ACD, the measure of angle three is congruent BCE, okay. So, A is congruent here, and this. These are alternate interior angles.

7

Participant 7: They are on alternate. . . well first they’re in between two parallel lines, that was given in two. So now you can form angle relationships and AC acts as the transverse. So, these are I guess definition of alternate interior angles. Alright, so then four is the measure of AC, ACD plus the measure of Angle 2 plus the measure of Angle BCE. BCE, is 180. I mean, they’re supplementary because they’re on a line, right, so this whole line is 180. I’m not sure what I would write here though.

9 Participant 9: And in the angle side relationship that means . . . Since these two sides are congruent, the angle opposite BC and the angle opposite AC are also both congruent. And then if you use the exterior angle theorem—

12

Participant 12: I’m just trying to remember, is there like a name for that

278

P# EU ESD EUSD postulate? You can substitute those two values, instead of measurement Angle ACD, I can write measurement of Angle 1.

13 Participant 13: The theorem says the sum of the measures of the interior angles of a triangle is 180 degrees. Complete the proof for this theorem. Given Triangle ABC, prove that measure of angle one plus measure of angle two plus measure of angle three is equal to 180 degrees. Fill in the missing reasons below. So, Statement 1 says Triangle ABC, and the reason is given. Statement 2 says through Point C, draw line DCE parallel to AB. The reason would be that there is only one line that goes through Point C that would make a line that is parallel to AB. Investigator: How do you know that, or where did you learn that? Participant 13: I learned it in one of my math classes in college. Investigator: In college? Okay. A geometry class, or education class, different math class? Education? Participant 13: Oh, education class.

16 I wrote “Given that AC is congruent to BC, and the Triangle ABC is isosceles, Angles A and E and Angle B

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P# EU ESD EUSD are congruent as well. The sum of the angles of Triangle ABC can be found by calculating Angle A plus Angle B plus Angle C equals 180 degrees. Or using substitution, Angle A plus Angle A plus Angle C equals 180 degrees, since Angle A is equal to Angle B. The exterior angle of a triangle is equal to the sum of the opposite interior angles. Therefore, Angle BCD is equal to Angles A plus Angle B. Or, through substitution, Angle BCD, is equal to Angle A plus Angle A.”

18 Participant 18: Reason; ABC through Point C, draw DCE parallel to here. What is this, parallel postulate? Investigator: What’s that? Participant 18: The parallel postulate states that for one line at any point not on the line, any line parallel, it doesn’t . . . there’s only one line parallel and passing that point not on the line. Can I just write that postulate?

19

I don’t think this is the proper way to say it.

23

Participant 23: Alright. Here . . . I’m gonna give a reason that I think would make sense for four. Investigator: Okay. Participant 23: Which is that the sum of . . . I don’t know, collinear, I wouldn’t say collinear, that’s not what I would say . . . The sum of angles that form a line is 180.

280

P# EU ESD EUSD Investigator: Alright. And when you say that’s a reason that you think makes sense, do you think other people would think that doesn’t make sense? Participant 23: It’s not that I don’t think it doesn’t make sense, I just think . . . I don’t know if that’s the official way we would describe how we form a line or how angles put together have a line . . .

24

Participant 24: And for ACD the measure of Angle 2 plus the measure of VCE do equal 180 because . . . I wouldn’t know the reason why.

24

[Regarding parallel postulate] I don’t know the proper, I guess rule or theorem it is, that it’s called.

25

[Regarding Option 2] I don’t know how to address the problem, but I think it has to do with exterior and interior angles.

Note. P# = participant number; EU = expressing understanding; ESD = expressing self-doubt; EUSD = expressing understanding or self-doubt.

281

Table D10


P# ML+ ML- 1

So, then I have 180 minus 2x plus x plus x is equal to 180 degrees. Oh, that doesn’t work. Can I use alternate, the remote thing?

2

Participant 2: Angle 1 plus Angle 2, plus Angle BCE is 180. And Angle BCE is congruent to Angle 3. Therefore, the measure of Angle 1, plus the measure of Angle 2, plus the measure of Angle 3 equals 180. I might have done something backward here. Investigator: Okay. Actually, not backward at all. You kind of restated stuff that you had already proven. Is there a word that you know from your studies in math or your teaching, anything that explains the process that you just did here? Participant 2: Congruence.

5

Investigator: Okay. So, is it fair to say that the measure of Angle ACD is the same thing as the measure of angle one? Participant 5: Yes. Investigator: Okay. And following that reasoning, what proper . . . What mathematical thing are we doing to go from Statement 4 to Statement 5? Participant 5: Congruence. I don’t know. Investigator: Okay. Participant 5: Equivalence. Investigator: Okay. And how are we using equivalence? Participant 5: I mean, substitution?

6 Investigator: Okay. Do you remember the term for the line that you draw in to help you with the proof? Participant 6: Auxiliary line?

6

Participant 6: So, the measurement of angle one is equal to the measurement of Angle ACD and the measurement of angle three is equal to BCE. That’s alternate interior angles. Investigator: How do you know that? Participant 6: I remember in middle school someone taught me to look for the Z.

8 Participant 8: For Statement 3 if I were to

282

P# ML+ ML- extend AB, since I know from Statement 2 that DC is parallel to AB, then the measure of angle one is equal to the measure of Angle ACD from the alternate interior angle theorem.

8 Participant 8: So, for the reason I would write parallel lines form congruent alternate interior angles.

9 Participant 9: Then by the exterior angle theorem, the sum of the non-adjacent interior angles is equal to, congruent to, no, equal to the exterior angle.

10 Participant 10: For three I put that they’re alternate interior angle, since we know DCE is parallel to AB. Then for four I put that we know that the measure of a straight line is 180 degrees. And then five would be substitution, since we know that the measure of Angle 1 is equal to the measure of ACD, et cetera.

12

Investigator: Okay. How do you know that those are alternate interior? Participant 12: By definition, if you have a pair of parallel lines and you have a transversal cut through them, interior means it’ll be inside the two parallel lines. It’d have to be between them. Investigator: Okay. Participant 12: And alternate means . . . I guess it’s kind of

283

P# ML+ ML- synonymous to opposite, so if one’s on the left inside of the top line, it’d be on the right inside of the bottom line.

14

Participant 14: Angle three. ACD plus angle two. Okay. I mean four is obvious because they add up to a straight line. Investigator: Okay. So, you can put that in for four. Participant 14: The language messes me up.

14

Participant 14: From there you can see different angle relationships, like alterior and interior angles, vertical angles.

15

Participant 15: Mm-hmm. So, measure of angle one is congruent to measure of Angle ACD. Measure of angle three congruent to measure of Angle BCE. The reason would be if two lines are parallel, the alternate interior angles are congruent.

18

Participant 18: What do they call the supplementary? What’s the word? Investigator: If two angles are on the line? Participant 18: Yeah, there was a word for this. Investigator: Linear pair? Participant 18: Linear pair, yeah. I used to say linear triple.

20

Participant 20: Oh. Then I know that Angle BCD is equal to the sum of A and B. Investigator: How do you know that? Participant 20: Because the exterior angle of a triangle is congruent to the opposite interior angles.

20

Investigator: Okay, so you wrote, “Since the opposite angles of an isosceles triangle are congruent.” You meant . . . Say again what you meant? Participant 20: That the angles opposite the congruent sides are congruent.

21

Participant 21: Okay. So, angle one is equal to ACD and angle three is congruent to BCE, so I’ll pick up a highlighter to show my parallel lines. Investigator: Okay. Participant 21: And I’m going to show my Z’s, so my Z’s are representing my alternate interior angles. And by definition, my alternate interior angles are congruent.

26

Then, we’re still trying to figure out these two. All right, so knowing from the, I believe the arc . . . What was it? The arc theorem?

28

Participant 28: So now what I would do is . . . okay, well . . . what would I do? Just making sure it’s B and not the whole thing. So, I would know that . . . I’m trying to visually look at

284

P# ML+ ML- this now. Oh! Well, I just drew a second isosceles triangle so now I’m making this, “Okay, well this line is now congruent.” And then visually, I know, that this angle, angle E is the same thing as Angle A. Now, writing like . . . I just . . . I know that they are just looking at the them because I drew the parallel line. I’m sure that there’s some way of explaining it. I’m not good at that. Participant 28: So now . . . okay, okay. Oh! There you go! I found it. Okay, so now I see BCE when I drew that mini triangle that this angle, so I’m going to call it BCE is equal to Angle B but also by alternating angles, that thing, and well, okay. This is going to sound really weird. Okay. So now, I am going to do something weird. I would make a line segment. I would make another line segment of equal value of- Investigator: Of BE? Participant 28: Of BE, and I know this isn’t the right way to do it and I would just extend D out further so that it equals AC and I would draw another parallel line and I’m . . . and then this is so weird that I’m doing this ‘cause I wouldn’t do this on a math test. I know that this isn’t something that . . . that this isn’t the way to do it.


285

Table D11


P# PMI 1 Oh no, I’m not allowed to prove that. I cannot. Because I’m trying to prove that so I can’t

make it my reason. 1 Investigator: Okay. So, you’re referencing a fact that in a triangle the measure of an

exterior angle is equal to the sum of the two remote interior angles. Participant 1: Yes. Investigator: If you were asked to prove that, could you do it? Participant 1: I think so, but I keep getting stuck because I know this, this, and this, the 3 angles in a triangle equal to 180 degrees. So, x plus x plus 2x minus 180 should equal 180 degrees. But when I do this, I get zero is equal to zero. Investigator: So, you’re getting an identity statement. Participant 1: Yeah, so there can be infinitely. . . Yeah. . . . So, now I’m stuck. Investigator: Okay. But do you agree that the measure of Angle BCD is twice the measure of Angle A. Participant 1: Yeah. Investigator: Because of that remote interior angles and exterior angle fact. Participant 1: It’s twice the measure of Angle A plus B. I don’t know. I’m confused.

2 Participant 2: Okay, so I see what’s happening here. If I extend this right here- Investigator: What are you extending? Participant 2: Line AB to make the other parallel line, so now I could see that they’re alternate, and I’m still a little . . . I know they’re alternate but I don’t think they’re interior or exterior.

2 Participant 2: Measure of Angle 1 plus measure of Angle 2, plus measure of Angle 3 is 180 degrees. It’s because all of the angles add up to 180, but I have to prove it from using the reasoning before. I can’t write that as proving it because that’s. . . Investigator: That’s what you’re trying to prove. Participant 2: Correct.

2 Participant 2: Okay, one, two, three. D, C, E. Okay, measure of Angle 1 equals measure of Angle 2, equals measure of Angle 3 is 180. Now we have measure of Angle 1 equals ACD. Measure of Angle 3 equals BCE. Measure of ACD plus the measure of Angle 2 plus the measure of Angle 3 equals 180. I need help.

3 Participant 3: Through Point C, draw a DC parallel to AB. Reason. I don’t know what the reason would be, you just do that. Investigator: Okay. That’s a pretty common response, actually. Do you know why you can do that? Participant 3: Well, they only give you that you have a triangle, A, B, C. Investigator: Okay. Participant 3: They don’t tell you anything about the triangle. Investigator: Okay. Participant 3: Well, you wanna draw it so that you can start talking about the rest.

286

P# PMI Investigator: Okay, so it’s used as like a tool? Participant 3: Yes. Investigator: Okay. Participant 3: I don’t know what the reason would be, I guess I would just write . . . I don’t know what I would write, actually.

3 Participant 3: So, I’d write this there. Wait, no I wouldn’t. [crosstalk 00:08:44]. Investigator: Why not? Participant 3: Because I’m proving that.

5 Participant 5: Okay. So, the statement says, “Triangle ABC . . .” And triangle, so that’s a given. Then to, “Through Point C, draw DCE parallel to AB.” So, the reason . . . I forgot. The reason is a justification, right? Investigator: Yeah, of the statement. Participant 5: Right. So, isn’t it just the definition of parallel lines? Wouldn’t I just write that?

6 Participant 6: The only thing that I’m unsure of is what to write for the reason for Option 2. Investigator: For Statement 2. Participant 6: For Statement 2. Thank you. Investigator: Okay. Participant 6: Through Point C draw DCE parallel to AB. I wanna say this is just a definition of a parallel line.

6 Investigator: And can I just ask you a question? You said addition postulate, what’s a postulate? Participant 6: I actually don’t know. It’s a word that I use a lot, but I’m not 100% sure what it means. Investigator: Okay. Participant 6: I think it’s just something that we all agree is true. I don’t know what the difference between a postulate and a theorem is.

7 Participant 7: But I feel like something is missing. Investigator: Why? Why do you feel like something is missing? Participant 7: I guess I thought the prompts would help me, but this may not be my preference in going through the proof. Investigator: Okay, so if you didn’t have the prompts, you might’ve gone through it differently? Participant 7: Yes. Investigator: I know you have a year of experience teaching high school geometry. Is this a proof that you’ve done with your students? Participant 7: I don’t remember off the top of my head, but I’ve definitely done similar proofs where we’re proving that a triangle, the sum of the measure of the interior of the triangle is 180, it just may not be this example.

10 Participant 10: I don’t know the reason for point 2. I don’t know how to explain it. [LATER IN INTERVIEW] Investigator: Okay. Awesome. So, for Statement 2. If you were writing a paragraph proof,

287

P# PMI would you need a reason for that? Participant 10: Not really, no. Investigator: So, you would just write into the proof, like draw a line. Parallel. Participant 10: Yeah. Well, I would . . . yeah, kind of. Investigator: Yeah. This is a little weird to have a reason for, because it’s like you’re justifying why you’re able to do that. Do you agree that you can draw a line through Point C that’s parallel to AB? Participant 10: Yeah. Investigator: Are there more lines that you could draw through C that are parallel to AB? Participant 10: No. Investigator: Okay. So, there’s only one. Participant 10: Yeah. Investigator: Okay. Is that ringing a bell, as something- Participant 10: One of Euclid’s five postulates things. Investigator: Ah. How do you know about those? Participant 10: I learned it in one of my classes. Investigator: Okay. In a college class? Participant 10: Yeah. College class. Investigator: Was it a math class or an education class? Participant 10: Math. Investigator: Okay. So, we mentioned Euclid’s postulates, and you think it’s one of those? Participant 10: Yes.

12 Participant 12: Through Point C, draw DCE parallel to AB. Reasons, I think we have to give a reason why that’s possible. Investigator: Okay. Participant 12: I think it is possible because, since it’s a triangle, I just don’t know how to state it as a reason. I’d have to think about that, but I know one of it is that because side AC and side CB are not parallel. Investigator: Okay. Participant 12: So, they intersect out of Point C. Since those two are not parallel, you can perhaps . . . Yeah. So, in any triangle, acute, obtuse, or right, you can always make a parallel line to the opposite side of that vertex.

13 Investigator: Okay. Can you explain to me the markings that you just made on the diagram? Participant 13: Oh. I have to fix that. Investigator: What did you have, and what did you change it to? Participant 13: I had two angle marks on angles ACD and angles BCE, but after you asked me about the markings, I noticed that if I left it that way, that would mean that Angle ACD and the measure of Angle BCE are congruent.

13 Participant 13: Statement 4 says the measure of Angle ACD plus the measure of angle two, plus the measure of Angle BCE is equal to 180. That’s the definition of supplementary angle. Investigator: Okay. You seem unsure about that. Participant 13: I think it’s because I always think of it like adjacent angles on a line are

288

P# PMI supplementary. But somehow, adjacent makes me always think that it should be two angles. Investigator: Okay, so you’re saying since there’s three, you don’t know if they’re still supplementary. Participant 13: Yeah. Investigator: Okay. Or, you don’t know if they’re still adjacent? Participant 13: I don’t know if they’re still adjacent.

14 Participant 14: And then three and one are I want to say, complementary. Investigator: Okay, which means what? Participant 14: Which means it adds up to 90. Investigator: Okay. Participant 14: But I don’t know how we prove it. Investigator: Okay. Participant 14: And then which means that, no something’s off. Because there’s no evidence to say that one and three adds up to 90, and either way that can’t be because then two is obviously does not look like a 90-degree angle. Even though I know we’re not supposed to base it on the picture. Participant 14: Let’s see, what can we say? For some reason I’ve always liked vertical angles best. Investigator: Okay. Yeah, I see that you marked all the vertical ones. Participant 14: Okay. Yeah, I’m not sure how we would prove this honestly.

15 Participant 15: Oh yeah. Yeah, but it’s hard to write reason for statement number two. Investigator: Why? Participant 15: I mean we don’t have a specific reason we can use for statement number two right? [LATER IN INTERVIEW] Investigator: Okay, so Statement 2, you’re not sure? Participant 15: Yeah. Investigator: Why? Participant 15: I don’t know what’s the reason for . . . like if we draw line parallel to AB. Investigator: Do you know what this line is called, that we draw into the diagram? Have you ever done a proof where you draw a line in the diagram to help with proof? Participant 15: No.

17 Investigator: Thank you. And you did proof one. How’d that go? Participant 17: I didn’t really finish. Investigator: Okay. What were you able to do? Participant 17: I gave reasons for three, four and five, but I was unable to give a reason for two and I kind of, by looking at the provided statements, I kinda disagreed with how they went about with the proof. Investigator: Okay, so you would’ve done the proof differently. Participant 17: Slightly differently, yes. Investigator: Okay. Do you have experience with proofs, where they give you the statements and you’ve gotta fill in the reasons? Participant 17: Yes.

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P# PMI Participant 17: From what? Participant 17: From practice Regents problems. Investigator: Okay. And I see you’ve left two blank. What bothered you the most about two? Participant 17: The statement given was ‘Through C, draw DCE parallel to AB’ and I didn’t have an adequate reason as to why I would just draw the line. Now, I know why the line is drawn, but, in terms of this format, I’m not comfortable with say . . . It’s not a given and it’s not anything mathematically that I would say to draw this line.

19 Participant 19: We can draw this line because of creativity. The line makes it easier to prove it. I’ll come back to that.

19 Draws one arc marking on Angle 1, Angle ACD, Angle 3, and Angle BCE as if they are all congruent.

21 Participant 21: Yeah, I wasn’t impressed with how they, like why with the fact that my students had to write draw BC parallel to AB. What was the reason that you could just, you know… Investigator: Do you know, the reason for that? Participant 21: That is a rule that you could draw any auxiliary line through a point. Investigator: Is that the reason that you used with them? Participant 21: I can remember? I think so. Investigator: Okay. So, you felt that statement was unnecessary? Participant 21: I don’t, not that it was unnecessary. I was more surprised because I had not seen something like this. Like this is something more of you just do it without explaining it.

23 Participant 23: Okay. I think for two you can say that . . . It’s weird to show the reason. Investigator: Why is two weird for the reason? Participant 23: Because it’s like you’re doing it for the sake of the thing. I don’t remember what our reasons would be for this.

26 Investigator: Do you know what kind of angle this is called when the vertex is on the circumference? Participant 26: Isn’t it just an inscribed angle? Correct? Investigator: Yeah. Inscribed. Uh-huh [affirmative], and is BCD an inscribed angle? Participant 26: BCD? No. Investigator: Okay. Do you know what kind of angle that is? Participant 26: Forgot the name of it. No. I don’t remember. Investigator: That’s called a central angle. Participant 26: Right. Investigator: Okay? . . . There’s a rule. Do you know about the rule for central angles? Participant 26: No.

29 Sure. So, the only thing that was confusing me was for number two, statement number two. Through Point C, draw DCE parallel to AB. I don’t understand how to write a reason for that. It just seems like we’re adding something on that we couldn’t use, so I don’t know what I would write as a reason.

Note. P# = participant number; PMI = pure mathematical issue.

290

Table D12


P# BAA 1 Investigator: So, you have this idea—

Participant 1: I think, but then I don’t know what to write. Investigator: So, you have trouble translating your thoughts into . . . Participant 1: Words. I can talk out the problem, but I don’t know what to write. Investigator: So, you can talk about why you think things work. Participant 1: Mm-hmm. Investigator: But writing it down mathematically is hard.

1 Investigator: Okay. How do you feel about this proof? Participant 1: Horrible. Investigator: Why? Participant 1: ‘Cause I can’t write down what I think. I’m becoming a math teacher.

5 Participant 5: Well, when you’re writing a proof, if you want to prove that this is isosceles, you have to think about well, what makes a triangle isosceles. So, by looking at this picture, I was trying to figure out well what do I need to show that will support this final conclusion, that AED is isosceles. So first, I said . . . The first thing that I noticed was, well, I have to show that there’s two congruent sides. So, I thought that would be enough, but then I realized I won’t be able to do that until I show that these two angles are . . . That maybe if I can show that these two angles are also congruent, then that would be isosceles. And then I realized well, if I can prove that these two triangles are congruent, I can go back and show that AE and ED are congruent and then it would have to be isosceles.

6 Participant 6: So, you need to prove that this is the isosceles so if this is true . . . Does it matter how I prove this? Investigator: What do you mean by that question? Participant 6: I think I can prove it, but the way that I’m thinking is some long, convoluted way. Investigator: Okay. Participant 6: So, I’m wondering if there’s a faster way.

6 Investigator: How you would check yourself? Participant 6: I would check myself from the end. Investigator: Okay. Participant 6: So, to prove that it’s isosceles I must have proved that two sides are congruent which I did in step five. Step five was based off the fact that I proved the two triangles are congruent which I did in step four. Step four is based off of side-angle-side, so I must have proved a side, an angle and another set of sides congruent. One side was given to us, AC and BD, yeah, one side was given to us. The other angles were proved in step three and the other side was EB, EC was also given to us, so I got side angle side and that fills everything else in, I think. These angles were based to the fact that they were an isosceles triangle which I proved in step two and I proved that based off of the givens and

291

P# BAA I had my given. Investigator: Okay, how do you feel about this? Participant 6: I think I got it.

7 Oh, I guess because since these two things are equal, it states that AB and CD are congruent, but I feel like I’m skipping stuff. I feel like I can’t just state this stuff. If my students did this, I would say that there’s something that’s missing, but I don’t know what it is.

7 Participant 7: I think a paragraph would lead to less error for students because this one isn’t so cut dry and it’s not a standard triangle proof where we’re just leading right to a side-angle-side, side, side, side. I feel those are very formulated. There are steps you take and then you’re just done, but this one requires many steps so a paragraph would allow students to explain the second part. This whole situation of AC is congruent to BD, which means that well, BC is actually congruent to BC, so if they both have this BC part and they’re both congruent, that means these two pieces are congruent together. Investigator: Okay, so you think that would be easier to explain through. . . Participant 7: The words. Investigator: The words. Participant 7: Yes, because I think the process of substitution, subtraction, and coming up with a vocabulary might be more difficult.

8 If I were to use this chart, I think the first thing I would say is, I don’t think I’m gonna use it, I’ll just write like this below, is that Triangle EBC is isosceles since EB is congruent to EC. I think even in the past when I had statement-reason charts on the side I would first try to figure out the proof by myself because then it made it easier to fill out the statement-reason chart.

8 Yes, it was very helpful. Once I draw my diagram and label it to come up with a goal before I attempt the proof.

9 I would use the chart . . . But when first looking at the proof I feel like it’s good just to have a clean sheet of paper . . . just because I feel like this restricts me . . . If you just thought of something and you wrote out like five reasons, and then you just thought of something, you would have to erase. And that just seems like demotivating.

10 Investigator: Interesting. But you’ve never written out a proof of why side-angle-side works, and why angle-angle-side works, but that angle-side-side doesn’t? Participant 10: I feel like we talked about it, but I can’t remember writing it. I mean, maybe I did, but- Investigator: As a future secondary teacher, do you think that that is something that you would want your students to know? Participant 10: Yeah. Because we just memorized everything. We never really understood why something was the reason it was.

11 I want to say I was just told that these were the methods. It was on our graphic organizer, and you just kind of . . . And they’re like, okay, these are the methods. And then, to never use angle-side-side.

292

P# BAA 11 Participant 11: I said, the reason I didn’t like geometry in high school, was because I

didn’t know what to write. Investigator: Okay. Participant 11: Especially for the reasoning part. I think the statement you just kind of, say what the next step is, but the reasoning, sometimes, there’s specific vocabulary that you have to use, and I wouldn’t always remember what to write.

11 I think, as long as I was comfortable, I would be okay teaching it.

12 There’s so many things that should be taught to make this more interesting for students to understand the reasoning behind it, but it’s just high school teachers, they have a lot to get through before the end of the year.

14 Investigator: Okay, awesome. Can you take a look at the next one, Proof 2? Participant 14: Why do they do this to children? Investigator: Ha, what don’t you like about this one? Participant 14: I don’t remember ever seeing something like this in high school.

14 Participant 14: AC is congruent to DB. Okay. And then EB is congruent to EC. So, I’m going to separate them. Investigator: Okay. Participant 14: But I already know this is something I wouldn’t prove. Investigator: What you mean the whole question? Participant 14: Yes. Investigator: Okay. Participant 14: Oh, they want to prove that AED is isosceles. Oh yuck.

14 Investigator: Okay. So, you seem to have a good idea of what to do. Does the difficulty lie for you in organizing it and writing it down in the mathematical language? What do you think is the turn off for you? Participant 14: I think the turn off for me is like structuring it. Investigator: In terms of order or? Participant 14: I think both. It’s like I see bits and pieces of it, and it’s like I feel, “Oh, this is why it is.” But then when it comes to putting it together, and this even happens with logic proofs, it’s like proofs in general. Organizing them and just putting them together, I have a hard time.

15 Investigator: Okay, do you think this one’s easy too? Participant 15: It would be hard for my students. Investigator: Okay, but for you, you think easy? Participant 15: Yeah. Investigator: Okay, what do you think would make it hard for your students? Participant 15: It’s overlapping triangles.

15 Investigator: Okay, do you know why . . . so for example you wrote here, side-angle-side, do you know why side-angle-side is sufficient or angle-side-angle, or side-side-side. Has it ever been shown to you why those are good cases? Participant 15: No. Investigator: Okay, do you think that you should know that as a teacher? Or do you think

293

P# BAA it is not necessary? Participant 15: I mean, it’s not necessary. Investigator: Okay, why not? Participant 15: To be honest, I don’t know why we need to prove two triangles are congruent, so why do we need to know, the reason behind the reasons? Investigator: Okay, so you’re saying you don’t really see the benefit of students doing this? Participant 15: I mean they can do proof, but like the part of proof two triangles are congruent . . . Investigator: You don’t see much value in that? Participant 15: Yes.

16 Investigator: Do you think that color is helpful in general in geometry? Participant 16: Yes. Investigator: Why? Participant 16: Because me writing all this down is like organizing my thoughts, with me highlighting things as a visual reminder of what I already know, helps me think about what I’m trying to then continue to prove.

18 Investigator: And how did you . . . think to draw those separately? Participant 18: I did that just because. I didn’t even finish the problem, that’s why. It just helps me to look at it, anyway, because it’s sort of a mess here. Investigator: Okay. So, what type of program or diagram would provoke you to draw separate- Participant 18: Every time I see overlapping shapes. Every time.

18 Participant 18: Okay, let’s see. How do I need to do this? Do I have to write my givens like everything? Investigator: If that’s part of the proof, yes. Participant 18: Aw, man.

18 Investigator: Do you feel like you need to be an expert in geometry even though you don’t teach geometry? Do you think it’s important? Participant 18: For math teachers? Investigator: Yeah, in general. Like should a math teacher, so your certification is in seven through 12? Participant 18: Correct. Investigator: Should a teacher who’s certified in seven through 12, maybe they only teach middle school, do they need to be proficient in geometry? Participant 18: Interesting question. Are you talking about someone specifically certified seven to 12, not like one through six or whatever? Investigator: Yeah, no a secondary certified math teacher. Participant 18: I think so, yeah. Investigator: Why do you think it’s important? Participant 18: Well, what are they gonna do if they get hired to teach geometry? Investigator: If they have to teach it, right.

294

P# BAA Participant 18: Yeah that’s primarily like the reason. For example if you told me not a seven through 12 teacher and said only an algebra teacher, I’m okay with that. Investigator: You’re okay with them not knowing geometry? Participant 18: If they can do their one job well, why do they need to necessarily know the rest? Investigator: Okay. Participant 18: It helps, sure, but let’s just say they were extremely proficient in algebra, and let’s say they were awful at geometry, it doesn’t detract from them being an algebra teacher. Investigator: Okay. Participant 18: I don’t think so.

21 Participant 21: So, my first statements will be my givens. Investigator: Okay. Because is that something you tell your students to always put the givens first? Participant 21: I always do. And the reason I tell them that it’s because it kind of gives them encouragement to get started on the proof. Otherwise, they would not know where to begin.

21 Yeah. So, then I know that BC is congruent to itself. I always tell my students, whenever you see lines that are overlapping, so parts of the triangle that are overlapping, you always want to think of reflexive. And that’s kind of again, a motivator.

21 Honestly, I think I did all right. But to be honest with you, sometimes with my reasoning, I don’t know if it’s sufficient enough based on the Regents rubric or based on what other teachers write, or based on the definitions, because I’ve learned so many different ways. I remember when I was in middle school, my teacher would just tell us write “by the definition of a midpoint” or “definition of a right triangle.” Whereas once I started tutoring, I saw teachers would be like, “No, you have to write it all out.”

Note. P# = participant number; BAA = beliefs and attitudes.

295

Table D13


Assessment Item 3

P# ICSN+ ICSN- 12

We can state that . . . I don’t know if we can go straight into it. Yeah, we probably could. We can say that AB is congruent to . . . I don’t know if it’s necessary, but just in case, I’ll probably do it. I can say that the . . . That AC minus AB minus BC . . . See, I can’t say minus, because then congruent sounds kind of off. All right. Congruent to BD minus BC?

18

I’m not confident. I’m not confident in my technicalities. Using measures versus equals. Measures versus actual segments and angles.

23 I’m subtracting AC minus BC is equal to, congruent . . . Oh no, no. I’ll say congruent, no, equal to BD minus BC.

Note. P# = participant number; ICSN = issue of correspondence between substance and notation (positive or negative).

296

Table D14


P# EU ESD EUSD 3

Participant 3: So, AC congruent to DB, EB congruent to EC, given. You have to use the subtraction property? Investigator: Okay, what does that mean? Participant 3: If I were to remove BC, AB would be congruent to CD. Investigator: Okay. Participant 3: I don’t know how to write that, I think . . . I would just write AB congruent to CD by subtraction property, and what I’m subtracting is BC. That’s definitely not worded right. Investigator: How do you know that it’s not worded right? Participant 3: It doesn’t make sense to somebody that was reading it.

5

AC and . . . Those are congruent. And then EB and EC . . . EB and EC are congruent. Prove that AED is isosceles. AED is isosceles. Okay. Well. Plan. I have to prove that AE and ED are congruent. I also . . . Yeah, if I can prove that AE and ED are congruent, then I can prove that it’s isosceles. I don’t remember if that’s enough. Is that enough? This should be that . . .

5

Participant 5: They are the same length because they overlap. Investigator: So, you’re saying on AC and on BD, BC takes up the same amount of space? Participant 5: Yeah. Investigator: Yes.

297

P# EU ESD EUSD Participant 5: What property is that? Investigator: Do you want me to tell you? Participant 5: I don’t know. Reflexive, something like that?

5

Okay. Opposite . . . Okay. So now that I’ve summed up these two are congruent . . . I’m not . . . Why, I don’t know why I know these two are congruent.

5

Investigator: Okay. And you’re definitely correct that you have to use supplementary angles. Why do you know that they’re congruent? Participant 5: Because if EBC and ECB are congruent, then subtracting from 180, ABE and ECD would be the same. Investigator: Excellent. Participant 5: I don’t know how to say that.

9 Let’s see what else would help us. Oh okay. Maybe this isn’t how you prove it but I’m going to go ahead and do that. So that would mean that angle EBA and angle ECD are congruent because they’re supplementary angles to the same angle measure.

9 Yes. And then corresponding parts of congruent triangles by that EAB and EDC are congruent. And then since there are two angles that are

298

P# EU ESD EUSD congruent, AED is an isosceles triangle.

10

Okay. We know that AC is congruent to DB. And from there, for AC, we know AC is equal to AB plus BC, and DB is equal to DC plus AB, and since they’re both equal, that means AB is congruent to CD. And I don’t know what to do next.

11

I don’t know what to put on the reason side.

12

Participant 12: And EB is congruent to EC, the reason is given. All right. I know for one part of it, I’m just looking ahead. I know I’d have to prove that triangle . . . I know one way to prove this is by first proving that triangle ABE is congruent to Triangle DCE by side-angle-side. Investigator: Okay. Participant 12: We already have one side. First, I can state that Triangle BEC is an isosceles triangle. Investigator: Okay. Participant 12: My reason would be that Triangle BEC has two congruent sides. So, that’s already a side right there. I can state that Angle EBC is congruent to angle ECB because I know that the base angles and isosceles triangle are congruent. Investigator: Okay. How did you identify those as the base angles?

299

P# EU ESD EUSD Participant 12: The base angles are always the angles opposite from the congruent sides. Investigator: Okay. Participant 12: So, that’s already an angle right there. The last thing, from what I remember, we’ll probably have to use the subtraction postulate. It’s kind of been a while, so hopefully I can state it properly.

14

Well, when it’s something that I teach, it’s easy since I know the language well enough from college and from other classes that are higher at like level, you pick up the correct language. Then it’s something I feel comfortable with and there’s always, I can look it up if anything. I can look it up and then with me it’s more like, “Okay, this is how the textbook would define it, how can I keep it mathematical but break it down for students.” But then for something that I don’t teach like geometry or higher-order math, it’s more like, “Would a professor think that this is enough to give me full credit?”

15 Investigator: Okay, what can’t you use? What are you not allowed to use? Participant 15: A-S-S. Investigator: Okay, do you know why? Participant 15: For A-S-S . . . I mean if you have two sides and a non-included angle, you can draw more

300

P# EU ESD EUSD than one triangle. The triangle is not unique. Investigator: And when did learn that? Do you think you knew that as a high school student? Like the reason that you can’t use A-S-S? Participant 15: No, when I taught pre-calculus, as a student teacher. Investigator: When you taught pre-calculus as a student teacher? Participant 15: Yeah. Investigator: Okay, so that came up with the Law of Sines? Participant 15: Law of Sines ambiguous case— Investigator: And the use case. Oh interesting, so now you’re using that knowledge as a geometry teacher? Participant 15: Yeah.

20 Participant 20: If I can prove that those triangles are congruent, then I can prove that DE is congruent to AE, which will give me an isosceles triangle because I have two sides that are congruent.

22

Participant 22: Which means they also have a common angle, B. I

301

P# EU ESD EUSD don’t know if that helped or made it worse for me. Investigator: The separating? Participant 22: Yeah, ‘cause now I have like this angle/side situation. I’m like what the hell do I do with that?

23 Investigator: Okay, and what are other methods that you know of to prove triangles congruent? Participant 23: You could use side, side, side . . . Angle, angle, side . . . There’s some other ones. Hypotenuse, leg if you had that’s a right triangle. Investigator: Do you know what case you cannot use? Participant 23: You can’t use A-S-S. Investigator: Do you know why? Participant 23: Because when you have two sides and a non-included angle there could be the ambiguous case and so you could have two triangles.

24

Okay. EB is congruent to AC. Prove that AED is isosceles.

24

I wouldn’t know where to start.

29

Participant 29: Since EB is congruent to EC, then Triangle BCE is isosceles, and the measure of Angle BEC is equal to the measure of ECB. I left the reason

302

P# EU ESD EUSD for that blank, because I’m not sure what those angles are called. I feel like it’s opposite, something like that, but I forgot that part. Let the measure of Angle BEC equal x. Then the measure of angle EBA equals 180 - x, and I said supplementary. Similarly, the measure of angle ECD equals 180 - x. Therefore, the measure of EBA is equal to the measure of angle ECB, and I said BC is congruent to BC, reflexive property. Therefore, AB is congruent to CD, and I put subtraction. I feel like that’s a longer reason, like it’s not just called subtraction. Triangle EBA is congruent to triangle ECD, by side-angle-side. Thus, the measure of EAB is congruent to the measure of EDC. Corresponding parts of congruent triangles are congruent. Triangle AED is isosceles.


303

Table D15


P# ML+ ML- 1

Cause I know if you subtract AC from DB, hold on. If you subtract AC from DB you get BC, which means AB is the same as CD, but I forgot . . . I think it started with an R in the name.

1

If you subtract these two pieces you should be left with AB and AC and since . . . when you subtract congruent sides, you’re left with congruent parts.

1

Since it’s just being moved over x amount, C is being moved over x amount from D and B is being moved over x amount from A, aren’t these two the same angles? I don’t know-

4

Yeah, so I’m trying to say that if these are congruent then these have to be because they are a part of their supplement. That’s fine?

4

Congruent parts of congruent triangles are congruent.

7

Alright, cool. Okay. In the diagram below, AC is congruent to DB. Prove that AED is isosceles, so we want the big triangle, is isosceles. Alright, so. AC congruent to DB. DB is congruent to AC. That’s given. I know that then we can state that BC is congruent to BC, but I think there is some. . . a ton of steps I feel like before that. Oh. I feel like before this, I have to say that AB plus BC and . . . is equal to AC. Then . . . EB is congruent to EC. Oh. CD plus BC is equal to BD. I know that if you . . . there’s some reason. There’s a reason where if you have these combinations you can state that BC is congruent because they’re pretty much shared between the two segments, the AC and the BD. So, I want to mark this here, I just can’t remember the reason.

7

Participant 7: Yeah, and then, you would have to. . . then you could state that ABE is congruent to ECD, right, then that proves that triangle ABE is congruent to ECD by side-angle-side and then you can use, oh, then you use CPCTC. Investigator: What does that mean? Participant 7: Congruent parts of corresponding triangles are congruent.

304

P# ML+ ML- 9

Participant 9: Okay. So, they told us that AC and BD are congruent, and both these lines share BC, so if you subtract BC from both of these, then we get that AB is congruent to BD. Investigator: Okay. Participant 9: I think there’s a name for it but I don’t remember. Investigator: A name for what? Participant 9: This . . . That if they share part of the line segment you can subtract it or something.

10

Investigator: Okay, and what did you say the reason for that was? Participant 10: Well, we know these angles are congruent because it’s an isosceles triangle, and then it’s a straight line, so they’re both supplementary angles. And then since they’re supposedly the same value, they’ll be congruent. Investigator: Okay. So, do you know how to formally state that as a reason? Participant 10: Something like supplements of supplementary angles are congruent?

10 Investigator: Okay. So, a few questions about what you said. What’s CPCTC? Participant 10: Corresponding parts of congruent triangles are congruent.

12 Reason is that . . . Corresponding parts of congruent triangle are congruent.

13

If I were to say that . . . Can I say that triangle ACE is congruent to triangle BDE because they have one leg with the same length, and they have angles of the same length? But . . . They share angle E as a reflexive property.

13 Participant 13: And then through congruent parts of congruent triangles are congruent, we can say that DE is congruent to AE. Investigator: Okay. Participant 13: And therefore,

305

P# ML+ ML- triangle AED is isosceles. Investigator: Okay. Participant 13: Oh, corresponding parts. Investigator: Where does the corresponding go? Participant 13: In the first. Investigator: Okay. What do you think made you catch that? Participant 13: Because I was reading it as I was writing it, and I was like, “How do I already know that they’re congruent parts?” Investigator: Okay. Participant 13: That’s what I was trying to prove, and then I was like, “Oh, corresponding.” Therefore, triangle AED is isosceles.

15

Participant 15: SAS. So, for this question we need to prove triangle AEC is congruent to triangle DEB first, using SAS. Investigator: Okay what’s SAS stand for? Participant 15: Side-angle-side. Then after proof two triangles are congruent, we can use CPCTC: congruent parts of congruent triangles are congruent.

16

I wrote down the given. And then I mentioned that BC is congruent to itself, and it’s reflexive, and that I know that AB is congruent to CD, but I couldn’t think of why.

17

I use congruent parts of congruent triangles are congruent to state that Angle A is congruent to Angle D, and then I stated that EAD was isosceles, because, when your base angles of a triangle are congruent, the triangle is isosceles.

19 Participant 19: After that, I can finally say that the side AE is congruent to the side ED by CPCTC. Investigator: And what does that stand for? Participant 19: Congruent Part

306

P# ML+ ML- of Congruent Triangles are Congruent. Investigator: Okay. Participant 19: Oh, hey, no no. No. It’s something else, not congruent. Investigator: Which one is not congruent? Participant 19: The first one. Investigator: The first C? Participant 19: The first C is not congruent. Oh, corresponding parts of congruent triangle are congruent.

20 Investigator: Wait, so what does CPCTC mean? Participant 20: Corresponding parts of congruent triangles are congruent. So, if triangles are already congruent, then every single pair of corresponding parts would be congruent.

24

Participant 24: The tiny line segments, CD and AB, they’re definitely equal. Investigator: How do you know that? Participant 24: Because AC is equal to BD, so it doesn’t, I don’t know how to explain, but it doesn’t really matter like how far they are on top as long as they’re on top of each other, like whatever extra line segments, they have to be equal. I wouldn’t know how to explain that.

27 Participant 27: Can I just write CPCTC? Investigator: You can write it, but can you say what it means? Participant 27: Corresponding parts of congruent triangles are congruent.


307

Table D16


P# PMI 1 I think Angle A and Angle D should be congruent but then I also have to . . . I think I also

have to prove that Angle E is not the same as Angle A and Angle D. Because if Angle E is the same then it’s gonna be equilateral.

1 Investigator: Okay. So, for an isosceles triangle do you need exactly two congruent sides or at least two congruent sides? Participant 1: Exactly.

2 Participant 2: Thank you. Okay. EB. First I’m marking. Then I’m showing congruence. Okay, I’m having a problem showing congruence because they’re on top of one another. Investigator: What’s on top of one another? Participant 2: AC and BD. Investigator: Okay. Participant 2: I’m not sure how to show that being that they’re on top of one another.

2 Participant 2: Because two sides have to be congruent for it to be isosceles, or does an equilateral triangle mean that it’s isosceles as well? Investigator: Can you say a little bit more about what you mean by that question? Participant 2: Okay, so if all three sides are congruent that means any two sides would be congruent, which means in fact that an equilateral triangle is isosceles?

2 Investigator: Okay. Do you remember anything from your studies or preparation for teaching about how to prove that triangles are congruent? Participant 2: That two triangles are congruent? Investigator: Yes. Participant 2: Oh. I did study it. I don’t remember it. Let me think a minute. Investigator: Sure. Participant 2: Two triangles are congruent. I don’t remember.

3 And then supplementary angles . . . Supplementary angles are congruent.

5 So, because AC and BD are congruent, BC . . . I don’t know. Yeah. I have no idea. . . I remember something where you can say that the sum of AB and BC plus . . . Equals the sum of BC and CD by reflexive property, or something like that.

5 Investigator: So, do you remember how to prove triangles are congruent? Participant 5: Angle-side-angle. Investigator: Ah. Is that the only one? Participant 5: Or side-angle-side.

8 I think at this point I would either use angle-angle or side-angle-side because I notice that both triangles have angle E. So, I would use the reflexive property for that.

8 The reflexive property can be used either for angles or sides and it’s just to show something that’s already given in the diagram.

308

P# PMI 10 Investigator: Another thing you mentioned is side-angle-side. So, you can use that to

prove that two triangles are congruent. Do you know why you can use that? Why side-angle-side is sufficient to say that the triangles are congruent? Participant 10: Honestly, no. Investigator: Okay. So that’s just something that you’ve kind of— Participant 10: It’s just ingrained. Investigator: Okay. Do you know the other congruence methods? Participant 10: Side-side-side, side-angle-side, and then angle-angle-side. Hypotenuse-leg, but that’s only specific cases. Not angle-angle-angle. Investigator: So, angle-angle . . . what can that give you? Because that’s something else. Participant 10: I don’t remember.

11 I’ve never seen anything like this.

11 And since it’s isosceles, I know that the interior angles are congruent.

11 Participant 11: Side-side-side, angle-angle-angle. I know I’m not supposed to use angle-side-side. Investigator: Okay, do you know why? Participant 11: I’m sure I’ve done it before, I just don’t remember it right now.

13 Participant 13: I just know that it’s an isosceles triangle. Investigator: Okay. How do you know that? Participant 13: Because in the given, it said that EB is congruent to EC. Investigator: Okay. Participant 13: Which automatically makes . . . It might not be isosceles! Investigator: Why not? Participant 13: Because what if it’s equilateral?

17 Investigator: Do you know, or have you ever proved why something like side-angle-side does work? Participant 17: Have I proved it? Yes. Do I remember it? I could probably figure it out, but I wouldn’t be able to say it right now.

24 Participant 24: So, in the diagram below AC is . . . similar . . . to DB? Or does this mean congruent, right? Investigator: What, this symbol? Participant 24: Yeah. Investigator: Yes. This means congruent. The similar symbol is like this. And this one’s congruent.

25 I believe that ACE is similar to BDE since AC is congruent to DB and EB is congruent to EC. So that makes these two similar. And knowing this makes me think that AE is congruent to DE which will make triangle AED be isosceles.

28 Participant 28: Well, if anything, I would say BEC is isosceles. Investigator: How would you know that? Participant 28: Well, . . . I mean, visually, it looks like an isosceles. . . . AED looks like an equilateral. Just visually it looks like it, and I know you can’t use visually in math; however, that’s just what I do, and the BEC looks like an isosceles just because the fact

309

P# PMI that it’s thinner than the equilateral triangle, than AED, and. . . all sides are equal. BC is smaller than BE and CE. So visually, that’s just how I look at it. And then what I would say is . . . well, this one’s a little tricky. . . . If it said AD is congruent to AE, then I would know those two sides are exactly the same, . . . and that would prove as isosceles, not an equilateral but then I would also feel that . . . well, I’m stuck.

29 Investigator: And what would you need to show about angles in order to show that a triangle is isosceles? Participant 29: Show that two of the angles are equal in measure. Investigator: Do you have any idea what those angles would be? Participant 29: . . . I feel like it’s going to be A and D. Investigator: Okay. Is there any reason why you’re leaning towards those angles? Participant 29: Honestly? Investigator: Mm-hmm [affirmative]. Participant 29: I know it shouldn’t be, but it’s the picture. Investigator: Just visually? Participant 29: Yeah, visually, it’s making me lean that way. And even though I know I shouldn’t, now I’m thinking of . . . It’s . . . I think if E weren’t where it’s placed, then it wouldn’t be isosceles, so maybe that is a good way of thinking. I don’t know. I am lost.


310

Table D17


P# BAA 1 And then I have to prove that the diagonals are perpendicular to each other. But I don’t

know how. Do I have to write this? It’s a waste of time. 1 Participant 1: So, I’m just stating that because . . . see this is my issue because I don’t

know where I’m going with this. Investigator: So, you feel like you’re— Participant 1: I feel lost. Investigator: Do you feel lost or overwhelmed or both? Participant 1: Both, because I have all these thoughts going in my head of what to do, but I don’t know how to organize it. Investigator: How were you, if you remember, how were you taught in high school how to organize everything that you’re thinking when you look at a problem like this? Participant 1: Yeah, but they were easier. They weren’t like this.

1 Participant 1: But it’s just wrong. Investigator: The way you wrote it? Participant 1: Yeah. Like there’s so much that goes on here and not much that goes down here. Investigator: Do you think that there’s more that goes on with a geometry proof than there is with solving an algebraic equation? Participant 1: Yes, much. Investigator: Why? Participant 1: Because I don’t know how to organize my thoughts.

3 Participant 3: Yeah, it looks scary at first. Investigator: It does. Participant 3: Usually triangles, they’re like easier, I feel, than parallelograms.

4 Participant 4: Oh gosh. Investigator: Yeah, so tell me about the oh gosh. Why? Participant 4: I just get nervous. Investigator: You get in general or is there something about what you saw on this page that made you nervous? Participant 4: No, I think it’s just persona., I just think I take too long, but I don’t know. I just—things just take a little bit longer for me than other things, but that’s okay.

10 Participant 10: I tutored geometry before, but it was all the first semester, which is just . . . a lot like triangles. Investigator: Okay. Do you think that in general people are more comfortable with triangles than other shapes? Participant 10: Yeah. Investigator: Why do you think that is? Participant 10: I think it’s because they just focus so much on triangles. The class that I observed at the place I’m supposed to student-teach, they did triangles from September all

311

P# BAA the way to the end of November, whereas they barely covered other shapes much. Investigator: Okay. So, were you able to gain any insight into why the teachers did that, or no? Participant 10: No. It’s just . . . that’s the curriculum. Investigator: Okay. So, they were just following the curriculum. Let’s say you were told that you were going to be teaching geometry this— Participant 10: Oh yeah. Investigator: This semester? Participant 10: Yeah. Investigator: As a student teacher? Participant 10: Two classes. Investigator: That’s so exciting! But you don’t seem excited. Is it your preference? Participant 10: I don’t mind teaching, I’m just worried. Investigator: What are you worried about specifically? Participant 10: That I don’t really know the material inside-out, and the kids are going to ask me something that I don’t know how to answer. Investigator: Okay. Are you more worried about that in geometry than in other subjects? In other courses? Participant 10: I guess. In geometry, the majority of the stuff I did learn, I guess, I just have to re-look at it, whereas in trig, it was some stuff that I’d never learned. It’s completely Common Core, which I didn’t take. Investigator: Okay. So, comparing geometry to, let’s say, Algebra 1. What would you be more confident with? Participant 10: Algebra 1. Investigator: Why do you feel more confident with that? Participant 10: I guess because almost every single thing you learn in algebra reappears in other math classes. Like proving a triangle is congruent. The only times I’ve ever actually studied that is in a class specifically meant for geometry. Otherwise, it never really comes up. Investigator: So, geometry topics are more unrelated. Participant 10: Yeah.

13 Investigator: Okay. Okay. How do you feel in terms of confidence about proving different types of quadrilaterals? Do you feel better about this than the previous proofs, or less confident? Participant 13: A little less confident because I think because I’m not confident with the properties of quadrilaterals in general.

15 Participant 15: No, I don’t think it’s a hard problem. But it’s hard for the students. Investigator: Okay, what kind of help do they need? Participant 15: First, they need to know how to prove a parallelogram is a rhombus first. Then they need to know, they need to prove these two triangles are congruent first. And since these two triangles are overlapping, so it’s hard for them to have enough information like Angle Side and Angle, that’s why it will be hard for them.

15 When I teach proof, I always tell them, statement number one, write down all the given information.

312

P# BAA 16 I have no freaking idea how to do this.

18 I think you could do a proof by contradiction of this. Investigator: How would you do a proof by contradiction? Participant 18: Assume that this side, BC and DEC are not congruent. Investigator: Okay. Participant 18: I’m not actually going to go through it. I don’t really want to think too much into it. I just feel like you could. Just like on a gut feeling. Investigator: You feel like you could do it, but you’re not going to do it. Participant 18: Yeah. Mostly because it’s a High School proof, it’s probably way more difficult. Investigator: So, you don’t think proof by contradiction comes up in a High School curriculum? Participant 18: Interesting reasoning you have there. Investigator: I’m asking. Are you saying it’s probably not necessary to do contradiction because it’s a high School problem and contradiction doesn’t come up in High School? Participant 18: Yeah. Investigator: Okay. Do you remember when you first learned proofs by contradiction? Participant 18: Most likely in 220 Math. Investigator: What’s 220 Math? Participant 18: Discrete Mathematics.

18 Participant 18: Yeah, the minimal requirements. Parallelograms have more requirements than trapezoids. Investigator: Okay. Participant 18: Including two sides have to be parallel. Alright then after this, I’m just gonna write this. Parallelograms, and I think there’s a split here and I forgot why. Is there a kite? There’s a kite in here, but that’s not really a parallelogram. Oh, there’s a quadrilateral, oh lord. Quadrilateral. Then you put a kite over here or you draw a kite. Which nobody really studies in high school.

18 We know we have a quadrilateral. We can back off on kites because we don’t care about those. I forgot what it is, because some kites can just be, what’s the defining characteristic about kites? If I remember correctly it’s two diagonals or a perpendicular. Then in that case some quadrilateral. I’m gonna erase my kite. 100% I forgot about it.

19 Participant 19: Because it’s a . . . I don’t know how to pronounce it. Investigator: Rhombus? Participant 19: A rhombus. So, rhombus has four equal sides and– Investigator: How do you know that proving that it has four equal sides is enough to prove that it’s a rhombus? Participant 19: Because, rhombus is also a parallelogram, but rhombus has one more property. I mean, each side has to be equal to each other . . . I mean, four sides need to be equal to each other. Investigator: Okay. How do you know that? Where did you learn that? Participant 19: When I was in China. Investigator: As a student?

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P# BAA Participant 19: Yeah. When I was in fourth grade, I guess. Investigator: Fourth grade. Okay. Participant 19: But I feel like this question is going to take me a long time to think about it because I have no clue what’s going on.

19 Investigator: Okay. So, what do you normally do when you get stuck with a proof? Participant 19: I usually think it over and over again. So, usually my teacher told me to skip it and come back later. But I would rather spend my time on it because I’m kind of like . . . I don’t want to get stuck on one question, I’d rather to finish it and then move on.

20 Investigator: Okay, so you have a very coherent and clear plan for how to prove this. How do you feel about writing the proof? Participant 20: I feel like it’s going to be a lot, but I can do it. Investigator: You can do it. Okay. Are you excited to do it? Participant 20: No. Investigator: Okay, why not? Participant 20: Because it’s going to be a lot of writing.

21 Participant 21: Okay. So, I write the statement and I write my reason. Investigator: Do you always use statement reason charts? Participant 21: Yes. Investigator: Okay. Why? Participant 21: I think it’s easier than a flow chart. Investigator: Okay. Participant 21: Paragraph. I haven’t actually honestly tried paragraph. Investigator: Okay. So, you don’t have experience with paragraph? Participant 21: No, I don’t.

21 Participant 21: Right? Because if you’re saying that it’s a definition, and you know the definition of a parallelogram. So why then write it out? That’s redundant. It might be, but it might not be… Investigator: So, it would be nice if you knew what was required. Participant 21: Right. Who made this up? I want to ask them this.

21 I can’t remember. I might be done to be honest with you. But to be honest with you, I choose not to teach my seniors this with the proofs of quadrilaterals because they had a very difficult time with triangle proofs.

21 Investigator: And you think the quadrilateral proofs are harder? Participant 21: Forget it. They’re going to look at this and say, what is this? Investigator: What do you think makes it harder? Participant 21: They don’t even know what shapes they’re going to look at. Investigator: So, they need to know properties of shapes that are foreign to them. Participant 21: They’re going to look at this. They’re gonna say, why are there two triangles making a boomerang? Okay, let’s be real. Investigator: Do you think that if this, were something you had done in more depth with your students, that you would have an easier time with the proof? Participant 21: Absolutely not. Investigator: Not?

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P# BAA Participant 21: Not. Investigator: Okay why? Participant 21: Because they do not care.

22 Investigator: Okay. So, are you done? Participant 22: I think. Investigator: Okay. Participant 22: Yeah. Investigator: What would you need to know, or what would you need in order to know for sure that you’re done? Participant 22: What do I need to show . . . wait, wait, what? Investigator: Like I said, “Are you done?” And you said, “I think.” So, what would you need in order to make you positive that you were done? Participant 22: Confidence, I guess. Investigator: Okay. Why do you think you don’t have confidence? You knew when you were done with the last one. Participant 22: Because this is something like, I don’t know. The last one was like . . . I guess triangle things are more comfortable, ‘cause it’s like the basic. And then like everything else is kind of like built up out of triangles. So, like, once you have a triangle similarity, or a triangle congruence kind of thing going on, it’s like, “Okay, great, I’m done. They’re congruent.” And then when it’s anything else like circles, parallelograms, you’re kind of like, “I’m using triangles to prove it, but I don’t know if in the end my triangle proof is like getting me to that statement.”

23 Investigator: What do you think is something that teachers or students would stumble with on this problem? Participant 23: I don’t know, maybe identifying this reflexive angle. Investigator: Angle C? Participant 23: Yeah. Investigator: Okay, do you think in general this proof or a type of proof like this about quadrilaterals is harder than triangles? Participant 23: Yeah, I think in general it is. Investigator: Why? Participant 23: Cause you have to be a little, kind of, creative. Something we were doing today in student teaching, we had a right triangle with an altitude to the hypotenuse and we were showing one of the right triangles formed inside were similar to the larger ones. Investigator: Mm-hmm [affirmative]. Participant 23: And students couldn’t re-orient the triangles. When they’re stacked on top of each other and when you have stuff going on it can kind of be visually, you know . . . Investigator: Overwhelming? Participant 23: Yeah.

27 Investigator: Did you just make an ugh? Participant 27: Yeah. Investigator: Okay, what is ugh about this? Participant 27: I don’t like rhombuses. Investigator: Why?

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P# BAA Participant 27: They’re just, I don’t know, complicated. They’re trying to be a square, but they’re not.

27 Investigator: Okay, and what do you mean when you say it’s a square turned on its side? Participant 27: Like, it’s a square, and then it’s turned like that. Investigator: Okay. So, there’s some relationship between a rhombus— Participant 27: Right. Investigator: —and a square, okay. Participant 27: Which is why I hate rhombuses. Investigator: All right. Participant 27: Because I never remember rhombuses.

28 So, what would be hard for me is so to speak the goal. What do I . . . what is . . . what are things that important that need to be stated? What are things have to be stated at the end and like these are the reasons why it is so I think I just . . . I don’t have enough confidence when I write it, that is this all I need or is there more information and sometimes what happens is I start . . . I will write things down and I’m like, “Oh wait! But that I know too. And that I figure out and do I need all this information? Is it too much information?” But I guess not . . . more information is better than not enough. And I just struggle with . . . like I have this thing where I need to know why things happen. That’s my biggest thing in math. You give me a formula. How did you come up with that?


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Table D18


P# EU ESD EUSD 6 Participant 6: I think if you can

prove two consecutive sides are congruent then you can prove that the entire shape is a rhombus. Investigator: Okay, are you saying you think that because you remember it or because that’s making sense to you? Participant 6: I’m saying that . . . Okay, so first I’m saying it because I remember it. It’s a vague thing in the back of my mind. Investigator: Okay. Participant 6: But now that I’m actually sitting down and thinking about it, if it’s a parallelogram that opposite sides are congruent. Investigator: Okay. Participant 6: So, if I prove that this side is congruent to this side, I’ve also proved that they’re all congruent. Investigator: Sorry, just for the tape can you say side means? Participant 6: Sorry, so if I prove that BC is congruent to CD then I prove that all four sides are congruent because BC has to be congruent to AD and DC has to be congruent to AB. So, I just proved the two sets are congruent to each other by just proving that one of each pair is congruent to each other.

7

Participant 7: Okay. So I like to draw whatever information I can get from each given

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P# EU ESD EUSD individually so I know sometimes that people will recommend writing all the givens and then writing it out but I think it’s a better process in terms of understanding, if I write down a given, right, for example ABCD is a parallelogram, and then you write the following one is that BC is congruent to AD, AB is congruent to CD, BC is parallel to BD, and AB is parallel to CD. It’s just information you can draw from it. So, I did this, so I have three givens and I have steps underneath each one ‘cause that’s the information I could take away from what they gave me. So now I’m pretty much at the point where I proved BC is congruent to DC which is what I said I wanted to do. I used that through triangle congruence and CPCTC. I’m just now trying to figure out how I’m working it back to say that all four sides are congruent. I know that because this is a parallelogram, opposite sides are congruent, and I just proved that consecutive sides are congruent. I want to state that all the sides are congruent, ‘cause I can’t just state that BC is congruent to CD, because consecutive sides being congruent is not a property of a rhombus so that wouldn’t help me. I need all four. Investigator: Well, what does it imply if the consecutive sides of a parallelogram are congruent? Participant 7: Well, I would say

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P# EU ESD EUSD then that the other consecutive sides are congruent. Investigator: Okay, and why would that be? Participant 7: Because of the definition of a parallelogram. If BC is congruent to AD, and AB is congruent. Oh, so can I do it through substitution? Investigator: Sure, how? Participant 7: Because BC is congruent to AD, right? So then, and I know that from a definition of a parallelogram, and then DC is congruent to AB. So, the same thing. Substitution. I could state that. . . so BC, so AD is congruent to AB, and this is substitution property. Then I would probably put some kind of statement that all four are congruent to each other. So, BC is congruent to DC, which is congruent to AD, which is congruent to AB. Definition of a rhombus, can I just put it there? I’m not sure. I feel like there’s something that has to go in between here. Investigator: So, if your reason was definition of a rhombus, then you would kind of be using what you’re trying to prove as the reason, ‘cause it says prove ABCD is a rhombus. So, if you say that’s the definition of a rhombus, that’s how I know it is one, it’s kind of assuming what you have to prove.

12 Participant 12: The key thing we want to prove is that . . . Rhombus has a few

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P# EU ESD EUSD characteristics, but the easiest one that I could think of right away is that ABCD, all the sides are congruent in ABCD. Investigator: Okay. If you prove that? Participant 12: That would be sufficient enough to prove that ABCD is a rhombus. Investigator: Okay. Can you comment about how you know that would be sufficient? Participant 12: All right. I remember from . . . Well, I actually learned this from . . . I refreshed my knowledge from student teaching, because the biggest misconception is that students have trouble remembering what’s a rectangle, what’s rectangle, square, rhombus, which is which? Investigator: Mm-hmm [affirmative]. Participant 12: The way I could think of it is that a square has characteristics of a rectangle and characteristics of a rhombus. So, the rectangle is where the right angles come from, and the rhombus is where the congruent sides come from. Investigator: Okay. Participant 12: I know that that’s true, because in a square, all the sides are congruent. That’s how I remembered it.

20 Now I’m thinking that if I can just prove that a pair of consecutive sides are congruent, then ABCD is a rhombus, since it’s already a parallelogram.

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P# EU ESD EUSD 21 Participant 21: So, okay, so

what am I want to prove that ABCD, is around this. So now I need to think of, how am I going to prove it to be a rhombus? I need to think about the properties. Right. What are some properties of a rhombus? All sides are congruent, diagonals, create 90 degrees. Right? And they bisect each other. Investigator: Okay. Participant 21: Sides are parallel. Investigator: Okay. Participant 21: Opposite sides are parallel and opposite angles are supplementary. Investigator: Okay. How do you know those characteristics? Participant 21: Because I just taught this unit.

21

Investigator: So, you’re, you’re wondering would it be sufficient for reason for you to write “definition of parallelogram.” Participant 21: Yeah. Investigator: Okay. Participant 21: I never know.

21

Participant 21: And now I think I’m done. Investigator: Why are you done? Participant 21: Because I’ve now shown that two consecutive sides of a parallelogram are

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P# EU ESD EUSD congruent. Investigator: Okay. And does that make it a rhombus? Participant 21: I think so. Investigator: Okay. Why do you think so? Participant 21: But I don’t think it’s sufficient.


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Table D19


P# ML+ ML- 4

Investigator: Can you just tell me what is the transitive property? Participant 4: It’s kind of hard to explain I think but . . . like it’s kind of like if I study then I’ll do my homework, then I’ll do my homework then I’ll pass the test, and then by transitive property, because . . . I don’t know how to explain it but then if I study then I’ll pass the test.

6

Participant 6: Which means . . . So, I proved those two angles. Oh, I have to say that Angle C is reflective. Investigator: What’s the word? Participant 6: It means that it’s congruent to itself. Reflective.

6

I guess I could just say angles in a triangle are 180 degrees.

12

Investigator: Okay. Great. Just a question, so in the first proof you did and then this one. When you got down to writing substitution as a reason, you seemed upset about it. Why? Participant 12: There’s definitely a nicer, more precise way to say it. I’m just trying to . . . Investigator: Why do you think that’s imprecise? Participant 12: So, you’re saying it’s right? Investigator: Yeah. That’s what you’re doing, right? Participant 12: I’ll be honest, I never saw that before. Investigator: You never saw it as a reason? Participant 12: Yeah, so that’s why . . . I’m pretty sure that has to be the statement. It’s just I feel like a reason would be . . . There’s I guess a more geometrically professional way to say it. Investigator: Okay, so it’s like you don’t feel that it’s up to the standard of mathematical language? Participant 12: No.

17

The other thing that I think might help, is Angle C. It’s not congruent to anything, but I just feel like I’m gonna use the fact that it’s reflective to itself in my proof, ‘cause I think it’s positioning us to prove that Triangle BEC is congruent to triangle DFC.


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Table D20


P# PMI 1 Participant 1: So ABCD. And then I’m thinking that I have to prove that it’s a square.

Investigator: What’s a square? Participant 1: The parallelogram is a square. Investigator: Okay.

1 Investigator: Okay. So, if you have a quadrilateral with four congruent sides, it’s a square? Participant 1: Yes. Investigator: Okay. Participant 1: Oh, wait. No. Investigator: No? Participant 1: Because all the angles must be 90 degrees. Investigator: In order for it to be what? Participant 1: A square. Investigator: Oh. So, is that a type of quadrilateral? So, you have a quadrilateral with four congruent sides. Participant 1: Mm-hmm, with four congruent sides. But it needs to have four right angles. Investigator: In order to be what? Participant 1: A square. Investigator: Okay, so if it has four congruent sides but not four right angles . . . Participant 1: So, it has to be a rhombus? Investigator: Is that a rhombus? Participant 1: I don’t know. Investigator: What other types of quadrilaterals do you know? Participant 1: A rectangle, a rhombus, square, a kite . . . This is horrible. Investigator: Why do you think it’s horrible? Participant 1: Because I really don’t like geometry. Hold on. I need to think. Investigator: You seem very perseverant even though you don’t like geometry. Participant 1: Yeah, because it’s interesting. I just don’t know what to think. Wait, so my different types of quadrilaterals. I only know the main ones. Investigator: What are the main ones? Participant 1: A rectangle, a square. I don’t know why I know kite, that’s so random. A rhombus. A trapezoid. Really don’t know the rest. Investigator: Okay. So, what would you need to know— Participant 1: For rhombus, I know that the diagonals must be perpendicular. Investigator: Okay. Participant 1: That’s really, and aren’t rhombuses squares? Something has to deal with a square ‘cause I don’t know why it’s coming to my head.

2 Participant 2: Prove ABCD is a rhombus. In a rhombus . . . I’m going to write down the characteristics. Investigator: That’s a great strategy.

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P# PMI Participant 2: All sides are equal. Opposite sides are parallel. Now, I know that it’s a pulled square. Investigator: A what? Participant 2: Pulled square. That would mean that angles are congruent? I’m not sure about that.

2 Investigator: You’ve noted that all sides are equal, opposite sides are parallel. It’s kind of like a “pulled (or slanted) square,” and that opposite angles are congruent. If you want to prove ABCD is a rhombus, do you think that you need to prove all of these, or one of them? Participant 2: Well, if I just prove all sides are equal then it could be a square. Now, is a square a rhombus? No, I’d say a rhombus is a square. No, that’s not true. I don’t think just one would prove that it’s a rhombus. Opposite sides are parallel. Well, that could be a rectangle. That could be a square. It could be a rhombus. Investigator: Anything else it could be? Participant 2: If opposite sides are parallel? Investigator: Mm-hmm [affirmative]. Participant 2: Rectangle, rhombus, square . . . Yeah, what do you call that, the trapezoid. No. No. Investigator: Why not? Participant 2: Only one pair of opposite sides are parallel in a trapezoid. Investigator: Do you by chance know, I’m not sure if you’ve heard— Participant 2: Oh, a kite. Investigator: Oh, a kite is a different quadrilateral. I just want to talk about what you said about trapezoid for a second. We’re digressing a little bit, but the common core, when common core geometry came out there was a lot of debate over the definition of trapezoid, and it was actually decided that a trapezoid is a quadrilateral with at least one pair of parallel sides, meaning that it could be one pair of parallel sides, or it even could be two. Participant 2: Okay, well that changes everything then. Investigator: Why does that change everything? Participant 2: Because if it could be at least two then it could be a rhombus. No, it could be a square. Wait, it confuses me, because it was into my knowledge bank for so many years that trapezoid only has one pair of parallel sides. Investigator: Exactly one. Participant 2: Right, so everything that came after that was confusing, and now what is a trapezoid and what is not a trapezoid became something that I never really delved into.

3 Investigator: Okay. What are you given about ABCD? Participant 3: Oh . . . it’s a parallelogram. Investigator: So, does that help with something else you were looking for? Participant 3: Yeah. So, then I already know that those are congruent.

5 Investigator: Okay. What are your thoughts when you see the next one? Participant 5: Oh boy. Investigator: Why oh boy? Participant 5: I don’t know what a rhombus is.

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P# PMI Investigator: Okay. So, can you write that down? Participant 5: I don’t know what a rhombus is? Investigator: Yeah, please. You can make your own if you want more. And what do you mean when you say you don’t know what a rhombus is? Participant 5: Because I . . . Is it a square? Is it a rectangle? I feel like this is something we’ve talked about in grad school, and I still don’t know. So, I picture a diamond in my head, but that’s an overgeneralization, right? Investigator: A diamond. Okay. So, I know that you teach algebra now this year for the first time and that you student taught in a middle school. Did you talk about quadrilaterals and their properties at all with your students ever? Participant 5: Because it’s not a big part of the— . . . Although it’s in the seventh and eighth grade curriculum, it’s not tested in the state exam very often. Investigator: Okay. So, it wasn’t a focus. Participant 5: So we . . . Yeah, it wasn’t a focus. Investigator: Okay. So is this . . . Does that barrier stop you from attempting this? Participant 5: I mean, I’ll attempt it, but . . . Investigator: Okay. But, so your initial planning step, what you did for the last proof would kind of be impossible here. Participant 5: I don’t remember . . . Yeah, I don’t remember how you prove something’s a rhombus. Alternate . . . I don’t remember. Investigator: Okay. Participant 5: So, I guess I’m stuck. Yeah, I can’t do it.

7 Participant 7: Because they both come from C. That reason would be reflexive, let’s write it out. So then, oh. So then could I prove these triangles are congruent using angle-side-angle? Investigator: What do you think? Participant 7: I feel like it’s okay. I feel like I’m hesitating because this is a right angle, but if it was just a regular angle I would say absolutely.

8 BE is perpendicular to CED. DF is perpendicular to BFC and CE is congruent to CF. All right so prove ABCD is a rhombus. So, the first thing I would do is recall the properties I know for a rhombus and maybe just jot it down just so I don’t forget. So, I know that a rhombus is a parallelogram. So that can be one of the first things I could try to prove.

8 Well, after I prove a parallelogram then I would just need to prove one thing that it’s a rhombus however I would have to be care which property of a rhombus I use to make sure it doesn’t overlap with let’s say a square. Because I know a square also has four congruent sides. So, I’m not sure if I would be able to use that because it could technically also be a square. But I believed diagonals perpendicular to each other I think that is a special case for rhombus and can’t be applied to a square or a rectangle.

8 And before I, I said something by accident how, “Oh this could be a square, how would I know?” But rhombus I believe, like if we were to form let’s say a tree diagram, after parallelogram would come rectangles . . . Anyway, so a square is a special, I don’t know if I’m saying this backward, but a square is a special type of both a rectangle and a rhombus.

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P# PMI 9 Participant 9: I think if you can show that the diagonals bisect each other.

Investigator: Okay. Participant 9: I think that’s like one unique feature of the rhombus. Yeah, I think that’s what I’m . . . No, but in a square, they also bisect each other. Investigator: So, when you say unique feature, what do you mean? Participant 9: That like compared to all the other quadrilaterals the rhombus is the only one that has whatever that feature is that makes it unique to that quadrilateral. Investigator: So, I just want to make sure I’m interpreting what you’re thinking correctly. You think that there’s some characteristic that will make the shape a rhombus and not any of the other quadrilaterals, and that’s the one that you need to prove to show that it is a rhombus? Participant 9: Yes.

9 Investigator: Okay. So let’s say you were tasked with designing a unit about quadrilaterals or proofs about quadrilaterals, what do you think some major takeaways would be for students? Participant 9: Well, what do you mean by take away? Investigator: Like, what are the important things that the student should walk away with? Participant 9: Oh, they should be aware of characteristics for each of the quadrilaterals. Investigator: Okay. Participant 9: And why . . . Now I’m just saying this just because we had this conversation, but why a square is a square, why it may or may not be considered a rectangle, because I don’t know. Investigator: Okay. Participant 9: Or why a rhombus isn’t a parallelogram. Because when I was a student, I always felt like a rhombus and a parallelogram are like they’re the same thing. Investigator: Why? Participant 9: Because they’re slanted quadrilaterals.

11 Participant 11: Okay, BE is perpendicular to CED, which means that forms a 90-degree angle. Then, DF perpendicular to BFC, that’s another 90-degree angle. And then, CE is congruent to CF. Prove that ABCD is a rhombus? I don’t even know parallelograms and rhombus . . . I probably did. Yes, okay. Investigator: Wait, tell me more about how you’re feeling. Participant 11: It says parallelogram ABCD, but now we’re proving that ABCD is also a rhombus. I don’t know, I guess it never really crossed my mind that a rhombus is a parallelogram. Yeah. Investigator: I just want to note the time down because that’s a very interesting statement. You haven’t seen a proof like this before? Participant 11: Mm-hmm [negative]. Investigator: Where you have to prove that one shape is in fact also another. Participant 11: Another shape, yeah. Investigator: Okay. Have you seen proofs about quadrilaterals before? Participant 11: Yes. Yes. Investigator: Okay. Do you know properties of parallelograms and/or rhombuses? Participant 11: I know that in parallelogram that opposite sides are parallel. Opposite

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P# PMI sides are congruent. Opposite angles are congruent. And that their diagonals are congruent. Investigator: Okay, and how do you know those things? Participant 11: From tutoring. Investigator: From tutoring, okay. Participant 11: Yeah. Investigator: And do you know properties of a rhombus? Participant 11: I think that their diagonals are not congruent . . . Investigator: Okay. Participant 11: Oh no, wait. No, no, no, no, I have to back out. Investigator: Okay. Participant 11: Wait, wait, hold on. Investigator: Sure. Participant 11: No. In a rhombus, all four sides are congruent. Investigator: Okay. Participant 11: But, in a parallelogram, diagonals are not congruent . . . Investigator: Okay. Participant 11: Because not all four sides are congruent, which means, if it’s stretched then the diagonals are not congruent. Investigator: Okay. Participant 11: Yeah. Okay. For a rhombus, since all their sides . . . Oh no. Can that be stretched? I’m not sure. Now, I’m just confusing myself. Investigator: Okay. Participant 11: Yeah. Investigator: Confusing yourself, between . . . Participant 11: I think the diagonal part is, I don’t know if they’re congruent. Investigator: Okay. Participant 11: Yeah. Investigator: What would you need to know, in order to prove this? Participant 11: I think, I’m thinking I can’t just prove that the sides are congruent, because that’s just a square. Investigator: Okay. Participant 11: And, I think, I have to prove that opposite angles are congruent. And then I’m also, it’s not . . . I already know it’s a parallelogram. I already know that . . . Investigator: You know that because it tells you? Participant 11: It tells me, yeah. Investigator: Okay. Participant 11: Since it is a parallelogram, I feel like I already know that their opposite angles are congruent. Investigator: Okay. Do you think it’s . . . Participant 11: I don’t know. Investigator: Easier or harder to make a plan for this proof than the last one? Participant 11: Definitely. I think it’s because I don’t know enough. I don’t know what I need to prove that it’s a rhombus, that’s a rhombus.

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P# PMI 14 Participant 14: Okay, so they tell us that BE is perpendicular to CED. So, attempt to draw

some right angles. ZF is perpendicular to EFC. Another pair of right angles. Okay. So, we have a bunch of 90-degree triangles. And then CE is congruent to CF. Okay. I don’t remember how to prove something’s a rhombus. Investigator: Okay. What would you need to know in order to start thinking about this, then? Participant 14: I always forget. So, my thing is to prove that it’s a parallelogram first. Investigator: Okay. Participant 14: Or do I have to prove, I always confuse is a rhombus a square. I always forget that. Like I think of my geometry teacher in my head, but I always forget. And then it’s also if it is a square, do I have to prove that it has exactly four right angles, which I don’t think I have to.

18 Participant 18: Okay. All squares are rectangles, but not all rectangles are squares. Do you want me to talk about properties of individual types of quadrilaterals such as . . . Investigator: Sure. Participant 18: Okay for example when in a rectangle all the angles are 90 degrees, congruent. Opposite sides are congruent, diagonals are congruent. Quadrilaterals let me think about that. All parallelograms have, no that’s not true. Maybe it is true. Yeah, I think it is true. Investigator: Maybe what is true? Participant 18: That the diagonal thing. I was gonna say . . . Investigator: What diagonal thing? Participant 18: All parallelograms, have congruent diagonals. Investigator: Okay. Participant 18: And I’m saying this on a hunch mostly. Investigator: Okay. Participant 18: Like if I really wanted to, I could say okay let’s try some cases, but I really don’t want to, and I think I’m right.

21 Investigator: Okay. So, you wrote down all these properties of a rhombus. How many of them do you have to prove? Participant 21: One. Investigator: Any one? Participant 21: No, because all sides are congruent. That could prove a square. Investigator: If something’s a square. Is it a rhombus? Participant 21: If something is a square is it a rhombus? No.

21 Investigator: So, you think you need to show that Angles A, B, C, and D are not right angles to show that this is a rhombus, not a square. Participant 21: Yes.

24 Investigator: And that a rectangle has . . . What’s something about a rectangle that you’d be willing to bet on? Participant 24: Opposite sides are parallel. Participant 24: And that one set of opposite sides are longer than the other side.

329

P# PMI 29 Participant 29: I mean, then we have that those two sides are congruent. I don’t feel like

it’s enough. Investigator: Why not? Participant 29: To prove that it’s a rhombus. Just because those two sides are congruent, we leave out the fact that they need to be parallel. Not those sides, but the opposite sides are parallel, and that the other two are also congruent. Investigator: Can you list all the givens in this problem? Participant 29: Yeah. So, the ones that are stated in the problem? Investigator: Mm-hmm [affirmative]. Yeah, anything you’re given. Participant 29: BE is perpendicular to CED. DF is perpendicular to BFC. CE is congruent to CF. Then, also, what I can deduce from that . . . Okay. Investigator: Are those the only givens? Or is there anything else that’s given? Participant 29: Oh my God. Parallelogram. ABCD.


330

Table D21


P# BAA 1 Participant 1: I don’t want to deal with circles.

Investigator: What did you just say? Participant 1: I don’t want to deal with circles. Investigator: So, you saw that one of the remaining two proofs has a circle? Why don’t you want to deal with that? Participant 1: ‘Cause I hate circles. Investigator: Why do you hate circles? Participant 1: I just don’t like this. Investigator: You don’t like this at all? Participant 1: Geometry. Investigator: Why? Participant 1: Because it’s hard. Investigator: You think it’s harder than— Participant 1: It puts me at a level of discomfort. Investigator: Okay. Yeah, I think it’s important to talk about that. Do you think that this is common, or you think it’s unique to you? Participant 1: Common. Investigator: Why do you think geometry makes people uncomfortable? Participant 1: Geometry puts people in a state of feeling lost and uncomfortable, and I feel like their thinking process is blocked. Investigator: Why do you think that geometry blocks people more than say, algebra? Participant 1: ‘Cause it’s a different way of thinking.

1 Participant 1: Yeah, I’m not doing this. Investigator: Which one? Participant 1: This one. Investigator: Proof number 4? You’re not doing that. Why not? Participant 1: I don’t know where to begin. Investigator: Okay. What about this makes you not know where to begin? Participant 1: The whole thing. I don’t know. I really just don’t know my circle facts. Investigator: So, you see a circle and that’s it? Participant 1: Yeah.

2 First of all, I don’t know what secants are, so that’s tripping me up here. If I knew more about secants I might want to delve more into here. It’s wordy, the problem. . . . Yes, I don’t know how to proceed because secant . . . I’m finished. I can’t get past that word right now.

4 I feel like I remember nothing about circles.

9 Investigator: Okay. How do you feel about this proof compared to the other ones you’ve done? Participant 9: I feel like it’s a circle. And that circles are complicated.

331

P# BAA 11 Participant 11: I feel like, even now, with the students I tutor, we don’t focus on circles

that often, either. Investigator: Okay. Participant 11: And it’s not the best advice, but when it comes to, I guess, passing the Regents it’s just, don’t worry about it. If you can get everything else right, then don’t stress it, because if you feel like you can get everything else right, then it’s fine. But I guess for a student who really wants to get everything right, then they would want to learn it. But most of the students I tutor, it’s their algebra skills that really get to them. Or sometimes it is the proofs, and mostly they only give triangles or quadrilaterals, so it’s . . . Yeah. Circles.

15 Participant 15: I don’t like circles. Investigator: You don’t like circles, why not? Participant 15: I don’t know, I’m bad at circles. Investigator: Why, what about them? Participant 15: It’s just so . . . I don’t know, just since I was in high school, I’ve been bad at circles.

16 Yeah, I’m struggling with the . . . I just feel like anything inscribed in a circle becomes really abstract for me, because it’s like a million different properties that you keep in mind and then the fact that there’s nothing like labels, concretely. I’m just like I don’t even know where to start. And then I’m looking, and in my head, I’m making assumptions that I know in geometry you can’t necessarily make.

18 Participant 18: Yeah. So, you have your equal, your similar, and your congruent. Investigator: Oh. Participant 18: Yeah. I mess it up all the time. Investigator: Why do you think you mess them up? Participant 18: Because I don’t care. I frankly don’t care.

20 Participant 20: I have no idea. I really hate circles, so please don’t make me— Investigator: You hate circles? Participant 20: Yeah, so please don’t make me do this. Investigator: So even before reading it, you’re turned off to the problem. Participant 20: It’s a circle. There’s some chords or something sticking out. I don’t know. Investigator: Okay, is this something that you have already taught this year, or no? Participant 20: No. Investigator: Okay, do you think you would be more prepared after teaching it? Participant 20: No, I would have to research it first . . . Oh, yes. I think after I researched and taught, and trial and error, I would be okay. Investigator: Okay, you would feel more comfortable. Participant 20: As of right now I do not feel comfortable.


332

Table D22


P# EU ESD 2

I don’t know anything about circles other than diameter, pi, circumference, chords I know.

6 Participant 6: I’m not just proving the HE and HD are congruent or the sum. I feel like this is harder because I’m proving the multiplication. Investigator: Okay. Participant 6: Which I think means I need to show similar triangles, and then set proportion and then cross-multiply or something. Investigator: Why do you think that? Participant 6: Because it’s the only way to get multiplication with triangles, or the only one I can think of so far—

12 Participant 12: I also know, like I said, the exterior . . . The intersecting angular thing, but that’s not going to help me here. I’ll just go back to it. Secants HBD and HCE are drawn. All right. I’m cheating a little bit so I can modify . . . The thing we want to prove, so I could cheat a little bit and work backwards. Investigator: Okay. Participant 12: I can modify that equation that I want to prove into HC divided by . . . The length of HC divided by the length of HD is equal to the length of HB divided by the length of HE. Let me just make sure that’s correct. HC divided by HD, HE . . . All right. I don’t know why, but that’s intuition saying that that should be one of the things I should do. Investigator: Okay. Participant 12: HC over HD, and HB over HE. I don’t know if . . . Looking at that from what I’ve been . . . Lately I’ve been teaching about . . . In student teaching, I’m looking into lessons of similar triangles. Investigator: Okay. Participant 12: That’s a statement of proportionality, and I could see its two triangles.

15 Participant 15: Wait, HE times HC, that means HE. . . Equal to HB over HC. That means we need to . . . ohhhhh, we need to prove these two triangles are similar. Investigator: How did you figure that out so quickly? Participant 15: From this. . .

333

P# EU ESD Investigator: So, you used the multiplication statement at the end? Participant 15: Yeah, though I didn’t read the theorem, the theorem is complicated . . .

21 Participant 21: I believe, I’m thinking I need to use similar triangles. Investigator: What tells you that right away like how do you know that? Participant 21: I just remember it.

23 Participant 23: I took something with . . . A class that had inversions with circles and stuff, and I remember we would do these kind of similar triangle stuff all the time. Investigator: Okay, that class makes you think of similar triangles when you see this? Participant 23: Yeah. Investigator: Okay. Participant 23: I definitely would just go for that approach. Alright, HE . . . Okay. See I’m gonna use similar triangles because then all I need to do is get the whole thing to a piece, which is right here, and then the whole thing here to this piece. If they are two similar triangles then I know that the ratios to the corresponding sides are equal and then I can multiply that, and get that.

29 I’m not sure. My instinct is telling me to make triangles. I’m thinking connect BC and DE. Anytime I see multiplying, I’m thinking this is probably going to come from proportions, and then proportions guide me to similar triangles.

Note. P# = participant number; EU = expressing understanding; ESD = expressing self-doubt.

334

Table D23


P# ML+ ML- 28

Oh, that’s a good question. What would be similar? Oh! Well, okay, this would sound, I don’t know if it sounds stupid but the distance from H to C, I think, would be similar to the distance from B to D if that makes sense or, I mean, from H to D. So, it’s almost as if this, the HC, is almost in ratio with the HD so that’s what I would assume ‘cause I don’t really know how to solve this. So, because if they’re in, I would say, similar ratio, I would also assume that HB is in ratio with HE and that’s what I would assume and that’s why I would say that they’re equal but I wouldn’t know how to . . . if there was any other reason, I wouldn’t know how. I don’t . . . see. I mean, I’m sure that there’s triangles involved in angles because the fact that they’re inscribed in the same angle, that means that their angle measure is the same and then . . . oh yeah.


335

Table D24


P# PMI 5 Participant 5: In a circle K shown below points B, C, D, and E. Line the circle with

secants HBD and ACE. I don’t remember any of the circle properties, with the secant, with what’s . . . There’s some formula where double this . . . I don’t remember. Investigator: Okay. Participant 5: Chords BE and CD are drawn. There’s something about if you draw chords inside of a triangle which angles are congruent . . . I don’t remember.

7 Investigator: Okay so why did this proof make you think of similarity because you haven’t mentioned that yet. Participant 7: I think because eventually you prove that the triangles are similar, which they definitely are because without knowing the theorems, I know that DBE and DCE, those two angles are congruent. I know that BAD and CAE are congruent which means that these two inner triangles, triangle BAD is similar to CAE. I don’t know if that can help me. Investigator: What do you know about similar triangles other than they have two congruent angles? Participant 7: Well, they have the same angle measure, just not the same side. . . the same side length. So, they can be different sizes but the same shape. That’s generally what students say. Investigator: Okay, and is there any relationship that has to hold between the sides? Participant 7: I don’t think anything more than a standard triangle, meaning that if you add up two sides, that sum has to be larger than the third, but I don’t think that’s what . . .

9 Participant 9: But they’re similar by angle, these two triangles, so that would mean that BD and CE are proportional. Investigator: Okay. What does that mean? Or how does that help? Participant 9: I’m not sure how it would help though. I just know that they’re proportional.

14 And then I know that there’s, I think it’s with a secant, some rule about the relationship between the angle created, and I think the segment outside. . . . I don’t remember.

24 Investigator: So, do you have any initial thoughts about this? Participant 24: I wouldn’t even know where to start. Investigator: Do you find that this one is more intimidating than the other ones? Participant 24: Yes. Investigator: Why do you think that’s the case? Participant 24: Because I can’t remember anything about secants. Investigator: Okay. Like their properties? Participant 24: Yes. Investigator: Okay. Do you know what a secant is? Participant 24: No.


336

Appendix E: Frequency of PRTs and GPTs in End-of-Section Exercises

Table E1

Practice Problems at the Conclusion of Every Section

Section

PRTs GPTs

n % n %

1.1 (Part 1) 2/10 20% 0/10 0%

1.1 (Part 2) 1/24 4% 0/24 0%

1.2 (Part 1) 2/13 15% 0/13 0%

1.2 (Part 2) 1/18 6% 0/18 0%

1.2 (Part 3) 0/11 0% 0/11 0%

1.3 2/24 8% 0/24 0%

1.4 1/34 3% 0/34 0%

1.5 2/45 4% 0/45 0%

1.6 (Part 1) 2/32 6% 0/32 0%

1.6 (Part 2) 2/9 22% 0/9 0%

1.7 2/31 6% 0/31 0%

2.1 0/25 0% 0/25 0%

2.2 1/24 4% 0/24 0%

2.3 (Part 1) 2/27 7% 0/27 0%

2.3 (Part 2) 1/8 13% 0/8 0%

2.4 4/20 20% 0/20 0%

3.1 7/23 30% 0/23 0%

3.2 7/35 20% 0/35 0%

3.3 5/18 28% 7/18 39%

3.4 6/36 17% 2/36 6%

4.1 3/32 9% 4/32 13%

4.2 2/28 7% 4/28 14%

4.3 1/36 3% 2/36 6%

337

Section

PRTs GPTs

n % n %

4.4 5/34 15% 1/34 3%

4.5 3/35 9% 2/35 6%

5.1 (Part 1) 0/14 0% 0/14 0%

5.1 (Part 2) 0/20 0% 2/20 10%

5.2 1/14 7% 2/14 14%

5.3 13/32 41% 4/32 13%

5.4 21/33 64% 5/33 15%

6.1 1/27 4% 2/27 7%

6.2 1/25 4% 1/25 4%

6.3 2/28 7% 0/28 0%

6.4 1/31 3% 1/31 3%

6.5 1/21 5% 1/21 5%

6.6 (Optional) 2/20 10% 3/20 15%

6.7 (Optional) 2/25 8% 4/25 16%

7.1 0/33 0% 3/33 9%

7.2 1/38 3% 5/38 13%

7.3 (Part 1) 0/26 0% 1/26 4%

7.3 (Part 2) 0/9 0% 0/9 0%

7.4 4/34 12% 1/34 3%

7.5 1/35 3% 0/35 0%

7.6 0/53 0% 1/53 2%

7.7 (Optional) 0/28 0% 0/28 0%

7.8 0/39 0% 1/39 3%

8.1 0/32 0% 1/32 3%

8.2 0/37 0% 4/37 11%

8.3 0/34 0% 1/34 3%

8.4 0/25 0% 0/25 0%

338

Section

PRTs GPTs

n % n %

8.5 0/42 0% 0/42 0%

9.1 0/52 0% 6/52 12%

9.2 0/37 0% 7/37 19%

9.3 0/27 0% 12/27 44%

9.4 (Optional) 0/41 0% 1/41 2%

9.5 6/38 16% 3/38 8%

9.6 0/39 0% 1/39 3%

9.7 0/29 0% 1/29 3%

9.8 0/28 0% 0/28 0%

10.1 0/27 0% 0/27 0%

10.2 0/40 0% 0/40 0%

10.3 0/41 0% 0/41 0%

10.4 1/33 3% 0/33 0%

10.5 0/28 0% 1/28 4%

11.1 0/33 0% 0/33 0%

11.2 0/31 0% 1/31 3%

11.3 0/29 0% 1/29 3%

11.4 (Optional) 0/43 0% 0/43 0%

11.5 0/26 0% 0/26 0%

11.6 0/26 0% 0/26 0%

12.1 0/8 0% 0/8 0%

12.2 0/51 0% 0/51 0%

12.3 0/22 0% 0/22 0%

12.4 1/20 5% 0/20 0%

12.5 0/32 0% 0/32 0% Note. All percentages are rounded to the nearest whole number.

339

Appendix F: Frequency of PRTs and GPTs in End-of-Chapter Problem Sets

Table F1

Review Problems at the Conclusion of Each Chapter

Chapter

PRTs GPTs

n % n %

1 5/38 13% 0/38 0%

2 1/28 4% 0/28 0%

3 4/36 11% 1/36 3%

4 3/29 10% 5/29 17%

5 1/30 3% 3/30 10%

6 0/36 0% 4/36 11%

7 0/30 0% 0/30 0%

8 0/28 0% 0/28 0%

9 1/35 3% 3/35 9%

10 0/30 0% 0/30 0%

11 1/43 2% 0/43 0%

12 0/30 0% 0/30 0%

Note. All percentages are rounded to the nearest whole number.

340

Table F2

Cumulative Review Problems at the Conclusion of Each Chapter

Chapters

PRTs GPTs

n % n %

1-2 1/29 3% 0/29 0%

1-3 2/29 7% 2/29 7%

1-4 7/34 21% 2/34 6%

1-5 3/38 8% 0/38 0%

1-6 2/29 7% 7/29 24%

1-7 1/29 3% 2/29 7%

1-8 0/31 0% 2/31 6%

1-9 1/35 3% 1/35 3%

1-10 3/32 9% 0/32 0%

1-11 1/36 3% 2/36 6%

1-12 0/36 0% 3/36 8%

Note. All percentages are rounded to the nearest whole number.

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