knowledge construction in high school physics: a study of
TRANSCRIPT
Knowledge Construction in High School Physics:
A Study of StudenüTeacher Interaction
A Dissertation
Submitted to the Faculty of Graduate Studies and Research
in Partial Fulfilment of the Requirements
for the Degree of Doctor of Philosophy
in Curriculum and Instruction at the
University of Regina
by
Warren Edward Wessel
Regina, Saskatchewan
May, 1998
Copyright 1998: W. E. Wessel
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A bstract
This study is a description and analysis of student leaming when
required to use vector mathematics to represent two dimensional situations in
the solution of grade 12 physics problems. Coupled with this exploration, the
role of a teacher as a facilitator in creating effective conditions and interactions
to facilitate student knowledge construction was critically analyzed. Nine grade
12 physics students volunteered to participate in a process of articulating their
reasoning and problem solving strategies over a sixteen week period in a
regular secondary school classroom setting. The participants were taught the
normal content of the Saskatchewan grade 12 physics curriculum by the
researcher who is an accredited, experienced physics teacher. Data were
collected by video recording of classroom sessions, interviews. student
assignments, and field notes maintained by the researcher.
Student leaming is described through a combination of excerpts of
student discourse and data collected from other sources during the study.
lnterpretation of student-teacher interaction is informed by a constructivist
perspective of student knowledge construction and conceptual development in
science education, and the personal teaching experience of the researcher.
Student leaming during increasingly complex use of vector mathematics is
described. The sequence of topics begins with vector addition and subtraction,
and problems requiring those functions for solution. Vector components are
then developed using a combination of classroom activities and interactive
discussion. The final topic developed is mornentum.
Students were found to have well developed experiential knowledge
which interfered with their construction of conceptual knowledge. Concrete
examples did not guarantee that students would develop conceptual
i
understanding of a given phenornenon. Students used algorithrns
indiscriminately and often did not know if their answers were reasonable. When
momentum was introduced, the students showed a variety of attempts at
constructing the concept. They did not appear to understand the cornplex
process of mathematical representation during the study even when they were
instructed about the process. Students confused the meanings of equal and
balanced which led to mistakes in writing vector equations representing
relationships between forces. They did not seern to understand why vector
mathematics had to be used in solving problems even when they correctly
employed them in calculations. Transfer of knowledge frorn mathematics
classes to physics classes was almost non-existent.
In the final chapter recommendations for changes to physics curkulurn
and instructional strategies are presented. Student difficulty in applying vector
mathematics to physics problem is explained in ternis of student difficulty in
understanding direction as a characteristic of sorne physics concepts and in
representing physics concepts using mathematical models. Suggestions for
future research include development of instruction to facilitate student
understanding of mathematical representation and metacognitive ski11
development by students.
Acknowledgements
I would like to acknowledge several people who assisted and supported
me during this project. My supervisor, Dr. Paul Hart, and cornmittee members,
Dr. Marlene Taylor and Dr. Fred Bessai provided extensive constructive
feedback which helped me shift my focus between the forest and the trees and
resulted in pushing the final work beyond what I could have achieved alone.
Committee members, Dr. Bev Robertson and Dr. Reg Fleming gave astute
feedback at timely moments assisting in the construction of the dissertation.
I must also thank my wife, Floralyn, for her patience, and my children,
Shana and Jason, for their encouragement during the long journey.
1 wish to acknowledge the receipt of an SSHRC Doctoral Award received
in support of my doctoral program.
I would also like to acknowledge the cooperation of the Board of
Education of the Regina Schooi Division for allowing me to carry out the
research project in their division and for granting me an educational leave in the
1992-93 school year. Lastly I would like to thank the nine grade 12 physics
students without whose participation the research would never have been
possible.
iii
Table of Contents
......................................................................... Absbact ...............................,.... i
Acknowiedgements .......................................................................................... iii
Table of Contents ............................................................................................. iv
List of Figures ................................................................................................... xi
................................................ . Chapter One Introduction to the Study 1
.................................................................................... Purpose of the Study 3
..................................................................................... Context of the Study 4
Rationale of the Study .................................................................................. 8
Significance of the Study .......................................................................... 11
Assumptions .................................................................................................. 12
Delimitations and Limitations of the Study ............................................... 12
................................................ . Chapter Two Review of the Literature
..................................................................... Student Leaming in Science
...................................................... Eariy views of Student Leaming
....................... Ausubel's and Piaget's Views of Student Leaming
................. Research Exploring Student Alternative Conceptions
Hear and Temperature ............................................................... Force and Motion ........................................................................
..................................................................... Gravity and Energy
P hysics Vocabulary .................................................................... ...................................................... Problern Solving in Physics
......................................................................... Vector QuantÏties
...................... Characte ristics of Student Alternative Conceptions
iv
........... lntegrating Alternative Conceptions in Views of Learning
................. lncorporating Alternative Conceptions into Instruction
.......... lnstructional Strategies and Classroorn Atmosphere
..... A Teaching Model and Practical Example from Physics
Student Assessrnent ............................................................... m e r Consequences of Alternative Conception Research .
... Teachers' Involvement in Developing lnstructional Strategies
Constnrct~sm in Science Education .......... .. ....................................... Beginnings of Constructivism in Science Education Literature ..
................................. Von Glasersfeld and Radical Constructivism
......................... A Constructivist View of Knowledge Construction
Social Construction of Knowledge ................................................... Methods of Studying Learning in Science ...............................................
General Research Methods ............................................................... ............................. Research Methods Informed by ConstmctMsm
............................ Researchers as TeacherdTeachers as Researchers
Chapter Three . The Research Method ................................................ 54
Introduction .................................................................................................... 54
The Development of the Project ................................................................ 56
Obtaining Approval .......................................................................... 56
The Choice of Prairie view Collegiate ............................................. 57
Selection of Student Participants ....................... .. ........................ 58
Records of the Study .................................... ... ............................................. 60
Videotape Records ........................... .. .............................................. 61
Persona1 Journal .............................................................................. 62
................................................................... Test and Quiz Resuits
........................................................................... Laboratory AcZivities
............................................... Demonstrations ................................... ............. Classroom Interactions between Students and Teacher
....................................................... Student 'Think Aloud" Sessions
....................... .. ....................* Problems Using Two Principles .... .. .................................. Interviews Used During the Research Study
Making Sense of the Data ..................................................... ................... Preparing the Transcripts ................................................................... Creating Meaning, Seeking Understanding ................................... The Next Stage in the Anaiysis ........................................................
.............. Chapter Four O The Research Environment and Context
.................................................................................................... Introduction
...................................................... The Setting of Prairie View Collegiate
The Student Participants ............................................................................. The Individual Participants .................................................................
.......................................... An Illustration of Participant Knowledge
................................................................... Students and the Context
................................................... ................. The Teacher-Researcher ..... My Contribution to the Environment .................................................
...................... . Chapter Five Constructing Meaning from the Data 92
Introduction .................................................................................................. 92
The Initial Phases of the Çtudy ................................................................... 94
The First Session O Friday Aftemoon ............................................... 94
Developing Rapport ........................................................................... Setting the Stage for Vectors in Physics ..............................................
Exploring the Place of Numbers in Physicç ................... .. ............. Chalk Brushes. Addition and Subtraction .................... .. ..............
Leaming Concems ldentified in the Initial Sessions ............................. Visualizing as a Problem Solving Aid ............................................ Sarne Words. Different Meanings .................................................... ldentifying My Error in Understanding Student Communication .
Using Discussions to Build Models of Students' Knowledge ............... .. .................................... Addct~on of Vectors ....... ....................................
Subtraction of Vectors ......................................................................... Comments about Student-Teacher Discussions ...........................
Student Questions as Windows to Student Thinking ....................... ....
"How do you know that you have to add them?" ............................ "Do you figure out subtraction and addlion the sarne way?" ...... 'But there is no changew O A Discrepancy ........................................ "1 don't understand what this meansn ............................................... A Previously ldentified Learning Blockage ...................................
The First Quiz ................................................................................................. What Did the Results Show? ............................................................. Feedback From Participants After Ten Days ..................................
Vector Components and a Bucket of Sand .............................................. "Hanging Masn Lab Investigation .............................................................
Student Confusion of Balanced with Equal .................................... lntroducing a New Vector Concept O Momentum ..................................
Students' Initial Conceptions of Momentum ...................................
vii
Reconstructing Student Conceptions of Momentum ................. .... 1 5 1
Student Approaches to Constructing Momentum .......................... 153
Three Students Discuss Physics and Vector Mathematics ............................ 159
A Final Vector Problem ................................................................................ 162
Why the Participants Chose to Use Vector Mathematics ............. 166
Some Final Student Thoughts ........................................................ 170
A Student Thinks Aloud While Writing a Test ....................................... 172
Students Discuss a Test on Mornentum .................................................. 173
Chapter Six O Insights. Implications and Recommendations ...... 175
lnsights Gained into Student Leaming Processes ................................ 175
The Participants View of Their Own Leaming ................................ 176
Students' Experiential and Conceptual Knowledge ..................... 179
Student Alternative Conceptions and Communication ................ Implications and Recommendations for Secondary School Physics ..
Amount of Content in Secondary School Physics ......................... Increasing the Relevancy of Physics to Students .......................... Connections Between Mathematics and Physics Cumcula ........
implications and Recommendations for instructional Strategies ......... ...................................................................... Small Group Instruction
Using Discussion as an Instructional Strategy ............................... Questioning and Discussion Skills O Leamed Processes ............ Using Incorrect Answers as Discrepant Events ............................. 196
Metric Units as Aids to Concept Development ............................... 197
............................................................ Implications for Students Leaming 198
Student Difficulty Using Vector Mathematics in Physics ....................... 199
viii
...................................... Mathematical Representation of Concepts
Recognizing Direction as a C haracteristic .......................... .. ....... Applying Vector Mathematics in Physics ........................................
......................................................................... 1 rnplications for Research
................................................................................ Conduding Statement
References ............................................................................................................ ..................................................... Appendix A: Approval for Research P roject
Ethics Approval . University of Regina
...................................................................... Application for ap proval
......................................... . Letter of approval University of Regina
Permission . Board of Education of the Regina School Division
............................................................. Letter requesting permission
................................................................. Latter granting permission
......................... Appendix B: Communication with participants and parents
.............................................. Letter of invitation to students and parents
.................................................................................. Project volunteer fom
.......................................... Letter of acceptance to students and parents
................................................................ Letter of rejection and thank you
.................. Parental letter of consent to participate in research project
...................................................... Appendix C: Sample Section of Transcript
.......................... Transcript from classroom session of 23 March, 1993
..................................................................................... Appendix D: Vector Quiz
. .......................................................*.....*...*,..*....... Chapter 3 Quiz 'Jectors
........................ Appendix E: Summary of Difficulties Exhibited by Students
..................................... Summary of Difficulties Exhibited by Students
List of Figures
Fiaure
5.1 a
5.1 b
5.2a
5.2b
5.3a
5.3b
5.4a
5.4b
5.5
5.6
5.7
5.8
5.9
5.10
5.1 1
Descri~tion Page
............................... Vectors B and J as initialiy drawn on board
..................................... Colleen's first step in her addition drawing
.......................................................... Alternative diagram that I drew
. ...................................... Diagram after Judy added a connecter .... .............................................................. Judy's initial vector drawing
............................................................. . Final version of 6 3 = AV
Initial diagrarn of car tuming corner .................................................. ...................................... Diagrarn showing vectors added by Anne
........................................................................ Anne's vector diagram
.......................................... Anne and Marie holding bucket of sand
............................. Ryan and Dean liftirig the bucket at low angles
................................................................ "Hanging Mas" apparatus
"Hanging Massn apparatus with vector representation of forces . .......................................... Vector components of forces Fi and F2
........................................................................ The force board set-up
CHAPTER ONE
INTRODUCTION T 0 THE STUDY
Over the last two decades many researchers have applied constnictivism
as an epistemological foundation for the study of student leaming in science.
Solomon (1 994) and Osborne (1 996) trace the beginning of viewing student
leaming constmctively to an article by Driver and Easley (1978) who argued
forcefully that student leaming in science depends to a much greater extent
upon prior experience and knowledge than previously thought. Since then
many researchers have argued for a constructivist view as a basis for
understanding student learning, designing instructional strategies, and
developing curriculum in science education.
Exploration of the relationship between student leaming and teacher
instruction has been the focus of many research studies. Bodner (1986), for
example, discussed viewing student Iearning as knowledge construction and
designing instructional strategies to explore student preconceptions and
misconceptions. He argued for a shift in the teacher's role from transmitting
knowledge to facilitating meaningful leaming by students. Similarly, Hand 8
Treagust (1 991) argued that instructional strategies be designed to guide
students in examining their own conceptions with a view to reconstructing thern
rather than having teachers transmit knowledge directly. Dykstra (1 996)
employed constnictivist learning theory to redesign his instruction to enhance
student understanding of first year coiiege level physics. Students in his
classes explored their conceptions and worked together to reconstruct them.
Novak (1 988) has argued that student leaming can be improved by
understanding student conceptions and developing instruction which enables
1
2
students to make their own connections between scientific conceptions and
knowledge already held. In a comparable rnanner Gunstone (1 988) maintained
that teachers should redesign instruction to assist students in constructing their
conceptions, but did not feel that only a single strategy could accomplish this
goai. Wheatley (1 991) described student leaming as construction of rneaning
and proposed the use of problem solving as the central focus of science
instruction. Other researchers have expanded the description of student
leaming in science to incorporate a combination of personal and social
knowledge construction (Cheung & Taylor, 1991 ; Driver, 1988).
Secondary school science teachers have also been involved in
developing instructional strategies and cumcula based on constructivist ideas.
For example, Driver (1 988) worked directly with teachers in an action research
project to develop science curricula and instructional strategies based on a
view of leaming in which students actively constructed knowledge. Von
Glasersfeld (1 996) emphasized the need for teachers "to construct a
hypothetical mode1 of the particular conceptual world of the students they are
facing" (p.7). Teachers, it seems, need to blend both the social and individual
aspects of knowledge construction in teaching science because learning in
science can be thought of as individual knowledge construction in a social
setting (Cobb, 1996).
The interrelationship between leaming, instruction and science
curriculum takes on a distinct perspective when viewed through a constructivist
lens. Teachers and students are viewed as working individually and
collectively to facilitate student knowledge construction by making connections
between classroorn experiences, and concepts and knowledge already held.
F rom this constructivist perspective, leaming in science involves students in
active construction of concepts and meaning making which result in meaningful
3
leaming where 'new information is linked with existing concepts, and integrated
into what the leamer already understandsn (Edmondson & Novak, 1993, p. 548).
In this study teaching and leaming were viewed as dialectically related
because they are essentially interdependent and inseparable. Instruction was
designed to facilitate student leaming; and, feedback about student leaming
rnediated instructional design. Pedagogical decisions were made as the resuit
of the teacher's assessrnent of particular student needs on a given day. The
instructional strategies were chosen because they best suited the leaming
needs of the participants. The sequence of classroom events was selected to ffi
the requirements of the students at particular times. The instruction would have
been modified to some extent with a different group of students.
Purpose of the Study
This study explored the leaming-teaching dialactic using methods
designed to investigate the interrelationship between students and a teacher
during specific leaming experiences with vector mathematical models in grade
12 physics. An attempt was made to understand student leaming processes by
the interactive exploration of their explanations about their struggle to
understand, and by closely observing their activities in a regular classroom.
The role of the teacher was also closely examined in relation to student
leaming. In other words, this study focussed on analyzing student knowledge
construction as represented by elicited verbal descriptions of their leaming
process during episodes of instruction in vector mathematics and their
application to problerns in physics.
The purpose of this study was to attempt to understand how students in
grade 12 physics actively construct their knowledge of the use of mathematical
models to understand certain cornplex physics concepts. The study also
4
examined teacher interactions with those students as a means of mediating that
leaming experience during part of a five month semester in a regular physics
classrocm. More specifically the study focussed on the following questions:
How do grade 12 physics students construct their knowledge of
mathematical models in grade 12 physics, when required to use vector
mathematics to represent or model two dimensional situations and to
solve physics problems?
How does a teacher interact with a group of grade 12 physics students
during the leaming process to facilitate their knowledge construction of
the use of vector mathematics in physics?
Context of the Study
Although students are expected to leam the same concepts,
understanding individual student leaming is complex because each person
experiences a different knowledge construction process. That is, students who
are instructed together construct their knowledge and understanding
individually while participating in a social leaming environment. From a
constructivist perspective, differences in leaming may be attributed to variations
in student history. K. Roth (1990) argued that because of variations in their life
experiences students do not arrive at class with identical personal knowledge.
Although students in this study had progressed through grade levels at much
the sarne rate in the same school, or similar kinds of schools. the prior personal
knowledge' individual students brought to class was assumed to be different.
Edrnondson & Novak (1993) have stated that meaningful leaming cannot be the
same process for each student because he or she has to link classroom events
' Students' prior personal knowledge includes personal experiences, previous fomal education and personal theones constructed to organite their knowledge. In this dissertation al1 this knowledge is viewed as fluid and dynamic,
5
with his or her currently-held concepts which are unique. In addition each
student's construction of knowledge is affected differently by interactions with
the teacher and other students in the classroom, and by instructional strategies
chosen by that teacher (Pope & Gilbert, 1983; Benson, 1989; Wheatley, 1991).
In physics, leaming requires considerable intellectual effort and presents
a unique set of difficulties for each student. To be successful students must
construct and reconstruct many concepts, and leam how to apply them to
problem solving situations and to their everyday experiences. For example,
new knowledge, that physics students may have to construct and resonstruct,
includes concrete2 concepts, such as mass and distance, and abstract concepts,
such as time, momentum and energy. To complicate the matter, leaming in
physics requires that students identify a concept and then represent it as a
variable in a mathematical equation. The theoreticai comerstone of the
physical sciences, and physics in particular, is the use of mathematical models
to represent relationships between concepts identified in natural phenornena.
Mathematical models are used to descnbe actions and interactions of objects
within the universe; but even more importantly, mathematical models can be
used to predict new interactions between objects. Ultirnately, mathematical
models direct the progtess of research in physical science as long as they
successfully explain empirical obsenrations and predict events accurately.
In high school physics classes mathematical models are typically
represented as rnulti-variable formulae, such as the well known forrnulae,
F = ma and E = rncz. Such models can be presented to students using a
variety of instructional strategies; for example, a teacher rnight employ
* Mass is seen by students as concrete because of their everyday and laboratory expenence with the concept. Expenenced physicists do not necessarily share this view because of their much more extensive knowledge and expenence. They rnay view mass as abstract because of their knowledge of mass-energy interaction.
experimental work, lectures, or cooperative leaming groups to introduce
students to these conceptual models. Typically, once preliminary instruction
has been completed, mathematical models are applied to physics problems
which can range in difficulty from simple to extremely cornplex. In tum, problem
solving generally requires that students use a formula, or model to predict a
future action or position of an object based on a given set of conditions.
During my twenty-three years of teaching experience with secondary
science students I have observed that comprehension of abstract mathematical
models and usage of formulas are key tasks that a majority of students find
difficult to master. In these areas of science cumcula, problems in student
leaming consistently occur. This observation of conceptual inadequacy,
particulariy between concrete and abstract applications. has been persistent in
most science classes throughout my career in teaching.
In particular, grade 12 physics students have consistently exhibited
difficulty in leaming the complex mathematics-related concepts that are present
as part of the grade 12 physics curriculum (Saskatchewan Education, 1992).
Although many examples of complex concepts exist in physics, the application
of various forms of mathematics as theoretical models of physics concepts is an
area that is of particular interest to me. This specific use of mathematical
models, common in both chemistry and physics, is difficult for students to
master, and challenging for teachers to instruct.
In my experience students seem to have limited difficulty with one
dimensional kinematics and dynamics; however, when students begin the
section of the physics curriculum involving the study of dynamics and
kinematics in two dimensions, most typically begin to fiounder. This floundering
is a crucial point in leaming vector physics and the precise focal point of this
research. To be successful in grade 12 physics, students must leam to
represent certain physics concepts using vector mathematics. While this
process of knowledge construction has proven to be difficult for most grade 12
physics students, during a five month semester most (although not all) of these
students do appear to develop a sufficient understanding of some aspects of the
knowledge necessary to solve some two dimensional problems using vector
mathematics. Given the identification of a potentially crucial point in student
learning during the semester of physics this study investigated student
development of knowledge during this period that is critical to leaming certain
physics concepts. Given the inadequacy of previous investigations using
quantitative measures to understand this leaming phenornenon (Driver &
Easley, 1978; Welch, 1985; Tisher & White, 1986; Lythcott & Duschl, 1990), this
study used qualitative methods that sought to capture student talk and thinking
processes which, in tum, could lead to a more complete understanding of how
students think, and construct knowledge in this area of physics.
In a preliminary study Hart & Wessel (1992) analysed and described
student-teacher interactions occurring during two tutorial sessions with four
students. The interactions were informa1 conversations during which I
attempted to assist them in analyzing their problem solving procedures and
identifying sources of confusion in solving the problems. This analysis of
interactions was used as an initial means of investigating research methods that
could effectively probe students' struggle to understand vector concepts of the
physics curriculum. The present study attempts to expand the investigation of
student leaming using more intensive data collection procedures over an
extended period of time.
"Teachers have important contributions to make to cumculum
development, the analysis of teaching and leaming, and to the improvement of
teaching-leaming contextsm (Roth, W-M., 1995, p.4). W-M. Roth argued that the
8
perspective of classroom teachers necessitated their inclusion in classroom
research because they could provide insights that could not be made by
extemal researchers alone. In addition he contended that their participation in
classroom research was necessary because they seem unable to relate to
findings of tradlional university-based research, or to translate them into
changes in pedagogical practice. Drawing on personal experience he
described the value of classroorn teachers and university researchers
collaborating on research projects. The combination can explore different
research questions than can extemai disinterested researchers working alone.
The use of altemative research methods allows teachers to examine problems
more directly relevant to their practice. Similady, Steffe (1991) has argued that
researchers should shift their perspective from one of detached extemal
observers to one which includes both teachers and students as participants in
research.
Rationale of the Study
The section in grade 12 physics which introduces vector mathematical
models provides an appropnate area for detailed study of student knowledge
construction and teacher-student interaction in leaming for three reasons. First,
the use of vector mathematics to represent physics concepts is an abstract and
complex process in which students are required to learn to complete the
physics course successfully. Physics curriculum mate rials provide appropriate
leaming tasks for examination of students' construction of mathematical models,
because most students on arriva! at grade 12 physics do not have much
understanding of vector mathematics and their application to problems in
physics. Although students have used some mathematical models since
elementary education, in grade 12 physics they are required to conceptualize
new more complex mathematical models.
An example of a simple mathematical model is the familiar formula for
area of a rectangle:
Area = length X width. (or A = i x w)
'While most students can employ this formula and similar mathematical models
with only occasional mechanical enors, few students even at the grade 12 level
are proficient when using matheniatical models that involve the application of
l e s familiar (and more abstract) rnathematics, such as vectors.
One example of a problem requiring vector mathematics for its solution is
the analysis of forces acting on an object in two dimensions. Vector
mathematics must be employed to calculate the resultant force acting on a
body. Few grade 12 physics students are able to solve this type of problem
before they have been instructed about the use of vector mathematics to such
problems. This consistent initial lack of proficiency by most students provides a
unique place in the physics classroom for an examination of student thinking
and leaming. This point in physics instruction where students must understand
vector mathematics is crucial for studying how students construct particular
knowledge about the application of mathematical models. Genuine personal
understanding by students is required for success in physics problem solving.
My central concern is how individual students come to this understanding.
Second, my experience shows that most students correctly develop
some ability to apply vector mathematical models during one semester of grade
12 physics. This developrnent varies considerably in a typical class of physics
students. Some students essentially master abstract concepts, such as two
dimensional force and momenturn concepts, during the eighteen to twenty
weeks they take physics. At the end of a five month semester they are able to
solve virtually any problem given them and appear to understand the concepts
1 O
adequately from my point of view as an experienced physics teacher. A second
group of students are able to answer typical problems correctly, but cannot
apply vector mathematics to new or atypical types of problems. Although these
students may achieve a fairiy high mark in the course, I do not believe that they
have developed as deep an understanding of principles as students in the first
group. A third group of students in a typical class is not be able to solve these
types of problems except by randomly ananging numbers from the problems on
the page. Members of this group frequently do not successfully complete the
course. The nurnbers Vary from class to class. but normal classes have some
members of each group. This expected range of student abilities should lead to
a nurnber of crucial incidents as the participants learn to use vector
mathematics in physics.
Third, I have received considerable feedback from my students and they
tell me I have successfully assisted many of them in developing their
understanding of vector mathematics in physics. This kind of feedback has
been provided by students during the time they were taking physics in my class
as well as by students who had completed one or more years of post secondary
education. While less able to foster leaming in the third group of students
described, I know that I have had success in facilitating members of the first two
groups to understand physics concepts. Although never formally documented I
have accumulated evidence for these claims during the years of my teaching
experïence. This research could enable me to describe in some detail how my
interaction with some students facilitated their construction of abstract, complex
concepts; that is, how they came to leam.
This specific area of student conceptual leaming in one part of the
physics curriculum provides a unique site for studying knowledge construction
in students. Examining student activity during lessons where students are
11
required to comprehend vector mathematics in physics should help us to
understand the process of knowledge construction by physics students.
Significance of the Study
The analysis of student leaming constructed from this study rnay be
useful to teachers of physics in developing their own understanding of the
process by which students construct knowledge of mathematical models in
physics. By understanding this knowledge construction process more
completely, physics teachers rnay be able to design more effective instructional
strategies to facilitate physics students in acquiring these complex abstract
concepts. In addition, descriptions coming from this study rnay assist teachers
to reflect on their interactions with students, and to improve their ability to
interact given a deeper insight into student stniggles, thus facilitating more
effective leaming by students. Although no rnagic formula exists, the study rnay
help produce a deeper sense of empathy or understanding of individual student
concerns, problems and difficulties.
The use of mathematical models is cornmon to al1 sciences. Formula
manipulation is required in most science classes at the secondary school level.
Problern solving in mathematics also requires similar mental strategies to those
used by students in grade 12 physics. The descriptions of leaming produced in
this study could be useful to teachers in these and other related fields. If
knowledge construction is a process that is similar in other types of leaming,
then the deeper understanding of knowledge construction in physics rnay
extend to a more thorough understanding of knowledge construction in other
subject areas, such as mathematics. Teachers rnay be able to develop a sense
of which skills and knowledge students should transfer to other subject areas.
Assumptions
Several assumptions about how student thinking and leaming. and
student-teacher interactions may be investigated are important to this study.
First, students are assumed to be active constructors of knowledge when
learning physics. A second assumption is that conversation, discussion and
other f o n s of verbal communication can provide meaningful information about
student thinking and knowledge construction. Closely related is a third
assumption that with practice students will be able to describe their thinking and
leaming with enough detail to allow a meaningful description of the processes.
Delimitations and Limitations of the Study
The study was delimited by the choice of only nine students for the study
group and restricting the data collection to their leaming during one portion of a
five-month semester. No attempt was made to correlate the leaming processes
of the participants with other factors such as gender or academic achievernent.
This study was restncted to the constniction of knowledge about mathematical
models in physics. The data were collected only when students were leaming
how to use vector mathematics in grade 12 physics and interpretations of that
data are delimited to leaming processes that were exhibited during that time.
Data collection was delimited by the choice of method. No follow up study of
the students has been conducted to this moment.
My role as an active participant in the the study is a limitation. As
teacherlresearcher I decided how classroorn instruction wouid proceed and
made spontaneous decisions as to which part of the activity would be
videotaped. The ability of communication to act as a window on student
thinking is also a limitation of the study. The range of learning processes that
can be described are limited by the students who participated. No attempt was
13
made to screen the choice of participants. The tirne spent on various topics was
limited by the necessity of completing mandatory sections of the C U ~ C U I U ~
because a provincial credit in the course would be eamed by the students at the
end of the study.
CHAPTER TWO
REVIEW OF THE LITERATURE
The purpose of this literature review is to provide a critical synthesis of
literature on qualitative studies of student leaming and knowledge construction
in science education. In addition, literature related to the involvement of teacher
interaction in student leaming is analyzed. First, I provide a detailed
examination of the literature in science education which describes how students
leam abstract science concepts of the type that occur in grade 12 physics in
Saskatchewan schools. This section provides a comprehensive summary of
research on student alternative conceptions in science because of its relevance
to this study. In the second section, the review overlays the notion of
constructivist leaming on recent research in science education. Student
leaming can be described in terms of principles of constnictivism, the current
epistemology used as a foundation for much current research in science
education. The third section focusses on qualitative approaches that appear to
have merit in investigating student learning in science. lncluded are research
methods that have evolved from the adoption of constnictivism as research
epistemology. The final section reviews arguments for including classroom
teachers as full partners in research studies as was the case in this study.
Student Learning in Science
Many teachers have accepted a transmission mode1 as appropriate for
teaching science because they view science in a tradlional manner; that is,
they believe science knowledge is unproblematic, science provides right
answers and truths in science are discovered (Carr, et al.. 1994). The
transmission model of teaching in science is deeply rooted in our cutture, in
both teachers and students (Roth, W-M. , 1993).
Leaming in science is typically a difficult task for students and this is
unlikely to change because of the complex structure of science (Duit, 1991).
Instead of reading or discovering the book of nature, scientists impose
constructs and concepts on observed natural phenomena to organize and to
understand thern better (Driver, Asoko, Leach, Mortimer and Scott, 1994).
Driver et al. argue the complexity in science lies in the study of the constructs
advanced to explain natural phenomena rather than in the phenomena
themselves. Carr et al. (1994) state that exploration of the history and
philosophy of science and inclusion of newer models of leaming from cognitive
psychology have prompted the science education community to focus on
student leaming in science and, as a result, have begun to change the view of
teaching science from a transmission model to one of student construction of
knowledge.
Millar (1 989) summarizes the problem facing teachers in science
classrooms:
... science as a school subject poses a formidable challenge to the teacher in maintaining the involvement of many pupils simply because the science covered at school is, alrnost entirely, a consensually agreed body of knowledge. There iç, therefore limited value in children taking away from science lessons ideas that diverge radically from the accepted ones. This means that science can come to look like the transmission of a body of knowledge that cannot be challenged by the leamer, and whose learning leaves Iittle scope for the creative involvement of the leamer. (p.590)
Earlv Views of Student Learninq
Before the last two decades, student leaming in science classrooms was
described in ternis of transmission from teachers to students. Direct
16
transmission models of student leaming began to lose favor because of their
inability to explain some important intellectual achievements, such as, creativity
and decision making (Gagné, 1985). Our thinking about leaming in science
has gradually changed because of developments in leaming psychology and
episternology. Cognitive psychologists began to describe cognitive functions of
students during learning; and, philosophers moved away from positivist and
empiricist attempts to establish truths toward a constructivist view of knowledge
building (Novak, 1988).
Discovery and inquiry leaming were among earfy attempts at curriculum
development built on a view of students as active participants in their learning
(Trowbridge & Bybee, 1996). Discovery leaming, pioneered by Bruner (1 961),
was used as the foundation for curriculum development and led to BSCS
Biology, CHEM Study and PSSC Physics which were the standard courses
from the 1960s to the 1980s in Canada, the United States and rnany other
countries. In discovery leaming classrooms students were expected to discover
laws goveming nature as a result of experiencing scientific phenornena in
laboratory settings. Students participating in these curricula did not discover
science concepts as was expected. Driver et al. (1 994) have argued
convincingly that students should not have been expected to discover laws of
science because those laws are social conventions communicated through the
social and cultural institutions of science. While not as successful as hoped,
these curricula represent a genuine attempt to involve learners actively in their
own leaming.
Ausubel's and Piaaet's Views of Student Learninq
If I had to reduce al1 of educational psychology to just one principle, I would Say this: The most important single factor influencing
leaming is what the leamer already knows. Ascertain this and teach him accordingly. (Ausubel, Novak & Hanesian, 1 978, p. iv)
This oftquoted statement of Ausubel expressed the importance of student
knowledge and foreshadowed an extensive research program in science
education exploring student altemative conceptions. Ausubel distinguished
rote and meaningful leaming based on the relatability of material being ieamed
(Ausubel & Robinson, 1969). Relatability depended on the non-arbitrariness of
connections a leamer made with knowledge already held and the
substantiveness of the material. Rote leaming is arbitrary, verbatim and non-
substantive; whereas, meaningful leaming is non-arbitrary, non-verbatim and
substantive. Edmondson and Novak (1 993) modified these definitions slightly
defining rote leaming as 'Yhe acquisition of new information without specific
association with existing elements in an individual's conceptual structure (Le.,
memorization)"; and, meaningful leaming as occumng %hen new information
is linked with existing concepts, and integrated into what the leamer already
understandsn (p. 548). This distinction in types of leaming is important because
in science education meaningful leaming is the goal of teaching science.
Piaget wrote about conceptual development in children. His work "can
be interpreted as one long struggle to design a mode1 of the generation of
viable knowledge" (von Glasersfeld, 1989a, p. 6). Piaget developed a
constructivist theory of cognitive development and cognition based on a series
of developmental stages (von Glasersfeld, 1989a). He observed his own
children while they attempted to solve problems and supplemented these
observations with interviews designed to explore their conceptual development.
In its time Piaget's work was unique because his method of research, which
involved describing leaming in a small number of children as they developed,
represented a departure from large sample quantlative procedures.
18
Von Glasersfeld (1 984, 1995) has written extensively about Piaget's view
of knowledge construction as adaptive and changing. Piaget argued that for
every experience, a person had certain expectations of outcome because he or
she tried to assimilate each experience to already existing knowledge. If a
leamer is unable to assimilate a particular experience, aien a perturbation
occurs. To reestablish cognitive equilibrium a leamer bnngs meaning tc new
experiences through a process called accommodation. This process requires a
leamer to restructure currently held concepts or create entirely new ones (von
Glasersfeld, 1989a). Assimilation and accommodation are regarded as
important rnechanisms in the description of leaming in science.
Studies explonng student alternative conceptions have provided data to
strengthen this view of leaming. Viewing student leaming as individual
knowledge constniction has helped me develop an understanding of student
learning as it appeared to happen in my classrooms and I have used this idea
to interpret some of the experiences described in this dissertation. This
constnictivist view of leaming is an important part of the theoretical foundation
of this study.
Research Ex~lor ina Student Alternative Conce~tions
As the importance of students' pnor or existing knowledge became
recognized, Driver and Easley (1978) were among the first to recommend more
extensive research to examine and describe student conceptions. They argued
for studies using naturalistic approaches, such as interviews and classroom
interactions with students, because these methods were better suited to
exploring individual student knowledge. Driver and Easley felt teachers
needed to know something about the range of student conceptions because of
their effect on classroom instruction. Researchers have examined student
19
conceptions in physical and earth science, biology, chemistry and physics. The
general goal was to descnbe student-held conceptions with an eye to designing
instructional strategies to assist students in reconstructing their concepts to be
more in line with those of the scientific community.
A variety of terms has been used to describe the conceptual knowledge
students bring to science classrooms. Expressions, such as "altemative
frameworksn (Driver & Easley, 1 W8), "preconceptionsn (Ausu bel, 1 968).
"misconceptions" (Driver, 1983), "personal models of rea lw (Champagne,
Gunstone & Klopfer, 1985b). "spontaneous knowledgen (Pines & West, 1986),
and "intuitive theories" (McCloskey , 1 983) have been used in science education
literature. Abimbola (1988) argued that researchers with an empirical view of
science tended to use terms that implied error in leamers' ideas such as,
rnisconceptions, misunderstandings and erroneous ideas; but, researchers who
view leaming science as conceptual development used terms such as. existing
conceptions, prior schemata, and alternative frameworks. He has argued
convincingly for the use of alternative conception. This terni is now preferred by
many researchers because this wording refers to 'experienced based
explanations constructed by the leamet' and 'conveys respect on the leamer
who holds those ideasn (Wandersee, Mintzes & Novak, 1994, p. 178). In this
dissertation alternative conception is used when refemng to knowledge of
physics concepts brought to class by the participants.
In an early study Boyd (1966) surveyed male college sophomores about
comrnonly held rnisconceptions using a true-false questionnaire. His data
showed no significant correlation between misconceptions held and the
number of science courses completed by the sophomores. He concluded that
misconceptions were strongly held and difficult to change. Boyd did not
interview students to detemine more details about their misconceptions. The
20
weakness of this study was that it did not provide details of the misconceptions
or rnake suggestions for new instructional strategies.
Since the 1960s research into student alternative conceptions has been
extensive. A variety of methods has Seen employed with groups ranging from
individual students to large scale studies involving thousands of students. In
1994 Pfundt and Duit produced a bibliography of more than 2500 alternative
conception refereences (as cited in Dykstra, 1996). A representative sample of
studies exploring topics related to physics is reviewed.
Heat and Temperature
Erickson (1 979) explored children's thinking about heat and temperature
using open-ended informal interviews of ten twelve-year old boys and girls.
Four laboratory tasks were explored and children's explanations were
examined in videotaped interviews. Soma students viewed heat as a matenal
substance which moved from object to object. Others saw heat as having
positive and negative qualities that cancelled each other when hot and cold
substances were combined. Another student view was that metals heated more
quickly than glass because metais attracted heat better. Lastly. students tended
to view temperature as a measure of the amount of heat in an object rather than
a measure of the intensity (Erickson's t e n ) of heat.
In a later study Erickson (1 980) used a Conceptual Profile lnventory to
examine conceptions of heat and temperature in a group of 276 students from
grades five, seven and nine. He found a range of beliefs about heat and
temperature similar to those in his eariier study (Enckson, 1979). Erickson
suggested that the instrument might be useful for teachers to ascertain prior
conceptions of their students. Once a teacher had reviewed the inventory for
his or her class, instruction could be designed based on the profile of
21
conceptions in his or her classroom. My crïticism of this approach is that
considerable time is required on the part of teachers and this process would
have to be perforrned for each new concept used.
Force and Motion
Osborne and Gilbert (1980), using a technique called interviews-about-
instances (IAl), explored understanding of force in forty students aged seven to
nineteen. Students were shown cards depicting familiar situations and asked
questions about the scientific concepts represented. Their results showed one
group of students confused common meanings of words with their physics
rneaning; a second group did not think of force unless motion was occum'ng;
and a third group viewed force as a physical quantity possessed by objects in
motion which ran out as they stopped. In a similar study Gunstone and Watts
(1985) examined concepts of force and motion in students nine to nineteen
years old. These authors found that students thought constant motion required
a constant force perpendicular to the direction of motion, the amount of motion
was proportional to the applied force and stationary bodies had no forces acting
on thern. Sadanand and Kess (1990) assessed knowledge of fifty-seven
college bound seniors who had elected to take a non-calcuius physics course.
The authors used multiple-choice questions expanded with written
explanations. They found similar alternative conceptions conceming constant
force and motion, but also found students thought reaction forces were less than
action forces.
Gravitv and Energy
Watts (1 982) used the IAl technique to investigate conceptions of gravity
held by forty secondary school students. He found that many students viewed
22
gravity as acting through air, and without air they believed there was no gravity.
Some thought gravity increased with height above ground and others thought
gravity acted on falling objects but not on stationary ones. Watts concluded that
instruction was needed to "bridge the semantic gap" (p.120) between students
alternative conceptions and those of physicists. In a related study Watts (1 983)
investigated student views on energy and summarized his findings in a series
alternative frameworks. Some students thought of energy as a human attribute,
while others viewed it as something stored in objects that caused events to
happen. Other views were that energy is associated with activity and
movement, and energy was a kind of fuel capable of doing things. Students did
not think of energy as conserved; rather, they saw it as a product that was
released like smoke. Watts argued that these descriptors of student knowledge
would be useful as starting points to design instruction to facilitate conceptual
change. Solomon (1984) complimented Watts on his research but argued that
he went too far in categorizing the frameworks so specifically. She felt that
meanings Vary from time to time in students and do not have the rigidity
descnbed by Watts.
Boyes and Stanisstreet (1990) assessed 1 130 boys and girls aged
eleven to sixteen about their understanding of the law of conservation of
energy. They used a survey instrument which provided definitions of the law as
students might interpret it. Students understood law most frequently as a legal
term rather than as a description of objects in nature. Conservation was most
frequently interpreted in the environmental sense of using sparingly or wiçely.
The authors concluded teachers should be aware of these misunderstandings
and plan their instruction to facilitate redevelopment of these conceptions by
students.
23
Phvsics Vocabulary
Jacobs (1989) worked with Crst year physics students to explore their
understanding of vocabulary used in physics. She examineci students'
understanding of words which were part of their everyday vocabulary but have
special rneanings in physics, such as, speed, velocity, mass and weight. She
found that student comprehension of the physics meanings was weak. Jacobs
argued this lack of understanding had implications for physics teachers
because confusion occurs when instructors use the more restricted meaning in
physics and students apply their everyday meaning.
Problern Solvina in Phvsics
Chi, Feltovich & Glaser (1981) employed clinical interviews to investigate
problern solving strategies of physics novices and experts. The authors
interviewed a number of physics novices and experts about their manner of
problem solving and demonstrated that members of the two groups went about
solving the same problems quite differently. Experts tended to group problems
according to underlying physics principles, such as conservation of energy or
conservation of rnomenturn; whereas, novices tended to group problems
according to their surface structure. For example. novice problem solvers
separated ail problems involving pulleys into one group. and those with inclined
planes into another. Experts spent more time than novices raflecting on a
problem and qualitatively formulating a solution prior to applying a formula to
calculate a numerical answer. Novices spent little time thinking about the
solution to a problem and tended to simply choose a physics formula that fit the
variables in the question. This study produced a useful description of
differences in problem solving processes used by novices and experts.
24
Vector Quantities
Aguirre and Erickson (1 984) interviewed twenty grade 10 boys and girls
about their conceptions of vector quantities prior to beginning formal study.
Their goal was to create a data base of conceptions to be used in cumculum
development. Students were given tasks involving position, displacement and
velocity of boats in a river. They were asked to solve various problems, such as,
how a location on a lake is described, and how fast a boat travels in a moving
strearn. The authors concluded students intuitively use some vector
characteristics in their solutions; for example, students knew the location of a
fishing spot on a lake had to be specified by distance and direction, and river
currents changed the velocities of boats. Aguirre and Erickson concluded that
students have "an intuitive set of ideas about the nature of vector charactetistics;"
however, these authors did not appear to consider that students were solving
the problems posed using their own understanding of what was required.
Students would be aware that when describing the location of a fishing spot on
a lake both a distance and a direction had to be provided, but I think Aguirre
and Erickson go too far in assuming that this experiential knowledge can be
interpreted as containing an intuitive understanding of vectors.
Aguirre (1 988) interviewed thirty grade 1 0 students about their
conceptions of vector kinematics using laboratory apparatus. He presented
three situations during interviews - a power boat crossing a river, a frictionless
cart moving across an inclined plane, and two orthogonally moving carts. He
found students used the ground as the predominant frame of reference when
describing motion and did not use other frarnes of reference when detennining
velocities of the objects. Students viewed component forces as acting
separately rather than together, and generally confused component and
resultant velocities. Aguirre pointed out that teachers should be aware of these
25
student preconceptions and recognized that simply telling students correct
answers was not sufficient instruction. He concluded that different instructionai
strategies need to be developed for teaching vector kinernatics to students but
did not provide specifics of the strategies.
Characteristics of Student Alternative Conceptions
The "Alternative Conception Movement" (ACM) (Gilbert and Swift, 1985,
p.682) has camed out research exploring student alternative conceptions in
most areas of science, including biology, chemistry, physics, and earth science.
Miller (1 989) stated that the ACM had made significant contributions to science
education research by helping us appreciate the complexity of the processes
involved when students leam science. The following is a summary of
characteristics of alternative conceptions as they appear in the literature:
Learners corne to formal science instruction with a diverse set of alternative conceptions about most natural phenornena. These concepts are used to explain events in manners that are very different from either adult or scientific explanations. Students can hold multiple views and explanations of a natural phenornenon. (Cheek, 1992; Driver & Bell, 1986; Gunstone, 1988; Stepans, 1991 ; Wandersee, Mintzes & Novak, 1994)
Alternative conceptions seem to be independent of age, ability, sex and cultural background. They are tenaciously held by leamers and are not usually modified by traditional instruction. Altemative conceptions frequently parallel the conceptions of eariier scientists and philosophers. (Cheek, 1992; Driver & Bell, 1986; Gunstone, 1988; Stepans, 1991 ; Wandersee, Mintzes & Novak, 1994)
Instructional strategies designed specifically to produce conceptual change have shown some success in facilitating construction of conceptions that match those of the scientific community; however, discrepant events during instruction do not always produce the cognitive changes expected, and the alternative conceptions may be maintained even when leamers answer questions correctly on tests. (Cheek, 1 992; Stepans, 1991 ; Wandersee, Mintzes & Novak, 1994)
Scientific concepts are often presented assuming that leamers irnmediately understand them; however, leamers' alternative conceptions interact wiai those presented during instruction in unpredictable ways producing unintended leaming outcomes. (Cheek, 1992; Stepans, 1991 ; Wandersee, Mintzes & Novak, 1994)
Children can hold contradictory conceptions at the same time. One set can be used to operate in science classrooms and answer science questions, while the other set is used to explain happenings in their experiential worid outside the classroom. (Cheek, 1992; Gunstone, 1988) Even after several years of science instruction many adults and science teachers hold the same alternative conceptions as students. (Stepans, 1991 ; Wandersee, Mintzes & Novak, 1994)
Alternative conceptions have their source in each individual student's cornplex experiential history, including direct obseivation of the worid, peer culture, and language, as well as, television and formal classroorn instruction. Each individual has a unique history; and, therefore, each holds a set of alternative conceptions that is different from other students. (Wandersee, Mintzes & Novak, 1994)
lntearatina Alternative Conceptions in Views of Learning
Science educators have developed a number of leaming models
incorporating principles generated by research on student alternative
conceptions. One general feature of these models is the inclusion of some
cognitive connection between new experiences and knowledge already held by
leamers. The nature of these connections varies but in al1 models when
meaningful leaming occurs students actively construct new knowledge in
conjunction with existing knowledge.
Driver (1983) and Driver and Bell (1986) descnbed students as
constructing knowledge in a manner similar to scientists who use experimental
activaies to examine natural phenomena. Students' alternative frameworks
help to organize their interpretation and comprehension of their experiences as
new knowledge is incorporated into current frameworks. The authors view
leaming in science as an individual and social constructivist process. Students
27
are required to construct and evaluate meanings through personal reflection
and interaction with others. Pines and West (1986) also described student
learning as a process of active meaning but, in their variation, students
interweave knowledge from fomal science teaching with their prior knowledge.
Osborne and Wittrock (1 983, 1985) and Wittrock (1 985, 1986) described
a generative mode1 of knowledge construction. Leamers are viewed as
generating cognitive links between their memory and new sensory input to
construct meaning for new events. Students generate the links by expending
intellectual effort. Teachers can facilitate this process, but not directly control it.
These models are closely related because they require that connections
be made between new knowledge and existing knowledge to create meaning
or understanding. Specific recommendations about teaching strategies are
necessary if these models and the fruits of alternative conception research are
to be brought to the classroom.
lncorporatin~ Alternative Conceptions into Instruction
The results derived from altemative conception research have important
implications for classroom instruction because student conceptions are formed
before instruction. These conceptions differ from ideas taught in science, are
strongly held. and are resistant to change (Gunstone, 1988). The value of
alternative conception studies is "not so much that they could or should
establish noms of conceptual development in leaming science, but that they
raise awareness of the possible perspectives pupils may bring " to science
classrooms (Driver and Easley, 1978, p. 80). Pope and Gilbert (1983) believe
that effective teachers need to understand their students' conceptions because
meaningful leaming can only occur when classroorn facts have personal
relevance to students. Ebenezer and Erickson (1996) believe teachers need to
28
be aware of student conceptions when planning instruction. The authors
stipulate that this task not be loaded on the shoulders of classroom teachers;
rather, teachers need to be an integral part of designing their application in
classrooms. The traditional instructional strategy of providing definitions of
concepts and statements of principles is not sufficient for leamers to perform
complex intellectual tasks required to leam in science (Reif, 1985). Students
should not be viewed as empty vessels or blank slates that can be filled by
lecturing (Gilbert, Watts and Osbome, 1982; Gunstone and Watts, 1985); rather,
they must be actively involved in their leaming (Millar & Driver, 1987). Teachers
should not respond to student demands for ''right" answers, nor should they
yield to the temptation to attempt to transmit knowledge directly through lectures
and textbooks (Roth, K., 1990).
Instructional Strateaies and Classroom Atmosphere
Driver and Easley (1978) maintain that the designers of instructional
strategies have to consider the individuality of leaming. They believe
classroom experiences need to be designed to cause conceptual conflict, but
that students have to be in a non-threatening, student-centred environment for
such conflict to produce successful conceptual change. Students need to
interact to clarify their own ideas and explore alternative ideas through
techniques such as srnall group discussion (Driver & Bell, 1986) and student
debates (Gilbert, Watts & Osbome, 1982; Roth, K., IWO). Other common
features of constructivist classrooms might include discrepant events
(Nussbaum, l985), experiences designed specifically to distinguish scientific
conceptions from everyday views, peer discussion and analogies (Driver, 1989:
Glynn, Duit & Thiele. 1 995). Gilbert, Watts and Osbome (1 982) and Gunstone
and Watts (1985) agree that classroom experiences should be designed to
29
produce cognitive conflict in students. Julyan and Duckworth (1 996) want
students to articulate their ideas, test them through experimentation and
conversation, and consider connections between their lives and concepts being
studied.
Posner, Strike, Hewson and Gertzog (1 982) conclude that "leaming ... is best viewed as conceptual change (p.212)" and Yeaching science involves
providing a rational basis for conceptual change (p.223).* Driver and Bell
(1986) and Gunstone and Watts (1985) concur that leaming in science can be
profitably viewed as conceptual change. K. Roth (1 990) advocates questioning
as a means of exploring student conceptions. In a similar way Posner et al. see
part of a teacher's role a kind of 'Socratic tutoi' interacting wlh students to
explore their concepts, analyzing students' alternative conceptions, and
assisting students in reconstructing their conceptions.
Teachers need to take student alternative conceptions into account when
choosing and designing instructional strategies (Duit, 1931). Duit maintains
teachers should portray science as capable of change and developrnent, not as
a list of eternal truths. Science teachers must inform their practice with both
views. Science concepts that are taught cannot Vary from the consensually
agreed body of knowledge, but science teachers cannot lose sight of the
conceptions that students already hold. Teaching should be designed to assist
students in exploring their conceptions and restnicturing them to match more
closely those of the scientific community.
Facilitating conceptual change in students takes considerable tirne, most
likely in the order of months or years (Gunstone & Watts, 1985; Driver &
Erickson, 1983). Driver and Erickson believe that this time is required because
leaming in science is complicated and because students rnust leam about
theoretical entities invented by the scientific community to explain natural
30
phenomena. White and Gunstone (1989) argue that changes in belief take a
long time and require considerable reflection by leamers. These authors
maintain that leaming this process (of reflection and conceptual change) takes
time and could be started at the elementary grade levels. They identify the
practical concem that most typical science teaching strategies frequently
produce short terni success because test preparation may be achieved by
leaming patterns to answers rather then understanding basic concepts. This
wony will not be easily overcome as students are reluctant to discard a process
that they have used successfully in secondary classrooms.
There seems to be little doubt that alternative conception research has
advanced the description of leaming in science classrooms. The research
infoms teacher experience more thoroughly than previous work; however, in
spite of these advances, a grand solution to teaching and leaming in science
education is not at hand. As Wandersee. Mintzes and Novak (1994) note the
leaming processes in science are complex. They caution against expecting a
quick fix in leaming of science, because "we are finding out that quality learning
takes much longer to occur than was previously thought. (p.20)"
Champagne, Gunstone and Klopfer (1985a) describe some instructional
strategies that theoretically conforrn with conceptual development. First, they
state that students should leam to work with multiple representations of a given
problem, such as rnanipulative objects, diagrarns, physical rnodels, data tables
and graphs. Second, students should have new experiences causing them to
bring their currently-held conceptions into confrontation. Their final
recommendation is that students should analyze problems qualitatively before
applying mathematics.
Teachers have the responsibility of inducing cognitive conflict in students
and then helping them to resolve the conflict. Driver et al. (1 994) argue:
If teaching is to lead students toward conventional science ideas, then the teacher's intewention is essential, both to provide appropriate experientiai evidence and to make the cultural tools and conventions of the science cornrnunity avaiiable to students. (p. 7)
Students can be shown experimental apparatus or demonstrations designed to
produce confiict in their understanding and create anomalous results. The
anomalous results produced are then dealt with by individual leamers in the
social environment of the classroom to resolve the conflict. Chinn and Brewer
(1 993), who have described student reactions to anomalous data in detail.
argue that:
The fundamental ways in which scientists react to anomalous data appear to be identical to the ways in which non-scientist adults and science students react to such data. (p. 3)
Science teachers act as guides to mediate for students in bridging between
their everyday worid and the communify of science.
Millar (1 989) has credited the alternative conceptions movement with
considerable contributions to understanding leaming in science; however, he
does not accept that a constructivist view of learning implies one typically
constructivist instructional model. He argues that al1 teaching strateg ies can
lead to student leaming and that regardless of the strategy restructuring of
concepts takes place in the heads of leamers. Millar wonders if the
constructivist movement will provide workable applications that can be used in
a class of twenty-five or more students. Some attempts have been made to
design instructional sequences based on conceptual change and constructivist
models of leaming. Examples of these are described next.
A Teachina Model and A Practical Example from Phvsics
Stepans (1991) proposed a five-step teaching model based on
alternative conception research. First, students make predictions about
32
physical setups; second, they discuss their beliefs and ideas with other class
members. Third. their predictions are tested using laboratory apparatus. Fourth,
conflicts are resolved by further discussion; and, finally , the new concepts are
extended to other physical situations Ri the lab.
Dykstra (1 996) has developed a different way of teaching physics to first
year coliege students. Although he makes no reference to Stepans (1991)
model, Dykstra's approach has considerable simifarity to it. Based in part on an
earlier work (Dykstra, Boyle & Monarch, 1992), Dykstra teaches classes of
about twenty-five physics students in a different way than most traditional
university instnictors. The first instructional step is to explore student
conceptions of physical quantities, for example velocity, force and acceleration.
This stage is complete when groups of students have recorded their predictions
about the way objects will behave in certain conditions. Students then perform
laboratory activities and evaluate their data using cornputer graphing programs.
Experimental results are compared with their predictions and discussed with
others in the class. Conflicts between results and predictions are resolved
through discussion led by Dykstra. He avoids judging any proposed solution to
the conflict and allows class members to decide on resolution. He reports some
success wlh his technique, as well as, sorne frustration among students who
are used to instructors providing answers directly. He believes the leaming
experienced by students is superior to that in traditional classes and also finds a
greater sense of personal satisfaction with the new technique.
Student Assessrnent
As new instructional strategies are developed to facilitate student
conceptual change, modifications to assessrnent strategies are required. They
should be designed to assess student conceptual understanding to prevent
33
students from memofizing patterns for answering questions (Gunstone, 1 988).
As an alternative assessment strategy, Gunstone suggests using predict-
observe-explain (POE) exercises which he has employed in his research. One
problem with using POE in classroom assessment is that it is labor intensive
and difficult to transfer to classrooms for teacher use with thirty or more students.
This technique was designed for an interviewer to interact with students as they
made their predictions and explanations. For classroom use some
modifications would be required; for example, groups of students could do the
POE exercises and then present the results of their deliberations to other
groups and the teacher.
W-M. Roth and Roychoudhury (1993) studied physics students in a
constructivist leaming environment to determine if student preferences for
leaming environment were dependent on their views on leaming. The authors
consider that constructivist leaming environments "emphasize the interaction of
leamers with their physical and social environmentsn (p.28). The classroom
was designed to have open-inquiry; that iç, students were in control of what to
investigate and which resources to use. Their findings were somewhat
contradictory to their predictions because al1 students found group work to be
beneficial but not for the same reasons. Those who viewed leaming as
acquiring knowledge transrnitted from teachers and texts liked group work
because it reduced the load. Those who thought of leaming as individual
construction of meaning used groups to discuss concepts and develop
meaning. W-M. Roth and Roychoudhury were surprised that the leamers they
described as constructivist sought assistance from teachers to assist in
resolution of problems because the authors considered this strategy to be
inconsistent with individual construction of meaning. I do not see this approach
as contradictory because students often view teachers as facilitating their
34
knowledge construction; therefore. they would be expected to ask teachers for
assistance. Answers to questions can assist students in constructing their
knowledge. especially when they experience a block in reasoning or can not
determine how to make a connection between two ideas. Most teachers are
aware of this process when they ask and answer questions.
The influence of marks as a measure of achievement on students has
been noted (Gunstone, 1988). He is aware that students respond to marks as a
goal of leaming but does not seem to grasp the full extent of their influence on
today's students. In a similar way W-M. Roth and Roychoudhury (1993)
recognize students are focussing on achieving high marks to gain entrance to
university. These authors express concem that this overriding goal causes
students to develop coping skills to do well on final exams, rather than develop
understanding of physics concepts. The role of marks is considerable in a
secondary student's academic career. High achieving students are constantly
aware of the mark value of al1 classroom activities.
Other Conseauences of Alternative Conception Research
Alternative conceptions research has influenced other areas of science
education. For example, Gilbert and Watts (1983) have proposed that
curriculum development in science could start by reviewing descriptions of
alternative conceptions and using them as a foundation for curricula. Driver
and Bell (1986) argue that a spiral curricula is required because of the length of
time required to achieve conceptual change in students. Spiral curricula revisit
concepts and allow more detail and complexity to be added on each cycle.
Driver and Oldham (1 986) argued that cumcula should incorporate conceptual
development as part of the documentation. They believe conceptual
development should be included as an integral part of each C U ~ C U I U ~
35
document rather than remaining extemal to curricula as an instructional
strategy . A few authors have suggested that students in science should be taught
metacognitive strategies to assist them in constructing and reconstructing their
concepts in science. Reif (1 985) suggests that we stnve to teach students more
generic skills about how to leam a new concept and knowledge related to it. He
believes that students coufd benefit frorn instruction aimed at teaching them
about metacognitive skills in general. Pope and Gilbert (1 983) take a slightly
different slant by advocating that leamers leam to reflect on their own views and
'recognize their roles as theory builders (p.193)." These authors are arguing
that students need to be aware that they construct their own theories and these
theories can be refined through reflection and additional experiences. Millar
and Driver (1987) accept that pedagogy has to account for the effect of leamers'
prior knowledge on leaming activities, but see pedagogy being designed to
empower people to act more effectively in their daily lives, in their involvement
with natural events and with technological artifacts.
Teachers' Involvernent in Developina Instructional Strateaies
A key participant in any science classroom is a teacher. Regardless of
how leaming is viewed a teacher is an integral part of the process of leaming in
science classrooms, and is responsible for implementing instructional strategies
that facilitate leaming by students. Gunstone (1 988) has argued that teachers
have reacted positively to alternative conception research Pecause 1 better
informs teachers' own classroom experience. The results and descriptions of
leamers are more consistent with teachers' practical experience than earlier
types of research. In spite of this acceptance by teachers, direct applications of
research results to classroom teaching have not been easy to achieve nor are
36
they prevalent.
Driver (1 988) suggests that instructional strategies should be developed
using a process of action research directly involving classroom teachers. The
resulting strategies would be tested in classrooms during their development.
She argues that this would be a natural development of accepting a
constructivist view of student leaming. If teachers are to adopt strategies
designed for conceptual change they must be part of the research programs
that develop them (Driver, 1989). Driver maintains that students cannot develop
scientific conventions by themselves; rather, they must be constructed with
assistance from teachers who are part of the scientific community.
K. Roth (1990) points out that teachers have to undergo their own
conceptual changes about teaching and student learning. These reflective
processes take teachers considerable time as they do with students.
Recognizing conceptual change is required by teachers, Ebenezer and
Erickson (1 996) make a plea for teachers and researchers entering into
collaborative teaching and research projects. They believe the most effective
means of promoting change in classrooms is to involve teachers in the design
of change.
Constructivism in Science Education
Beainnin~s of Constructivism in Science Education Literature
The adoption of constnictivisrn as an epistemological base for research
in science education has taken place over a number of years. In the United
States Kuethe (1 963) and Boyd (1 966) explored student misconceptions in
physics but made no reference to individual student knowledge construction,
nor any reference to constructivism. About the same time Ausubel (1968)
described meaningful leaming as an individual process requinng new
37
knowledge be integrated with currently held knowledge. While he made
significant contributions to understanding leaming, he does not seem to have
been described as a constructivist in the literature.
Solomon (1 994) credits "Driver and Easley's (1 978) mernorable articlen
with creating the tools necessary "for the accelerated rise of constnictivism in
science education (p.3)." Osborne (1996) recognizes the same paper as
initiating the view that successful leaming in science depends more on prior
experiences than on cognitive levels of development. Driver and Easley had
not yet tumed to constructivism as an epistemological foundation for leaming in
science.
Magoon (1977) is frequently cited (see for example, Cheek, 1992; Driver
& Oldham, 1986; Gilbert & Swift, 1985; Gunstone, 1988) as one of the fint to
advocate a constructivist mode1 of leaming to direct educational research.
Describing limitations of the traditional approach to educational research,
Magoon argued,
There are some good indications that educational research may have reached a crisis stage with regard to its major Fisherian experimental design tradition, and perhaps that the paradigm has never worked. At any rate schooling, teaching, and leaming go on without being explicable via traditional approaches. (p.653)
In his view Piaget's research and publications, Chomsky's work with linguistic
developrnent in children, and Kuhn's (1970) descriptions of paradigms in
science were the driving forces behind the shift to a constructivist view of
educational research and leaming. He viewed educational research as
undergoing a paradigrn shift similar to those described by Kuhn in science.
Posner, Strike, Hewson, and Gertzog (1982) argued that no well
articulated theory of conceptual change had yet been described. They
described the process in leamers as analogous to Kuhn's paradigrn shift and
38
incorporated Piaget's processes of accommodation and assimilation to explain
how concepts changed. Pope and Gilbert (1983) appeared to agree with
Magoon and Posner et al.,
Recently we have seen a paradigm shift within psychology and education resulting in a renewed interest in the individual's active processing .... An emphasis is now placed upon the active person reaching out to make sense of events by engaging in the construction and interpretation of individual experiences. (Pope & Gilbert, 1983, p. 1 94)
Pope and Gilbert traced the constructivist position to Kelly (1969, cited in Pope
& Gilbert), concluding that he drew on constnictivist principles to formulate his
Personal Construct Psychology.
Osborne and Wittrock (1 983), and Pope and Gilbert (1 983) took the
position that leaming in science could best be viewed as knowledge
construction with leamers having an active role in the process. Osborne and
Wittrock (1 983, 1985) incorporated individual knowledge construction into their
generative leaming model to explain the generation of links between stored
mernories and new experiences to explain alternative conceptions by students.
Driver and Erickson (1 983) argued that viewing students as actively
constnicting knowledge was based on a "constnictivist episternology" (p.39).
but made no reference to constructivisrn as described by von Glasersfeld (1 984.
1988, 1995). Strike and Posner (1 985) described an epistemology similar to
constructivism, and Driver and Bell (1 986) referred to a uconstructivist view" of
thinking and learning in science; however, none of these authors made any
reference to von Glasersfeld's writing on constructivism.
Driver and Oldham (1986) cited von Glasersfeld directly in their
description of a constructivist approach to cuniculum development. As well, von
Glasersfeld (1 984) was cited by Bodner (1 986) in his article describing a
constnictivist model of knowledge and its implications for teaching. Driver
39
(1 988, 1 989) drew on von Glasersfeld's view of constnictivism as a foundation
for viewing individual knowledge construction, but argued that his view was not
sufficient to descnbe social aspects of leaming in science. In a similar rnanner
Millar (1 989) acknowledged the value of constructivism in describing individual
knowledge construction, but argued that constnictivism was not a suficient
explanation for the social aspects of knowledge construction in the scientific
community. Wheatley (1 991) drew on von Glasersfeld's work when advocating
the adoption of constnictivism as an episternological base for science. He
rnaintained that constnictivism fulfilled many requirements for understanding
leaming in science.
Cheek (1 992) asserted that von Glasersfeld's version of radical
constructivism should be adopted as a theoretical foundation for Science-
Technology-Society (STS) education. By the 1990s the constructivist learning
model was being desctlbed in literature aimed at practising teachers (for
example, Y ager, 1 991 ) and in teacher education texts (for example, Trowbridge
& Bybee, 1996). In the following section a description of constructivism is
p rovided .
Von Glasersfeld and Radical Constructivism
Ernst von Glasersfeld is a leading theorist and proponent of
constructivism. He has written prolifically during the last three decades, having
published more than a hundred papers on constructivism and related subjects
(see for example, von Glasersfeld, 1984, l988a, 1995). Von Glasersfeld (1 988)
argues that the most fundamental presupposition in describing teaching and
leaming is that of knowing. He concludes that the transmission model of
leaming has gone wrong because it was based on an inadequate model of
knowledge. For progress in understanding education to be made, a different
40
model of knowing and leaming is required.
Radical constructivism is "an approach to or a theory of knowing" (von
Glasersfeld, 1995, p.1) which does not treat knowledge as a search for tmth as
a refiection of a worid independent of the knower. The key to understanding the
terni consfrucfivism used in this dissertation and most science education
literature (see for example, Bodner, 1986; Cheek. 1992; Driver & Bell, 1986;
Driver & Oldham, 1986; Gunstone, 1988; Wheatley, 1991) is expressed in the
following two principles (von Glasersfeld, 1 995):
1. Knowledge is not passively received either through the senses or by way of communication. Knowledge is actively built up by the cogn izing su bject. 2. The function of cognition is adaptive, in the biological sense of the t en , tending towards ft or viability: cognition serves the subject's organization of the experiential world, not the discovery of an objective ontological reality. (p.5 1 )
Although most educators accept that students construct their own knowledge as
descnbed in the first principle (Cheek. 1992; Wheatley, 1991), the second
principle is by no means universally accepted by science educators. It
stipulates that human cognition organizes an individuals experiences rather
than discovenng objective knowledge. Acceptance of this second principle
requires a fundamental shift in epistemology. Von Glasersfeld (1 993) has
stated that radical constructivism is postepistemological because "it posits a
different relationship between knowledge and the real outside worldn (p.24).
Most people (scientists, philosophers and lay persons) have been
metaphysical realists in that they believe the real world is directly observed with
their senses (von Glasersfeld, 1984). Most persons simply assume their senses
have direct access to the extemal worid and do not generally think of their
senses as providing impulses which their minds convert into an experiential
reality. This assumption has worked well for most individuals and allowed them
to function successfully in their daiiy activities.
A Constructivist View of Knowledae Construction
Constructivists view knowledge as stored mentally in the form of
concepts which are not pictures or replicas of reality. They are mental
structures which are used by their constructor to operate successfully in his or
her experiential world. Concepts are not viewed as true; rather, constructivists
use the term viable to descnbe knowledge that remains useful to a knower in
the experiential world (von GlasersfeldJ 981, 1984, 1988b). Von Glasersfeld
(1 981 ) describes viability:
We construct ideas, hypotheses, theories, and models, and as long as they survive, which is to Say, as long as Our experience can be successfully fmed into them, they are viable. (In Piagetian ternis we might Say that Our constructs are viable as long as our experience can be assimilated to them.) Briefly stated, concepts, theories, and cognitive structures in general, are viable and suwive as long as they serve the purposes to which they are put, as long as they more of less reliably get us what we want. (p. 91)
Von Glasersfeld emphasizes that even when "perpetuated viability" of concepts
is experienced, this repetition of success can not be interpreted as a
correspondence to the real worid, or as truth in a traditional sense.
As long as people can successfully function using their currently-held
concepts, they are said to be in equilibnum. Von Glasersfeld (1989b) describes
this state of cognition:
Equilibrium refers to a state in which an epistemic agent's cognitive structures have yielded and continue to yield expected results, without bringing to the surface conceptual conflicts or contradictions. In neither case is equilibrium necessarily a static affair, like the equilibrium of a balance beam, but it can be and often is dynamic, as the equilibrium maintained by a cyclist. (p. 5)
AI1 people have experiences which do not fit immediately with currently held
42
concepts. When they become unable to function comfortably in these new
circumstances, a state of disequilbrium is said to exist. When in disequilibrium
leamers atternpt to reestablish equilibriurn through either assimilation or
accommodation. Which of these mental processes is employed depends on
how different the new experience is.
Assimilation is used when a leamer is faced with new experiences for
which a course of action can be recognized within his or her current knowledge;
for exarnple, a person might experience a new animal and place it within his or
her concept of dog. Assimilation allows us to recognize different instances as
similar, even though no two experiences are ever the same; and, small
differences in experiences must always be ignored. Von Glasersfeld (1 989b)
described the process as an experience is assimilated.
There are always small differences that I consciously or unconsciously disregard in order to establish the permanence of an individual identity. This disregarding differences is an essential component of the process of assimilation, the process that enables us to extemalize experiential items. (p. 4)
In spite of Our ability to ignore some differences between experiences, some
can never be assimilated because differences are too great to ignore.
When a new experience is sufficiently different that it cannot be
satisfactorily assimilated, leamers use accommodation to remove their
disequilibrium. Accommodation requires, at a minimum, some restructuring of a
current concept. In other cases, entirely new conceptions and categories of
knowledge have to be created to accommodate the new experiences. A person
might see a wolf for the first time and try to assimilate 1 within his or her concept
of dog; however, the differences would be too great to ignore and a new
concept of wolf would be constnicted. The significance of the differences could
be determined as result of individual experience or interactions with other, more
experienced persons.
A constructivist perspective provides a different manner of viewing
human beings as leamers. Three assumptions about leamers are important to
constructivists (Magoon, 19ï7). First, what human beings already know has
important consequences affecting human leaming and their interpretation of
what happens around them. Second, control of leaming is within individual
leamers. Environment and the external world constrain but do not determine
actions of leamers. Third, humans develop knowledge through constnicting
meaning from communications rather than by directly absorbing it from the
surface elements or words in conversation and print. Von Glasersfeld (1989b)
argued:
If the objective (of education) is to lead the children or students to some form of understanding, the teacher must have some notion of how they think. That is to say, teachers must try to infer, frorn what they can observe. what the student's concepts are and how they operate with them. Only on the basis of some such hypothesis can teachers devise ways and means to orient, direct, or modify the students' mental operating. (p. 16)
When viewed from a constructivist perspective student leaming activity during
class becomes important to teachers. To a constnictivist, student verbalizations
of ideas and concepts function as a window ont0 student conceptualizing,
thinking and concept development. Teachers can use student dialogue to
attempt to understand how a student is thinking about a particular concept in
science or physics.
From a constnictivist perspective the function of teacher instruction is
viewed differently than from other models of leaming. Rather than being seen
as transmitters of knowledge, teachers are viewed as facilitators of student
knowledge construction. Von Glasersfeld and Steffe (1 991) recommend that
teachers work to develop skills to create conceptual models of individual
44
student leaming to aid them in assisting students with their leaming. Teacherç
can use these conceptual models in choosing instructional strategies to provide
individual assistance to students in their knowledge construction.
Social Construction of Knowledae
Even though Driver and Easley (1978) were credited with beginning the
drive to constmctivism, they did not see an individual constnictivist model as
sufficient to explain leaming in science. Because science is a consensually
agreed upon body of knowledge, the authors argue that students cannot
independently discover the rules and definitions of the scientific community.
Driver (1988) has continued to emphasize that science is public knowledge that
is better described as 'carefully checked construction (p. 136)" than as
discovery. Leaming in science involves individuals being initiated into the ways
of seeing of the scientific community (Driver, 1989). Without the presence of a
teacher as member of the scientific community students would have no way of
knowing a particular viewpoint was shared with the scientific community.
Driver, Asoko, Leach, Mortimer & Scott (1 994) view leaming science as
involving a combination of personal and social processes. "Individuals must
engage in a process of personal construction and meaning making (p.8)" before
they can enter "into a different way of thinking about and explaining the natural
world" and become socialized in the practices of the scientific community.
Learners have to acquire rules to manipulate the symbols of science, a process
which is impossible without contact with the community of scientists or their
representatives. Concepts leamed in science classrooms must be similar to
those of the scientific community, because there is little value in students
carving away ideas that are significantly different (Millar, 1989). Leaming in
science can be more thoroughly understood when viewed as enculturation into
45
the scientific community combined with individual knowledge constmction.
Cobb (1 996) argues that both individual knowledge construction and
enculturation occur when leaming a body of knowledge located in a community.
His view is that the researcher interprets data frorn a particular perspective but
does not deny the existence of the other perspective. He concludes "that the
sociocultural and constructivist perspectives each constitute the background for
the other (p.48)." In a similar way Fosnot (1996) maintains that sociocultural
and individual constructivist processes are interwoven. She views them
mutualiy affecting each other. lndividuals do not act alone; we are social beings
and, as such, interact wlh others to construct mutually shared knowledge and
meaning.
Von Glasersfeld (1993) has responded to critics who have argued that
radical constructivism does not take into account the role of social interaction in
leaming. He states he has "always maintained that social interaction is a
powerful influence in the construction of knowledge. But we were busy devising
models for al1 the elementary constructing that has to be done before a
cognitive organism can begin to know and interact with others. (p. 24)"
Methods of Studying Learning in Science
General Research Methods
Welch (1985) complied an extensive review describing the state of
research in science education and made recommendations about future
research programs. He described a variety of studies that investigated the
effect of teacher behaviors on student outcornes in science and has concluded
this focus had produced very little in the way of improvement in classroom
teaching. As a result, noticeably absent from his list of recommendations 'is
46
further study of teachers and teaching behaviors, something that even surprised
me" (p.441).
Welch recognizing that student behaviors during leaming had been
under-researched, remarked, "If one thinks of the students as the prirnary actors
in the leaming process instead of the teachers, then the study of appropriate
behaviors seems highly desirable" (p.443). He noted the lack of research on
student behaviors and suggested a great deal could be leamed by investigating
students while leaming science. He did not recommend appropriate methods
for studying student leaming nor which behaviors should be the focus of study.
Welch (1985) made two other points conceming the state of research in
science education. First, he maintained that one important finding in cognitive
psychology was that learning is influenced by previously-held student
knowledge and stated 'cognitive researchers believe that understanding how
children leam will lead to improved instructionn (p.436). A second point that he
makes, almost in passing, is, 'It is difficult to separate student behaviors from
teacher behaviors because they often occur simultaneously" (p.431). The
identification of students and their interactions with teachers in science
classrooms seems obvious in hindsight, but the traditional behavion'st model of
research was predominant at that time in the United States. Welch's reference
to 'student behaviors" when recommending research focus on student leaming
in science does not fit with the recomrnendation that he is rnaking.
White and Tisher (1 986) reviewed research investigating education in
the natural sciences and, like Welch, identified studies investigating teacher
behaviors as a major component of science education research to that time.
Their review included descriptions of research studies exploring the effect of
advance organizers and behavioral objectives on student performance in
science. These authors maintained that the emphasis in science education
47
research was changing. They argued that educational research was shifting
from studies of classroom management and teacher skills to processes
employed by students while leaming cognitively complex skills and knowledge
in science. Tisher and White emphasized research related to cognitive
psychology more than Welch had done, probably because the former pair are
part of the Australian research community where the hold of behaviorism was
never as strong as in the American research communrty of which Welch is a
member.
Research Methods lnformed bv Constructivisrn
One dnving force behind the shift to constructivist based studies in
science education has been research findings that informed teacher practice.
For example, research studies have identified the failure of some students to
apply principles leamed in science classes to their everyday experiences. As
described by Gunstone (1 988),
This is particularly shown by the relatively common finding that students successful on standard foms of science achievement tests can fail to use this leamed science to interpret everyday phenornena and analyze usual situations. (p.73)
Research examining externally observed behaviors, such as performance on
tests, had not created a picture of leaming in science classes that was
meaningful to teachers. Because of this inability new methods of investigating
and describing leaming in science were developed. They were designed to
explore student leaming in order to create more detailed descriptions of student
leaming. Newer rnethods have been used to investigate cognitive processes
such as, student thinking, conceptual development and concept organization in
memory. Cognitive psychologists began by investigating rnemory, then
expanded their targets to explore cognitive processes such as, problem solving
48
and thinking (Gagné, Yekovich & Yekovich, 1 993).
Research studies designed to explore student leaming in science have
often used some fom of interview. lnterviewing tediniques and subsequent
data analysis have proven to be time consuming; and, as a result, the nurnber of
research participants in such studies is more lirnited than in traditional statistical
studies in science education. Interviews are usually recorded on audio or
videotape, transcnbed, and analyzed. Descriptions of student leaming
constructed from interview data have been rewarding and provided both
researchers and teachers with a better understanding of student knowledge
construction in science classrooms. Interviews have been carried out using a
variety of aids to assist students and interviewers to focus questions and
answers. In al1 variations interviewers are allowed to ask questions which help
students expand their verbalizations, but are not supposed to judge student
responses or teach during the interview (Delamont, 1992; Glesne & Peshkin,
1 992).
Osbome & Freyberg (1985) used a variation of interview referred to as a
think-aloud interview. In this format students are given typical science problems
to solve. While developing their solutions they verbally describe their thinking
and reasoning. A second variation known as an interview-about-instances (IAl)
uses pictures which are examples of instances and non-instances of a
scientifically accepted view of a concept (Osborne & Freyberg, 1 985; Osbome 8.
Gilbert, 1980; Watts, 1982). Students examine each picture and are asked to
explain their thinking and reasoning as they decide about each picture.
A third variation of interview used in science education research is
referred to as predict-observe-explain (POE) (Champagne, Klopfer & Gunstone,
1980; Gunstone, 1988; Gunstone & White, 1981 ). This style of interview is
employs a mechanical apparatus which is designed to demonstrate some
49
scientific principle. Students are asked to predict the resutt when the apparatus
is operated to explain the reasoning used in reaching their prediction. The
demonstration is performed and the results are discussed by student and
interviewer.
A different approach of probing of student understanding and thinking
used concept mapping and V-diagrams (Novak & Gowin, 1984). 60th
techniques have students create diagrams of their concepts and
interrelationships between them. These conceptual aids help students
visualite and describe relationships between between concepts as they
CU rrently are onderstood. The pictonal images show interconnections between
concepts and provide a permanent record at a given time. The maps and
diagrams can be used by interviewers to probe student understanding of the
concepts. Concept maps produced at a later date c m be compared to early
ones to provide a visual picture of concept development. W-M. Roth (1995)
used concept maps differently in his physics classrooms. Working
collaboratively, groups of students discussed relationships between various
physics concepts and created concept maps as a fom of review for the unit.
Walsh, et al. (1 993) used a qualitative method called phenomenography
to investigate students' understanding of relative speed. The authors used this
method to create data that could be interpreted to provide descriptions of
different ways students experience relative speed. They presented a number of
students from grade 12 physics courses and first year university physics
courses with four or five problems to solve and had thern describe how they
arrived at their solutions during individual interviews with a researcher. Walsh,
et al. documented that some students could solve problems mathematically and
obtain correct answers, but were unable to provide an appropriate explanation
of the principles of physics that were applicable to the problems. The authors
50
recomrnended that teachers introduce a new topic with a discussion designed
to have students explore their everyday experiences for genuine examples of
the new physics principle.
Researchers as TeachersKeachers as Researchers
Many researchers have advocated the inclusion of teachers into
research programs as participants because of the importance of the link
between teachers and educational research (Driver, 1989; Ebenezer &
Erickson. 1996; GolbyJ 989; Roth, W-M. 1995; Steffe, 1983, 1 991 ; Tobin,
Tippins & Gallard, 1994). White and TÎsher (1 986) concluded that the great
energy and effort expended in the pnor decade on research on education 'did
not spill over into seeing that the results affected practicen (p.897). They
expressed considerable disappointment at this outcome because some very
interesting results have been produced during that time. The authors argued
this failure to carty research results into classrooms is partly due to the
exclusion of teachers frcm active research roles. They concluded that for
changes to occur at the practical level in classrooms, future research projects
should include teachers in researcher roles. In a related position von
Glasersfeld and Steffe (1991) added their voices in support of teachers
participating in classroom research. They argued researchers should be active
teachers in classrooms because many critical events occur during teacher-
student interactions at unpredictable times.
Baird, Mitchell and Northfield (1 987) maintained that to bridge the gulf
existing between educational theory and practice teachers need to be
encouraged to act as researchers. They believe that research involving
practising teachers produces outcornes that are recognized as more relevant to
classroom instruction. They argued that much research to that time ignored
51
complexities in regular classrooms and did not draw on teacher expertise to
establish research questions meaningful to teachers. Golby (1989) supported
this contention by stating that teachers have to expand their role active
researchers and research should focus on local knowledge that is not
generalizable in the traditional way.
Butzow and Gabel(1986) repotted the results of a survey of three
hundred seventy high school science teachers conceming their interests in
research in science education. The survey showed problem solving and
meaningful leaming were of considerable interest to this group of classroorn
teachers; however, the same group considered students' beliefs about science
and alternative conceptions to be of little concem or interest. As the authors
pointed out the teachers surveyed seemed to be unaware of the connection
between problem solving, alternative conceptions, and students beliefs in
science. Butzow and Gabel predicted that teachers would accept research
results if actively involved in the research, rather than passively receiving the
findings of others research.
Wandersee, Mintzes and Novak (1994) argue that one goal of research
in science education should be to catalogue critical junctures in student
leaming. These junctures would identify topics on which teachers could focus
their instruction to facilitate conceptual change. They authors imagine
researchers in classrooms following students in classes and labs to determine
critical junctures. They believe future researchers in science education should
have a minimum of fve years teaching experience, but do not recommend
teachers acting as researchers in their classrooms even though this step seems
obvious.
In one atternpt to use constnictivism as an epistemological foundation for
a research method, Cobb and Steffe (1983), and Steffe (1991) have described
52
the "teaching experiment." Steffe argues that the unique feature of this method
is that:
... the researcher acts as teacher .... because there is no intention to investigate teaching a predetemined or accepted way of operating .... Based on curent interpretation of the child's language and actions, the experimenter makes decisions conceming situations to create, critical questions to ask, and the types of leaming to encourage. These on-the- spot decisions represent a major modus operandi in teaching experiments and the researcher has the responsibility for making them. (P-1 77)
Steffe advocates an approach to research that is prïmarily concemed with
understanding individual student leaming, and recognizes that this approach
requires that a researcher be directly involved in the interaction that occurs
between teachers and students. Cobb and Steffe suggested three reasons why
researchers should act as teachers. First, most other research methods do not
sufficiently value student conceptions and that when they are considered some
of the most interesting insights about student leaming have been found.
Second, clinical interviewers do not focus on critical moments when cognitive
restructuring occurs; however, teachers are aware of these moments and focus
on thern as they teach. A third reason for researchers acting as teachers is
because student leaming is greatly dependent on the context in which leaming
occurs and that context includes a teacher.
Research methods suggested by constnictivism are complex which
reflects the complexity of leaming and teaching in education. Research in
science education has developed its understanding of leaming to a point where
investigations of the dialectic between teacher and students are necessary to
produce a more complete picture of learning in science classrooms. Teachers
need to act as researchers because they are an integral part of the leaming
process. Students have to be viewed as participants in research because their
53
learning is the central focus of the research. Distinctions between researchers
and teachers will continue to blur as research studies explore leaming in
science classroorns.
CHAPTER THREE
THE RESEARCH METHOD
Introduction
This study was designed to assist in developing a more complete
understanding of the complex process by which grade 12 physics students
construct their knowledge of mathematical models when required to use vector
mathematics to represent (or model) two dimensional situations and to solve
related problems in physics. I also sought to understand how my interactions as
teacher of those students helped to create condlions to facilitate or impede that
knowledge construction. This combination of objectives required a unique
research method which would capture the complexity of the interaction between
teacher and learners.
In spite of Delamont's (1992) caution against selecting qualitative
methods because they are "harder. more stressful and more time-consuming
than other typesn (p.viii), a method was needed that would provide a clear
picture of student leaming within a classroom setting. The data recorded could
then be transformed through analysis into a description of the teaching-leaming
dialectic. A naturalistic (Lincoln & Guba, 1985) method involving a group of
students to whom I taught grade 12 physics was developed to provide a variety
of data sources. This method can be described as a process of mutual
construction of meaning by students and teacher. Meaning making was two-
fold; first. students constructed meaning as they learned physics; and, second, I
constructed meaning about teaching and leaming as the project proceeded.
The theoretical foundation of this research project was constructivism;
and, therefore this study is located in the constructivist paradigm (Guba &
54
55
Lincoln, 1989). Guba and Lincoln (1994) argue that research based within the
constructivist paradigm should employ a methodology that is hermeneutical and
dialectical. While they acknowledge that the issue of quality criteria in
constructivism is not well resolved, they contend that the quality (or goodness)
of this type of research should be judged using criteria, such as trustworthiness
and authenticity. which are consistent with its constructivist roots and
hemeneutical dialectical structure.
To ensure the trustworthiness of this research several design features
were incorporated. Assurnptions about the study were cleariy identified in
description of the research project. Data collection occurred over petiod of
sixteen weeks and several means of documentation were used dunng the
project, including video recording, test arid assignment results, and the
researcher's personal journal. The description of understanding constructed in
this research was compared to similar studies in science education on
alternative conceptions, including work on vector mathematics in physics (see
for example, Aguirre, 1988; Aguirre & Erickson, 1984). The hermeneutical
dialectical research method was described openly for the scrutiny of the reader
and a sincere effort was made to explain the complexity of the process with
which the description of the teachingeaming dialectic was constructed.
To ensure the authenticity of this research a thoughtful attempt was
made to present the voices of participants, teacher and students, in a balanced
rnanner during the construction of the meaning from the data. Readers are able
to judge the authenticity of the results for themselves using the extensive
verbatirn excerpts from the data which were included to support the
interpretations of classroom events. Weaknesses in my teaching were
acknowledged and incorporated in the description of meaning making. The
implications. insights and recommendations described in the final chapter
56
contain rnany aspects of Lincoln and Guba's (1 994) criteria for authenticity such
as enlarging personal constructions. improving understanding, and catalyzing
future action.
Guba and Lincoln (1 989) assert that for qualitative studies the onus is on
the readerç of the research to transfer the descriptions appropriately to other
situations. While the descriptions of leaming which were constructed in this
research are most applicable to the particular circumstances of the study,
applications of the results can be made to other situations; however, such use is
the responsibility of the reader.
In this study I taught participating students portions of the approved
curriculum of Saskatchewan Grade 1 2 Physics (Saskatchewan Education,
1992) for which they received provincial credit upon successful completion of
the course at the end of the semester. The research data were collected in an
ordinary classroom to maintain as much authenticity of student-student and
student-teacher interactions as was reasonably possible. Even in the
classroom setting the group of students took a few days to adjust to other
aspects of the research study. A laboratory or other setting would have made
this adjustment period longer, and may have affected interactions between the
participants in the study.
The Development of the Study
Obtainina Approval
As required by the University of Regina, approval was sought and
granted from both the Human Subject Research Ethics Review Cornmittee of
the University of Regina, and the Board of Education of the Regina School
Division. Documentation of these procedures are found in Appendix A.
Following this step I contacted the principal of the collegiate where I intended to
57
conduct the research. I explained the nature of my study and solicited support
which he offered enthusiastically. The principal stated that this type of research
was important because it directly involved a teacher in a classroorn study. As
my work proceeded I kept the administration informed of al1 aspects of the
project, including provision of the names of participants and our room location
in the school.
The Choice of Prairie View Colleaiate
This research study was conducted at Prairie View Collegiate' located in
a middle-sized city in Western Canada. When the study was conducted, Prairie
View had a population of approxirnately fifteen hundred students and eighty
teachers. In the year of the study the Grade 12 graduating class to which al1
student participants belonged consisted of about 250 rnembers.
Between forty and fifty percent of the Grade 12 students in Prairie View
Collegiate selected physics in their final yeaP. Secondary students believe thai
studying physics requires considerable use of mathernatics. While many
students did not enrol in physics because of this mathematics requirement,
others opted for physics specifically because of extensive use of mathematics3.
Although most high school students sense the need to use mathematics in
grade 12 physics, they do not understand why mathematics is so important in
current physics curricula because they do not fully comprehend how
mathematics is used to represent concepts in physics or other sciences.
Classes at Prairie View were scheduled in two five-month semesters.
The first sernester ran from the beginning of September to the end of January;
the second running from the start of February and finishing at the end of June.
' Prairie View is a pseudonym for the school where the research project was camed out, This estirnate was from enrolment figures at Prairie View Collegiate. ' Personal knowledge gathered over rny years of teaching.
58
Data were collected over a twelve week period d u h g the second semester in
the 1992-93 school year. At Prairie View each day was broken into five one-
hour periods. AII students and teachers followed the same tirnetable every day
of each semester. Students were randomly assigned to classes which on
average contained between twenty-five and thirty-five students.
Selection of the Student Participants
An extensive record of interactions between participants was necessary
to produce authentic descriptions of classroom activity. These, in tum, were
required for the exploration of student knowledge construction and teacher
interaction which were the main objectives of my study. I selected ten student
participants for the following reasons. First, records of student-student and
student-teacher interactions, as well as, many other aspects of classroom life
were going to be preserved by a variety of procedures, including videotape and
written student responses. Given the interactions cornplexity, these types of
data collection would only be manageable if participation was limited to about
ten persons. Second, individual student commentary had to be readily
identified and easily transcribed. Tracking participant interactions would be
impossible in a full sized class of thirty or more students because the video
camera could not provide picture and sound records with sufficient detail and
clarity. Third, in a large group less sustained interaction and discussion
between students, and between teacher and students would have occurred.
Participants for this study were chosen from a regularly scheduled
physics class taught by with Sarah Russell.' I had spoken with her earlier in the
school year about working with a group of her students in my research study.
She granted me permission to discuss with her students the possibility of
* A pseudonym.
59
participating and was pleased to be able to assist with the study. All students
would be taught by her for the first month of the semester during which time she
would teach them the sections of the physics curriculum that preceded the unit
requiring the use of vector rnathematics.
I met with her students on the first day of the second semester and
explained the nature of my research project and the level of cornmitment that
their participation would require. At this time a letter and "interest fomn (see
Appendix A) were distributed to al1 members of the class. Students were
requested to share this letter with their parents or guardians. Those who were
willing to participate were asked to retum a cornpleted interest form to me. They
were given about ten days to consider their decision during which time I was
available to answer any questions and address any concems that they raised.
Ten students were chosen from the thirteen who volunteered to
participate in the project. My selaction of participants was made to provide a
balance of approximately equal numbers of male and female participants.
Participants were not selected because of academic ability and they
represented a range of ability and personaiity. Students with good
communication skills were required because of the necessity of describing their
thinking and knowledge development during the project.
Volunteers who were not chosen to be participants were each given a
"tank-yod letter (see Appendix A). The ten students who were selected for the
study were given a letter formalizing their acceptance and a parental
permission form (see Appendix A). All participants retumed their signed
parental permission forrns which were kept in my confidential project file. We
met as a research group at the end of the fifth week of the semester. About two
weeks into the classroom sessions of the study one participant dropped grade
12 physics from her academic program and left the project. She was not
60
replaced, and my research study was completed with the nine remaining
students.
After completing the data collecting portions of the study, students were
going to retum to their regular classroorn; however, for two reasons 1 decided to
continue teaching them in the small group until the end of the semester. First,
they would not have easily blended back into Sarah Russell's class due to a
slightly different progression through the physicç cumculurn. Second, the
participants believed that the small group instruction and discussions were
beneficial to their learning. i decided to continue teaching them because this
was one way that I could reward them for participating in my research. This
decision was fortuitous because I was able to assess their understanding of
vector mathematics usage in physics problem solving over a longer period of
time than was originally planned.
Records of the Study
Data collection procedures used in this study were designed to capture
spontaneous and planned contributions by students as they leamed about
physics concepts. A comprehensive record of interactions that occurred in the
classroorn durhg instruction was required. As an active participant in the
research I was unable to record written notes in a field journal during classroom
sessions. Because of this factor a number of other data collection procedures
were used during classroom sessions. This variety assisted in providing
"between method triangulationn which Delamont (1 992) describes as "getting
data on something with more than one rnethod" (p.159). By using a variety of
strategies to probe student leaming it was more likely that the data could be
used to produce a deeper and more meaningful understanding of student
leaming and teacher interaction that were the goals of this study.
61
Several student activities normally employed in my physics instruction
were used dunng the study. These created a number of planned and
unplanned situations in which students had an opportunity to describe their
struggle to leam physics concepts. Each activity produced a rich vanety of data
which was used in the analysis. The individual strategies are described below.
Videota~e Records
Videotape recordings provide continual records of events (Glesne &
Peshkin, 1992; Erickson, 1992). In spite of their continuity Erickson describes
them as incomplete because of decisions about the location of the camera and
where to focus it. Most classroorn sessions in the study were recorded using
video cameras. The videotapes provided a permanent record, both visual and
audio, of many interactions that occurred between students, and between
students and teacher. These videotapes constituted the primary source of data
for the study.
lnitially a small conference room and then a regular classroom were
chosen as locations for the study. In both locations only a single video camera
could be used to record the activities. The views were stationary, because the
camera could not be moved without disnipting the instructional flow of a
session. This type of video record would have been suficient for the study but a
further alteration to the recording procedure was made.
Prairie View Collegiate had one classroom which was equipped for
video telecasting to distant parts of the province. This room was equipped with
five video cameras, each of which could be focussed at a different area within
the classroom. One camera was free-standing and could be aimed at any
location within the room. A second camera was capable of being focussed on a
desk top at close range and served a function similar to an overhead projector.
62
The three remaining cameras were stationary; two were focussed on the white
boards at the front of the room, and the third was mounted at the side of the
roorn to provide a wide angle shot of the class. AI1 cameras were controlled by
a single handheld remotecontrol unit which was used to activate any single
camera at a given time. The activated camera simultaneously fed to a television
monitor located at the front and a videotape recorder which preserved the
images.
Each classroom session was recorded using cameras focussed directly
at what I considered to be the most important activities in the class. Changing
cameras was done simply and unobtrusively with the handheld remote control
device. With a little practice I developed an ability to switch cameras at
opportune times. The majority of the classes were recorded on VHS video tape
producing a permanent record for later analysis. The only classroom sessions
not recorded were those during which students wrote tests or quines, or were
watching a video on the television monitor.
The video recording served one additional purpose. 1 previously
indicated that I was unable to keep a written field notes due to my active
participation in the study. As I became more comfortable with the recording
procedure, I found that I could make brief verbal notations and comments on the
tape as a session proceeded. These comments could be made unobtrusively
as part of the conversation in the class. Later they served as reminders that I
considered a particular event as important or interesting.
Personal Journal
Although not able to keep field notes I kept a personal journal during the
project. Within an hour after each daily session I wrote a log of my thoughts and
impressions of the day's events. My journal entries included some background
63
information for each session, my description of each day's events, and rny
reflections about what had happened and how I would proceed the next day. I
described student reactions and some questions or comments that had been
made during the class. My journal entries included personal reflections on
student leaming, the nature of physics, and teaching physics.
This journal seived as a record of my reflections about my teacher-
researcher role in the project and on my planning for future classroom sessions.
Notes sumrnarking discussions with teachers interested in my project were also
included. These conversations were important because they showed me that
other teachers had sirnilar concems to those I was exploring.
Test and Quiz Results
Among the documents collected in this study were tests, quines, lab
reports and other assignments used to assess student progress and establish
final grades. I used student responses to problems on tests in three ways. First,
student responses provided an impression of how well each student had
constructed concepts presented in class and identified curricular areas that
required additional instruction time. Evaluating student leaming in this manner
is a common practice of mine and was used during this study. lnterpretation of
test results helped me to create mental images of students' knowledge
constructed as a result of their work Our physics class.
Second. on tests and quines I included some questions or scenarios to
which students were requested to respond by expressing their thoughts and
ideas about particular aspects of physics. Their responses were not used to
produce their final grade but were valuable to me in assessing their
understanding of some concepts in physics.
Third, I photocopied a number of incorrect student test responses on
64
separate pages which were distributed to the participants for analysis. They
were asked to explain where the student had gone wrong in his or her thinking
and application of physics principles. This procedure was recorded with the
overhead camera as well as room cameras providing permanent records of
student comments about their own and each others' work.
At the end of the semester each student wrote a final examination which
assessed student knowledge construction on al1 topics and concepts covered
during the semester. Because I taught the participants for the remainder of the
semester this examination provided an opportunity to examine their
performance at a time distant from the initial instruction period.
Laboratow Activities
A cornmon instructional strategy in teaching physics is the use of
laboratory activities as experiential assists to student leaming. Laboratory
investigations require that students manipulate equipment which provide
concrete applications of physics principles involved. While working w l h
equiprnent students have an opportunity to interact with each other in a less
structured manner than often occurs in regular classroorn sessions. A variety of
laboratory investigations are available to explore principles involved in using
vector mathematics in physics. Two such investigations were chosen as part of
my instruction during the study.
The first involved measurement of static forces and calculation of vector
components of those forces; the second employed 'explodingn carts to examine
one aspect of momentum. These activities were perfonned by al! student
participants working in groups ranging in size from two to four persons.
Students worked together du ring the experimental phase, and then interpreted
(together or alone) the results and wrote individual lab reports. Student actions
65
and interactions dunng these investigations were recorded on videotape. The
free standing camera was moved and set-up at various locations to capture
records of some activity of al1 participants.
Demonstrations
I used a number of demonstrations to increase the relevancy of physics
for the participants. While some of my demonstrations were carefully planned,
others arose spontaneously during the course of a session. I used these
activities as practical illustrations of physics principles intended to assist
students in making connections between physics concepts, their curent
knowledge and their everyday experiences. In this research I used these
demonstrations in a manner similar to prediction-observation-explanation
(POE) (White and Gunstone, 1992; Gunstone, 1988). In this type of event an
experimental demonstration apparatus is set-up and students are asked to
predict what will happen when it is operated. Once their predictions have been
made, the apparatus is operated and obsenred. Finally students were asked to
explain if what happened was consistent with their predictions. White and
Gunstone used POE events with individual research participants, but in this
study I used such events with al1 participants in a single group. Students were
asked to predict what would happen and why. Next a demonstration was
performed to determine the accuracy of their predictions. This procedure was
used on several occasions.
In al1 my demonstrations I atternpted to use apparatus that students were
familiar with from their everyday Iives, because I wanted to provide connections
between the physics principles and the world that students knew. Among
demonstrations used during the project were: lifting a bucket of sand with ropes
at various angles; dropping objects to determine which one hits the floor first;
66
propelling rulers to show direction of flight; and, applying multiple forces to a
desk. I performed some demonstrations because they had to be executed in a
particular manner to ensure results; others were performed by students. AI1
demonstrations were recorded on videotape providing a permanent record of
student and teacher comments and interactions.
Classroom Interactions between Students and Teacher
While teaching the participant group I frequently used interactive
questioning and discussions as instructional strategies. As well, conversational
interactions occurred frequently between rnyself and students, and among
students. These interactions were somewhat different from those that occurred
in my regular classes because of the smaller group size. lnteractions and
discussions were not more frequent; rather, they could continue for a longer
time and, consequently, explore a topic in greater depth. This apparent depth
resulted from each student having more opportunity to express his or her
thoughts and to ask follow-up questions. Unfike in larger classes where many
conversations can not be heard throughout the room, students were physically
close enough together to be able to hear most conversations. Participants
appeared to be less inhibited in the smaller group and seemed less concerned
about proposing an incorrect answer or an incomplete explanation.
As teacher I made decisions conceming the flow of the lesson through
my choice of instructional strategy. I continually evaluated how much time to
spend on a topic, which new strategy to adopt in class. and when to explore
students' ideas in more depth. These decisions were a normal part of my role
as teacher; however, they did make the progress of each class personal and
unique. My professional pedagogical decisions were an integral part of the
progress of each class session and influenced the records that were produced.
67
Student "Think Aloud" Sessions
As in regular physics classes, problem assignments of various types
were given to the participants to do either during class time or as homework.
These assignments were used to provide students with applications of physics
principles in a variety of problem solving situations. Usually students worked on
them alone or in srnall groups. Before working through them as a class or
posting solutions in class students were to have tried solving each question.
They did not have to solve a problem successfully but they did have to make a
genuine attempt.
When the time came to take up a vector problem assignment, students
were chosen to place their solutions on the white board. Alone or in pairs,
students wrote their solutions to problems at the board. While writing they were
asked to "think aloudn and describe in detail their thinking at each step in their
procedure. Other participants could ask them questions about what they were
doing or how they came to decide which step was next in their sequence.
These sessions were videotaped.
Because of special circumstances, on one occasion I used a think-aloud
procedure in a quite different manner. A single student was placed in a
separate room with a tape recorder, and asked to verbalise her thoughts as she
wrote a test. The results were recorded on audiotape.
Problems Usina Two Princinles
In physics many problems can be solved by applying more than one
physics principle. For instance, the speed and time of fiight of an object thrown
up or dropped vertically can be detemiined using conservation of energy or
kinematics. Methods based on quite different pinciples can produce correct
solutions to certain problems. Neither method is more correct or appropriate.
68
All students were given two problerns on separate sheets of paper with
instructions to solve thern using two different principles. The questions had
been worded in a manner that directed students toward the different solutions.
Each student had to write his or her responses individually with no
collaboration. Once answers had been obtained using both principles, students
were asked to indicate in writing which rnethod they thought was correct and
which they understood better. Students were also asked to explain how it was
possible that two different principles could produce identical answers. The
written responses were collected and kept on file as part of the permanent
record of this project.
Participant intewiews are a very cornmon procedure in qualitative
research and have been used in science education research to examine:
leamer concepts (White & Gunstone, 1992), problem solving procedures (Chi,
Feltovich & Glaser, 1981), children's rneanings of words (Osborne & Freyberg,
1985), concept mapping and Vee-diagrams (Novak & Gowin, 1984), and a
variety of alternative conceptions (see for example, Wandersee, Mintzes &
Novak, 1994). During this project informa1 interviews were used on two
occasions.
The first intewiew occurred the day after the class had written a test on
momentum and only three students were present. Their tests had been graded
and retumed to them. Because they had written the answers only twenty four
hours eariier, the memory of their thinking dunng the test was quite fresh. I
interviewed them together for approximately forty-five minutes regarding how
they went about answering questions on the test.
The second occasion was at the end of the study when I interviewed the
69
participants about how they viewed the events of the previous fourteen weeks.
As a focus for this final interview students solved a force board problem
consisting of three spring scales attached to a central ring balanced in a state of
static equilibrium (see Figure 5.1 1). Their task was to work independently and
calculate the reading that would be registered on the covered scale.
Their understanding of this force board problem served as the beginning
for the interviews. At the students' request these interviews were conductea
with two or three participants at a tirne. I agreed to this arrangement because
dunng the project participants seemed more communicative about their thinking
processes when other students were present. They seerned able to use each
others' expressions to aid in describing their own thinking about physics
concepts. These interviews were open-ended and unstructured; participants
were free to discuss any topic that they wished which related to Our physics
class. At times I helped them expand soma responses by asking follow-up
questions. Each interview took between twenty and twenty-five minutes and
was recorded on videotape using a single camera.
Making Sense of the Data
Preparina the Transcripts
The first step in analyzing the data was the preparation of transcripts from
the videotapes of classroom sessions. 1 did the transcribing personally and the
process was time-consuming. Between k e and seven hours were required to
produce a transcript for each hour of taped classroom activity. Although
transcription was slow, doing the work myseif proved to be valuable because
the process was rich with opportunities for reflection about the study.
Transcription was done using a word processor which proved to be an
extremely valuable tool because the participants' speech emerged in bits and
70
pieces as videotapes were viewed repeatedly.
A transcript of thirty minutes of classroorn session (about ten pages) is
provided in Appendix C. The sample shows the coding developed for
organizing the data. The videotape location of each session is indicated in the
identification code which shows the tape and time where a session can be
found. Time indicators were placed at approximately two minute intervals
throughout the transcript to aid in the relocation of specific data. In the
transcripts speakers were identified with the initial of their pseudonyms. An
asterisk (*) following an inlial indicated that a speaker was on camera; an initial
without the asterisk indicated a speaker is heard but not viewed directly on the
videotape.
When verbal discussion was not present (or was too iow to be audible)
but some activity was occurring, I wrote notes in the transcript describing
student activity. These descriptions were typed in regular font and enclosed in
parentheses. On other occasions I wrote reflections and thoughts about
activities as I viewed them. These comments were typed in italics and placed in
parentheses. Care and time were taken to create transcripts that accurately
displayed spoken words and actions of the participants.
Creatina Meanina. Seekina Understandinq
Once the transcripts had been prepared the next steps were to try to
make sense of the data and create meaning for the events that had occurred in
the classroom. 1 read and reread the transcripts and my journal entries for each
classroorn session and augmented this process by reviewing video records to
reacquire a sense of the events anc! atmosphere in the classroom. lnitially I
read and assessed data from only one classroom session at a tirne but as the
analysis progressed individual days flowed together to produce a more
conünuous picture of classroorn events.
The steps in this stage of analysis are not easily described
chronologically because I repeatedly reexamined the data until I felt ready to
write about the experiences. It was necessary to dwell in my data (van Manen,
1990) for some time before I began to make sense of the data. I had to
construct meaning for the research from the hundreds of items which 1 had
collected. Transcripts and journal entries were read and reread until I could
begin to interrelate the experiences that had occurred. The first step in
constructing meaning was the identification of events and factors which seemed
important because they shifted the flow of a classroom session in a particular
direction. Some affected flow for only a few minutes, while others had influence
over longer periods. Examples of these events and factors included questions
from students, comments by participants indicating confusion, and
dernonstrations that caused students to wonder about and to reconsider
concepts which they thought they understood.
When I detected one of these events or factors. I placed it in a category
reflecting its source. I listed five sources of these shifts in the flow of events;
students, teacher, instructional strategy, school routine and researcher
reflection. My classification system involved a combination of color-coded
highlighters and sticky-notes. I identified significant sections, decided on a
category for each and highlighted them in the appropriate color. When the
highlighting process was cornplete, each transcript was reread and notes
describing why each item was placed in a particular category were written and
attached at the appropriate place on the transcript. Each note was color-coded
matching the highlighter color. This colorful classification system made location
of pam'cular types of incidents straight forward.
Initially, identifying these events and factors seemed to be an important
72
stage in the data analysis and considerable time was spent creating categories
reflecting causes of the shifts. As my analysis progressed this classification
could only be seen as the initial stage in rny conceptualkation of the complex
process. This system tumed out to be useful in thinking about events during the
analysis but was not of much use in developing the descriptions of leaming that
were created later. The approach was reductionist and too mechanistic
removing the richness from the data and losing most of the complexity of the
relationship between student leaming and teacher instruction. Separation of
individual contributions was not an effective means of data management
because of the complex dialectical nature of the relationship.
The Next Stacie in Analvsis
After spending considerable tirne expecting that the categories and
subcategories would directly lead somewhere, I eventually concluded that they
could not fumish a frarnework for further analysis; however, they did provide an
opportunity to explore the data in detail, and gradually helped me to understand
something of the complexity of the descriptions that I was trying to create. At this
time in the analysis I felt that I could begin to write about the incidents which
happened while the participants were leaming to use vector mathematics in
physics problem solving.
A sense of the setting and context that were established during this
research study are necessaiy if the reader is to understand the descriptions of
the interactions between myself and the participants presented later. These
descriptions can best be appreciated if the reader has a thorough
understanding of the environment in which the classroom activities took place.
The study took place in one secondary school which had its own characteristics.
To this is added the flavor of the participants; both the students and I brought
73
Our own contributions to the environment. Without detailed understanding of
the complex nature of the setting created by these components, the descriptions
of learning provided later may not be understood as cornpletely. The next
chapter describes the setting of the study.
CHAPTER FOUR
THE RESEARCH ENVIRONMENT AND CONTEXT
Introduction
The data collected during this study consisted of approximately two
hundred pages of transcripts from videotapes of classroom activities combined
with about the same number of pages in my personal journal. These two
sources were supplemented by tests and quines written by students, video
records and written student comments. The research questions were used to
focus the analysis which was fomulated to create understanding of the
dialectical relationship occumng between my instruction and student leaming,
and to constnict detailed descriptions of the leaming processes that students
engaged as they learned to apply vector mathematics to problem solving in
physics.
The data gathered were dependent on various components contributing
to the environment in which the research was done. Prairie View Collegiate,
which was the physical setting, had a character, routine and momentum of as
own. These indirectly influenced student responses because the school setting
affected their lives in ways that could not be isolated from the activities in our
classroom. The effect of the school is, perhaps. the easiest to describe and is
probably the least complex component of the environment.
The contributions of the participants to the research environment are
harder to describe because they are more complex and interrelated. The nine
student participants and myself had different personalities and life experiences
which combined to create a unique atmosphere in our classroom that affected
instruction, leaming and the research results. Student participants brought to
74
75
class baggage which influenced their interpretation of my instruction. They
came with preconceived thoughts and ideas about the nature of learning and
teaching, and alternative conceptions of the physics concepts within the physics
curriculum. The baggage that I brought to class was no l e s influential on the
flow of the instruction and leaming. My pedagogical expenence was
responsibie for instructional strategies that I chose, and iimited the choices of
response that I employed in a given situation. The ten individuals who
participated in this research created a small community that was a major
component of the research environment. The descriptions of the
teachingheaming dialectic constmcted from the data are partly the resub of the
environment that was created by the school and the participants.
In this chapter I describe the school and the physical setting that it
created. Next, character sketches of the student participants and their
contribution to the environment are described. Following these, my
contributions as teacher-researcher are presented. When these three factors
are cornbined they provide a partial picture of the messy reality that made up
the setting for this research study and the data that were recorded. The reader
needs to understand a bit about this part of the analysis to appreciate fully the
descriptions that make up the next chapter.
The Setting of Prairie View Collegiate
As a teacher I had become somewhat desensitized to the routine and
background noise within Prairie View; however, acting as a researcher during
this project reawakened rny sensitivity to certain aspects of secondary school
culture. Several features of the school routine inserted themselves into the
research study causing interference that was unpredictable in nature and effect.
This research project was designed to capture as accurately as possible
76
student knowledge construction in authentic classroom conditions. One result
of this arrangement was that the study classroom could not be isolated from the
activities in the school.
The inertia possessed by a large secondary school is considerable
because its routines are established by tirnetables and programs taught in the
school. Our classroom was disturbed by school activities in a manner similar to
classrooms of other teachers. Sessions could be shortened or cancelled on
short notice to accommodate special events in the school. Momentary
interruptions that created pauses in normal instructional activity were common,
usually happening two or three times per hour session. Interruptions included
intercom announcements which were unscheduled but which completely
dominated the classroom atmosphere. Even when announcements were of no
interest to my class, the sound interrupted any instruction or leaming. Knocks at
the door were another common distraction and included requests to see
individual students, to sel1 tickets for school sporting or musical events, and to
collect paper for recycling. Most requests were an important and worthwhile
part of the school culture, but ail interrupted the flow of instruction and leaming.
On numerous occasions students made requests to bave the room; for
example, students requested permission to go to the bathroom, to see a
guidance counsellor, to get a book or calculator, or for other pressing personal
business. Students were away from classes because of illness, commitments to
music or drama programs, sports, club functions, and SRC responsibilities. The
rnajonty of these absences were considered acceptable by the administration
because students were not "skipping" classes. In spite of this acceptability
students were away from physics class and not benefiiing from presentations,
activities or classroom discussion that were an integral part of instruction and
learning in Our class.
77
Our classroom contained additional sources of distraction and
interruption, including a fax machine, telephone and electronic gear used for
distance education broadcasts. The phone rang and the fax machine printed
messages without waming. Occasionally students answered the phone and
took messages for teachers of the distance education programs. As time
passed we became accustomed to the rhythm of the room and the commotion
ceased being so great a source of distraction.
A final frustrating effect of schooi routine was the bel1 (actually an
electronic beep) that signalled the end to class. Regardless of what was
happening at the end of a session the signal came for students to move to their
next class. On several occasions a few extra minutes wouId have been useful
especially when a critical incident occurred close to the end of the session;
however, students had tc proceed to their next class when the bel1 rang.
Students may have been punished had they been late for their next class. Fear
of punishment is not a motivating influence for leaming that might occur after the
bel 1.
The Student Participants
Ten grade 12 physics students, all but one of whom continued to the
study's completion, acted as participants in my research. Student participants
are referred to by pseudonyms which provide anonymity but preserve gender
because sorne aspects of the analysis may be more meaningful if the gender of
a participant is known.
As with most students the participants came to physics class with a range
of knowledge about vector mathematics and its application to physics problems.
At one end of the range was Dean who had not taken a geometry-trigonometry
course and had Iittle knowledge of vectors. At the other end were Judy and
78
Anne who had taken grade 12 geometry-trigonometry and had superior
academic records in both science and mathematics courses which they had
completed. The remaining participants fell between these levels with varying
degrees of knowledge about vector mathematics.
The nine participants who completed the project ranged in ability at
physics as shown by their final grade 12 physics marks which varied from 50 to
86 percent, including four marks over 80, three marks in the 70s, one in the 60s
and one in the 50s. This spread was coincidental because no consideration
was given to probable achievement when the participants were chosen. This
variation was fortuitous because it allowed for considerable variety of
interactions between students and teacher during the project.
The Individual Participants
From the beginning Judy exhibited confidence in her ability with
mathematics and physics which was rnaintained throughout and verified by her
final mark. She tended to listen carefully to discussions and inte ject after they
had developed, usually making her arguments clear and to the point. Anne,
who achieved the highest final mark in the class, was reserved at the beginning,
but as her confidence grew, she asked more questions and participated freely
in discussions to enhance her leaming. She was very conscientious about
completing her work and preparing for tests which was reflected in her final
grade. Lisa was fairly quiet and never particularly outgoing. Although she
asked few questions, she responded well when questions were directed to her.
As time passed she interacted with her peers more frequently and responded
more spontaneously. Her final physics grade was third in the class. Colleen
asked many questions and commented openly from the first day. She was
verbally expressive while leaming, frequently asking questions and fomulating
79
her ideas as she talked and discussed concepts. She openly showed her
frustration when struggling and her pleasure when successful.
Kevin and Ryan were extremely outgoing and showed Iittle sense of
personal embarrassrnent regardles of what they were doing. They appeared
extroverted and were frequently centres of attention, never hesitating to
comment or ask questions. Either student would take part in dernonstrations or
other activities without hesitation. Although they achieved marks in the 70s,
they did not understand the concepts as well as either Judy or Anne. Todd was
very quiet from the beginning and never did contribute openly to the class
except when asked directly. He showed only sporadic interest in the topics
discussed. His final grade was about the same as that of Kevin and Ryan.
MarÏe was outspoken and readily contributed to class discussions and activities.
She interjected without waming and usually followed discussions closely. She
did not appear particulariy concerned about her achievement in physics and
finished with the second lowest mark; her final mark was in the low 60s. Dean
was much weaker in physics than any of the other students. He tended to sit
quietly, except during the la& activities in which he patticipated fully and
seemed to enjoy. His behavior in class was such that he did not draw attention
to himself. He never asked questions or volunteered answers, and responded
laconically when asked questions directly. His final mark was just a pas.
I had not previously taught any of the student participants but they
exhibited the usual range of personalities and variation in leaming ability I have
come to expect over the years. Student personalities are important because
they effect interactions with others and classroom behavior. Individual student
leaming ability was important because it influenced how students viewed and
acted on my instruction in physics. Similarities to previous students were
important because my ability to identify individual problems with knowledge
80
construction and to modify my instruction to assist them was the result of my
practical experience with many students.
An Illustration of Participant Knowled~e
It is impossible to find a single classroom episode that illustrates the
knowledge about physics which participants brought to class; but the following
incident helps to show something of the rnanner in which the participants
approached problems in physics. It focuses on a demonstration in which I
placed a single momentum cart on a table. This device had a spring-loaded rod
that can be released to fire outward. If placed in contact with sornething (for
example, a wall or another cart) the spring rod causes the cart to move, but
standing alone on a table the moving rod should cause only a srnall jerk of the
cart leaving it in place. Students were asked to predict the results before the
spnng-loaded rod was triggered. They made the following responses.
C - I think it will move. Tchr - Which way? This way? (right) This way? (left) C - I don't know. K - It should basically stay ... it might go that way (left). R - I think it will stay. Tchr - Ryan thinks it will stay. R - Well, it will move a little bit. Tchr - What do you think Anne? A - I think it's going to go that way (right). Tchr - Anne thinks it's going to go this way. This way (right) for
Anne. This way (left) for Kevin. OK. Right in the middle for Ryan . . . basically the middle.
K - It will move a bit. Tchr - Shall we try and see? Several - Yeah. Tchr - OK, let's try it. (Spring was released. The cart jerked and
remained where it was.) Pretty exciting, eh?
This conversation was illustrative of student knowledge and confidence
in a number of ways. First, students did not appear to have thought about this
81
event and did not know what would happen. Second, students' predictions
were based on application of previous experience and hunches rather than on
physics principles. There was no evidence they were using physics principles
in their analysis. Third. they hedged their predictions by adding that there might
be a little bit of movement. This qualification allowed them to claim any small
amount of motion confirmed their prediction, or was not covered by their
prediction because their knowledge was not quite precise enough. By adding
this hedge they were less likely to be cornpletely wrong, but they did not have
the certainty associated with a definitive prediction. To them this was an
acceptable compromise.
Later in the same class a question came up about which falling cart
would hit the floor first, a heavy one or a light one. Students should have been
aware of the scientific view because it is part of the curricula of the prerequisite
courses for grade 12 physics. Kevin indicated quite clearly, "the one with the
more weight will fall faster." This view is consistent with Aristotle's interpretation
that the rate at which an object falls depends on its mass. Dunng the same
discussion Colleen argued an object with a larger surface area would hit
ground first. and Ryan thought they would hit at the same time because of
"Newton's Law." These responses illustrate several characteristics of
alternative conceptions that were summarized earlier.
These students did not hold conceptions that are consistent with those of
the scientific comrnunity. Kevin's view that heavier objects fall &ter is
consistent with Aristotle's view and intuition (McCloskey, 1983). Ryan made a
more accurate prediction but it should have been attributed to Galileo. Colleen
showed that she had leamed something about falling and the surface area of
the object but did not have the relationship clear enough in her mind that she
could apply the principle to this situation. The students remernbered fragments
82
of knowledge that they had been taught over the yearç. Their knowledge was
incomplete and they were uncertain how to apply what they did remember.
These data are typical examples of alternative conceptions held by students.
Students and the Context
Students frequently brought to class small backpacks which held some of
their personai belongings. Contents were as diverse as calculators, books,
paper, magazines, clothing, pencil cases and food. A small part of each
student's identity was contained in these backpacks. In addition to these
material objects students carried with them a more important form of baggage.
Student participants came to this physics class with between seventeen and
twenty years of life experience which was partially revealed as they functioned
and interacted. Each student possessed beliefs and understandings about his
or her abilities and skills, the nature of science and physics, and how he or she
leamed. The participants arrived with physics concepts in various stages of
construction. To understand individual student struggles while leaming physics
I had to develop an awareness of each student's knowledge by constructing "a
hypothetical mode1 of the particular conceptual worlds of the students" (von
Glasersfeld, 1996). At times students felt they understood a particular concept
but were unaware of the incomplete state of their knowledge. On other
occasions students did not make connections between their currently held
concepts and those being discussed in class. As well, students were much less
aware than I was of the effect their own knowledge had on their leaming. None
of them completely understood the influence of their prior knowledge. but we
could not ignore its influence. To assist them in leaming meaningfully I had to
facilitate reconstniction of their concepts and to relate the physics concepts to
their currently-held knowledge.
83
The terni baggage is used in this dissertation as a general term to
describe al1 knowledge and experience that participants brought with thern to
the classroom. Baggage could be interpreted as having negative implications,
but that view is not intended here. On the contrary baggage is used with
respect for the owner because of the considerable effort expended by each
person in acquiring his or her knowledge. Without their prerequisite knowledge
of reading, writing and mathematics, and other life experiences leaming physics
in our classroom would have been impossible. I expected students to have
acquired certain knowledge and skills before enrolling in grade 12 physics. Its
curriculum had been designed to build on a foundation of concepts and skills
leamed previously.
How a student is viewed by his or her peers and teachers contributes to
self-image and has an influence on student leaming. A student's self-image
had an effect on the frequency with which he or she responded to questions or
took risks in class. This was evident from the first day when the students
expressed concem about the video camera making a permanent record of their
responses which 'other people could see." Even greater concem was shown in
our final location because of the presence of the monitor that continually
displayed their actions. A student's belief about his or her ability at physics was
also a factor in student perception of self-image. When students doubted their
ability at physics, they did not respond to questions or contribute to discussions
as readily as when they were more secure about their ability. Many times
students answered, "1 don't know" or "1 don't understand that" rather than risk
giving an answer that may have been incorrect or made them appear foolish.
Dean was not at al1 confident in his ability in physics and tried to remain out of
the discussions. His lack of confidence was so great that he never volunteered
a single answer during the study. On one occasion when asked if vector
84
mathematics were used in buying and selling Anne responded, "1 donrt know
what a vector is." Her grades in mathematics were superior and I expected her
to be aware that vectors were not used in commercial transactions. Only after
the student participants began to accept that we were creating a non-
threatening atmosphere did they begin to risk making rnistakes.
Students anived at class with a mixture of beliefs about the nature of
physics but their ideas were not well formed. When asked "What is studied in
physics, or what does it do?". their responses indicated that they did not
understand the purpose of physics. Judy responded with "How things move.
Why they move that way; like gravity. We're doing acceleration." Colleen's
reply was less coherent, "Like why something falls at a different speed; like
something that you take for granted. I don't know." Participants frequently
exhibited the notion that knowledge was discovered by physicists in nature and
that science knowledge was absolute or true knowledge. When 1 asked them if
physics concepts were human inventions or were they really out there. Judy
responded with "They're really out theren and in follow-up said 7ime has
always gone by. It's not just now." Kevin had some understanding of
acceleration from driving a car but argued 'acceleration is going faster and
deceleration is going slower."
In most classes the participants asked questions of me and discussed
with their classrnates their understanding of physics concepts and problems.
Both processes were seen by students as useful in helping them leam. These
activities showed a little of what students thought about teaching and leaming.
Seeking answers directly from me indicated that they expected me to transmit
knowledge directly to them, but their discussions showed that they knew that
they could receive assistance from others with their construction of physics
concepts. When l took part in the discussions as an equal they saw me more as
85
a facilitator of concept construction rather than a transrnitter of knowledge. I do
not believe that they had a clearly formed understanding of learning and
knowledge construction. only that these activities could help them leam.
Students' Iife experiences included phenornena they had witnessed in
person, viewed on television. read about, heard about, or acquired in other
rnanners. This knowledge was drawn on by students for examples relevant to
physics concepts being developed. Students asked about sluations they had
experienced and for which they were seeking explanation or understanding. At
times student generated queries were difficult for me to relate to the discussion
at hand because connections between their queries and physics were not
always obvious. Their atiempts to make connections between life experiences
and physics concepts helped me to develop a sense of the progress that
students were making in understanding a particular concept.
The contribution of the student participants to the environment was not in
isolation from the school setting. Furthemore, the relationship between the
students and myself was syrnbiotic in nature because we worked together for
our mutual beneft. We created the biotic part of the context together as a result
of our interactions in the classroom.
The Teacher-Researcher
I had been a staff mernber at Prairie View Collegiate for nineteen years
when I undertook this study. My post-secondary education included bachelors
and masters degrees in chemistry and a bachelors degree in education with
specialties in chemistry and physics. Over my twenty-five year career my
teaching assignments have included junior and senior secondary physics,
junior and senior secondary chemistry (including International Baccalaureate
higher level chemistry), and middle years science classes. I had taught grade
86
12 physics for nine years prior to beginning my research study. In addition to
my classroom teaching for the last few years I had been department head of
science at Prairie View and had spent several years as chairperson of the
Science Curriculum Advisory Cornmittee for Our school division. This
cornmittee consisted of al1 science department heads in the division and was
responsible for providing advice to division level administrators conceming
science curricula. I had acted as a cooperating teacher for several inteming
teachers in physics. When the new science cut-rÎcula were implemented
provincially in the 1990s, 1 acted as pilot teacher and implementation leader for
two different courses. This variety of leadership roles provided opportunities to
interact with other experienced science teachers, to identify common concems,
and to provide supervision and guidance for younger teachers.
During my professional career my philosophy of science education has
undergone considerable evolution. To provide a brief snapshot of this
development I will briefly describe three components within rny philosophy - science, teacher and student. Because of my practical experience and theory
development my view of each has evolved considerably over the years.
When I began teaching my view of science would have been described
as a positivist. My thinking was that scientific laws and relationships were
discovered in nature by scientists, and a major goal of scientific inquiry was to
discover knowledge which was a perfect match (or truths) for the world around
us. At the beginning of rny teaching career I did not hesitate to confirm for
students that atoms really did exist even though I could not provide proof. My
understanding of the nature of scientific knowledge has changed. I now view
scientific developments as human constructions designed to describe natural
phenornena not as they are, but in a way that has rneaning for humankind. I no
longer state that atoms exist; rather, I explain how and why the atornic model
87
was developed and indicate that it is successful in organizing natural
phenornena for human purposes. I have corne to view mathematical models in
science as useful representations of relationships between concepts that we
identify in nature rather than definitive statements of truth about them.
As my view of science and its processes changed so did rny view of
teaching and instruction. lnitially I thought of teaching as simply describing
ideas and concepts for students. and assuming that I could be more effective by
making my descriptions of concepts and principles more clear. I felt that good
teaching only required logical presentations and clear explanations. In spite of
my attempts I was never able to have students achieve the success that I
expected. No matter how clearly and logically I presented concepts in physics
and chemistry, al1 students did not achieve the understanding of these concepts
that was my goal. As my experience grew I began to see students as working
with the ideas and concepts that we discussed in class, and trying to construct
personal understanding and meaning for them. 1 began to wonder how to get
inside their heads to understand how they leamed about physics concepts. I
realized that Our mutual questions and discussions created windows that
helped me to formulate strategies to assist them in leaming. My view of my role
as teacher has shifled from someone who transrnitted my knowledge directly to
students to a facilitator of knowledge construction by others.
As my view of scientific knowledge and instruction changed so did my
understanding of student leaming. lnitially I thought of students as more or less
directly receiving information that was presented in class. From the beginning
of my practice very few students exhibited signs of leaming in this manner but
some time passed before my understanding of student leaming changed. I
began to see leaming as a more individual process because no instructional
strategy that I employed was equally effective with al1 students. Each individual
88
seemed to be leaming in his or her own manner and rny assistance had to be
designed for each student's difficulties. l came to realize that students did not
receive knowledge directly from my teaching; rather, they reassernbled various
components of my instruction to fit their own needs. On occasion their
reconstructions were almost unrecognizable when compared to what I had
intended. After many years of practical experience the only generalizations that
I could make were that students seemed to grasp concepts more easiiy when 1
could relate thern to their own experiences. and that handssn activities and
concrete exarnples illustrating physics principles seemed to promote leaming
for most students.
Other experiences helped me to reform rny understanding of teaching
and leaming physics. F irst, when acting as a cooperating teacher for teacher
intems, I had an opportunity to observe their struggles teaching physics
concepts and to watch student-teacher and student-student interactions in a
different manner than I could do as teacher at the front of the room. lnteming
teachers exhibited difficulties teaching physics concepts similar to those I had
experienced over the years.
While I was stmggling with my philosophy of education, my daughter was
enrolled in grade 12 physics. Although she had been taught by a teacher in
another school, she stniggled with the same physics concepts that my own
students did. When I worked with her, she exhibited sirnilar difficulties to those
that I had observed in my own students. Around this time I met regularly with
other senior science teachers for reflective discussions. The experience with
my daughter and these conversations led me to understand my observations of
students were meaningful to other teachers and that we shared mutual
concems and diff iculties with teaching. These experiences convinced me that
student leaming difficulties that I encountered were common with other teachers
89
and students, and that none of these experienced teachers was able to provide
definitive solutions to Our problems.
By the time I looked into a doctoral program in education I had developed
a deep interest in examining the relationship between student leaming and
teacher instruction. This desire was fostered by an inability to improve my
students' performance in class beyond a certain level. I seemed to have
reached a poslion where little improvement was being made using my current
teaching strategies and learning models. I needed to understand leaming and
teaching differently if I were going to enhance the understanding of physics
exhibited by my students. Research literature in science education provided me
with a vocabulary to descnbe my understanding of and frustration with student
leaming and teacher instruction. In particular, aspects of cognitive psychology
and alternative conception literature informed my practice in a meaningful way.
When constructivism was added as a philosophical base for cognitive
psychology and alternative conceptions, I found that some educational research
literature was indeed relevant to my practical classroom concems.
Mv Contribution to the Environment
I arrived with baggage from a lifetime of experiences inside and outside
the classroom. This baggage was a major contributor to the research
environment. My understanding of teaching physics was paramount, because
in grade 12 physics I was trying to facilitate student leaming of physics.
Students had to construct concepts that were as close as possible to those in
the physics community. The concepts that the students constructed could not
be substantially different from those of physicists. Students' concepts would be
less sophisticated (especially mathematically) than those of practising
physicists, but would have to reflect accepted physics principles rather than
90
intuition or folklore.
In a previous çtudy (Hart & Wessel, 1992) 1 described three blockages
that students exhibited while leaming grade 12 physics. I considered these to
be significant and Iooked for similar or related blockages exhibited by the
participants. My examination of the alternative conception literature focused my
attention on looking for examples from the participants that matched those
documented in the literature. Because I was working from a constructivist base I
planned my instruction to be consistent with viewing students as constructing
knowledge rather than receiving it.
The events occurring during a given lesson were determined by my
choice of instructional strategy for that session. An alteration to a strategy
produced frequent changes in atmosphere, for a lesson or even brief segments
of a lesson. My pedagogical decisions determined the course taken during a
particular classroom session. Different instructional strategies would have
produced a different data. For example, when I used lecturettes or stories,
classroom activities were teacher-centred. When students were asked to use
visualization in solving or understanding a problem, the atrnosphere in the
classroorn shifted to be more student centred. If students asked questions or
made comments about a concept under discussion, then they detemined the
flow of a session. Their questions and comments may have been of general
interest, or inquiries intended to clarify their concept developrnent; but, in either
situation students had considerable responsibility for establishing the
environment in which leaming would take place. It is important to remember
that the data produced were dependent on the instructional strategy used and
would have been different had an altemative instructional strategy been used.
Another responsibilWRy that fell to me was the development of a tnisting
classroom atmosphere and good rapport between the participants and myself.
91
Good rapport and trust between us was required for students to take the risks
necessary to develop and construct the concepts required in grade 12 physics.
A trusting atmosphere and relationship were required for students to express
their thoughts and ideas in the manner that was necessary for succeçs of my
research.
To understand the descriptions in the following chapter it is necessary
that the reader have a sense of the environment and setting in which the study
was made. The student participants and myself, together with the school, acted
interdependently to produce the environment. Without being aware of some
aspects of the creation of the environment and setting, the interpretations that
are made in the next chapter cannot be fully appreciated.
CHAPTER FlVE
CONSTRUCTING MEANING FROM THE DATA
Introduction
As I waited in the srnall seminar room for the arriva1 of the ten student
participants, a variety of thoughts flooded through rny mind. Even though my
study posed a problem of considerable interest to me, I was uncertain how to
capture the complexity of student leaming that would occur in the next few
weeks. The qualitative method that I intended to employ for investigating my
research questions included a variety of means of capturing data. As is
common in qualitative studies, I was prepared for some evolution of my method
as the result of the emerging nature of the data, reflection generated by these
data, or research experience I gained. My joumey was about to begin, but rny
destination was by no means clear.
The most important feature of the study was the dialectical relationship
between participant leaming and rny teaching as they leamed about the use of
vector mathematics in physics. Over the years I had become aware of the
complexity of this interaction but this was the first time that I had trkd to capture
it for detailed analysis. Care had been taken in designing the project to ensure
that participants received instruction similar in manner to what I used in regular
grade 12 physicç classes. I did not intend to radically change my instructional
approach to teach these participants. The course content was consistent with
the provincial CU rriculum guide and essentially what the participants would
have received had they been assigned to my regular classes. My purpose was
to explore within this normal teaching context the dialectic between teaching
and leaming, and to describe this interaction as fully as possible.
The two major differences between my study group and a regular grade
93
12 physics class were the number of participants, and how well I knew them at
this time in the course. First. ten students participating in the study allowed
more interaction among the participants and with me than occurs in a regular
class of about thirty students. These circumstances facilitated data collection,
and allowed each student to be more expressive dunng his or her learning. As
well, the smaller number allowed me to focus more intently on individual
student cues, including verbal statements and nonverbal indicators.
Second. I was not as acquainted with this group as I would have been
with a regular class of students after about a month of teaching because I had
not worked with these students previously. Being less familiar with thern could
have posed a problem because the rapport developed between students and
teacher normally has considerable impact on their interactions. From my
experience a sense of trust and understanding among participants is helpful in
facilitating effective leaming in a secondary physics classroom. A few days
working together would allow the class to become comfortable with me and I
with them. The degree of comfort established between students and teacher is
intangible, but both know when it is present. Although unobservable, good
rapport benefds both teachers and students with instruction and learning. A
sense of good rapport became noticeable after four or five days as is shown in
the data for the initial phase of the study. At the beginning of that time more
teacher-talk than student-talk was observed; however, by the end of the first
week the participants appeared to be more at ease, and had begun to interact
more openly with me.
In this chapter I present selected portions of the data and describe the
meaning that was constructed from this research. The firçt few days of the
project did not go as smoothly as the remainder and are described in some
detail. The most important outcome during this time was the development of
94
rapport between students and myself. The second phase of the research
occuned after the students and I seemed more mutually cornfortable and
appeared to understand each others' goals for the project.
Next I describe my development of the vector concepts in grade 12
physics. Instruction began by examining the purpose of numbers and
mathematics in physics. We reviewed addition and subtraction of vectors, and
then examined applications of these functions to problem solving in physics.
The principles of components of vectors were developed using demonstrations
and lab activities. The problem of identifying direction as a property of certain
quantities emerged frorn the study of components and is described. Momentum
was introduced using lab activities and problem solving assignments. Student
understanding of mornentum is descnbed in detail. On the surface their
leaming appeared to be effective; however, when their knowledge was probed
more deeply, their !irnited understanding became clear.
The final section of the chapter discusses a force board problem (see
Figure 5.1 1) which students had to solve several weeks after the concepts had
been taught. Their solutions to this problem raised a nümber of questions and
concems about student leaming. The chapter concludes with student
responses to probes of their understanding of V ~ ~ ~ O U S concepts in physics.
Throughout the chapter I provide text from classroom data to illustrate my
construction of meaning of the dialectic. I include comments throughout which
are constructed to express my thoughts about particular developments and
deeper insights produced by the research.
The Initial Phases of the Study
The First Session - Fridav Aftetnoon
In most secondary schools Friday aftemoons are times when students
95
"gear down" from the hectic week. During Friday aftemoon classes teachers
often have difficulty getting students to focus on academic activities. The
participants were in this state of mind when I firçt met them. Only twenty minutes
remained before the bel1 would send them to their final class of the week. The
duration of this initial meeting was fomiitous because I expected only to make
an initial acquaintance with them and did not plan to begin formal instruction
during this time. Students showed signs of nervousness because we were
beginning a new type of leaming experience together.
The participants entered the roorn and each found a seat around a table.
The sense of nervousness in the room faded when I introduced myself and
casually talked with them to leam a little about each of them and to allay some
of their concems. Ryan wanted to know, "How did you corne up with ten
(participants)?" The process by which they had been selected was explained,
and they gave a small cheer in recognition of their selection. They seemed
genuinely pleased at having been selected to participate in my research
project.
Although the participants had been informed that a video camera would
be used during the study, they expressed some apprehension about its
preuezûe. Their greatest fear was that they might do something embarrassing
and that event would be permanently recorded. I was able to alleviate some of
their unease by explaining that, for the most part, I would be the only person
viewing the videotape records. Self-image was important to these students,
and how they appeared on camera was a concem throughout the project. Fear
of embarrassrnent could have adversely affected student self-esteem and
participation. Relationships of trust and cooperation were necessary for
students to participate in a manner required to produce data for this research.
As part of this initial conversation, I asked why they had decided to
96
volunteer for my research project. Any illusion I had that they might be taking
part because of an interest in the process of student leaming quickly
evaporated. Their reasons for choosing to participate included "... because a
friend had volunteeredn, and seeing an advantage in being part of a small
leaming group rather than a large regular class. Participants generally thought
that srnaller groups facilitated their leaming. They saw smaller groups as less
intirnidating, as well as, allowing more opportunities to ask questions and
discuss physics concepts with each other and the teacher.
The last few minutes of our first meeting focussed on discussing some
ideas and concepts that we would explore during the next few weeks. I began
by asking questions such as the following, What does physics attempt to do?",
What is a mathematical model in physics?", and What is a physics concept?" b
These questions were not the type that students were used to thinking about or
even discussing in physics classes. These concepts were part of the
prerequisite courses for physics, but few students had reflected on the
relationships about which I intended to engage them in conversation.
By initiating discussions about these questions, my intention was to have
students begin thinking more deeply about their cuvent knowledge of physics,
how it had been acquired and how it was structured. This instruction was
intended to encourage students to begin a different kind of search for
understanding about their structure of knowledge in physics. If students came
to understand the interrelationships in physics, the purpose of vector
mathematics in physics might, I thought, be clearer to them. Once the function
of vector mathematics was recognized, leaming these new physics concepts
should be easier for students.
My teaching experience has shown that instruction which relates physics
concepts directiy to the experiential world of students tends to be more effective
97
in facilitating understanding of those concepts by leamers. As a result I attempt
to bring everyday exarnples of physics phenomena to my classroom. To be
most effective such examples must be representative of experiences students
have had. To facilitate this approach I incorporate in my physics lessons some
knowledge of cunent events in their wortd including, television programs, and
everyday items such as, curling rocks, billiard balls and satellites. Even some
cartoon characters on television are often useful because some cartoon humor
depends on knowing that certain actions portrayed are contrary to some laws of
physics. In addition students frequently bring to class physics-related examples
about which they are curious. Student illustrations can create teachable
moments and more engagement in class activity.
The first physics concept that we reviewed was Galileo's law of inertia
(Newton's first law of motion) which states that a body remains at constant
velocity until an unbalanced force acts on that body. Students frequently hold
alternative conceptions about this phenornenon (Gunstone & Watts, 1985;
McCloskey, 1983; Osborne & Gilbert, 1980) because within their practical
experience objects stop moving when no longer pushed or pulled. The
following dialogue illustrates an attempt to stimulate the kind of student
reflection for which 1- was hoping. I asked, "Does it surprise you that when you
let go (of) a (curling) rock, 1 continues to move?" Ryan responded, "It doesn't
surprise me. If you really think about it though." He was familiar with the
behavior of curiing rocks on ice but had not thought deeply about their motion;
however, when Ryan reflected briefly about the situation he expressed a sense
that the motion might be more complex. Like most students I have obsewed in
secondary schools Ryan had not spent much time thinking about the application
of physics principles to phenomena in their worid outside the physics
classroom.
The participants amved with some preconceptions about leaming and
teaching established in their minds. For example. they knew that during this
project they would Iearn how to use vector mathematics in physics and had
predetermined ideas about the type of instruction used in physics classes.
Some early student comments illustrated these preconceptions. Marie asked,
"Should we take notes in this class?" and also wanted to know if "we would be
following Mrs. Russell's outline of the course?" These two questions illustrate a
little about her view of student learning and the role of teacher in that process.
Her first question indicated that she thought note taking was an expected task in
class, probably based on previous experience. The second query indicated
that she knew the physics curriculum had a structure or order but was aware
that my outline might be different from Mrs. Russell's.
The first few sessions with students in the project were not unlike initial
contact with other classes of secondary students. We had to becorne
acquainted with each other, and to leam a bit about each other's values,
interests and expectations. I had to gain the confidence of the participants, and
work out logistics for using technology in the project. Each change I made
caused some disruption and concern among students. The first few days had a
number of these peculiarities which had to be accommodated by both teacher
and students. It seemed that they had to understand my goals and to believe I
was personally concemed with their success in physics.
Students' confidence about their knowledge of vectors initially appeared
to be low and Penny's lack of confidence interfered with her ability to answer
questions, as demonstrated in the following.
Tchr - ... Let's Say we're talking about a velocity 50 krnh east. In that expression, which is the direction?
C - East R - East Tchr - Are you sure? P - NO Tchr - What do you rnean, no? P - l guess not.
In the expression "50 kmBi east" Penny seemed uncertain as to which part
represented direction, even though 'east" seemed obvious, because east was a
direction in her everyday language. In fact, she probably did know this was the
correct answer, but was too nervous to focus on supporting her response. She
was unable to relax sufficiently to form a reply, partly because rapport between
us had not yet developed to a point where rny comments and questions were
non-th reatening.
By the second week of the study a number of changes had occurred. We
had moved to Our final location. The new room created some turmoil because
of the technology but we acclimatized quickly as we learned how to operate
cameras and recorder. The rapport between myself and the students had
grown rapidly. Development occuned faster than in a regular classroom mainly
because with fewer students I could interact with each person every day. They
quickly realized that I was sincerely interested in their leaming and that ! would
provide assistance in any way I couid. They were beginning to accept that the
project would not interfere in their Ieaming. The cameras which initially were a
major concem faded into the background as students became less concemed
about them.
Setting the Stage for Vectors in Physics
Explorina the Place of Numbers in Phvsics
A major goal of physics is to study natural phenomena with the intention
of identifying interrelating concepts in those phenomena. Theoretical
explanations describing these interactions are then proposed, usually in the
f o n of mathematical relationships or models. Few students in grade 12
physics seem to understand this theory-building process in any depth because
of its complexity and lack of formal instruction aimed at explorhg the process.
In most high school physics classes mathematical relationships are provided
directly, or derived in a simplistic manner. Given this reality I provide students in
my classes with opportunities to foster their examination of the application of
mathematics to represent physics concepts which are familiar to them.
To assist students to understand part of the relationship between
numbers, mathematics and real worid concepts I began by discussing area a
concept with which they have had considerable experience. Most students
express a belief that they understand area and its mathematical formulae. The
following interaction illustrates my initial approach, and an example of the
student reaction to the ideas presented.
Tchr - ... but what were some of those things that you considered were concepts in physics? Distance, time, motion, velocity, al1 that stuff you gave me, right? I want to show (you) a little bit about the relationship between the real world and mathematics You guys have done such an incredible job of leaming this stuff that you've blended it iogether. For instance, if I ask you what's the area of this table.
M - Length times width. Tchr - And how do you get the length? M - Measure it. Tchr - With what? A - A ruler Tchr - Sure, and the width? R - Measure it Tchr - With? R - A niler. Tchr - And you would take those numbers and M - Multiply Tchr - Multiply. Isn't that incredible. And then you'd have the area, right? A - Yes.
Student participants quickly provided a correct formula which allowed them to
101
calculate the area of a rectangular table in the classroom. They knew how to
obtain length and width, as well as, what to do with these measurements once
obtained. They appeared to understand the concept and the calculations
involved.
At this point I wanted the students to explore a little more deeply the
relationship for which they had so rapidly produced a formula. When I asked
them, "Do you know what you did there (in the area calculation)?". Ryan replied,
'math." While this response was not wrong, it did not convey any of the
complexity present in the operation. When I asked other students to describe
their concept of area, Anne replied with a definlion, "the amount of surface
occupied." Her response, while technically correct, did not really indicate that
she grasped the idea that area is a concept abstracted from our observations of
the world. At this time I was unable to assist Anne in describing her concept of
area more fully.
This discussion was very difficult for these students to engage because I
do not believe that they had ever thought much about the relationship between
the "formula" for area and the "concept" of area in the real world. At this time Our
discussion was not going well as indicated by their low tones and laconic
responses. This type of discussion was unlike any in which they had previously
been involved and even their voice level indicated their sense of uncertainty
and frustration. Had this discussion occurred after we had known each other for
a longer tirne, the interaction would probably have been more free flowing. The
students had to take considerable risk because they were unsure what the
answers should be or what I expected in their responses. Their contributions to
the discussion were tentative indicating they were not yet secure enough with
me or the situation to risk expressing an uncertain opinion.
I asked the participants, %hy are numbers used in physics?" to which
they replied.
A - Quantitative. Tchr - Why? R - Because it's harder. A - It's harder (chuckle) K - It's exact.
These responses are brief but illustrate something of the level at which these
students were operating. That mathematics might be introduced to make
physics more difficult indicated that students viewed physics as having been
made complicated by design. While this answer had been made partly in jest,
in my experience many students seem almost to believe this reason. The
"quantitativen response simply described the nature of numerical data, rather
than expressing why numbers were used. The "exacr response indicated a
conception that using numbers was more precise perhaps producing a more
accurate and precise answer. This sort of question is not nonnally explored in
most high school physics classrooms. Few students have reflected much about
such ideas, and were only able to articulate their thoughts and ideas in very
brief responses.
1 next asked them to speculate about why humans had invented
numbers, counting, and other mathematical operations. I wanted them to think
about what situations rnight have prodded humans to invent numbers. The first
response came from Marie who responded, "t detemine population." The
second response was "like foodn from Kevin by which he meant that food might
have had to be counted. Other suggestions with no expansion were "Adam and
Even and "abacusn. Ryan suggested that perhaps marks on a cave or dwelling
wall might have been used to keep track of the number of animals in a person's
possession. While these responses were not extensive, they indicated that the
participants were thinking about these questions. They also conveyed that such
1 O3
questions were difficult to respond to partly because they had limited
experience with this type of question.
To assist the participants in discussing symbolic systems, numbers and
their development were discussed. They knew that numbers are international
symbols and essentially independent of language, but they had never reflected
on that characteristic. When asked whether other symbols existed that were
independent of language, students responded with music (which I had not
thought of), and sign language (which is not independent of language). They
were unaware that symbols for chemical elements were identical in al1
languages, even in those that did not use the same alphabet.
The intended purpose of these discussions was to have students think
about their currently-held concepts and knowledge. These discussions and
questions were intended to prod them into reflecting on their conceptual
knowledge, how it was stnictured and how it might be modified. I was trying to
bring their ideas to the surface to provide me with a means of connecting my
instruction with their experience. Discussions like this, along with other student-
teacher interactions occurred frequently during the remainder of the research
project.
These discussions occurred very early in the project and were quite
teacher-centred because the students were not yet openly volunteering
answers. As the study progressed students became more willing to talk openly
and began to take some pleasure in exploring this new type of questioning.
The participants were struggling to make connections between their
experiential knowledge acquired over their lifetimes and the conceptual
knowledge we were exploring in these discussions. This conflict occurred on
many occasions during the study and is discussed throughout this chapter.
1 O 4
Chalk Brushes. Addition and Subtraction
Before beginning to use vector mathematics in physics I typically have
students explore some relationships between numbers and concrete objects. I
have developed a demonstration using chalk brushes (or other concrete
objects) to illustrate how mathematical models using addition and subtraction
have been created for concrete objects. I try to show students that certain
operations with chalk brushes can be represented on paper (or the chalk board)
by mathematical expressions. I began by writing on the board mathematical
expressions such as, "2 + ln, u4 - 2n and "2 - 3". Students realized that the first
two expressions, "2 + I n and "4 - 2", could easily be represented using chalk
brushes in the processes of combining and taking away. They understood
these relationships and wondered why I spent time on something that seemed
so obvious and simple to them.
When asked for a numerical answer to the expression "2 - 3n, they readily
responded, 'negative one (-1)". When questioned about how I might
demonstrate this operation using chalk brushes, they suggested throwing away
three chalk brushes from a pile of two. When an imaginary third chalk brush
had been thrown away, they were satisfied even though they recognized that a
"negative one chalk bnish" did not exist in any real manner. What they failed to
understand was that a 'negative one chalk brushn had no meaning in the type of
operations that were performed with chalk brushes. The mathematical
expression could be written but had no counterpart in manipulating concrete
objects.
I was trying to illustrate for the participants that they had within their
minds a reserve of mathematical knowledge from which they had to select
specific pieces required tu solve physics problems. They did not seern to
realize that most of thern had leamed far more mathematics than was needed
1 O5
for grade 12 physics. I wanted them to understand that in physics they required
only a small part of their mathematical knowledge. Mathematics is chosen by
physicists to provide accurate descriptions and predictions of relationships
between abstract concepts identified in natural phenornena. Physicists do not
use vector mathematics to rnake comprehension of physics more diHicuit, rather
vectors and other complex f o n s of mathematics are used because they
enhance understanding of some natural phenomenon. By stimulating student
reflection on the use of familiar mathematics they may be more likely to
understand a need for using vector mathematics in more complex physics
situations. I have introduced vector mathematics in physics using this approach
for several years because grade 12 students do not corne to class wlh any
appreciable understanding of the relationship between physics concepts and
their representation by mathematics
After exploring historical reasons for inventing numbers and mathematics
I asked students to suggest some situations where numbers were important or
useful in their everyday lives. The first suggestion was to measure time
accurately in the operation of a microwave oven. Judy and others had
experience with microwave ovens, and knew that precise time measurement
was necessary to produce edible food. Buying and selling were the next
suggestions put forth. Students knew that addition, subtraction, multiplication
and division were required in commercial enterprises, and accepted that some
knowledge of percentages and negative numbers would be useful; however,
they did not see a need for square roots or calculus in commercial transactions.
When pressed to explain why calculus was not required by shopkeepers,
Colleen responded that, "It would be impossible", and Kevin replied, "because
they donPt need it." These two responses indicated that Colleen and Kevin were
beginning to understand that mathematics are employed as functional tools in
1 O 6
circurnstances outside mathematics classes. Shopkeepers and physicists do
not use more complicated mathematics than are necessary for a particular task
they have in mind. The former require only anthmetic; but, for the latter, vector
mathematics, calculus and other advanced mathematics are necessary.
While setting the stage for the introduction of vector mathematics in these
sessions, some concems about student leaming were identified. For exarnple,
student use of mental imaging and confusion about the meanings of words
used in physics and lay language were recognized in the initial sessions of the
study. Confusion in communication between the participants and myself was
also seen in these early sessions.
Learning Concerns ldentified in the Initial Sessions
Visualizina as a Problem Solvina Aid
A concern that I had and which has been identified in the literature (see,
for example, Chi, Feltovich & Glaser, 1981) was students' inability to visualize a
problem as they were solving it. Part of my response to this concern has been
to instruct students how to approximate answers to problems before beginning
to calculate answers. For example, when I asked students how long it takes for
a person to hit water after diving from a ten metre diving tower, most students
were unable to suggest an answer until they were instructed to close their eyes
and imagine the event happening at a local pool. Aiter they carried out this
instruction, most came up with a good approximation of the actual time.
Another example of this instructional strategy occurred after the
participants had calculated an answer for a velocity-distance problem. The
problem asked for the average velocity of a car which travelled 500 metres in
2.60 seconds. The answer, which they easily calculated, was 192 metres per
second; but, students were unsure how fast a velocity this represented because
1 O7
of unfamiliarity with metres per second as velocity units. When asked to
visualize this velocity to produce a mental image of how fast it was, most
students were unable to do so until reminded that a football field is about 100
metres long, and the car would have tmvelled almost two football fields in one
second.
Mental imaging of problem situations can be a very powerful aid in
solving physics problems when used suitably. Unfortunately students did not
always amploy this problem-solving aid at appropriate times. In the two
examples described they were capable of perfomiing mental imaging but
needed practice at the technique. The participants frequently did not employ
their powers of imagery to solve problems nor did they appear to apply physics
principles when attacking a problem. Students frequently applied formulae
without any consideration as to whether the resulting calculation produced a
reasonable answer, or whether a chosen formula represented the fundamental
principles necessary to solve a problem. These classroom observations were
consistent with the findings of Chi, Feltovich & Glaser (1981).
Same Words. Different Meaninas
For effective communication to occur both speaker and listener must
have similarly constructed meanings for the words used. In physics classes
effective communication is no less important than in other subject areas;
however, physics teachers frequently use common words with different
meanings than usually associated with these words. Science education
Iiterature descnbing alternative conceptions, reviewed eariier, has numerous
examples of confusion that can anse from differences in meaning (Jacobs,
1989). The participants arrived at class with extensive knowledge acquired
from their life experiences, including some understanding of concepts used in
1 O 8
physics, but their perception of such concepts was usually quite different from
that required for use in physics.
One example of confusion about word meanings can be shown with the
ternis speed and velocity. ln the Word Finder@ Thesaurus (Clans Works 4.0,
1996) these two words are synonyms. Participants could use these words
correctly in everyday contexts and brought these cornmonplace meanings to
Our physics claçsroom. In a physics context, however, speed and velocity are
not synonymous, and cannot be used interchangeably. The following
interaction resulted from my attempt to help students comprehend the manner in
which these words are used in physics.
Tchr - We use the two words interchangeably in language. What's the difference in physics between speed and velocity? Or is there any difference? Judy, what did you Say the difference was?
J - Velocity is a vector. Speed is not.
Judy described the difference between speed and velocity by stating that
velocity was a vector quantity and speed was a scalar quantity. She was aware
of the difference in the two quantities in physics, and was able to explain that
velocity was concemed with direction which is a distinction that is highly
significant in physics but very diffîcult for students to understand because of its
abstract nature. Others in class did not see a need for velocity to be
represented by a vector, nor a need to apply vector mathematics in these cases.
An additional complication, which was more difficult to describe, is in one-
dimensional physics problems vector mathematics do not appear to be needed
even when directions are involved. Correct answers to these types of problems
can be detemined using a form of algebra.
The participants did not see the difference between quantities that must
be represented by vectors and those that can sirnply be represented by scalars.
1 O 9
They could list some exampfes of each, but did not fully comprehend that some
variables were dependent on direction. In their lives they have very limited
experiences in which direction is a factor that matters. While direction is
important when driving cars one does not require vector mathematics to
navigate a car successfully. Students' experiential knowledge of the world was
firmly established in their minds and was overriding my attempts to create a
need for conceptual representation using vectors.
A second example of a physics concept that was complicated because of
its use in cornmonplace language was acceleration. Students' experience
driving cars provided them with some practical knowledge of acceleration. One
peddle in most vehicles is called the accelerator and is used for making a car
go faster. The following interaction between rnyself and Kevin illustrates a
typical level of understanding of acceleration among grade 12 students.
Tchr - The brake, is it an accelerator? K - No, it's a decelerator. Tchr - Ah, it's a decelerator, what's the difference between
acceleration and decele ration? K - Acceleration is going faster. Tchr - Oh, acceleration is going faster, and deceleration is? K - Going slower. Tchr - Going slower. How do you define acceleration? A change
in speed? From a physics standpoint the brake is an accelerator.
Kevin had in his mind a clear distinction between acceleration and
deceleration; however, physicists do not make this distinction between
speeding up and slowing down. They interpret both situations as accelerations
with the difference being indicated by using a notation for direction. In the real
world students did not view brakes and accelerators as doing the same thing
because each has a profoundly different effect on the car.
Another source of confusion with vector mathematics occurred as a result
110
of the use of the ternis addition (or adding) and subtraction (or subtracting)
when working with vectors. These two processes were the only operations with
vectors that were required for problem solving in grade 12 physics. These
terms were confusing because students already had clear and fimly
established concepts of addition and subtraction from anthmetic and algebra.
When vector mathematics were used, new and very different operations for
addition and subtraction had to be leamed and understood. While
considerable care was taken to distinguish algebraic addition and subtraction
from vector addition and subtraction, students did not make this distinction
easily or compietely. Some students still had a tendency to employ algebraic
addition or subtraction when solving vector problems as was demonstrated
during the first quiz (see appendix D) where some students used algebraic
addition and subtraction to solve a problem involving vectors.
The students' confusion is tied to their prior knowledge and their inability
to understand the manner in which physicists use mathematics to represent
various concepts identified in the world. The students have clear meanings for
addition and subtraction which they automatically bring to bear on problems
using vector addition and subtraction. They have considerable difficulty in
setting aside their prior knowledge and adopting a different meaning for a
concept that has worked well for some time. This conflict between knowledge
that students already hold and the new knowledge that is being presented is a
fundamental obstruction to overcome when new conceptual representations are
being introduced to physics students in secondary school. To construct the
conceptual knowledge of the world of physics, students need to leam more
about the structure of their own knowledge.
Identifvina Mv Error in Understandina Student Communication
Clear communication between students and teacher is vital for proper
facilitation of student knowledge construction; however. many instances of
confusion between the participants and myself occurred in our conversations
over the course of the project. One instance was illustrated in the following
interchange between Colleen and myself conceming representation of vectors.
During this discussion it became clear that I did not understand exactly what
she was talking about. This lack of understanding on my part did not allow me
to provide appropriate responses to assist her knowledge construction.
Tchr - In mathematics when they have something that has a nurnber, and a direction. What do they use to represent them?
V' - I can't remember. Tchr - You can't remember. Can you remember, Colleen? C - What do they use for what? Tchr - What do they use for a vector? C - Arrows. Tchr - Arrows. Right. So an arrow, this wasn't meant to be a hard
question, you see. This is an arrow, of course, this is a line. How do you make it into an arrow?
C - Put a head on it. Tchr - Put a head on it. How long do you make it? C - Over top of the letters Tchr - 1 don't understand. C - How long do you make the arrow? Well isn't it right over top of
the letters? Tchr - Well, you could put it over A2 in vector A, but how do I know
that I shouldn't have made A this long? C - Proportional to the number. Tchr - The length is proportional to the magnitude.
This exchange illustrated a situation where we were talking about different
things which 1 did not realize until I read the transcript. I did not understand that
Colleen was talking about the small arrow placed over a letter in a common
' When used in direct quotations V indicates a voice which could not be identifid. When representing vectors in the te* of this dissertation bold face letters (A) will be used
rather than a letter with a small amw over top because this second representation is aifficult to create on the word processor.
112
representation of vectors. I was expecting an answer relating to the length of
arrow used to represent some vector quantity. Eventually she was forced to
think about the length of the arrow because of rny question; however, there was
no way of knowing if Colleen resolved her confusion when she gave her final
response because I did not follow it up. If she did not realize that we were
discussing different arrows, she probably felt somewhat fnistrated and confused
at the end of our discussion. Had I understood what question she was
answering, I could have easily pointed out the source of our misonderstanding.
This brief interaction illustrates something of the importance of student
meaning and teacher meaning being close together. Wiihout similarity of
meaning effective communication is impossible and leads to confusion and
frustration in both teacher and students. The meaning of my question was not
transmitted in the words that 1 spoke, rather it was constructed by Colleen as
she heard them. This bief section of transcript showed the importance of
student constructed meaning being identical to teacher meaning if effective
communication is to occur. The strength of a constructivist description of
communication and meaning making is shown in the interpretation of this brief
episode.
Using Discussions to Build Models of Students' Knowledge
Addition of Vectors
Interactions and discussions with students were extremely valuable to
me. While carried out primarily with one student, reactions and inte jections
from other class members helped to provide a more thorough picture of the
class members' level of understanding of vector mathematics. A rather lengthy
interaction with Colleen illustrated how I went about creating a picture of her
current knowledge of vectors and their functions. The interaction took about ten
113
minutes of class time and began when I drew two vectors, B and J (see Figure
5.1 a) on the board. I asked the participants to use their curent knowledge of
vector mathematics to add B and J.
Tchr - ... Can one of you go to the board and very carefully explain just what you're doing? Colleen?
C - I don? remember. I did very bad in vectors. Tchr - Well, I'm not surprised. 1 don't mean that penonally or
anything. No, no, no, vectors are tricky. So you don't know how to add vectors.
Figure 5.1 a Vectors B and J as Figure 5.1 b Colleen's first step initially drawn on board in her addition drawing
My initial response was very pooriy worded and could have been construed as
a reflection on Colleen's ability. I immediately tried to clanfy my intention
because I realized how this comment could have been negatively interpreted. I
had meant that vector mathematics were difficult and that it was not surprising
that someone had trouble with them. I did not mean that I thought Colleen was
incapable of doing vector mathematics. The conversation continued,
C - Just join the lines. I thought that you were actually adding them, like when they are in the square brackets. You were actually adding them.
Tchr - I don't want to do that. All I want to do here is draw diagrams, because in physics.
C - You mean like on the board.
Tchr - Well, go do 1. Show me what 6 + J means. C - Do I have to draw a triangle? Tchr - Sure do that. What does it mean when I Say that I add them
together? C - Well, just connect the ends. Tchr - Then do it. Put them up there. But don't erase mine (J and B).
Don't use mine, redraw them somewhere else on the board.
Colleen showed that she had some understanding of how to add vectors, but
could not cleariy describe the process. She associated "drawing trianglesn with
addition of vectors, probably because diagrams for addition of vectors
frequently contain triangles; however, she was not really sure why triangles
were there, only that she expected there to be one.
C - Do I have to draw a triangle? Tchr - Just stop there. It's not wrong, I just want you to tell me what
you did. Why did you do that? C - I just drew the ends together. (see figure 5.1 b) Tchr - Why? C - Because that's what you're supposed to do. Tchr - To do what?
At this point I wanted her to explain in more detail what she had done and why.
Students were not experienced enough at talking through their reasoning in
these situations, yet such verbalization appeared to help others in class see
how a particular student worked through a given concept or problem. She
continued her work at the board.
R - Add them. C - 'Cause you're supposed to make triangles. Tchr - Let's start with. You put them together. Why did you put
those two ends together? C - Well, I don't know. You're supposed to. Tchr - You put a head and a tail together though. Is there a reason
why you did that? Why didn't you put the two heads together? Or the two tails?
C - When you put the two heads together you get the wrong angle. I don't know.
Tchr - I don't know. (1 was repeating her answer here)
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C - 1 don't know either. Tchr - But you drew it that way, and that's right actually.
Ryan's brief interjection was an indication to me that he was listening and
engaged in the discussion. Colleen had drawn the correct diagram for the
addition of 6 + J (see Figure 5.1 b), but I wanted her to provide a more detailed
description of her thinking and reasoning. I told her that her drawing was
correct because she was becoming unsure and nervous in front of the class,
and may have considered that my questioning was an indication that she was
not proceeding correctly.
C - I don't know. It has something to do with the end of one is fixed. The end of this one is fixed, and then you just take this and rnove it over.
Tchr - You moved 8 over here, right. C - I moved J over. Tchr - But you moved thern both. You moved them both. Because my
original drawing is over there. You rnoved B over there, a close facsimile. I don't think it's quite as long as my vector B but that's al1 right. And then you put 1 in position, you put J in a particular position. I can imagine four different places that you can put it, and you picked that one. Why that one?
C - I don't know.
Judy entered the discussion at this time and explained that the diagram had
been drawn in one particular orientation because of the definition of vector
addition. Colleen retumed to her desk leaving her diagram on the board. I
added a second diagram (see Figure 5.2a) to facilitate further discussion about
reasons for drawing one particular diagram for vector addition.
Tchr - Why didn't you draw that diagrarn? (Figure 5.2a) C - I don't know. I just know that you're not supposed to draw that. Tchr - Yes. you know you're not supposed to, and that's fair
enough. You know when you add vectors you join them. In what way?
C - A head to a tail. Tchr - Head to tail. That's a definition. It's very difficult for you
even to answer a question why. Why? Because that's what
you're supposed to do. Is that right, Lisa? L - Yeah.
Colleen knew that she was supposed to draw one particular diagrarn but was
unable to explain why hers was the correct one. She appeared to have
memorized an algorithm for solving vector addition, but not to have developed
understanding of the procedure. Colleen's diagrarn (Figure 5.1 b) showed only
vectors B and J, no resultant vector (or surn) had been drawn.
Tchr - However, you've done this. This represents 6 + J, right. And thafs what it says, however, there's no answer there. So what part of the diagram represents the sum of the two.
C - The angle.
Figure 5.2a Alternative diagram Figure 5.2b Diagrarn after Judy that I drew. added a connector.
I could not interpret her response "the anglen in a way that was meaningful to
me. Colleen was uncertain about how the sum of B and J was to be
represented on the diagram that she had drawn. Further prompting with
questions was of no assistance in moving Colleen toward completing the
diagram. Her responses seemed to indicate that she was at the limit of her
knowledge about vector addition. Colleen was uncertain how to proceed and
was not willing to nsk more attempts at the solution. She had retumed to her
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desk which was a signal to me that she wanted someone else to take over. At
this time Judy reentered the discussion.
Tchr - Which angle? (Judy indicated with her hands at her desk what must be done to complete the vector diagram.) Judy, just go draw it on the board. (Judy went to the board. She drew a line to complete the diagram, see Figure 5.2b.) That line there. Now is that a vector?
J - No, it's a connector. Tchr - No, it's a connector. R - Put an arrow head on it, then it will be a vector. Tchr - Ah, put an arrow head on it, then it would be a vector.
Should it be a vectof? If I take a vector plus a vector should I get a vector? Or if I take a vector plus a vector shoufd I get a number?
J - You should get a number. Tchr - Are you sure? J - Oh, no! You should get a direction too. Tchr - You should get a direction too. Now which way does the
direction go?
As mentioned earlier Judy was regarded as a strong student and appeared to
have a sound command of rnathematics, yet she was unsure about how to
answer the question conceming direction until she reflected briefly. Ryan's
comment indicated that he was engaged in the discussion and had followed
what was happening. Clues of this sort were constantly present when
discussion was allowed to be free flowing, but no teacher could attend to ail of
thern even with considerable teaching exparience. Too many things are
happening simultaneously in normal classrooms for teachers to attend to every
response and interaction.
At this stage the students were unsure at which end to place the head of
the vector in Figure 5.2b. Some had been previously instructed how to
construct the correct answer, but lacked the insight to be able to transfer that
knowledge to the current situation. I could have directly told them the answer
without allowing discussion. Such an approach would have been quicker, and
I l 8 was definitely a temptation to speed things up. I resisted because if they used
their own knowledge to amve at a solution, a much better chance existed that
they would achieve the insight to transfer their conceptual knowledge from
mathematics class to physics problems. This instructional decision was based
on rny understanding of a constnictivist model of student knowledge
construction described earlier.
Judy and Colleen talked quietly (too low to be heard on the videotape)
with each other attempting to detemine the direction of the resultant vector.
Colleen used her hands to show the direction, but Judy was not sure that they
had amved at the correct answer. When asked whether the arrow should point
towards B or towards J, Judy drew an arrow head on the vector but was
uncertain whether her choice was correct. Ryan, Penny, and Colleen also
discussed direction, but were unable to arrive at the certainty required in
mathematics where consensus is not sufficient to determine action. These inter-
student discussions were rapid and animated but not clearly recorded on
videotape.
At this point Colleen uttered. "1 don't remernber, but I don't think it's
correct". As well, Judy was still unsure whether her diagram was correct.
Somewhat in frustration Colleen then suggested that the entire diagram was
incorrect and had been frorn the beginning. I inte jected with the following.
Tchr - No it's not. No, ifs not. What you have drawn is correct so far but the head doesn't go by B. Where does the head go? Since it doesn't go by 6, it's got to go by?
C - B y J . Tchr - By J. A - So it goes like this.
My reason for inte rjecting at this time was that they had become quite frustrated.
I wanted them to stniggle to detemine the answer but not to give up trying.
Even though Anne made her first contribution to the discussion at this time she
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had been following the work at the board and the interactions. This c las
discussion had allowed participants to view each other struggle with vector
addition, and to see other approaches used in leaming a new concept.
We spent several minutes reviewing the mechanics of vector addition as
they had been developed in Our discussion of the preceding problem. My
summary was intended to help participants review their construction of these
new concepts and make connections with their previous knowledge. The exact
conceptual development of each student was not known but I hoped that this
synopsis would assist them in reviewing their own understanding of the process
of vector addition.
This section describing the development of vector addition requires a few
summary comrnents because several aspects of the dialectic between student
leaming and my instruction came to the surface. The process of vector addition
was the first procedure to be explored because it is more frequently used in
grade 12 physics and seems to be the more straight forward of the two functions
(addition and subtraction). In spite of this apparent simplicity a great deal of
conceptual knowledge has to be acquired by the students. Participants who
had leamed about vector addition earlier had memorized a procedural rule as
was indicated by their response, "place the vectors head to tailn. Although they
were able to produce a diagram, they had not conceptualized what it meant to
add one vector to another. This rnemorization of a procedure without
understanding could be interpreted as a initial (or primitive) stage in graphical
representation; however, there is not sufficient data to establish such a
sequence at this time.
Leaming conceptual knowledge of this nature is a very delicate process
and can be disrupted by even srnall impediments. If clear communication is
lacking, teachers are not able to respond appropriately to student inquiries.
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Confusion between teacher and students can anse over small details, such as
length of anows or meanings of words. After analyzing the participants'
struggle to learn about vector addition, the question of effectively facilitating
their leaming about vector addition has to be considered. Board wotk and
diagrams were used extensively in this development, but there may be other
means (for example, rnanipulative apparatus) of assisting students to develop
conceptual understanding of vector addition. The concems raised here are
examined further in the next section which describes student struggles to leam
about vector subtraction.
Subtraction of Vectors
The mechanics of vector subtraction was explored in a discussion that
lasted about twenty minutes. I moved through this section more rapidly for two
reasons. First, I had spent considerably longer on this section than I had
allotted in my course timeline. The prescribed grade 12 physics curriculum had
to be completed for provincial credit and this requirement was unchanged for
this research project. Secondly, the mechanics of vector addition and
subtraction were required to solve certain physics problems, but by themselves
did not illustrate direct applications in physics. By moving through the
mechanics and introducing physics problems which required vector
mathematics to solve, students would experience concrete examples of vector
applications. Working with applications in problem solving would assist
students in developing more understanding of how vector mathematics is used
to solve certain types of problerns in physics. The application of vectors to
physics problems might also facilitate leaming by the students as they began to
see how vector representation worked and how they could be used to solve
problems that could not be solved by algebra.
We began Our conversation with an exploration of how a diagram to
show vector subtraction should be arranged.
Tchr - What do you do if you want subtraction? How do you put the vectors together, if you want to subtract? Obviously, Ys not head to tail, because if it were head to tail? What do you think Judy, do you remember how to do it?
J - No, maybe. It might be tail to tail.
Judy was chosen to work at the board for a reason of which students were
unaware. She would likely be able to draw a correct diagram in the shortest
time. By deliberately choosing Judy to work at the board class progress was
sped up without appearing to msh. She drew the correct diagram with two
vectors in proper tail to tail position (see Figure 5.3a). I instructed her to
continue with the subtraction operation.
Figure 5.3a Judy's initial Figure 5.3b Final version of vector drawing B - J = A V
Tchr - OK, now you draw that. OK, I won't ask you to put the head on it (the A vector) yet, because we didn't decide whether we wanted to subtract B from J, or J from B. I should have indicated up here that we were taking B - J. And this equals some other vector.
P - Does it matter if it's 8 - 3 or J - B? Tchr - Yes, it does. Yes, it does. That's an important question because
J - B =, this is going to be something else. I start with the same diagrarn though. This is 6, this is J; and, of course. same line. It can't be the same answer though, can it. 6 - 2 is not the same as
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2 - 6. Where do we put the head? (long pause) No guess, Anne? A - No. At opposite ends. Tchr - Yes, one has got to be at the bottom, and the other must be at the
top, right?
Penny had asked an important question inquiring whether the order of vectors
affected the answer in vector subtraction. I drew on an analogy of numerical
subtraction to show her that the answers would be different and the importance
of drawing a correct diagram. The "long pause" lasted about a minute and
would have been longer without my directed question to Anne. She understood
that the two subtractions, 6 - J and J - 6, would produce vectors that were
180" opposite in direction, but did not know which subtraction produced which
answer. The discussion picked up again with several students participating.
R-IsayyouputitonJ. Tchr - You Say you put it on J, any reason? Or just because you like it
there? R - Because 6 is on top and J is taken away so Tchr - Which one are you taking away though? Go back up to the
question, where it says. To start we're talking about the left diagram, right?
R - No, we're talking about that one. Tchr - Yeah, this one here on the left side. (They clarify which
diagram) I've got B - J and J - B, but just deal with the one that Judy drew, B - J. What am I taking away?
R - J Tchr - OK, and you want to put the head down at J. R - I don't know. Maybe it should be on top. I don't know. It should be on
top. Well, you're taking the J, minus the 6 and you're going towards the B, so it should be on top.
Tchr - I don't know if the reason is right, but the answer is. lt goes here. (See Figure 5.3b)
In vector su btraction the correct resultant vector does not seern as intuitive as
with vector addition where resultant vectors look visually correct. Du ring this
time Colleen and Ryan were intently discussing what the answer to the problem
should be. Their conversation was not captured on tape and not noticed until
the tape was reviewed. When the participants were allowed to interact with
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each other as they were struggling with difficult concepts. conversations
between participants erupted spontaneously. They sought to discuss their
struggles over the concepts with each other. At times more than one
conversation occurred simultaneously. Even with only nine students many
interactions occurred that could not be captured on tape because they were
over (andor unnoticed) before a camera could be focussed on them.
Transcnpts of classroom activity do not completely capture the nature of these
interactions. The halting quality and slow progression could not be effectively
transferred from video record to transcript page. Pauses and b rief inte jections
occurred frequently; many utterances were too faint to be recorded.
As with vector addition I reviewed our discussion to summarize what we
had accomplished. Ryan had chosen the correct direction for AV, but I was not
convinced that his reasoning was valid. Even if Ryan understood his reasoning,
1 was also quite certain that little chance existed the others had followed his
thinking. My summary focussed on two points. First, I showed them a method of
detenining the correct vector direction using the "An sign (symbol for "change
in") which they used frequently in science. Then I showed them an alternative
method for vector subtraction which employed the addlion of a negative vector.
This approach seemed more complicated but provided an alternative that might
be useful to some students. Some mathematics teachers used this alternative
rnethod and showing students might have provided a bridge between physics
and mathematics.
This description of students struggling to leam about vector subtraction
raises concems and issues as did the previous one on vector addition. Vector
subtraction is more difficult to conceptualize because students do not normally
have available experiences that require vector subtraction to provide full
explanations of events. For students, leaming about vector subtraction wlhout
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applications is very difficult because the process seems to be rules and
diagrams with no reason for doing the process. They view it as making physics
hard for no apparent reason. From the teacher's point of view the problem is
how and when to introduce the rnechanics of vector addition and subtraction to
the students, and how to bridge the gap between the mathematics and using
them for representing the relationships between physics concepts.
Commentç about Student-Teacher Discussions
Student-teacher discussions of this type reveal a great deal about
student leaming. The descriptions produced by analysis illustrate the type of
image that can be created from as few as ten minutes of discussion between
students and teacher. For example, I had expected that the participants who
had leamed about vector addition in previous courses would be able to answer
the addition problem correctly in a few seconds. In spite of their previous
experience they struggled for more than ten minutes finally ending in frustration
when Colleen suggested that the whole solution was probably wrong. The
knowledge that I wanted them to retrieve was a fundamental function in vector
mathematics, yet they were unable to bring it to the surface even with directed
prodding through questions. They had not leamed the concepts of vector
addition in a way that pemitted them to transfer the concepts to different
situation. The responsibility for the state of their knowledge should not be
placed on the shoulders of the students, rather it falls on the mathematics
currÏculum and the instructional strategies used in those and other classes.
A second consideration was that for me the answer to the vector addition
problem seemed so obvious because the vector diagrams are intended to act
as aids to the process. When vector diagrarns are viewed with understanding
as visual representations of vector addition and subtraction, the answers
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emerge from the diagram; however, students did not appear to see diagrarns as
serving this function. Instead of using the diagrams as a problern solving aid,
these students seemed to view diagrams as separate problems. If this situation
is an example of an initial stage of graphical representation, then students
exhibiting that stage could be described as not visualiu'ng the diagrams as
displaying the cornponents of a problem (here vector addition). They do not
visualize the connection between the problem and the diagrarn. The
participants were unable to make this connection in spite of instruction
designed to assist them in doing so. Alternative forms of representation that
might assist them, such as manipulatives or cornputer simulations, need to be
explored.
Interactive discussion between students and teacher was informative, but
can be perceived to have drawbacks. This instructional strategy was time
consuming and appeared to cover less content than other foms of instruction
might have done. Because of the extensive content in the physics curriculum,
teachers are often reluctant to spend long periods of time in discussion with
students at the apparent expense of content. This concem of excessive content
will be explored in the final chapter.
For the participants a sense of frustration occurred because answers
were not provided directly in the manner to which they have become
accustomed. They have not reflected on their previous learning experiences
sufficiently to realize that they had been told how to answer these problems in
earlier mathematics classes and were now unable to use that knowledge in our
physics class. They had made little transfer of the knowledge from mathematics
classes to physics class.
This lack of transfer from other classes to physics is an important piece of
data. Students appeared to have leamed this mathematical knowledge in a
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rote manner. They did not display any characteristics of having leamed it in a
meaningful way, nor had they made connections betweeo their conceptual
knowledge of vectors and physics problems where it needed. No one seemed
able to show that he or she was capable of applying vector rnathematics to
physics situations, or to show rnuch evidence of being able to recall what had
been leamed previously. Students showed clearly that they had not leamed
vector rnathematics in a rneaningful way because they displayed little evidence
of understanding the mechanies of vector addition and subtraction.
Providing answers directly was a great temptation because this strategy
appeared to cover material more quickly and content in the grade 12 physics
curriculum is extensive; however, if the frequency of discussions were reduced,
a valuable leaming resource for teacher and students would be lost. For
example, one value to students waç that they viewed how others struggled as
they leamed new concepts. Judy was known by her peers to be a strong
student but was observed to struggle with vector addition. Without this type of
interaction other students may have viewed her as never struggling to acquire
difficult concepts. In my experience students have often expressed a feeling
that strong academic students were naturally gifted and never had any struggle
when leaming complex concepts. A second value of these interactions was that
they allowed me to formulate my responses to student inquiries in a way that I
thought would best assist them in overcoming their individual leaming
difficulties.
Student Questions as Wlndows to Student Thinking
"How do p u know that vou have to add them?"
During one class Marie asked, "How do you know that you have to add
them?" Most participants appeared to have developed a little understanding
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about vector addlion and subtraction, but when faced with applying vector
mathematics to physics problems they were often unsure how to do it. Marie
expressed her uncertainty about whether to use vector addition or subtraction.
Students needed to know how they could identify which operation was
necessary for a particular problem. They tended to look for word dues as signs
of the choice to be made rather than to decide on the basis of fundamental
principles of physics. This approach had been successful in their past
experiences in science classes and they were not willing to abandon it.
Physics experts study problems and mentally estimate solutions before
beginning calculations (Chi, Feltovich & Glaser, 1981). They look for underlying
principles which are important to the problem, and only then do they use
mathematical fonnulae. My students wanted to identify cues in the problern
which identified the calculation required without analyzing for underlying
physics principles. These cues were used to detemine which physics formula
was to be used without any consideration of its appropriateness. They did not
view solutions as based on physics principles, rather many seemed to view
problems as mathematical puzzles which have a trÎck for solving. This
approach had achieved success in the past and they are trying to use a
successful process in a new situation.
"Do vou figure out subtraction and addition the same way?"
Near the end of one class Kevin asked, "do you figure out subtraction
and addition the same way?" I replied that once we had an appropriate
diagram the same mathematical and geometric functions were used to
determine the answer. I assumed that this was an appropriate reply to his
question and would alleviate his confusion; however, when in follow-up he
asked, "So does it matter which way they are," I became more aware of his
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actual concem. My problem was that only two or three minutes remained in the
class and this was not sufficient to respond to his question.
At the beginning of the next class we explored Kevin's question. We
reviewed vector addition and subtraction and looked at different operations with
the same vectors to compare the results. This review was similar to the
previous instruction but took on fulfer meaning for the students because of
leaming that had already occurred. Review of topics in this way is designed to
help students acquire more thorough understanding because some knowledge
construction had previously taken place as a result of their first introduction to
concepts.
During this particular discussion Kevin, Judy, Ryan, Marie, Anne, and
Lisa participated directly in the conversation with me. In addition, considerable
interaction took place between students at their desks. These conversations
could not be recorded on videotape because they were quiet discussions
between two or three students as they explored the concepts being discussed.
"But There is No Chanae" - A Discre~ancv
Important leaming situations were created for students when physics
explanations of familiar events seemed to be counter to students' intuitive
understanding of those phenornena. When effectively used these situations
can provide students with impetus for reconsidenng and reconstructing their
insight into some physics concepts, One such discrepant interpretation
occuned when we studied changes in velocity of objects going around a corner
at constant speed. I introduced a problem of this nature before the students had
enough expertise to solve it successfully because I wanted the ideas to
percolate in their rninds for a few days. I thought that this sequence would
produce a better opportunity for them to see sorne discrepancy between their
129
understanding of this situation and the physics view of the same situation.
The situation in the problem was simple and based on personal
experiences of most students. Students were asked to imagine driving along a
highway at a constant speed of 130 km/h and then going around a corner
without changing their speed (see figure 5.4a). They were asked to state what
change in velocity occurred as their car went around the comer. Part of the
conversation went as follows.
Tchr - The car is travelling at 130 kmh east, then the car tums and goes 130 krnh south. The speedometer reads 130 krnlh. And, over here he looks at the speedometer, and it says 130 krnh. You notice that i fs called a speedometer. Not a Velocimetet', it's a speedometer. He went around this comer, it's a nice banked comer, he didn't have to slow down, he didn't have to speed up. He's got it on %ruisen. Now, I'm going to ask you. And, I don? want the answer. What is the change in velocity?
R - There is no change. Tchr - And Ryan said. There is no change". I heard that. Your
initial reaction is, "there is no changen.
Even though they had been wamed that this situation was tricky, Ryan
answered with considerable certainty that There is no change." He spent only
a few milliseconds thinking about the question before volunteering an answer.
This initial response that "tere is no change" has occurred every time that I
have used this type of question in this rnanner. Grade 12 physics students do
not see tuming a corner at constant speed as a situation where any change in
velocity occurs. Their confusion rested primarily on two related factors. They do
not comprehend that direction is an important attribute of some physical
characteristics and do not understand the difference between speed and
velocity. Confusion over this problern was resolved during the next few days.
At this time I intended for the students to understand that the physics solution
was quite different from the one they intuitively expected. This instructional
strategy provided them with time to think about certain problems and illustrated
130
that I would not provide an answer the firçt üme that they struggled with a
particular concept. I wanted them to realize that leaming was their
responsibility and that I would help thern when that help was required.
"1 don't understand what this means."
A few days after introducing the problem conceming the car tuming a
corner at constant speed (figure 5.4a), I asked Anne to work through the
solution for the class at the board. Getting her to go to the board required some
persuasion because she knew this problem was difficult and was unsure of how
to formulate the answer. Once at the board she proceeded with the problem
quickly providing essentially no explanation for what she had written on the
board. Her firçt step was to draw vectors VI and V2 on the diagram at the
board (see Figure 5.4b).
Figure 5.4a Initial diagram of car Figure 5.4b Diagrarn showing tuming corner vectors added by Anne
With assistance from the other students Anne decided that she needed to
perform the vector operation, V2 - VI = AV. Figure 5.5 shows the diagram that
she drew to assist with this calculation. She chose to calculate AV using
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Pythagoras' Theorem. Other students calculated answers as she required
them. While Anne was working at the board Judy asked me, "Can you use sine
of 45" to get the answer?" She had calculated the correct answer using a
different procedure from Anne and thought her approach was mathematically
sound, however, needed verification that it was correct. When I confinned that
indeed her method was correct, Anne's reaction (while still at the board) to what
Judy had done was, "Should I do that?"
Uncertainty and insecurity in physics students was exhibited in two foms
by Judy and Anne. In the first case, Judy was secure enough with her
knowledge that she tried a different solution to a problem. All she required was
confirmation that her process was valid. In the second case, Anne showed a
different form of insecunty when she inquired if she should do the same thing as
Judy. Even though class members had worked with her as she proceeded
through her solution, she was willing to abandon a successful procedure for an
alternative one because of uncertainty. She lacked sufficient understanding of
the pnnciples to have confidence that her work was correctly headed toward a
solution.
After further discussion the correct answer was determined to be
184 km/h [S45OW]. At this time Kevin remarked, ''1 don't understand what this
means." His comment was neither unusual nor unexpected. On many
occasions my students have calculated a correct answer to this type of question,
but have been uncertain what the answer represented. In this case students
knew the car's speed had not changed, but did not seem to make the
connection that a change in direction produces a change in velocity. Kevin
followed up his first comment with, "So, why is there a change in velocity?" He
had understood what I meant by a change in velocity, but still felt the answer
arrived at was contradictory to his intulive understanding of a car tuming a
comer. It seemed to be contrary to his everyday experience.
To help the participants achieve some understanding of comering I had
them reflect about an common experience by having them visualize travelling in
a car that was going rapidly around a comer. Everyone knew they would feel
as if they were thrown outward against a car door. Part of the interaction
between the participants and rnyself is presented in the following.
Tchr - What happens when you are driving and go around a comer real fast?
R - You move. Tchr - Where? R - To the side Tchr - So you get thrown to the outside. J - You stay where you are, and the car moves. Tchr - Ah, two different things. She says you stay where you are, and the
car tums. R - The seat slides undemeath you. You keep going straight. Tchr - You keep going in the same direction, and the car wants to tum.
And because you are usually strapped into it. you tum with it. So, how do you get tumed though? To make you turn, what do you have to do?
Vs - Tum the wheel Tchr - No, that's the car. To get your body. Your body is going in a
straight line at 50 k m h straight down the road and you want it to go in that direction. What do you have to do to it?
K - You have to apply a force. Tchr - There is no other way that you can do it. Imagine you are in a car
with no doors and no seat belt as a passenger, and that you have a nylon suit on and plastic seat (covers), nothing to hang onto, and you
have essentially no friction, imagine you corne to the corner at 50 km/h and the petson tums the comer.
C - You'd go flying out of the car. Tchr - You'd go flying out of the car, or if you are watching it from above.
Sornehow you have a hot air balloon and you watch this (situation) from above, what does it look like?
R - He falts out. Tchr - He falls out, but what would it (the motion of the passenger) look
like from above? K - He kept on going in the same direction. Tchr - OK, so it would look something like, the car goes around the corner
and the guy just keeps going along and skids along and bounces (on the road). It's not that the car is going along in a straight line and he gets thrown out. Right, he is going in a straight line. What was Newton's first law?
L - A body goes on in a straight line. Tchr - Sure, a straight line. That's what is happening. If you are going in
a straight line, the only way you can tum is (by applying) a force. In this case it is obvious that to get around the comer a force was applied. How do you apply the force when you go around the corner?
A - Through the steering wheel. Tchr - Sure, you tum the steering wheel. You do it through the wheels.
And if fi's on a highway they help you out a bit because friction can only carry so much. They tum the road up (banking).
K - That's why they do that (bank the road)?
This passage illustrated part of my attempt to assist students to
understand changing direction as a change in velocity. Tuming a comer in a
car is a phenomenon which they have al1 experienced. First they explored the
situation using their own knowledge and understanding of the experience.
Then I asked them to visualize a particular example of the situation where they
pretended to be on a seat wearing frictionless clothing. They knew that when
the car tumed a comer they would be left behind as it tumed out from under
them. Finally they were instructed to imagine viewing this spectacle from a
location above the car and to describe their motion as it would be seen from
above. They did this quickly and accurately. The final step in this procedure
was to make a connection between the phenomenon and physics principles,
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and to show banking roadways as a real life extension of the situation.
A Previouslv ldentified Learnina Blockage
Hart & Wessel (1 992) identified and described a blockage in student
leaming in which students were unable to distinguish whether to use sine and
cosine laws, or definlions of sine and cosine when solving certain physics
problems. This blockage appeared during the present study when Colleen was
asked to solve a problem about displacements. She replied, m i s is like the
cos law, or something like that." In the particular problem directions were due
South and due East which resulted in a right angle triangle in the vector
diagram. Because of this arrangement definitions of sine and cosine were
sufficient to solve the problem; however, Colleen knew about the more complex
"cos law" from her geometry and was using it unnecessan'ly.
Some physics teachers with whom I have discussed this concem deal
with it by forbidding students to use the sine or cosine laws in physics insisting
students solve al1 problems using components of vectors. This instructional
approach may reduce student errors but, in my opinion, does not help students
to transfer knowledge from mathematics classes for use in physics class.
Physics teachers may understand why using components is a superior
approach to problem solving in physics, but for students to adopt a teacher's
decision on expert authority does not necessarily lead to student
understanding.
The First Quit
One aspect of my role as teacher was to assess student knowledge and
provide a final physics mark for each student. About ten days into the project 1
gave them a teacher-prepared quiz (see appendix D) designed to assess their
135
knowledge of the concepts. I included some open-ended questions to provide
me with feedback from students. The quiz was similar to many that I have used
in my grade 12 physics classes consisting of four questions similar to those
done in class. The marks ranged from 47 % to 95 % with a class average of
77%. No one wrote a perfect paper, although in a larger class I would have
expected about ten percent of the students to do so.
What Did the Results Show?
Student performance on quines and other assessments provides
feedback about how completely they have constnicted knowledge on various
topics. The results on this quiz indicated to me that students did not understand
these concepts as completely as I might have believed based on Our
discussions. Their errors were similar to others I have seen over the years and
ranged from simple errors in calculation to major conceptual misunderstanding.
In spite of having examined solutions to similar questions in class, students
skipped steps or did not include al1 appropriate details in their solutions.
Students made a variety of procedural and conceptual errors in
answenng the questions on the quiz. For example, some students drew
accurate real-world diagrams but did not cse them to set up their vector
representation. in other cases students used algebraic addition or subtraction
to add or subtract vectors. Other students drew vectors using different scales in
the same diagram, a tendency identified previously (Hart & Wessel, 1992).
Another rnistake centred on student inability to identify correctly which angle
should be used in describing a vector's direction. Some students applied
Pythagoras' theorem when no right angle existed in their drawing and one
student interpreted a one-dimensional problem as a two-dimension question.
The errors and mistakes assisted me in establishing an image of
136
individual knowledge construction. The students' responses on the quiz
indicated that al1 had some level of incomplete concept development. Students
had made some progress in constructing and understanding concepts
necessary to apply vector mathematics to these physics problems, but al1
showed some aspects of incomplete concept development. Individual written
quiz results provided feedback to me which was used to assess each student's
knowledge construction and which guided me in planning instruction for the
next few classes. Attempts would be made to revisit and review concepts that
students applied incorrectly to solve problems on the quiz.
Feedback From Partici~ants After Ten Davs
The feedback conceming their participation in the project indicated four
points of interest. First, when asked, "What do you consider the best part about
participating in this study?", the majority indicated that the small number of
students in class was the main benefî. Some representative cornments were:
M - With a smaller group, it's easier to ask questions and get help very easily .
C - It is easier to leam a subject in a class of less [sic] people .... we are able to voice our questions more freely.
L - There is also more one-on-one communication. J - Being a smaller group there seems to be more group work and
we've been able to help each other. R - We get to express what we think and then you or other people in
the class can correct us or help us. A - We are not afraid to voice our suggestions and thoughts.
In open response students made a firm plea to continue the class in the small
group through to the end of semester. The original plan was to send them back
to Sarah Russell's class at the end of the study. I was surprised at the strength
of this request after such a short time. Next, students were asked what I could
do to improve their leaming and understanding of physics. Their responses
137
included:
T - Better explanations on how and when to use what (cos law, sine law). I am confused more than I thought.
R - I would like to do more dernonstrations so I can get the feel of p hysics.
J - More real world stuff, demos make it easier to visualize. C - ... more hands-on work.
They saw a need for more physical demonstrations and activities to illustrate the
conceptual knowledge that they were leaming. They believed from their
previous experience leaming that such activities assisted thern in
understanding abstract concepts. Students were also asked, "What do you feel
the most important thing that you have leamed about the use of vectors in
physics during this study?" Some comments were as follows:
M - ... with the real world diagrams and then the vector examples, these diagrams enable me to understand that vectors really do occur in the real world.
C - It makes the class a lot more interesting when we can relate it to real life.
A - Vectors are different and don? give you the answer you expect. We can relate it to real life.
As I graded their quizzes I noted key points to review with them. They
needed to exert more care when drawing vector diagrams to scale. length of
line and angle of direction had to be more accurately drawn, even in sketches.
Students exhibited considerable confusion about how to use direction in their
vector calculations. This was the first instance showing their confusion about
direction; others would follow. The ramifications of this confusion are discussed
in the final chapter of this dissertation.
Vector Components and a Bucket of Sand
Figure 5.6 illustrates an activity I used to provide the participants with a
concrete example of the effect of angle on the force required to lift a bucket of
138
sand. Using this activity as a focal point rny intention was to lead them through
discussion of this experience to apply vector components to represent the
forces in this situation. Initially two participants, Anne and Marie, stood on
desks above a bucket containing about
lift by pulling on ropes tied to its handle.
15 kilograms of sand which they could
They performed their first trial with
0
Distance Apart
Figure 5.6 Anne and Marie holding bucket of sand
angles and q close to 90". so that they were essentially lifting the bucket
vertically with the ropes. The next step was to widen the distance between the
desks to reduce angles 01 and 02. As they were reduced the magnitude of the
force required to lift the bucket increased and the task became more
challenging. When the angles reached about 35O, Anne and Marie were able to
lift the bucket only with considerable effort. One of them was hanging ont0 a
door frarne allowing her to pull with greater force. They were not surprised by
these events but were unable to explain why the force required increased as
the angles were reduced.
Figure 5.7 Ryan and Dean lifting the bucket at low angles
139
A later stage in the activity is shown in Figure 5.7. Ryan and Dean
replaced Anne and Marie but were not allowed to stand on the desks making
the angles, B1 and 0 2 , very small; as a result. the forces required to lift the
bucket were very large. In this set-up 01 and 02, were between 4" and 8" and
the force required by each student was between 500 and 1000 Newtons. They
were unable to pull with this magnitude because their shoes slipped on the tiled
floor. Essentially Ryan and Dean could not Iift the bucket under the conditions
set for them. When Ryan began pulling on his rope he quickly realized that he
was unable to lift the bucket and intuitively raised his amis to increase the angle
of the rope.
Tchr - Why do you want to lift your hands? R - Because it's easier. Tchr - Why does it make a difference? R - 8ecause the higher it is. Tchr - Oh, because the higher it is, what? D - It's easier to pull. A - The angle is greater. K - It's a smaller angle. Tchr - But why? You automatically know that you want to lift, you
want to get your hands up this high. But why does that make it better?
R - They had it up here. Tchr - They were able to pull at a different angle. R - Yeah, but that was up here. Tchr - They were at a different angle. But, a low angle is nard?
Zero angle, what if it were right at 0" angle? K - Then you can't do it. L - It's impossible. Tchr - You can't lift it at zero angle?
They understood that the smaller the angle the greater the force required to lift
the bucket and knew if the angle were reduced to O0 then lifting the bucket
would be impossible. They had some practical knowledge of this apparatus.
Tchr - What about as the angle goes up? Now, can you describe why that is the case? Remember there is obviously an angle. You are pulling with a certain force, and yet when the angle
goes down you have to pull with a greater force to lift it (the bucket). That does seern strange. Does the weight change?
K - No. It's harder to counteract the gravity. Tchr - It's harder to counteract the gravity, so the gravity must be
getting bigger, is that right? K - No. I fs staying the same. Tchr - No, i fs staying the same. Why is it gravity doesn't change,
the weight of the bucket doesn't change. What's happening? J - You're pulling 1 straight up instead of, like Tchr - I was pulling on the rope straight up. J - You're pulling your force directly opposite to the force that's being pulled on it, that's pulling it to the side.
(Judy gave some signs of knowing her answer was Wear and not clearly stated. She smiled and shrugged her shoulders.)
Tchr - I'm not saying ifs wrong. Can you be clearer on that? J - No. Tchr - What do you think Todd? T - I'm not thinking. I don't know. K - Because in order to Iift the bucket. You need a certain amount
of ah, like, weight. You need a certain amount of force to lift it up, and when you are pulling out the force is going out instead of up.
In this part of Our discussion participants were attempting to verbalize
their thinking and understanding of the relationship between the required force
and the angle of the ropes. They seemed to understand the qualitative
relationship between force and angle but were not considering any form of
mathematics to help them in expressing that relationship. They exhibited no
systernatic approach to organizing their data, nor to examining relationships
between the two variables (the angle and the applied force). In previous class
work we had gone over the mathematics necessary for this representation, but
no one brought them up. Worst of ail, Todd seemed to have wfthdrawn hirnself
from considering solutions and was simply not thinking about the relationship.
This episode was frustrating for me because the mathematics required
seemed so obvious. Vector components had been introduced earlier, but these
students showed no sign of applying those mathematical concepts to this
141
situation. As I had obsenred in many ottier classes, these students did not know
how to examine relationships between two variables in such a way that led to
mathematical representation. In science we commonly represent relationships
with mathematical expressions; but, there was little indication that the
participants understood the process that I was trying to help them work through.
The difficulty exhibited here by the participants can be explained partially
in ternis of knowledge construction. The type of problem they were trying to
solve was unlike any they would have seen before because they were being
asked to constnict a mathematical representation for the experience in this
activity. To build such a representation several difficuit steps were required.
First, they would have to identify the relevant physics concepts in this
phenornenon, including the angle of the rope, the force applied to the rope, the
weight of the bucket and the force of gravity. They seemed aware of these and
could describe something of the relationships between some variables. Next
the students would have to identify the cause and effect relationships between
the variable within this phenornenon. Finally, they would have to represent
interrelated variables by some mathematical relationship which mathernatically
described the relationships between the variables, in this case vector
mathernatics.
This type of knowledge construction is extremely difficult and does not
nomally occur in novice leamers in physics because they have not developed
a body of conceptual knowledge at this early stage in their study of physics
(Gagné. Yekovich & Yekovich, 1993). The participants could not have
successfully perfonned this task without additional practice and instruction
because they lacked the kind of conceptual knowledge in their mincis that was
required to perform this type of operation. In order for them to have been
successful, a much different instructional sequence would have to have been
provided for them.
As our conversation continued I prodded students more directly to
connect this experience with vector mathematics as a means of representing
this phenornenon. Their verbal description became clearer and more
sophisticated as Our discussion continued.
Tchr - But if you are then pulling on the thing at an angle why does it lift at all?
J - Because you're using a greater force. K - Because you are still lifting it up a bit. You are lifting it like up
and out, not just straight out. Tchr - You are lifting it up and out, not just straight out. K - Just use that vector math stuff. Tchr - What do you think Lisa? L - I think that when you are pulling out at the angle that gravity's
pulling it straight down, so that the angles are going out like this, but the gravity's pulling straight down, that's why it takes a greater force to pull it that way. (She used her hands to indicate directions)
Tchr - So, it takes more force, and yet does gravity get stronger'? L - No, it's going straight down. Ifs not going out to the sides.
Kevin interjected "use that vector math stuff" but did not follow up with an
explanation of how to use it, nor did any of the others make the connection
alluded to in his comment. Their verbal description grew increasingly accurate
but still contained no reference to vector components.
J - Part of your force is going !rom you pulling this way, and the other person pulling.
K - It's balancing off. J - Part of it is. Yeah, part of it's balancing off that way, and part of
it's pulling opposite gravity Tchr - So, part of it's pulling in what direction? J - Towards each of the people that are pulling. Tchr - Towards each of the people that are pulling, and those two
pieces that are pulling out that way. J - Balance off. Tchr - Balance off, and what about the parts that don?? J - The gravity's pulling down Tchr - Right K - And the extra force is pulling up.
1 43
Their verbal or qualitative description was quite precise but contained no
hint of using vector rnathematics or cornponents as a means of representing this
situation. An additional ten minutes of discussion still did not prompt students to
employ vector components. The students had considerable ski11 at describing
the situation in a qualitative manner but never made the transition to
mathematical representation. They were still operating at a concrete level of
thinking even though they have considerable expenence with mathematics and
its use in problem solving. Their level of conceptual development did not yet
allow them to operate with abstract mathematics in this area of physics. They
needed more experience and time to reflect about the whole idea of
mathematical repreçentation used in physics. They could find the area of table
using mathematical representation but were not able to employ a different, less
familiar mathematics in a new situation.
At this time partly because of time limits I instmcted them directly how to
use vector mathematics to describe the relationship between force and angle of
application. I reverted to a more direct form of instruction because I had run out
of ideas to continue the discussion. I was unable to create a different way of
approaching the problem in the classroom. When I instructed them more
directly about applying vector mathematics in this representation, a change
occurred in student demeanor. They reverted to a more receptive role in their
leaming rather than being actively involved in it as they had been in the
discussion that had just ended. They answered questions and made
calculations but now expected me to guide them quickly and efficiently. This
transition from active to receptive leaming occurred so smoothly that I did not
realize it had happened until I read through the transcript of the lesson. Both
the students and I reverted to our more traditional roles. Ofd habits die hard
especially when they are cornfortable.
"Hanging Mass" Lab Investigation
In the classroom the participants carried out a 'hanging mas*
investigation which was analogous to the sand bucket demonstration. The
apparatus (see Figure 5.8) used laboratov objects in place of the bucket,
scales were incorporated to measure the forces, and protractors were used to
measure angles. This simple apparatus allowed students to obtain sets of
numerical data experimentally which could be used to investigate the
mathematical relationship between the components of the force and the weight
of the hanging object.
Students worked enthusiastically at this activity with little prompting.
They established two groups, examined and assembled the apparatus, and
made the required measurements wth no guidance from me. The asked
essentially no questions of me during the experimental phase of this activity.
The first group consisted of Ryan, Anne, Colleen and Marie; the second was
made up of Kevin, Judy, Todd and Lisa.
The first group started to work with the equipment almost immediately.
Anne and Marie adjusted and manipulated the objects, strings and spring
scales while Ryan examined the protractor. Colleen sat at a desk near the
activity and prepared to record data as it was produced. The protractor created
some confusion for Ryan because it had two scales. Anne and Marie helped
him determine from which scale readings should be taken. The group rapidly
produced three sets of data and then moved the stands in their apparatus
further apart to provide additional set-ups and a wider range of data.
The second group spent a brief time talking before beginning the activity.
lnitially Judy actively adjusted the equipment while Kevin and Lisa recorded
measurements from the spring scales. Todd watched attentively but was not
actively involved in the experimentation. After they had recorded several sets of
145
data, Kevin remarked that. Yhey (the data) al1 seem to be the same." To remedy
this concem they moved the stands further apart as the first group had done.
Some of their measurements had been taken while the strings supporting the
objects were touching the tables which made the readings invalid. They
repeated those set-ups and recorded new data rather than changing their
numbers to match expectations. They appeared to understand that the
measurements had to be repeated when a mistake in the experimental
technique is found.
Figure 5.8 'Hanging Massn apparatus
Both groups worked well together and understood which measurements
they were to make and record. They knew how to change the apparatus set-up
to produce a range of experimental results; however, neither group attempted to
estimate or calculate expected results. No one tned to predict results before
making measurements even though they seemed to understand the procedure
and the apparatus. Both groups perfomed the experimental work quite well,
appearing to know which measurements to make and how to operate the
apparatus.
146
Their interpretation of the data was not camed out with the same
understanding and certainty with which they had made the measurements.
Confusion between mass and weight was evident as they analyzed the data. In
this activity the wei~ht of the object was balanced by the vertical components of
the forces registered on the spring scales. Both groups of students had some
difficutty in determining the weight of the object until I reviewed the concepts for
them. Some students requested my help in finding mathematical errors while
analyzing results; others requested confirmation of some aspect of their work as
they progressed. For example, Marie knew one of her answers was incorrect
but could not find the error. Rather than spend additional time finding the
mistake herseif she requested my assistance in locating it. While the students
analyzed their data, they helped each other work through the difficult areas in
calculation. On-task discussion was frequent as they were interpreting the
measurements.
Student Confusion of Balanced with Euual
Figure 5.9 shows the "hanging mass" apparatus and the vector
representation of the forces involved in the calculations. FI and F2 represent
the forces in the strings and measured by the spring scaies. Fg represents the
force of gravity that is acting on the object supported by the strings. The
participants made these representations with little assistance from me and
appeared to understand what they had done. The components of FI and F2
are shown in Figure 5.10 which was drawn without my help. The participants
appeared to understand why they drew the diagrams and what they meant.
Confusion in student thinking became apparent as they tried to write
vector equations to represent what they knew about the system. For example,
1 47
they knew that the vertical components of FI and F2 balanced the force of
Figure 5.9 "Hanging Masn apparatus with vector representation of forces
Figure 5.10 Vector components of forces Fi and F2
gravity acting on the object. They represented this relationship by the incorrect
equation, Fly + F2y = Fg, which says that the two vertical components eaual
the force of gravity; however, this equation does not correctly incorporate the
directional aspect of balanced forces. To the students this equation seemed to
read correctly because they substituted equal for balanced. Some time and
discussion was required for thern to see that their equation was incorrect and
that the relationship had to be represented by Fiv + F a + Fg = 0, or FI,, +
148
Fzv = - Fg. This error in representation resulted from not fully understanding
the place of direction in vector representation of forces and other vector
quantities.
A similar problem arose with the horizontal components of FI and F2.
Anne and others knew that the horizontal components, Flh and F2h balanced
each other because the object was stationary. Anne wrote the following
equation as to express this relationship, F1h = F2h. Again using the concept
of equal in place of balanced makes this equation seem correct. This equation
was then manipulated mathematically to become Flh - F2h = O. This error
resulted from student inability to understand the manner in which direction
effects a relationship and how direction was represented in vector equations,
and is another example of the necessity meanings for concepts held by
students being identical to those of the teacher. This problem was identified
from the transcripts and was not noticed during the actual class.
lntroducing a New Vector Concept - Momentum
Students' Initial Conceptions of Momentum
The decision about when to introduce new topics or concepts to a group
of students is never simple. As was true for regular classes, a balance between
fulfilling the requirements of the mandated physics curriculum and the level of
understanding developed by students had to be maintained for the research
participants. The Pace with this group was about average with no cause for
concern or for rushing to the next topic. While certain that these students were
still struggling with the use of vector mathematics in physics problem solving, I
also believed that introducing a new concept involving vector representation
might assist in their development by providing a different view of the use of
vector mathematics The next concepts were impulse and momentum which
are vector quantities following logically from Our cuvent work. The following
discussions with students illustrate the introduction of momentum to the
partici pan ts.
Tchr - How rnany of you have heard of impulse andor momenturn? J - I've heard of momentum. Tchr - What's it about? Do you know? A - You stop applying force but it (an object) keeps going. Tchr - Oh I see, I hadn't actually thought of it that way. You stop
applying force but it keeps going. 1 kind of like that idea. What does momenturn depend on?
J - Velocity. Force Tchr - Momentum is proportional to velocity, but what else? J - Mass Tchr - What else can you tell me about momentum? How would you
change it? J - Add more velocity or more mass to it (object). Tchr - Yes, we could do that, but supposing you have a fixed object,
Say a baseball or something like that, and you want to change it's momentum.
K - Add a force. Tchr - OU, add a force. Add a force or apply a force? T - Apply a force. Tchr - Sure, apply a force, that's al1 1 is,apply a force. What other
factors? (ten second pause) J - Whether 1's a constant force or something like that. A quick force? Tchr - A quick force or a constant force. So what really are you saying
is the difference between thOs8 two? J - One is partly pushing it and one is pushing it. Tchr - So what's the difference? J - One is going to run out. Tchr - What's going to run out J - The force. Tchr - The force is going to run out? The force runs out. What an
interesting idea. J - Well, it will slow down again. Tchr - Does it? Why would it slow down? J - There's friction. Tchr - What if l'm in outer space? Let's pretend there's no friction and
it (object) is in outer space, and you give something a little flick. J - It will go.
Tchr - It will keep going? 3 - Yeat-t.
This transcnpt was included to illustrate something of rny atîempt to explore their
conceptions of momentum and to bnng these conceptions to their awareness.
Participants came to the classroom with some understanding of the concept
because the word momentum was in everyone's vocabulary; however, each
had a different rneaning and comprehension of the concept. The most common
use of the terrn was in reference to a change in the flow of action in sporting
events. These excerpts illustrate something of the incornplete structure of their
knowledge and understanding of mornentum.
Anne's conception was that momentum is the property of an object that
keeps it going after the application of a force has stopped. Judy knew that
momentum was somehow related to mass and velocity and felt that force was
also involved. Her view of increasing momentum by adding "more velocity or
more mass" to the object was theoretically correct but did not explain how to
rnake those changes. Kevin understood that an applied force is required to
change the velocity of an object, an idea covered earlier in grade 12 physics.
Judy's ideas of quick and constant forces were not clearly explained and
she was unable to clanfy her distinction even with further discussion. Her idea
that the force was going to run out was similar to Aristotle's views on force and
motion. and consistent with other novice physics students (McCloskey, 1983).
Judy seemed to be aware her understanding was inconsistent because she
knew friction slowed objects but that they will move essentially forever in space.
Momentum was a very difficult concept for these students and most in my
experience. They did not come to class with much understanding of momentum
because their everyday experiences had generated little need for its
construction. Without identifying a need for momentum they (and most other
151
people) did not identify the concept in nature as physicists have done. They
have operated successfully until grade 12 physics without understanding
momentum or its vector nature. Producing a need from their experiences is a
challenging task for classroom teachers. Without such a need in students,
meaningful leaming is hard to achieve because connections to their currently-
held knowledge are diffÏcult to generate. The connections to currently-held
knowledge have to be made to the conceptual physics knowledge that students
hold rather than to their experiential knowledge. This task is difficult to achieve
successfully because their conceptual knowledge is not well established.
Reconstructina Student Conceptions of Momentum
Assisting the participants to reconstruct their knowledge of momenturn to
be more similar to that of physicists was a difficult task. Most had heard the term
momentum used in reference to sporting events, or with respect to objects
tending to keep moving, but these were the usual extent of their knowledge of
momentum. Most grade 12 physics students have not constructed the concept
of momentum as the term is used in physics. My goal as teacher was to help
them rnove from where they were conceptualiy toward where they needed to be
to use it in the study of physics.
Over a period of two or three days I introduced the participants to the
fundamentals of momentum as it is used in physics. The relationship between
momentum, mass and velocity was examined, as was its vector nature. The
mathematical formula, p = mv, was explored and problems related to
experiences in their lives were solved in class. I attempted to relate the new
ideas to their concepts of momentum described earlier to assist them in
reconstructing their conception of rnomentum. The instruction was intended to
help them in reconstructing the concepts of momentum with which they arrived.
The participants had completed a laboratoiy activity using spring-loaded
carts to investigate conservation of momentum. After the activity was completed
they had taken part in a discussion about it and made the calculations required
to complete the investigation. Their results and measurements were properly
interpreted to illustrate the expected result that momentum was conserved in
this set of conditions. This lab activity seemed to have produced the anticipated
results and al1 appeared to be in order until I inquired of them the significance
and meaning of their experimental results.
Tchr - But what does it show you if it (momentum before and after the spring release) equals 0.
C - They were stopped. Tchr - No. They didn't stop, they were moving. C - They're both in motion. M - They're both moving with the same speed. Tchr - Except that is not always true, because if there was a heavier
cart it moved slower than a lighter one. C - It doesn't make sense. That doesnY make any sense though,
because sometimes the cars had different speeds, so even if they take off in different directions they don? cancel.
Tchr - They weren't going the same speeds. C - And they weren't going the same velocity. Tchr - No, they weren't going the same velocities either. Were the
masses the same? C - NO. Tchr - What happened if you had a heavy cart, did it move slow or
fast? C - Slow. Tchr - Slower, so you have a big mass times a little velocity. If it
(cart) was a lower mass, what was its velocity, higher or lower? A - Higher
Colleen was confused because the total momentum of the carts was zero
even though they were moving. lnitially she responded that her experimental
results rneant the carts were stopped. A few seconds later she changed her
response by indicating her results meant the carts were moving. In spite of her
confusion Colleen knew the two carts moved away from each other when the
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spring was released and their velocities depended on their masses; however,
she did not understand how the carts' mornenta totalied zero or what was meant
by 'rnornentum is conserved. She did not appear to have incorporated the
vector nature of momentum in to her conception. Colleen and the others could
correctly predict the motion of carts with unequal masses and seemed to
understand an application of the conservation pnnciple to exploding fireworks.
In spite of appearing to understand the phenomena at a qualitative level, they
were unable to make or understand the mathematical representation of
momentum using vectors. Students had not developed a functional
understanding of the concept (or its mathematical model) that could be used in
problem solving.
Student Auproaches to Constructinu Momentum
The following conversation which lasted fifteen to twenty minutes
illustrates some students' attempts at constructing the concept of momentum.
K - Well, what exactly is momentum? C - What is mornentum? Yeah, how do you define it. Tchr - Well, you tell me. C - How can you conserve momentum when things are moving? Tchr - That's a good question. You tell me, what do you think momentum
is ? C - I thought mornentum was sornething moving. K - Momentum is. Is it a force? Tchr - Then what do we cal1 a force? K - It's a specific force. Tchr - Remember in physics we identified concepts. We've identified
force and we have a pretty good idea of what forces are. K - It's like an interna1 force. Tchr - But it's not like a force. C - It's a force that works off the friction. Tchr - But is it a force? C - I don't know. Tchr - But if it were a force, why don't we cal1 it a force? There's no point
in thinking up a new word if it's the same thing. If it were the same thing as force, physicists would just cal1 it force.
C - It just doesn't make sense. Tchr - How would you use the word (momentum)? A - Well, you could Say, it kept going because it had momentum. Tchr - OK, it kept going because it had momentum. J - Is momentum motion, or is momentum like a number? Tchr - What is it? M - Tell us. Tchr - But my answer to you is going to be a mathematical expression of
m a s times veloctty. M - But what does that tell you? Tchr - Exactly, what does that tell you?
Of interest here was the variety of approaches students used as they attempted
to construct the concept of momentum. lnitially Kevin and Colleen simply
requested a definition of momentum. I did not give one because I was quite
sure I could not provide the type of definition they were requesting. They
wanted a description or a lis? of characteristics which would help them visualize
or identify momentum in some concrete manner. They were not looking for a
mathematical formula or a statement that momenturn is the mass of an object
rnultiplied by its velocity.
Colleen expressed her idea of momentum as "something movingn. Her
version of momentum helps me to understand her confusion about the results of
the previous experiment. She could not understand how total momentum could
be zero when the carts were moving. Kevin, in contrast, tried to construct this
new concept by relating momentum to his concept of force, arguing first for a
"specific forcen and later for an "internat force." He was unable to provide a
definition or description of either t e n . His view may be interpreted as
consistent with seeing forces as imparted to objects and running out as the
object cornes to a stop.
Judy wanted to know if momentum was motion or a number. She was
exploring the possibility that her understanding could be improved by
quantifying the concept. She believed that seeing how mornentum was
measured could help her to construct the concept. Although Judy was engaged
and actively attempting to understand momentum, Marie clearly was frustrated.
Her appeal to me to LYell us" the answer irnplied it was a secret that I was
keeping and could tell them. When I explained that the only simple answer that
I can give them was in the form of a mathematical formula, she recognized that
this type of answer was not what she wanted. I did not know of any way to give
them an answer directly so I redirected our discussion to explore momentum
differently. I had them imagine a scenario where they would have to stop some
object moving toward them by standing in front of the object and being pushed
backward.
Tchr - Supposing something (an object) were moving towards you on pavement and you have to stop this thing. What is going to detemine how far you slide when this thing nins into you?
A - Mass and velocity. Tchr - Both together, or just one of them? C - Both. J - Both. Tchr - Could you figure out the sliding distance wlh just the
velocity or the mass. J&C-NO. Tchr - How would you describe that characteristic of mass and
velocity? C - Momentum. Tchr - OK, so momentum is that sort of. Now you said it's how long
it keeps 1 going. I might Say how much "oomphn there is in it. J - With the kid coming toward you and he has momentum. He
has a mass times velocity. Tchr - Right. J - So what's the momentum equal to, is that a big number or a
little number? Tchr - Well, why don? you figure it out? C - OK, Say he's going really fast. Tchr - How fast? 10 m/ç? That's pretty fast. C - Yeah, and he's like a 100 pounds Tchr - 50 kg, OK? C - Yeah. Tchr - OK, so 50 times 10 rn/s would be 500
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Colleen, Anne and Judy seemed to understand both mass and velocity of the
object were required to determine how far back they would be driven. Judy was
explonng momentum in ternis of the property that a child on a tricycle had as he
was coming towards her. She wanted a quantifiable way of expressing
momentum and came up with 500, but had not included the units with the
number. Their inclusion would have provided additional information; however,
the units were Nms (Newtonseconds) with which the students had no
experience. To assist them further in exploring momentum, I had them consider
a variation of the first scenario. I had asked them to imagine that they were
stationary on roller skates, grabbed ont0 a passing person on a bike and pulled
along.
Tchr - Assuming you were on roller skates and a kid (on a bicycle) comes along and you grab ont0 him. Right, you hold ont0 him. What happens to you and him?
J - You'd gain some mornentum. Tchr - You'd certainly gain some mornentum, you start moving in
the direction he's moving in, right? How fast do the two of you go, if you hold on to him?
M - You'd slow down. Tchr - Why? M - Because there's more mass. Tchr - That's right, more mass, srnaller velocity.
Their thinking was correct in this thought-experiment. They predicted
some momentum would be transferred to thern and the combination would
move slower than the person alone on the bicycle. In spite of understanding
this visualized situation they were unable to describe momenturn in a way that
was meaningful to them. At this tirne the bel1 rang and ended Our discussion.
This extended discussion was non-scripted and spontaneously
generated as the result of the interaction between the participants and me. My
questions and scenarios were produced as needed and not planned in
advance. During Our interaction I was aware of the participants' struggles to
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understand mornentum and tried to assist them in dissolving their confusion. In
spite of rny awareness during the discussion only after studying transcripts did
the range and variety of struggle become apparent. My detailed image of
student learning did not emerge without considerable analysis and
interpretation. Many events happened too rapidly dunng the discussion for me
to reflect on their significance.
The transcript of this discussion was useful for a number of reasons. The
first was to show how, on the surface, one instructional strategy appeared to
produce learning. Participants had carried out a traditional momentum cart
activity. Videotape records showed they conducted the activlty and appeared to
understand what they were doing. They understood what results they expected
to get because on several occasions they repeated trials which they recognized
as flawed. The post-lab discussion and submitted report sheets indicated they
understood that mornentum is conserved in this situation. To that point
instruction was routine and seemed to produce the results expected from the
sequence of activities and post-lab discussion.
Second, when probed more intensively about momentum, a lack of depth
of student understanding came to the surface. Students had not constructed
sufficient understanding of momentum to be of functional use in physics. They
had performed the laboratory activity and understood the procedure but were
unable to make the connections necessary to produce meaningful learning.
Students exhibited a f o n of rote leaming as a result of this lab activity, although
it did not initially appear that way.
The third value of this conversation was the manner in which it showed
the variety of approaches with which these students attempted to construct their
concept of momentum. Colleen's concept of momentum involved individual
objects moving and she became confused when she could not understand that
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total momentum of a system of objectç was zero even though objects were
rnoving. She was seeking logical consistency but was unable to achieve it.
The experimental resuits confused her even though she actively participated in
the activity. Kevin wanted a definition of momenturn but when one was not
forthcoming tried to describe momentum in ternis of force a concept he already
understood. This approach was unsuccessful because momentum is not a
force and his terms 'intemaln and "specificn were never clearly fomulated. Judy
attempted to construct the concept by exploring how momentum is measured.
She used this approach because she understood other concepts in terms of the
units in which they were measured. Lastly, Marie wanted me simply to tell them
an answer. Her request resulted from frustration but did indicate that she
thought understanding of momentum could be achieved by direct
communication.
This discussion lasted about twenty minutes and provided at least four
examples of knowledge construction among the nine participants. In addition
an evaluation of a laboratory approach to instruction was produced. The
surface results indicated that leaming had occurred as expected in the
traditional sense, but a deeper analysis of the conversation showed that
leaming was superficial and few connections were made to student
experiences.
Connections between their experiences and the conceptual ideas in
physics were never made as readily as I hoped they would be. They did not
see the need for conceptual representation of parts of natural phenomena and
did not understand the process, although they could mimic it under certain sets
of conditions such as the momenturn cart activity. Throughout the project I used
opportunities as they arose to have the participants describe their leaming and
understanding of the connections between mathematics and physics. In the
following section three students and I discuss physics and mathematics in an
attempt to gain some insight into their thinking about their own leaming.
Three Students Discuss Physics and Vector Mathematics
Three weeks into the study because of a school activity only Kevin,
Colleen and Dean were in class which provided an opportunity to discuss with
them their understanding of vector mathematics and its applications in physics.
Our discussion explored how they leamed vector mathematics in their math
classes and how that experience transferred to and connected with our work in
physics. This unstructured interview lasted about thirty minutes. To begin ouf
discussion I asked thern, "what do you know about vectors?"
C - They have direction and magnitude. Tchr - Is that right? C, K & D (al1 together) - Yes. Tchr - What does that mean? C - They have a number value but they also have a direction value. Tchr - OK, so you've got a direction and a magnitude. K - The things we measured. We measured force. C - We rneasured force or length or something. Tchr - But why do we use them (vectors) for force or length (actually
displacement)? C - What do you mean? Tchr - Well, why, why do we use them? C - The vectors? Tchr - Yes. K - There's certain applications. And stuff like that. Ships and
airplanes. C - When you went out for a jog, and you want to find out how long
you have to jog home. You use vectors. Tchr - You use vectors to jog? Don't you just run out and corne back?
Colleen described vectors as having magnitude and direction and the
others unanimously agreed with her. This response seemed to indicate some
understanding but could be interpreted as repeating my frequent reminder that
magnitude and direction were required when describing a vector. They added
that vectors were used in applications such as, "ships and airplanes" and ÿvhen
you jog." Both examples were taken from problems that they knew required
vector mathematics to answer. Neither use that they suggested explained why
vectors were used which had been the intent of my question.
Tchr - Which came first, vectors or the physical world? C & K - The physical worfd. Tchr - Why do you Say that? C - Because vectors are a part of mathematics that you use to help
in the physical worid. Because forces were here before mathematics were here, and gravity was here before math was here.
K - The knowledge was first. Tchr - Where did the math come frorn? K - People developed it.
When the relationship between vector mathematics and concepts in the
physical worid was explored their responses were fundarnentally restatements
of ideas expressed in earfier classes. Their responses indicated they had
atîended to classroom instruction but not that they had developed much
understanding of the place of vector mathematics in physics. Next Our interview
examined their knowledge of vectors learned in geometry-trigonometry and its
application in physics.
Tchr - But do you have to know about gravity to know about vectors? Didn't you come here knowing about vectors and not knowing how they were applied?
C - I knew how they were applied. I took trig. Tchr - You took trig, but which. In math which did you leam first, a reason
for them, or did you leam how to do vector mathematics first? K - How to do them. C - We leamed the mathematics first. Teachen don't show you a
reason for anything. K - They just Say R'II be on the test. C - Yes. Tchr - Well, that's true, but why did they have you leam vectors?
What about vector rnathematics? C - It's part of the course. Tchr - But why did they even bother to put R there? Sure it's part of
the curriculum, but would you bother to put something like vectors in a curriculum?
C - I don? know. Tchr - What do you think Dean? D - I have no idea. Tchr - No idea. D - NO. Tchr - Did you take them in math? D - I didn't take them in math. Tchr - So, this little blast is the first time you've ever seen them. D - Yes. C - What we did in math is different than what you do in this class. Tchr - How, how is it different? C - Well, there they did like a ship was going this direction, and then
another ship was going in this direction (indicated the directions with her hands) and then (the question asked) will they pass in front, or behind, or will they collide?
K - Well, it was the same idea.
Kevin and Colleen appeared to have developed little understanding of why
vectors were included as part of the geometry-trigonometry curriculum and had
transferred essentially nothing from that class to this physics class. This
discussion occurred after earlier written feedback and was consistent with
opinions expressed at that tirne. We discussed differences in instructional
approaches used in grade 12 geometry-tngonometry and grade 12 physics;
however, Dean was unable to compare approaches because he had not taken
grade 1 2 geometry-trigonometry . Tchr - Were the rules of the vectors different? C - I don't know. We did 1 differently. We used square brackets or
that s td . K - We didn't do the math. C - We only did it for like a week. Tchr - What exactly do you remember about them teaching about
vectors? I don't care about the problems right know, just the vectors. What did you leam how to do?
C - 1 didn't leam anything.
Tchr - Did they talk about adding and subtracting vectors? K - NO. C - Yes
K - Not really. C - We did but it was different than what we do here. There was
square brackets and stuff The notation was different. Tchr - Square brackets? C - Yeah, like we had square brackets, an A, then a BI and then. I
don't remember how to do it. Tchr - When you came out of the math part of vectors what do you
think you knew? C - Not very much, I didnJt understand it.
These sections of dialogue showed that Colleen and Kevin had not
developed much understanding of vector mathematics during their math
classes. They were unable to describe what they took in much detail, or how
the mathematics approach differed from that in physics. They could not reach
agreement on whether or not they had leamed about addition and subtraction
of vectors and they feit that the vector notation was different.
These students struggled at length to respond to my questions and
follow-up probing of their understanding of vector mathematics in physics.
When I first read the transcript, I considered that my questioning technique was
not good enough to produce the depth of response for which I had hoped;
however, on reflection I think these sections are indicative of the knowledge the
participants held at the time. I do not think that they had deep understanding of
vectors or why they were used in certain physics problems. These students
have not studied philosophy and generally did not reffect deeply on their world
view. They assumed a world was out there and that their knowledge told them
about that world. At this time these students had done the best they could do.
They did not currently have the ability to examine the relationship between
physics and vector mathematics more deeply.
A Final Vector Problem
The participants and I worked together for sixteen weeks from beginning
163
of the project to their final examination. The research classroom sessions
occupied the first nine weeks and video recording ceased at the end of that
time; however, I continued to instruct the nine participants for an addlional
seven weeks unül the end of semester. In our final week together I used
laboratory apparatus to create a problern for them to solve individually. A force
board problern was set up which they were to solve. They were instructed to
describe in writing how they made their prediction of the scale reading, how
they calculated their solution and why they used whatever method they chose.
In addition they were asked to inciude some comrnents about physics concepts.
Following their classroom work I interviewed them in groups of two or three
about the problem and their comrnents about physics.
A diagram of the set-up of the force-board problem is shown in Figure
5.1 1. Three spring scales were attached to a central ring as illustrated. Scales
SI and S2 were set to 10.0 Newtons and separated by an angle of 60". Scale
S3 waç covered so that its reading could not be seen. Students were to
examine the set-up and estimate the reading on scale S3. Having completed
the estimation they were instructed to calculate the reading on scale S3 by
whatever method they chose.
Students recorded their estimates of the reading and an explanation or
the reasoning behind their particular choice. Marie wrote, "15 N. I figured that
the total force should be larger than each initial mass of 10 N, but cannot be
more than the sum of them together. So I picked a number halfway between 10
and 20." Ryan stated, WO N. I did not use any physics pinciples. 1 chose this
estimated answer because I felt from past experience that it would be close."
Kevin estimated 22 N and explained, "It will take at least 20 to balance off the
two 10 N forces." Todd estimated 20 N and argued. "that are two forces pulling
164
against one force." Lisa estimated 25 N and suggested, 'lt is taking 10 N each
to pull it towards the sides at 60" so it should be double that including more
because sorne force is lost with the two putleys." Dean responded that, "1 got
20 N as the answer because I figured that if two forces are acting on one point,
the forces should be added to get the total force." Three others made estimate
in the range of 15 to 20 N and gave reasons similar to Marie's.
Figure 5.1 1 The Force Board Set-up
These responses were interesting and personally frustrating because
none of the students referred to the directional dependency or vector nature of
force. Lisa was the only student to mention the angle involved and her rationale
was incorrect. Ryan's use of expenence provided a reasonable answer, but
apparently saw no connection between physics concepts studied previously
and his estimation. Dean simply added the two forces algebraically and either
did not consider that the 60" angle might be of importance, or had no idea how
to include it in his solution.
As part of this exercise students were asked to describe their method of
calculating a reading for the third spring balance. Judy succinctly explained
that "drawing a vector diagram will show the total force." Marie wrote that "we
165
did the exarnples in class (where) we took the cosine of the angles and
multiplied it with the mas." Her method produced a value of 10 N which was
not correct and her reference to mass indicated some confusion in her thinking.
Five other students solved the problem by drawing diagrarns and making
calculations without any explanation as requested. The remaining two provided
one-step descriptions with few details.
Five students correctly calculated answers which were verified by
cornparison with the reading on scale SJ. When asked to comment on their
problem solving approach, Todd commented, '1 donPt know if it is the correct
method to do this but I sure got a close answef, and Judy replied succinctly,
"my method was correct." Lisa's method produced an incorrect answer but she
still felt, "the formulas that l used I believe is the correct method for doing this
question." Her error was the result of an incorrect angle in her vector diagrarn
rather than an error in calculation. She had confidence that her approach was
correct in spite of her incorrect answer.
Marie's initial attempt produced an incorrect answer. She wrote, "I'm not
satisfied, this was a wrong calculation. I should go back and make a more
appropriate diagram and use more numbers and use vectors." Her second
attempt produced a calculated answer of 17.2 N which was the correct value;
however, she comrnented '1 don't really know what I just did. But I thought that
maybe a vector diagram would help me get the right answer and it did." She
included no fumer explanation or rationale for her work.
Anne's method of calculation produced an result of 10.0 N for the scale
(S3) reading. lnstead of drawing a diagrarn of the force board set-up, she used
an incorrectly drawn vector diagram to organize her work. The actual scale
reading showed her calculated answer was wrong. She redid her calculations
and got a more reasonable value. In evaluating her own work she stated, ''1 had
the wrong angles in my diagram and I did not use my real world diagram to my
advantage."
Whv the Partici~ants Chose to Use Vector Mathematics
In interviews I attempted to have the participants explain in greater detail
(than the written explanations) why they chose to use vector mathematics to
calculate their solution to the force board problem. Lisa, Judy and Marie
expressed their reasoning and rationale in the following.
Tchr - What were your estimates (of the reading on scaie S3)? M, L, J - 15,20 Tchr - Why didn't you just Say it was 20? 10 + 10. J - Because the way it was set up you knew that everything had to be;
the ring in the middle wasn't rnoving anymore, so the forces had to be the same pulling out in each direction.
M - They were balanced J - But two were pulled a little bit closer, like a little bit more to one side,
so then to make 1 even out you had to have more of a force on this one. But you couldn't have twice the force because it wasn't going directly from the other side.
Tchr - And then mathematically what did you have to do to figure out the answer?
All - We had to use vectors Tchr- How did you know that you had to use vectors, Marie? M - I don't know, because we always use it for solving this sort of stuff.
Like you knew it wasn't 10 + 10 = 20, so you had to do something eise.
J - It would not make sense to use 10 + 10. You'd know that isn't right. Tchr - Lisa, how did you recognize that vectors had to be used? L - Just from the examples we had used before, and because I didn't do
this until the next day. People were talking about it. And I saw "forcen (in the question). And the way you had it set up on the board, you had the force going this way, this way and this way, so.
Tchr - How did your answers compare (to the actual reading) M - It was 17, the reading, but I got 17.2 Tchr - OK. Did you consider the .2 to be a significant e m r in your
thin king? M - No. J - No, because the little things you were using weren't absolutely
accurate; like you couldn't read them precisely. Tchr - What would you have considered a bad answer or something you
would have had to do some further thinking about? M-18or16 L - l got 18.6 Tchr - 18.6, that was your answer? L - Yeah. Tchr - That's off by 1.6, so did you go back and redo the calculations? L - I checked it over but I didn't figure out how to fix my vector thing. Tchr - So, what do you think the error was? L - My angle. Tchr - In your case 1 was stnctly an angle errofl L - Yes. But I used the same formula as everyone, I just had a angle that
was off. Tchr - Your vector diagram probably isn't quite right. L - Yes. It's (the difficult part) taking from the reai wodd and putting it in
the vector worid. You have to change; it's the shape of it. Tchr - But you're not supposed to change the vector, just put 1 in a
different place J - Well, like the confusing part is when you take; you have a line going
this way, and this is 60" or whatever and you add something to, like you donPt know. It's hard to figure out which part went where.
L - Which head and tail go together. J - Well, I even got that part, ifs just that when you move it you had. Well
here you have one going up, or over this way and one going down and you move them this way, then where did the angle go to; which?
Tchr - Which angle is which? J - Yes.
Even though these three students used vector mathematics correctly,
they exhibited considerable difficulty in explaining why they used them to
calculate the reading on the third scale. They understood that simple addition
(10 N + 10 N) was not the correct approach but expressed their reasons as
feelings and beliefs rather than analysis based on physics principles. Judy
recognized three forces were acting in two dimensions and that they were
balanced but was unable state reasons based on the directional nature of force.
Lisa recognized "force" in the question as a quantity that is nomially
represented by vectors but did not appear to work frorn fundamental physics
principles.
These three seemed to choose vector mathematics because they
recognized vectors are used in similar questions, not because of an
understanding of the basic pnnciples involved. Their struggle with the
representation process was obvious from their comments. They indicated
difficulty in moving from the real worid to the vector diagram necessary to solve
the problem. They appeared to be following an algorithm for representation
even though my instruction had been designed to provide them with an
understanding of the process. There is no sense that they understand the
process of representation in this excerpt of transcript. I was unable to get any
sense in our discussion that their understanding was any more complete than
they were able to express in the transcripts.
In this next section of transcript Dean and Todd attempt to explain their
approach to determining the reading on scale Sa.
Tchr - What did you predict for the answer to the set-up? O - 20 T-20 Tchr - 20. Both of you had 20. OK, what did you do with the
calculations? T - On the real calculations? Tchr - Yes, what did your answer corne out to? T - 17.32 (Newtons) Tchr - OK, and what about yours Dean? D - I got the same thing (17.32 N). Tchr - (Both of) You estimated 20 which is just 10 + 10. T - That's why I just added it on. Tchr - OK, you just added the two, you didn't think about anything else. T - Around there, yeah. Tchr - Did you think it would be exactly 20? T - No, not exactly 20 - it could be lower or higher. I wasn't sure of that. D - I thought it would be 20. Tchr - OK, that's fair enough. And how did you think it would corne to
be 20? D - I thought both forces were 10 and pulling on something. I thought 1
would just double. Tchr - OK, it would just double. When you did the vectors then, did you
expect the answer to give you 20?
D - Yeah. Tchr - OK, so when it gave you 17.3 were you a little surprised at the
answer? D - I thought it might be wrong, but I thought that was what was logical. Tchr - Why did you use vectors Dean. and not just add 10 + 10.
Because you did you drew diagrams and ... D - No I didn't draw a diagram, just a calculation. Tchr - OK, but why would you go through a vedor calculation when you
could see that 1 O and 10 is 20? D - I don't know, I just thought of doing vectors. Tchr - What did you (Todd) use? T - I used vectors. Tchr - Why? T - I was pretty sure 20 wasn't the right answer, so I used vectors to get
a more doser answer. Tchr - But why vectors? Why not just add 10 and 10 that adds up to 20,
especially when rny guess is 20. T - Ifs too easy. Tchr - Was that your only thinking? T - No. Well, you said using diagrams (in the instructions). So once 1
saw that I thought vectors - diagrams - vectors. Tchr - But you didn't think of the forces pulling in two dimensions. T - Oh yeah, I did think of that actually ... and the vectors pulling at 10.
10 and you said force ... and the force is in vectors
Both Todd and Dean estimated the reading on the scale would be 20 Newtons.
Todd hedged his prediction by indicating "not exactly 20, it could be higher or
lower," but Dean clearly thought the answer would be 20 Newtons. His view
was that the two forces would require a force equal to the sum to balance them.
Although Dean thought the answer would be 20, ha still perfonned a
vector calculation to confin his prediction. He did not draw any diagrams to
represent the force board set-up or the vectors that he used. His only reason for
using vectors was expressed, "1 don't know, I just thought of doing vectors." He
showed no indication in this discussion that he used physics principles
anywhere in solving this problem and does not appear to understand the
representation process necessary to use vector mathematics in solving this type
of problem.
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Todd's rationale was difficuit for him to express. He estimated 20
Newtons as the answer but thought vectors would produce a closer (more
precise) answer. His rationale included his thinking that the question would be
too simple if al1 one had to do was add 10 + 10. He refers to hints in the
question such as "diagramsn and "forces" which led him to think of vectors. As
with the others his rationale is based on recognition of the type of problem and
the quantlies involved, rather than an analysis of the physics principles.
These two students did not appear to use physics principles in solving
this problem. Dean seemed unaware of the directional nature of force and
Todd only has the slightest awareness that direction had to be included in the
solution. They predicted solutions by using persona1 experience and intuition
or by thinking about previous examples in class. They appeared to have very
little understanding of the process of mathematical representation that I had
been instructing them about over the previous 16 weeks.
These rationales for using vector mathematics to answer the force board
problem were disappointing for me. I was hoping for more evidence of
understanding the process of representation by vectors that 1 had tried to assist
them in constructing during the last sixteen weeks we had been together. I had
attempted to design my instruction based on a constructivist view of student
leaming. The instructional strategies do not seem to have produced the
success for which I was hoping.
Some Final Student Thouahts
At the end of the study 1 asked the students to describe which physics
concepts they considered easy and difficult to understand and why they were
easy or hard to leam. I was looking for student insight into what made physics
concepts diffÏcult to leam and hoped that the students might provide some
thought from the student perspective that would help in understanding their
struggle. The responses below were writîen by the participants before the final
interviews.
C - It is easier to understand concepts that you can visualize or see. Things like time and temperature are not easy to visualize but they are so common that they are easily understood.
A - The things that I understand most are things that I use and can apply to everyday life. Time we find okay because it's been with us al1 Our Iife. Our minds have already got seconds etched into them.
K - Speed is easy to understand because we deal with it everyday in things such as dnving.
J - Mass is al1 around us; everything has mass. Since you can feel mass, it's easier to understand. Wih weight you just multiply mass times gravity; weight is what you are actually feeling everyday.
D - Time was easy to understand and leam because I have been dealing with it al1 my life, and leamed about 1 in other classes like math and chemistry.
L - I found that force was pretty easy because I could imagine a clearer picture of what force looks like.
M - (concepts were easy to leam) because you could get hands-on (exarnples). Like related, like you did wlh vectors ... you took them, you brought a physics thing into the real world.
Concepts that students found easier to understand appear to have some
common characteristics. Most concepts in this group could be visualized by
students, or were readily identified by students in their experiential world. Most
easier-to-learn concepts could be illustrated with concrete examples in the
classroom or demonstrated through hands-on experiences. Lastly, students felt
sorne concepts, such as time and temperature, had been part of their
experience for so long they were uetched" in their rninds.
The participants provided exarnples of concepts which they found difficult
to understand, and described in writing what they believed made these physics
concepts more challenging to leam.
C - The stuff that I did not understand was stuff that was hard to visualize. If I can visualize or see something then I will understand it.
A - The concepts that I found difficult are things that don't have a concrete explanation. For example, there is no constant explanation for momentum. I also have a problem with concepts that are not "hands onw.
R - For mornentum and energy, at firçt I did not understand any of the concepts because it was confusing me. I do not know why it confused me. It was very fmstrating at first because I did not know how to apply all the different formulas.
K - I found momentum difficult because 1 is a concept that you cannot see, and one that isn't used very often.
J - Since you can't explain mornenturn and visualize it, 1 becomes diffÏcult to understand. After doing problems and using the fomula you begin to get a feeling for it.
D - Momenturn was difficult to learn because I get the formula for it mixed up with other formulas in other concepts.
L - I found rnomentum difficult to understand and learn because I couldn't develop a picture of what momenturn looked like. I couldn't figure out how to apply this to something in real life.
Students were unable to visualize concepts which they considered to be
difficult, or were unable to understand what characteristic was being abstracted
from or identified in their environment. For example, Dean was unable to
correctly separate the fomula representing momentum from other fomulae. He
could not identify the correlation between a letter in a formula and the feature in
his experiential world that was being represented. The most consistent
complaint was that they could not identify some concepts that physicists isolated
in nature.
A Student Thinks Aloud While Writing a Test
On one occasion Anne wrote a test alone in a separate room where she
recorded her thinking on audiotape as she answered the questions. This
procedure was only of limited success because of Anne's inexperience at
expressing her thinking in this manner. The test consisted of eight questions
and took her about an hour to complete. For extended periods Anne made no
comments at ail. At other times the only sounds were sighs, loud exhales and
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page flipping al1 of which did not provide insight into her thinking during the test.
At no time did she make any reference to basic physics pnnciples she
might have been using. I could not tell whether this absence was due to an
unawareness of them or an inabiliity to verbalize them. She described stages in
her problem-solving, such as real worid and vector diagrams but not in the
detail I had hoped for. She seemed to select her problern solving procedure by
the fom of a question. In one case she calculated a correct answer and
commented, "That's too easy." She did not have enough confidence in her
knowledge to know it was correct and seemed to be evaluating her answer by
the apparent difficulty of the calculations required. In another problem she drew
a vector diagram to determine a third force but was unsure of the direction of the
third vector.
A single attempt at recording thinking during test writing was not enough
to evaluate this method; however, the incomplete snapshot of her work was
consistent with the view of student thinking emerging from other data sources.
This method has potential for exploring student thinking but the logistics of
working with large numbers of students are difficult to solve because each
student must be alone to record his or her thinking. Students would need
practice and feedback to leam how to provide more useful descriptions of their
thinking than was achieved in this case.
Students Oiscuss a Test on Momentum
Approximately fcur weeks after beginning the unit on momentum the
participants wrote a test assessing their knowledge. The next day only Colleen,
Marie and Anne were in class so we discussed their performance on the test
which had been graded and they had in front of them. Unfortunately the audio
record of our conversation was lost due to a technical glitch. The only record of
1 74
this conversation was my journal notes written immediately afier Our discussion.
I asked them how they went about solving physics problems. AI1 said
they began by identifying the variables frorn the question and writing them
down. Then they scanned the formula sheet looking for a formula containing
the variables. Lastly, they substituted the numbers into the formula and made
the calculations. None of them attempted to identify physics principles
goveming the problem, nor did they try to estimate an answer for the problem
before beginning the steps described.
A weakness of their algorithmic approach was dramatically
demonstrated by Anne. One problem involved a cannon firing a shell which
resulted in the cannon moving backward until stopped by friction. Marie and
Colleen got the correct answer (about two metres) but Anne's work produced an
answer of about 37,000 metres. Her error was the resuk of substituting an
incorrect speed for the cannon into the mathematical expression she had
written. The error in substitution was not surprising under pressure of a test, but
her inability to detect such an enormous enor was disturbing. When I pointed
out the magnitude of the error al1 three recognized that 37,000 metres was an
unreasonable answer. When asked about her work, Anne replied that she
simply did not think about answers after completing calculations. Marie and
Colleen were not surprised at her response because neither of them
considered whether an answer was reasonable after finishing a calculation.
These three students used formulae uncritically in problem solving. They
did not spend much time thinking about the nature of a problem nor the
underlying physics principles. They did not estimate answers nor evaluate
them to see if they were reasonable. These were not the problem-solving skills
that I was aiming for when planning my instruction for the classes.
CHAPTER SIX
INSIGHTS, IMPLICATIONS AND RECOMMENDATIONS
My interest in this research project resulted from a desire to develop a
deeper understanding of student leaming as they constructed knowledge of
vector mathematics in physics and the role of teacher instruction in that process.
The descriptions of student learning in physics presented in the preceding
chapter produced a number of insights about the leaming of the participants
which are more generally applicable to student leaming in physics and in
science. In the first part of this chapter these insights about student leaming
processes in science are discussed.
This research experience has been transfomative and provided me with
a deeper insight into my practice and a more thorough understanding of the
leaming processes of students. The results have led me to suggest several
radical recommendations for classroom teaching of physics and science. In the
second part of the chapter implications and recommendations for physics and
science curricula and instruction are discussed. In the final section of this
chapter areas which require additional research are discussed.
lnsights Gained into Student Learning Processes
Although 1 had a general sense of student leaming processes because of
my previous teaching experience, the analysis of and reflection on the data, and
writing the descriptions of the classroorn sessions created new insights and a
more complete understanding about how students leam in grade 12 physics.
The analysis and writing processes were necessary to achieve these insights
into student leaming and the role of instruction in that leaming. The
176
participants' views of their own leaming are described in the first section. Next
a discussion of their experiential and conceptual knowledge and their struggle
to develop conceptual knowledge is presented. In the last part implications of
student alternative conceptions and communication are discussed.
The Partici~ants' View of Their Own Learninq
From the first day the participants arrived with a desire to leam about
physics and throughout the project generally looked fomrard to being in class.
They were motivated by a variety of factors, including personal interest and
knowing that grade 12 physics was a prerequisite for certain career choices. At
no time during the study did I deliberately spend time motivating them to
engage in leaming physics.
The students had some perceptions about how they learned and the role
that teachers had in that process. For the most part they were receptive
learners at the beginning of the study and were not comfortable being actively
involved in their own leaming. They expected me to provide instruction in a
direct manner. either through notes or by directly answering their questions.
Their expectation was that the notes and answers would explain the physics
concepts and that they would develop understanding from these explanations.
Initially they were uncornfortable when the instructional strategy did not fit their
receptive style of leaming.
When the students became comfortable with being activeiy involved in
their leaming, they were incredibly active. Although there were only nine
students in the class, interactions between participants were numerous.
Conversations erupted spontaneously among students and between students
and teacher. They discussed ideas with each other and openly argued about
interpretations of meanings of definitions, concepts and problems. These
1 77
conversations were so frequent that they were impossible to record on
videotape because cameras could not be switdied quickly enough to capture
al1 of them. Some discussions between students occumd quietly during other
activities and often were not noticed until the videotape records were reviewed.
The high frequency of interactions about physics was a revelation to me. The
participants were more talkative during their attempts to understand physics
concepts than I expected. All students were involved in these interactions.
Wïhout the videotape records the large number of these leaming interactions
would not have been identified.
The participants thought that natural ability was an important factor in
their success or lack of success in physics. They believed that some of thern
had more ability than others to do physics. On occasion individuals assumed
that they did not have the ability to use the mathematics which they believed
were required in physics. While a range of ability did exist among members of
this group of students, they were not always correct in their assessrnent of their
own ability or of others. Their main tool for assessing ability was the marks that
they received on their tests and assignments which they undoubtedly viewed as
a measure of ability as well as a level of achievement.
Their understanding of the structure of physics knowledge led to some
confusion for them. They perceived that there were knacks or tricks to doing
questions and searched for a systern of steps to follow that would lead them to
correct answers. The participants wanted an all-purpose algonthm (or even
several) which they could use to solve al1 types of questions. They appeared to
view scientific theories as algonthrns which can be used to answer problerns.
Theones were seen as providing understanding of phenomena in the worid by
answering their questions. This interpretation is consistent with their reliance
on experiential knowledge to understand phenomena and with their lack of
1 78
development of conceptual knowledge.
Their use of algorithms was nondiscrirninatory; that is, they picked an
algorithm that matched the variables in the problem. They demonstrated. for
example, considerable difficuity answering questions involving kinetic energy or
mornentum because both concepts depend on mass and velocity. They looked
for a formula that contained m a s and velocity and solved the resulting
equation, rather than basing their analysis of a problem on underiying physics
principles. Since both kinetic energy and momentum formulas contain an "mm
and a 'V, they frequently chose the incorrect formula unless some hint was
identified in the wording of the problem. Once they employed an algorithm and
determined an answer, they did not reflect on whether the answer was
reasonable or not. The cannon recoil question provided an excellent example
of this blind usage of formula. All three students who discussed this question
indicated that they never thought about an answer to a question once they
substituted numbers into a formula and made their calculations.
Their reliance on algorithms was not dependable because they did not
always remember the algorithm. This approach to learning did not produce the
depth of understanding that was required to transfer knowledge from one
subject area to another. A poignant example of this concem was identified
when we explored vector addition and subtraction. Some participants had
memorized algorithms for adding and subtracting vectors in mathematics
classes but none was able to rernember them accurately. Students were
unable to reconstruct the vector operations because they had essentially no
understanding of what vectors were, or what was meant by addition and
su btraction of vectors.
The participants were aware of difficulties and inconsistencies in their
knowledge construction. When something presented in the class was not
1 79
understood, they asked questions in an attempt to reduce or eliminate their
confusion. lnitially they were somewhat reticent to talk about their confusion;
however, as they became more at ease, they openly discussed their confusion
with me and the other students. When leaming about momentum they were
aware of contradictions in their understanding. This awareness was present in
spite of the evidence provided by work submitted that they did understand.
They appreciated the non-judgmental atmosphere of the classroom,
including rny own reactions to their struggles and the reactions of their
classrnates. There were very few cases of students "putting each other downn
during the months of the study. In reviewing the tapes and transcripts no cases
were identified where students had to be reminded not to discourage each
other. They had trouble describing their thinking as they leamed but without a
supportive classroom environment they would never have attempted to do so.
The participants employed a variety of strategies to construct physics
concepts but did not appear to attack knowledge construction in a planned or
coordinated manner; that is, I do not think they had identified principles of
leaming that they applied to the process. While other examples occurred the
most clearly documented case illustrating different strategies of knowledge
construction were exhibited dunng the discussion on momentum. Several
different individual attempts at constructing the concept were evident. The
participants accepted some responsibility for developing understanding of the
concept and thought that 1, as teacher, could help them in that effort.
Students' Ex~eriential and Conce~tual Knowledae
The participants relied on their experiential knowledge to an enormous
extent when leaming physics. They had not developed conceptual knowledge
that was useful in physics and did not seem to understand the process of using
180
conceptual knowledge to explain and understand natural phenornena.
Although they used mathematical formulas in calculations, they did not
understand the process of representation that was used to create the formula.
When the mathematics became more complex, they did not trust a model to
provide interpretations of situations; for example, when asked to calculate the
change in velocity of a car which went around a corner at constant speed, most
did not consider that there was a change in the velocity. Others did not
understand wnat the answer meant when the change in velocity was calculated.
The mathematical model was not useful in assisting thern to understand the
situation. A second example of the lack of use of conceptual knowledge was
displayed when they solved the final force board problern at the end of the
study. Even those participants who used vector mathematics properly were
unable to state an adequate reason for using vector mathematics in their
solution. Their choice was determined by intuition and previous examples
rather than deciding that vector mathematics were required to represent the
properties of forces.
The participants did not reflect to any extent on the application of physics
principals in their everyday experience. They had not thought about the action
of curling rocks or the place of numbers in science until asked to do so dunng
the study. lnitially I was concerned that my interaction was not skilled enough to
reach the Iimits of their reflection about such concerns; however, I no longer
think this to be the case. The participants had not been taught to reflect about
how we leam or what knowledge is. Nor had their life experiences provided an
intrinsic need to begin to think in a philosophical manner about knowledge
construction. Generally the students seemed to be positivist in their view of the
world and believed scientific laws were discovered in nature. Their belief was
that the concepts were really out there, rather than being constructed by
f8 l
hurnans to organize their world. On occasion they talked as if they had a
constructivist view of the wodd. but they did not understand the ramifications
that such a viewpoint had for learning science and other subjects.
The participants worked well with laboratory apparatus. On several
occasions they demonstrated their ability to operate laboratory equipment
skillfully. They understood what the equipment was meant to do and the
measurements that they were supposed to make during the experimentation;
however, this understanding of the operation of laboratory apparatus did not
appear to translate into understanding at a conceptual level. This outcorne was
disappointing because a traditional argument for the use of laboratory activities
in al1 science classes is to provide concrete exarnples of concepts that are
being studied. These concrete examples were expected to help students
develop more understanding of the concepts involved.
The bucket and ropes activity demonstrated their lack of understanding of
the process of mathematical representation of concepts identified in everyday
situations. They were able to describe the relationship between the angle of the
rope and the force needed to lift the bucket with considerable accuracy in a
qualitative manner; however, they made no headway in representing that
relationship using vector mathematics. The ability to perform this difficult
representation process was never demonstrated by any participant. Their
struggle with conceptual knowledge, its use and development, was ongoing
throughout the study.
When asked what concepts were easiest to leam, the participants listed
those that they could visualize or identify in their everyday experiences. They
were unable to visualize concepts which they thought of as diffÏcult to leam and
they wanted me to provide 'hands-on" activities and practical examples of
abstract concepts. They thought that if they could understand how a concept
182
was used in their experience, then they would be able to understand the
concept in physics. In spite of this belief the students did not demonstrate
development of conceptual knowledge during the hands-on activities that were
performed in class.
Although the participants at times exhibited some characteristics of
meaningful leaming, more extensive probing of their understanding revealed
that they had mainly achieved rote leaming. The leaming that occurred as the
result of the mornentum cart lab was a good example of this type of mimicry.
The rnanner is which they manipulated the equipment and lab reports submitted
indicated an understanding of some aspects of momentum. Their lack of
understanding was identified only when they tried to answer questions which
probed their conceptual development. The instructional sequence produced
the expected results but these did not accurately indicate the level of student
understanding. The assessrnent items used during this activity, student
observation and submitted report, did not correlate well to their conceptual
understanding of momentum.
Student Alternative Conceptions and Communication
As was expected the students showed confusion over the use of words
which are used interchangeably in lay language but have separate and distinct
rneanings in physics. Separation of vector addition and subtraction from
algebraic addition and subtraction was difficult for the participants who, at times,
used the algebraic operations when using vectors. Equally apparent was their
confusion between the terms balanced and equal when using vector
components. They wrote equations for relationships between cornponents
which indicated that they thought balanced meant equal. They looked only at
the magnitude of the component vectors and did not consider the directions of
183
the components as significant.
Confusion over the meaning of pairs of words provide strong evidence
for the necessity of clear communication between teacher and students. While
this need is always assumed by teachers, the apparent insignificance of an item
that can cause a breakdown in communication can not be underestimated.
Confusion can anse over seemingly minor points resolting in leaming
blockages which produce faulty knowledge construction or block it aitogether.
The most important feature documented in this research is that many of these
causes of confusion are not identified in the classroom as instruction occurs.
Students may be aware of some blockages, especially those that stop learning
completely, but are unaware of others because knowledge construction
continues but in a wrong direction. In this latter case knowledge produced is a
form of alternative conception.
Student transfer of knowledge about vector mathematics between
mathematics classes and physics was almost nonexistent. No student in this
group brought sufficient understanding of vector mathematics to be of practical
use in physics. Some had acquired algorithrns for addition and subtraction of
vectors but could not recall them completely. They brought only a poorly
developed concept of what a vector is and no one came with conceptual
understanding of what it meant to add or subtract vectors. Teaching the
rudiments of vector mathematics in physics classes is likely to continue for the
foreseeable future until a different approach is used in teaching these concepts
in mathematics classes.
This inquiry has reinforced rny understanding of the value of using
student questions and comments to build models of their knowledge
construction and conceptual development. Students ask questions and make
comments on the basis of what they think they understand about a concept.
184
The structure of their knowledge is indirectly revealed in the way that they
phase their questions. By using their questions and asking others I was able to
explore something about their knowledge development. Student responses on
tests and quiues, and work at the board pmvided additional sources of data for
development of these models of student knowledge construction. For teachers
to develop these models of student knowledge construction interactions among
students and teacher have to occur openly. In a classroom which is highly
teacher-centred, this type of model development is not possible, because
students do not have opportunlies to talk about their developing concepts with
the teacher or each other.'
Implications and Recommendations for Secondary School Physics
Secondary school physics curricula contain a considerable number of
concepts ranging in dificulty for both teaching and leaming. One group of
concepts, which include tirne, speed and mass, is easier to teach because
students have frequently experienced these concepts in their everyday lives. A
second group of concepts, such as weight and velocity, is more difficult for
students to leam because their meaning in physics is different from the meaning
in everyday use. The differences in meaning can be small but important and, as
a result, difficult for students to cornprehend. A third group of concepts,
including momentum, energy and entropy, is very challenging for grade 12
physics students because these are substantially different from meanings, or
essentially unknown, in everyday usage. This last group contains abstract
concepts which are very difficult to illustrate concretely because they are best
represented as mathematical models. - ' Appendix E contains a summary of leaming difficuities exhibited by the student participants in this study. The purpose is to provide the reader with a reference list of the leaming difficulties described in this research study.
185
Abstract concepts in the third group are currently a part of the physics
curriculum in secondary schools and are likely to be retained. The nature of
these concepts cannot be changed. but instruction can be modified to assist
students in achieving something closer to meaningful leaming instead of simply
mernorizing forrnulae and definitions. Different instructional strategies c m be
developed to facilitate student construction of conceptual knowledge. Leaming
about this group of concepts will never be a simple matter, but students are
more likely to develop better understanding if they leam more about the
structure of physics knowledge and the process of mathematical representation
than is currently expected in secondary school curricula.
Secondary physics curriculum guides usually describe student leaming
in ternis of outcornes or objectives, but do not provide guidance to teachers in
promoting conceptual developrnent in students. Physics and other science
curricula are not designed to have students explore the relationship between
science concepts and mathematics or the process of mathematical modelling.
Science and mathematics courses are developed with little attempt to
coordinate content in mutual support of each other. Some mathematics courses
may be required as prerequisites of physics courses, but students can not be
assumed to understand the process of mathematical representation used in
science as a result of completion of certain mathematics courses.
One objective of physics curricula is to have students understand the
nature of science knowledge and the processes of science/physics
(Saskatchewan Education, 1 992). To achieve such understanding student
leaming should be meaningful and new knowledge should be connected to
what the learner already knows. This study's research results indicate that
reducing the amount of content in physics curricula, making content more
relevant and meaningful for students, and increasing transfer between
186
mathematics and physics would be constructive changes in assisting students
to achieve meaningful leaming within secondary çchool physics. My
recommendations for change are presented in the following sections.
Amount of Content in Secondam School Phvsics
Throughout my teaching career and this research project I have been
aware that students did not develop the depth of understanding of physics
concepts for which I was aiming. Because of pressure to cover the content in
the grade 12 physics curriculum, additional time was not spent helping students
develop a conceptual knowledge base to inform their expenential knowledge. I
have found it impossible to help students develop conceptual physics
knowledge in the time allotted; however, the length of time spent on a given
concept is not the only issue. Alternative teaching strategies and leaming
experiences must be developed to increase success. Using the same
instruction in class for a longer period to time will not increase student
knowledge development. When curricula are designed with coverage of
concepts as a major driving force, the pressure to move on to the next topic or
unit dominates teacher decision rnaking. Until a change in curriculum focus is
made the pressure to "cover the course" can not be ignored by teachers. Good
pedagogy should direct teachers to ensure an adequate level of student
understanding before rnoving ont0 a new concept or unit; however, good
pedagogy is rarely the driving force in these decisions because of the
overwhelming pressure to cover the content.
Physics cumcula are norrnally constructed as a sequence of concepts
and principles to be leamed by students. The sequence can Vary to some
extent. but the emphasis in rnost physics curricula is on content rather than
processes of physics. If students have memorized definitions of physics
187
concepts, then they are assumed to understand the processes of physics as
well. Students are left on their own to connect their current knowledge and the
physics concepts. as well as. construct an understanding of the processes of
physics. A great deal of confusion and incomplete knowledge construction is
the result of instruction designed primarily to transmit content to students.
The function of mathematics and mathematical representation in physics
is assumed to be understood by students if they can answer problems and 'do
the math". This research has illustrated the inaccuracy of this assumption.
These students did not demonstrate understanding of the process of
mathematical representation even when instruction was designed to enhance it.
Little time is allotted to examine this relationship in most physics, science and
mathematics courses. Without making the connections between physics
concepts and fundamental processes of physics, students can not achieve
adequate understanding. New curricula in physics have to reduce the number
of physics concepts explored and allow students more time to develop
understanding of the processes and relationships in physics. If changes are
made only in grade 12 physics, then success is unlikely. Changes to the
curricula of science and mathematics courses taken pnor to grade 12 physics
are required to ensure better exploration of the mathematicslçcience
relationship. Much of their knowledge of the place of mathematics in science
and physics was obtained in the courses taken earlier. Changes in teaching
science and mathematics in earlier grades would provide the background for
application of mathematical models necessary to physics. The rush to move on
and cover the content in physics might be alleviated if understanding of the
processes was learned earlier in students' formal education.
188
lncreasina the Relevancv of Phvsics to Students
The abstract nature of many physics concepts creates considerable
leaming difficulty for most students. To master the concepts in grade 12 physics
students have to expend considerable effort in constructing new knowledge,
reconstructing their currently-heid knowledge, and making connections
between the two. This research showed that students consistently had
considerable difficulty making such connections on their own. Because of
students' inability to make cognitive connections instructional strategies need to
be designed with the aim of assisting students in connecting newly acquired
concepts to their currently-held knowledge.
Without students achieving understanding of physics principles and
concepts there is little rationale for students taking grade 12 physics. For most
students little content appears to be remembered for more than a few weeks or,
at most, a few months. Having students understand a few principles deeply and
seeing connections to their life experiences is a more sound pedagogical
position than covering a large amount of content but knowing that the students
will remember little in the future and will be unable to apply these principles
anywhere but in the physics classroorn. Secondary school physics teachers
should shift the focus of instruction away from covering the curriculum to helping
students develop a more complete understanding of a few concepts and an
ability to apply them to phenornena in their lives outside the classroom. The
current emphasis on moving through a series of concepts without assuring
understanding should no longer be acceptable teaching practice. To some
extent al1 secondary school science classes suffer from the same concem and
al1 could beneffl from a similar shift in emphasis.
If a decision is made to focus on better student understanding in grade
12 physics classes, changes to instructional strategies are necessary. In the
189
following suggestions some principles of instruction designed to facilitate
student understanding are described. These suggestions represent starting
points for teacher instruction, not finished products.
When mathematical formulas are introduced eariy in classroom
experiences, the participants treated the formulas as algebraic problems and
lost sight of the concepts that are represented. Evidence of this weakness was
displayed when the three students discussing the momentum exam indicated
they never thought about the answer once the calculations were cornplete.
They showed no indication of understanding the principles underlying the
problem, rather they followed an algorithm to arrive at a solution. To alleviate
this weakness a different sequence of instruction is required.
When introducing a new concept instruction should first explore students'
current understanding of the concept and then identify the concept in
phenornena in a qualitative manner. Much less ernphasis should be placed on
mathematical formulas than is traditionally done. Experiences should be
provided for students to assist them in identifying a concept in nature and
determining other concepts that are related to it in cause-and-effect
relationships. For example, when exploring acceleration students would
initially examine their own conceptions of acceleration acquired during their
lives, especially their experience driving cars. By first examining their own
conceptions students would becorne more aware of their current understanding
of the concept, here acceleration. Classroorn experiences would be created to
demonstrate the limitations and inconsistencies of their concept and to guide
them to reconstructing their concept to resolve their dilemma. Helping students
identify situations where their knowledge fails to produce understanding may
create an impetus for rastructuring their conceptions to match more closely
those of the physics community.
190
After students had a sense of their own conception and had compared it
with the scientific view of the same conception, they would explore variables
which cause changes in the concept; in the case of acceleration, m a s and
force. This initial exploration of cause and effect relationships would be
qualitative in nature rather than using mathematical representations. Students
would gain through experience and discussion an understanding that
acceleration increases as force increases and decreases as mass increases.
When students understood the qualitative relationship between the concepts,
they would determine a means of representing the qualitative relationship using
mathematics. A possible vehicle for this stage could be laboratory problems
which require numerical accuracy for satisfactory solution. If used at all, physics
fomulae would be the end point of concept developrnent rather than the
starting point. The traditional use of fomulae in secondary school physics
could be eliminated altogether because this research strongly indicates
formulae act as part of an algorithm and can be manipulated correctly without
understanding the relationships represented.
The ideas expressed here require field development in classrooms and
would evolve with teacher and student experience. Students would have to
leam to operate within this approach to instruction. If this strategy were
introduced in eariier grade levels, by grade 12 physics students may have
leamed to view leaming in physics as concept development rather than
memorization of facts.
On different occasions in the research, students demonstrated their
inability to recognize reasonable answers to problems (for example, Anne's
inability to recognize the answer to a test question as incorrect). This lack can
be interpreted as a manifestation of students not understanding relationships
between concepts in a qualitative manner. To estimate answers to problems
191
students need to understand the fundamental relationships between variables
before they can decide if an answer is reasonable. Estimation skills need not
be developed in physics alone, rather they should be part of instruction
throughout secondary mathematics and science courses, as well as,
elementary math and science courses.
Students will develop estimation skills in science and physics only if they
understand science and physics principles at a fundamental level and are
provided with experiences where estimation is crucial to success. Leaming to
apply algorithms in solving problems does not produce the level of
understanding required to develop estimation and approximation skills.
Laboratory activities can be modified to contain estimation components that
would be performed before measurernents are made and venfied after the
measurements are taken. Restructuring problems in physics to include
estimation and verification of answers can be done using existing resources.
Connections Between Mathematics and Phvsics Curricula
Students require extensive mathematical skills and knowledge for
success in physics classes. In spite of these needs little parallel organization is
apparent between the two curricula. Topics in one subject area seem to be
taught without regard for what has transpired in the other. Students in this
research project agreed unanimously that their mathematics instruction did not
provide them with sufficient understanding of vector mathematics to use them in
physics applications.
Discussions and professional development between science and
mathematics teachers could produce results if they focussed on interrelating
topics common to both cunScula. Potential benefits exist for both subject areas.
First, students could identify explicit connections between subject matter in the
192
two areas. These connections would make some mathernatics concepts more
relevant by providing practical applications for seemingly abstract principles.
Second, such discussions could beneft both mathematics teachers and physics
teachers because they could compare instructional approaches and refer
directly to each others' subject in their own classes. They would develop an
understanding of how various topics had been taught and which concepts were
most important in each others' classes. Lastly, they could discuss problems that
they identified in teaching certain concepts, and perhaps provide mutual
support for each others' teaching.
Some topics identified by this research which would benefit from mutual
discussion include direction conventions for vectors, vector components and
problem-solving applications of rnathematics. If direction conventions and
vector components were used in the same manner in both subject areas, then
students would not have to perfom the mental gymnastics currently necessary
to apply concepts from one subject area in another. To some extent problem
solving skills might carry over from one class to another benefiting both students
and teachers.
At a more radical level a different option for curricula development could
be explored. The bamers existing between subject areas are artificially created
and are present for convenience rather than out of necessity. Consideration
should be given to eliminating the barriers created by subject areas. Science
and mathernatics could be taught as a single subject. Other barriers are no less
artificial. Science subject areas such as, biology, chemistry and physics could
be removed leaving an integrated study of science and mathematics. I have
little doubt that such a radical change would not be readily accepted by many
teachers and administrators but it is obvious that some radical change is
needed if we are striving for meaningful leaming in science students.
193
Implications and Recommendations for Instructional Strategies
Small G r o u ~ Instruction
The choice of instructional strategy depends on a number of factors,
including teacher preferences, the concept or principle being developed,
classroom facilities, available resources and the group of students being taught.
Although not designed to compare instructional strategies. this research
showed that some choices facilitate student leaming better than others. While
no single instructional strategy should be considered as a panacea for leaming
difficulties, the small group instruction used in the study appeared to have
several advantages.
The participants unanimously agreed that this type of small group
instruction was beneficial to their leaming. Among reasons stated were that the
small group provided an atmosphere where they were l e s concemed about
persona1 embarrassment, and each felt that he or she had sufficient opportunity
to express his or her opinions and concerns. I observed, as well, that they were
more active in the discussions, and were generally more attentive to class
activities. I have tried to create a similar non-threatening environment in my
regular classes but in a large classroom with twenty-five to thirty students 1 is
impossible to allow al1 persons as much time as they would have in a small
group to contribute their ideas to general discussions.
It has been shown that girls, in particular, benefit from small groups and
less competitive classroom environments than occur in most regular science
classrooms; however, I have no doubt that al1 participants appreciated working
in Our smaller group. All students preferred to remain in our small group rather
than retuming to Sarah Russell's larger classroom. I do believe that the girls
would have been at more of a disadvantage than some boys had they retumed
to the larger classroom. While I can not imagine Kevin and Ryan shrinking into
194
the background in any class, I can readily picture Judy, Anne and Lisa
participating much less frequently in a larger group of students. Dean and
Todd, as well, would have been negatively effected by such a move because
both would have faded even more into background, and participated even less
frequently in the activities in a large classroom that they did in our small group.
Usina Discussion as an Instructional Strateay
Throughout the project much of the instruction was orchestrated through
student-teacher and student-student discussions (some would cal1 the
approach, Socratic questioning). Although this strategy appeared to be quite
time consuming, severai objectives was accomplished during these
interactions. First, conversations with students assisted me in developing a
mental model or image of how each student was constructing his or her
knowledge of physics concepts. These models provided a background against
which to formulate individualized responses for each student's inquiries.
ldentically worded questions from two students could require different
responses if the image of their concept development was different. Second,
students benefited by listening to and taking part in these interactions, because
they were able to experience part of each others' struggle to leam. Stronger
students were sometimes perceived to be naturally talented in physics leading
others to believe that a strong student did not have to work through his or her
own confusion to achieve understanding. Classroom discussions helped to
make frequent mental struggle seem to be a natural part of leaming. Third,
everyone heard al1 student questions and inquiries, and was involved in the
resolution of each. Student-student dialogue contributed to individual leaming
because similar conceptual difficulties were experienced by more than one
student. At times a participant who had already worked through some diffÏculty
195
in leaming was able to identify the source of another's confusion and help to
dissolve it.
The time-consuming nature of participant discussions may have been
perceived as a disadvantage. Considerable care had to be exercised to ensure
that each student had an opportunity to make his or her contribution to each
discussion. On occasions when progress was agonizingly slow, I was tempted
to answer questions and relieve concerns by providing "correct" answers. In
spite of pressure to complete the curriculum t resisted the temptation as much
as possible because on most occasions transmitting correct answers did not
produce the meaningful leaming for which I was aiming. For classroom
discussions to be successful extended time was required because students
needed to reflect at length about the issue being discussed. They had to
reconstruct their knowledge and this process could not be rushed. While I was
at times able to catalyze their restructuring process by providing experiences to
help them to understand, each student had to perfonn the restructuring
individually. When I attempted to speed up the restructunng process by
providing answers directly, students rapidly reverted to a rote leaming mode as
was shown on occasion in the transcripts of the classroom sessions.
Questionina and Discussion Skills - Learned Processes
The research record showed that most students took some time to
develop the skills necessary to becorne fully involved in classroom discussions.
They had to gain experience with my style of questioning and interactive
discussion. Transcrïpts from the eariier classes showed that the participants
were initially quite passive and that they did not expect to parücipate so actively
in their own leaming. They needed time and experience to gain a sense of the
value of being personally involved before they readily contributed to the
196
discussions. Initially, wait times were long and student responses were laconic.
I had to exercise considerable patience while waiting for answen and refrain
from answering questions or moving on to other students to reduce tension.
Only when the participants realized that I was not going to provide answers
directly did they change their approach to this style of interactive instruction.
Dun'ng the first few days students were not at ease, and neither was 1. They
were not used to having so much responsibility for their own learning.
Usina Incorrect Answers as Discre~ant Events
On one occasion I displayed to the class incomplete solutions that
students had written to questions on a test. These solutions had no identifying
marks because this procedure was intended to be anonymous. In one solution
a student had drawn a "real woridn diagram but little other work was shown.
When asked to provide the remaining parts of the solution, the group did so
quite rapidly. When another solution was displayed, Kevin said, 'l screwed up,
it should be ..." Kevin identified the solution as his and was the first to find his
enor. It was surprising that he had been able to identify his mistake and provide
a correction so quickly. The final incomplete solution displayed was for a
problem that generally had been poorly answered. Only one student had
solved the problem correctly on the test but together the group worked out a
correct solution which they had not been able to do individually during the test.
The performance and reaction of the students in this exercise was
informative about their thinking. When students knew with certainty that their
answers were wrong, they reexamined their problem solving method. This
knowledge of error seemed to be a powerful motivating force in this exercise.
This approach was tried on only one occasion but it may hold considerable
potential as an instructional strategy.
197
In spite of the success of this attempt I do not think tests and exams are
the best vehicle for this purpose because nomally they are the final items in a
section. The process would be more useful if used before testing so any
concept or skill development resulting would be assessed on the test and not
developed afîer it. If situations could be created where students solve problems
to predict specific results and then test their solution in some practical manner
using apparatus, the effect of this process might be tapped. Having students
test their solutions in a laboratory setting would provide a stronger opportunity
for reconstruction of the concept than is provided by simply looking up answers
in a text.
Metric Units as Aids to Concept Development
In physics, quantities have SI Metric units associated with them and
measurements are considered incomplete if that unit is not expressed.
Traditionally, most teachers expect students to include units with their
calculations and in their solutions to problems, and do not give full credit for
solutions unless appropriate units have been included. In this study the value
of including units in problem solving was frequently underestimated by the
participants. They did not seem to understand that units were required as part
of a measured quality, or that units could be useful in problern solving by
providing clues to the solution.
This research study provided additional support for this traditional belief
about the pedagogical value of units. On one occasion Judy attempted to use
unes to enhance her understanding of nature of momentum. Later in the
semester the participants were unable to understand that work and energy were
related until they saw that both were measured in Joules. The units helped
them to understand the relationship between work and energy. While not so
198
useful in every case encouraging students to be aware of the units of a quantity
provides them with an addlional tool for use in their development of the
conceptual knowledge that is part of the content of physics.
Another reason for including unes during the calculations and in the
solutions to problems is related to the process of representing physics concepts
in mathematical relationships. If students are allowed to drop the units during
calculations, there is a tendency for them to forget that the mathematics is only a
representation of the physics concepts. lncluding appropriate units helps to
remind them that mathematics is a tool used to express the relationship
between various concepts used in physics.
Implications for Students Learning
Students are responsible for their own learning and must expend
intellectual effort to engage learning activities (Driver & Bell. 1986; Osborne &
Wittrock, 1985). Novak (1985) agreed the responsibility for learning can not be
shared and must be consciously pursued by students. This view that students
are responsible for their own leaming seems easily defended, yet the
experience in the first week or two of the study with the participants strongly
indicated they were used to passiveiy receiving knowledge from books and
teachers, and that they had assumed almost no responsibility for their own
leaming . When asked to describe how they leamed about physics concepts or
solved problems the participants struggled to explain what they were doing or
thinking. They had not thought to any extent about how they leamed nor their
own place in the process. This situation was not changed much dunng the
study because they did not have sufficient time to leam metacognitive strategies
or develop an understanding of leaming, especially considering the lack of
199
such focus over their past twelve years of formal education.
Metacognitive processes can be as simple as awareness of techniques
that assist rnemory, or as complex as the awareness of one's knowledge and
modifying its structure or content (Gagne, Yekovich & Yekovich, 1993). Several
science education researchers have argued that metacognitive strategies
should be taught to students so they realize that they change and construct
concepts in their minds (Duit, 1991 ; Gunstone, 1 988; Gunstone & Watts, 1 985;
Roth, 1995). Students need to understand and control their memory to increase
their success at leaming complex concepts in science. This research supports
the position that students would benef~ from understanding more about how
they learn and how metacognitive strategies can help them reconstnict their
p hysics concepts.
These strategies require tirne to develop and should be introduced early
in their education. Students need to leam that they are actively leaming and
that teachers can not transmit knowledge to them directly. Each student
constructs the concepts individually in the social environment of the classroom.
Teachers can assist through their instruction in concept development by
providing relevant expenences for students; however, each student is
fundarnentally responsible for his or her own knowledge construction. Students
need to know as eariy as possible that they are responsible for what they learn.
The appropriate grade level where introduction of metacognitive strategies
should be made is an area for further research, but it is likely students could be
successfully introduced to such strategies and begin to take responsibility for
their leaming at a much earlier age than secondary school.
Student Difficulty Using Vector Mathematics in Physics
This research project has identified three elements which help in
200
understanding student difficulties in constructing and representing concepts
requiring vector mathematics. First, the students generally did not grasp the
processes of mathematical representation and model building that are a major
part of physics. Second. students did not intuitively identify direction as a
fundamental characteristic of some physics concepts and measurements.
Third, students struggled with problem solving which needed vector
mathematics because most had not developed an understanding of what
vectors are, or what types of concepts they are used to represent. These
elements are discussed in the following sections.
Mathematical Re~resentation of Conce~ts
The detailed analysis of student thinking made in this research confirmed
that the participants did not understand the process of mathematical
representation of physics concepts well enough to apply the process to new
concepts and situations. Because of this research and my teaching experience
I strongly believe the majority of secondary school physics students do not have
a fundamental understanding of this important process. Students leam to
manipulate fomulae that are part of the course curricula but, in general, do not
understand the relationship between the fomulae and the concepts which the
variables in that formula represent. Students treat physics fomulae as
algebraic expressions to be manipulated mathematically rather than
representations of certain quantities abstracted from nature. To some extent I
have fostered this attitude in students by providing algorithms for problem
solving and clues in the questions to help students choose the correct pathway
to the solution. In this study I attempted to provide the student participants with
a different view of this relationship by looking at the place of numbers and
mathematics in science, but the results of the study are strong evidence that this
20 1
step was not enough to create understanding of the representation process.
Additional changes and different instructional strategies are urgently needed.
The enhancement of student understanding of the process of
rnathematical representation cannot be achieved in grade 12 physics classes
alone. Students need experience with the principles of mathematical
representation much earlier in their fomal education than the last year of
secondary school. Courses in math and science taken before grade 12 physics
will have to begin to develop these skills and ideas. The cumcula of those
courses will have to be restructured to provide students with primary
experiences constructing rnathematical representations, rather than obsewing
them as secondary experiences from a teacher or text book. Classroom
experiences could be formulated so that their successful solution is dependent
on students developing mathematical representation for the concepts under
investigation. Com puter software and g raphing calculators have potential to
provide simulations of this process and to perform those mathematical
manipulations in which students tend to get bogged down. The use of
cornputers and the lnternet in assisting students in constructing physics
concepts is an area for further research.
In mathematics classes manipulatives are used to provide hands-on
experiences that show the relationship between mathematics and things in the
students' world. The use of manipulatives in elementary mathematics classes
assists students in coming to understand mathematical functions; however. in a
sense using manipulatives to represent mathematical operations is the reverse
of the process we want students to achieve in the secondary sciences. The
place of manipulatives in developing the process of mathematical
representation in science has to be expiored in more depth.
Post-secondary educators would also benefa from students who had a
202
better understanding of the process of mathematical representation. In post-
secondary science courses the representation process is essentially the same;
however, more complex mathematics, such as statistics and calculus, are
required to represent the relationships between concepts with accuracy.
Students entering subject areas such as biology, chemistry, ecology and
economics, as well as, physics would benefit from a more complete
understanding of the use of mathematics in representation.
Recoanizina Direction as a Characteristic
I reflected for some time about the students' inability to identify direction
as a significant characteristic of certain physics concepts before gaining even a
hint of insight. In their lives most experiences and problems did not require the
awareness of direction that is needed in physics. All of the participants had
driven cars which seems to be an experience requiring some knowledge of
direction; however, closer examination reveals that this was not so. When
beginning a trip a driver must start out in a particular direction, but after
choosing the correct road few navigational skills are required to arrive at a
destination. The participants did not live in rural comrnunities where reference
to direction is more cornmon; nor had any of the participants navigated boats on
large bodies of water where an awareness of direction is more crucial for
successful arriva1 at one's destination. Students tended to see direction as a
means of relating positions on the earth and not to characteristics of certain
concepts in physics. Their conception of direction was not the same as that of
practising physicists.
The participants memorized algorithms to solve problems that involved
direction and used clues in the problems to identify which algorithm to apply.
These clues were nonnally present in the questions as part of the written
203
description. Educators have assumed that successful problem solving of this
nature would lead to the development of understanding as time passed. Over
the years this approach appeared to be an effective way to teach students
because they successfully answered problems. This research study has helped
to show that this instructional approach did not produce the depth of
understanding which was traditionally thought to be created.
This lack of identifying the importance of direction in physics concepts
adds to the inability of students to understand vector applications in physics.
Without identifying direction as a fundamental characteristic of certain quantities
students cannot be expected to see any reason to use vector mathematics in
solving problems; and, vector mathematics will make little sense to them until
they are able to understand why direction must be part of some mathematical
representations. The participants did not benefit a great deal from separate
instruction about vector mathematics in geometry-trigonornetry classes as was
shown by their unanirnous surprise that vector rnathematics could be used to
represent anything in their lives. None of the students in this research had
developed enough understanding of vector mathematics in their mathematics
classes to be able to transfer that knowledge to Our physics class.
In a sense the confusion is the result of students' alternative conceptions
of direction. Students have a conception of direction in their vocabulary and
use this meaning in the physics environment. Their meaning is based more on
using direction to describe the location of some object or destination with
respect to some fixed point. For example, a car is located to the left of the
doorway, or the city is north of the United States. They do not understand the
concept in the rnanner that is required for success in physics in that they do not
associate concepts such force and velocity with having direction. Wihout a
more appropriate comprehension of direction as physicists use it, grade 12
204
physicç students will continue to struggle with the use of vector rnathernatics.
Conceptual development strategies, as described earîier, may produce some of
the reconstruction required for successful problem solving involving direction,
but more study and research are required before a practical classroom solution
can be developed.
A~plvina Vector Mathematics in Phvsics
The students did not understand the process of rnathematical
representation to any great extent, nor did they understand that direction is a
fundamental characteristic of many physics concepts. These two factors
combined to make the use of vector mathematics even more difficult for the
participants and most grade 12 physics students. The participants lacked a
perception of any need for vectors or vector mathematics. They did not have a
sense of why they had been taught about vectors in other classes nor could
they describe any practical applications when we talked about vectors early in
the study. Some were able to perforrn addition and subtraction using
algorithms but they did not exhibit understanding of the mathematical principles
involved in these processes. This deficiency was illustrated when they drew
vector diagrams to help with adding and subtracting vectors as part of solving
problems. Most students did not view these diagrams as aids which showed a
resultant vector; rather, they saw the diagrams as separate problems.
The research results show that the difficulties expenenced by students
leaming to apply vector mathematics are very complex. Three elements of the
stniggle have been descibed, mathematical representation, alternative
conceptions of direction, and not understanding the function of vector
mathematics. A simple solution to this problem does not exist because of the
complexity of the leaming processes that have to be achieved by students. The
205
three elements must be dealt with together and successful resolution can not be
achieved in one five-month semester in grade 12 physics. Solutions to student
difficulties in applying vector mathematics in physics have ramifications for
science and mathematics courses at earlier grade levels. Students rnust be
assisted on three fronts: first to understand the representation process; second,
to develop a different conception of direction; third. to develop an understanding
of the purpose of representing certain concepts with vectors. Resolution will
take considerable time and innovation to create instructional strategies and
experiences to accornplish these goals.
I have discussed the three elements separately but any solution will have
to incorporate their interdependent nature. While the concems described are
fundamentally cognitive in nature, they must be addressed in curricula to some
extent because curricula largely determine what is taught in science
classrooms. Resolution will have to start much earlier in science and
mathematics education. Elementary and middle years science teachers have to
begin to provide experiences that develop student understanding of these ideas
and relationships. Students need opportunities to test their own knowledge in
real-life experiences and then to reconstruct it in light of them. Most science
teachers do not have the arsenal of instructional strategies and experience
necessary to create these experiences for students because the type of
instruction that I am advocating had not been used to any extent in science
education. This will also have ramifications for teacher education programs.
Implications for Research
This study has identified several areas requiring additional study.
Research and development are required to explore and design the type of
instructional resources recomrnended. Classroom activities and experiences of
206
the kind required to assist students in developing understanding of the
representation process and the use of vector mathematics in physics require
extensive research and development by research teams composed of
classroom teachers and academic researchers.
Working with expen'enced practising science teachers when developing
these instructional strategies would allow researchen to receive constructive
feedback on the practicality of the design. Because of their practical classroom
experience professional teachers would be able to contribute to the design of
instructional strategies in meaningful ways. They are the ones who will
eventually irnplement instructional changes. Making them part of the
developrnent process increases the probability of instructional changes leading
to conceptual knowledge construction actually occumng. Research projects
involving small groups of physics and science teachers willing to act as
participant researchers could form the core of such a research program. Some
classroom teachers already have concems about the physics curriculum and
the type of leaming that occurs as a result; they already recognize the limits of
the current cumculum and the manner in which it is taught.
Some form of action research is a likely candidate for the development of
instructional strategy envisioned. The action research cycle described by
Zuber-Skerritt (1 996) of strategic planning, implementation of the plan,
evaluation and critical reflection seerns to be an appropriate procedure for
developing new approaches to teaching vector mathematics in physics. The
proposed research can be viewed as practical action research or emancipatory
action research depending on the outcome. If the research results in changes
to the curriculum and the manner in which physics and other science curriculum
are taught, then the research would be considered emancipatory. If changes
are made only to teachets instructional strategies but not to the system in a
207
more general manner then the action research would be practical (Zuber-
Skerritt, 1996).
lnvolving classroom science teachers in this kind of research program is
crucial to its success. Practising teachers are the ones that have to implement
the solutions to the concems raised by this research and they must be
participants in the development of new approaches to instruction. Their
practical knowledge of instruction will be an important element in designing
alternative strategies because they have a good sense of the kind of solution
that can be applied to a classroom of thirty or ço students. Collaboration with
academic researchers who facilitate the development of instructional strategies
can provide assistance and background knowledge as required. Such a
program has great potential for initiating change in science education.
Another area of potential research is focussed on the relationship
between conceptual and experiential knowledge development in secondary
science students. This project showed that experiential knowledge does not
guarantee that a student develops conceptual knowledge about those
experiences even when they are instructed how that conceptual knowledge
should be constructed. Concrete examples are not sufficient to create a need in
the students to constnict abstract representation of those phenomena. More
research on the student construction of abstract conceptual models must be
done to shed light on this complex process. Constructivism provides some
description of the concem but has not provided a definitive pathway to solve the
problern. The instruction used in this study was developed using a
constructivist understanding of knowledge construction but clearly did not
provide the depth of conceptual knowledge anticipated. This result is not
sufficient to force the abandonment of constructivisrn as an description of
leaming, but clearly using strategies based on constructivism in one grade 12
physics class is not enough to change students ways of leaming.
Teacher education is another area of potential research because
preservice teachers' knowledge of science is the result of current science
education C U ~ C U ~ and practices. Elementary preservice teachers generally
have few post-secondary science courses and little confidence with the content
of science cumcula. Ample opportunities exist to assist them in reconstnicting
their current science knowledge and help them gain confidence with the
science subject matter. Secondary preservice science teachers have
completed a number of post-secondary science classes and generally have
considerable confidence with the subject matter in their field. In spite of this
confidence they hold alternative conceptions about a number of concepts in
science; for example, the relationship between laws, theories and experiment
are not clearly understood by many, nor are the origin of beta particles in
radioactive disintegration wessel, 1 995). Science education students can be
shown the strength of instruction designed to reconstruct knowledge in a way
that produces meaningful leaming. If preservice teachers experience
alternative instructional strategies, they are more likely to attempt their use in
their own classes.
Research projects could be designed to evaluate the satisfaction and
sense of understanding that teachers and students feel when instruction is
designed to promote meaningful leaming. Dykstra (1996) has argued that the
sense of personal satisfaction he achieved when teaching first year college
physics students was sufficient for him to continue teaching in that manner. His
students also experienced a sense of empowerment about their own leaming.
Evaluating the sense of personal satisfaction of both teachers and students is
an area of potential research.
Research must be conducted to determine appropriate ways of
209
integrating elementary and secondary curricula across subject areas. The splits
between mathematics and science are artificial and do not help students see
connections between subject areas. Mathematics and science curricula would
mutually benefii from research designed to bring areas of common concem
together in a manner to reinforce each other. Secondary cumculum
development provides many areas for potential research.
Concluding Statement
In the discussion of rny research project I described some leaming that
occurred in one of my physics classes. While the description is unique and is of
the student participants, many of the details are representative of other leamers
in my teaching experience. The description shows something of the personal
and individual nature of leaming. Although only nine students took part in this
project, several different approaches to leaming are used by them as they
struggled with the cornplex concepts in physics. My understanding of how they
leamed was enhanced by using a constructivist interpretation of the process.
My narration of my instructional decisions was also presented with a
constructivist interpretation. The detailed picture of the days of leaming and
instruction is incomplete but does illuminate rnany facets of this dialectic in a
manner that had not previously been attempted.
210
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Appendix A
Approval for Research Project
223
HUMAN SUBJECT RESEARCH ETHICS REVIEW COMMllTEE
Application for Approval of Research P rocedures
Section 1: Identification and Purposes
1. Date: 1 5 Januarv, 1993
Name of Applicant: Warren Edward Wessel
Address: 21 34 Cunnina Crescent
Reuina. Saskatchewan. S4V 0M8
Title of Research: Teacher Interaction and Students' Construction
of Knowledge in Grade XII Phvsics
2. If the project will be part of a thesis, or class requirement. give the name of the supervisor:
Dr. E. Paul Hart
Department or Faculty:
Facultv of Education
3. Purposes: Give a brief outline of the main features and variables of the research problem. lnclude a brief statement which describes the significance and potential benefits of the study.
The purpose of the study is to examine the cognitive processes that secondary school physics students use when learning vector concepts in grade XII Physics (Physics 30). In addition, the study will examine the effect of teacher interaction on learning. The results should help educators to understand student knowledge construction. The study could also lead to a deeper understanding of adolescent learning.
Section II: Subiects
1. Briefly describe the number and kind of subjects required for data collection.
Between six and ten grade XII Physics students will be required as participants in this study.
2. What information about the research problem and their role in the project will potential subjects be given?
All information about this study will be provided for the students who participate in the study. Nothing will be withheld from them.
3. How will the consent of the subjects to participate be obtained?
Participation in the study will be voluntary and contingent upon written parental consent. (sample consent form attached)
4. What will the subjects be required to do in the course of the project?
The students will receive normal instruction in grade XII Physics, but in small group sessions. Students will be encouraged to participate in group discussions about their learning. Students will be video-taped and asked to discuss sections of the taped sessions. In addition they will be interviewed in an attempt to have them describe the problern solving (and thinking) processes they use to learn about vectors in physics. They will also be asked to keep a personal journal during the course of the study.
5. What assurances will the subjects be given and what precautions will be taken regarding the confidentiality of the data or information which they provide in the study?
The students will be informed that al1 personal data will be presented anonyrnously within the dissertation and that the original data will be archived and available only to the researcher and supervising cornmittee.
6. Will children be used as a source of data?
Yes 4 No
If yes, indicate how consent will be obtained in their behalf.
Parents of the students who are considering participation will be informed in writing about the purpose and nature of the study. They will also be informed that participation in the study is on a volunteer basis. After actual participants has been chosen, consent in writing will be obtained from their parents. In addition a meeting with the volunteer participants and their parents will be held to deal with any concerns that they may have. (sarnple letter of introduction to the students and parents attached.)
7. Will the researcher or any member of the research team be in a position of power or authority in relation to his subjects? (For example: A teacher doing research and using her class as subjects or a counsellor collecting research data from his clients.)
Yes d No
If m, indicate how coercion of subjects will be avoided.
The researcher is an experienced Grade XII physics teacher and will teach and evaluate the participants in the normal manner. Participation in the study is voluntary, and a student can withdraw from the study at any time he/she rnay chose to do so, return to regular classes, and be evaluated by another teacher.
8. Will deception If any kind be necessary in the project?
If y-, explain why and indicate how subjects will be debriefed after the study.
Section III: Access to Data and Findinas
1. Who will have access to the original data of the study?
The doctoral candidate, Warren Wessel, and his supervising cornmittee will be the only perçons with access to the original data of the study.
2. Will subjects have some access to the findings of the study?
The participants will have access to data as it is collected and to the results of the study. A summary of the project will be sent to each participant upon completion of analysis of the data. The study is part of a dissertation which will be available to the public through the University of Regina library.
3. What wiil the final disposition of the original data after the study is cornpleted?
The data and video tapes will be kept in confidential files, then will be archived for a period of five years, after which time al1 data will be destroyed.
Signature of Applicant(s):
Signature of Advisor or Instructor
U N I V E R S I T Y O F R E G I N A 227
OFFICE OF ASSOCIATE VICE-PRESIDENT AND DEAN \Y/ FACULTY OF GRADUATE STUDlES AND RESEARCH -=
TO: Warren Edward Wessel
FROM: Dr. G.W. Maslany. Chalr
DATE:
Re:
January 29, 1993
Teacher Interaction and S t u d e n t s ' Construction o f Knowledge in Grade XI1 -
Physi CS.
Please be advised that the cmmiftee has considered this proposal and has agreed
Acceptable os submiited. (Note: Only those appffcotions deslgnated In this way have ethlcal opproval for the research on which they are based to proceed).
Acceptable subject to the following changes and precautions: (Note: kese chonges must be resubmltted to the Cornmittee and deemed acceptable by it prior to the initiation of the research. Once the chonges are regorded os acceptable a new approval form will be sent out indicoting tt k acceptable as submmed.)
3. Unacceptable to the Cornmittee as submiîted. Please contact the Chair for advise on whether or how the project proposa1 might be reln'sed to become acceptable (ext. 4 16 1 /5l86).
c: Applfcant Academic Unit Head (Ef hics 1 .Docl
21 34 Cunning Crescent Regina, Saskatchewan S4V 0M8
19 January, 1993
Mr. Brian Malley Assistant Superintendent J. A. Bumett Education Centre 1600 - 4th Avenue Regina, Sask. S4R 8C8
Dear Brian,
I am writing this letter to request permission and to seek your support for a research project that I wish to conduct during the second semester of this school year. 1 want to conduct the project with a small group of grade XII physics students at Campbell Collegiate. The project is part of the requirements for the doctoral program in which I am enrolled at the University of Regina. The purpose of the project is to study the thinking and problem solving strategies that grade XII physics students employ when leaming about vectors and vector mathematics in physics.
l have enclosad the application which I submitted to the Ethics Review Committee at the University of Regina. Ethics approval for the project is pending. As part of the package I included a permission form intended for parental consent, and a draft copy of a letter that I will give to potential student participants and their parents. These may be a particular interest to you and the Regina Board.
I am looking foward to conducting the project and believe that it may produce some interesting results. I hope that it meets with your approval. Should you have questions, or require further details and clarification, I am available at either of the phone numbers below. Please do not hesitate to contact me.
Yours tnily,
Warren E. Wessel - 789-7346 (home) - 791 -8380 (Campbell)
THE BOARD OF EOUCATtON OF THE
REGINA SCHOOL DIVISION NO. 4 OF SASKATCHEWAN
J A Bunitt Eduutiori Cantm 160d 4th Ave.. Rsgina. çask. SJR 8C8 (306) 791-8200
January 25.1993
Mr. Wanen Wessel 2 134 Cunning Crescent Regina, Saskatchewan S4V 0M8
Dear Warren:
Please accept this letter as approval to proceed wÏth your research study, Teacher interaction and Students' Construction of Knowledge in Grade XII Physics*, as soon as you have approval from the University of Regina Ethics Committee.
Enclosed is a copy of the policy of the Board of Education with respect to research. Please note in particular the following conditions:
1 )participation by students is voluntary 2) confidentiality must be maintained 3)a copy of your thesis must be provided to the Board of Education upon
completion.
Because you are conducting your study with students, parents must be infomed and indicate their permission. I note that this fom has b e n submitted for approval to the University of Regina Ethics Committee.
Best wishes with your research and we look forward to seeing the results.
Brian Malley " / Assistant Superintendent Curriculum Support Services
Appendix B
Communication with participants and parents
Campbell Collegiate 102 Massey Road Regina, Saskatchewan S4S 4M9
February 5,1993
Dear Student (and ParenVGuardian),
The purpose of this letter is to invite you to take part in a research project at Campbell Collegiate during part of the second semester - February to June 1993. The project will be conducted by Mr. Warren Wessel a teacher at Campbell Collegiate and a doctoral student at the University of Regina. The study will require a group a six to ten grade XII Physics students from a normal Grade XII Physics class.
The purpose of the project is to examine the thinking processes and problern solving strategies that students use while leaming to use vectors in grade XII Physics. The participants in the study will be taught the vector section of Grade XII Physics by Mr. Wessel separately from the remainder of the class. The instruction will be the same as in any other class that he would teach. This is not an experirnental teaching technique, rather the study is intended to investigate how students leam to use vector mathematics during the course of Grade XII Physics.
The classroom sessions will be video taped for analysis. Students will be encouraged to reflect on their leaming and discuss their thinking and problem solving processes with each other and Mr. Wessei. In addition to clarifying thought processes for the researcher, the reflective process should prove valuable to each student. Through this reflection each student will benefit as a participant in the project.
In addition to the taped sessions at times through the project students will be interviewed to help describe their thinking processes and to help the researcher understand more clearly how they are leaming about vectors. Students will also be asked to keep a personal joumal describing their experience and to share that journal with Mr. Wessel.
The results of this project are intended to provide a deeper understanding and clearer picture of the thinking and problem solving processes that grade XII students use in leaming about vectors in physics. The resuits may help to design more effective strategies for teaching complex concepts to students.
The actual participants will be chosen from students who volunteer for the project. If more than are required volunteer, a research group of six to ten will be chosen randomly from those volunteering. Participation is strictly voluntary and a participant can withdraw from the project at anytirne and return to the regular classroom.
I hope that you will consider being a participant in this project, as I believe that it will be interesting and beneficial to al1 who take part. If you or your parents have any questions, please do not hesitate to contact me at either of the phone numbers below.
This project has the approval of both the Regina Board of Education and the University of Regina.
Yours truly,
Warren Wessel
Phone: 789-7346 (home) 791 -8380 (Campbell)
Proiect Volunteer Form
Please retum this form. onlv if vou wish to participate in the ~roiect.
Please return the form to Mr. Wessel by Friday, February 12, 1993.
Name: (please print)
I have read about the proposed study that Mr. Wessel will be carrying out wlh
my Physics 30 class and would definitely like to be one of the participants in the
study. I have discussed the participation with my parent(s) or guardian(s) and
they agree to my participating if I am chosen fom the volunteen.
(signature)
March 1, 1993
Dear and Parents,
Thank you for volunteering for my research project. You have been chosen and will be one of the participants, if you have not changed your mind since you initially volunteered. We will now definitely begin on the 4th or 5th of March and will continue until about Thursday the 8th of April. During that time you will be taking the class from Mr. Wessel in the libraiy seminar room. Each day you will corne directly to that room.
In order to complete the paperwork of this project you must now have your parents complete the enclosed permission form and return 1 to Mr. Wessel as soon as possible before the project begins. This from must be completed before you can take part in the project.
Again let me thank you for agreeing to take part in this project and let me congratulate you on being chosen to take part. I am looking forward to working with you. I am sure that we will find the study mutually beneficial.
If you or your parents have any questions or concems, please do not hesitate to ask me or cal1 me at one of the numbers Iisted below.
Yours truly,
Warren Wessel
Phone: home 789-7346 S C ~ O O ~ 791 -8380
1 March, 1993
Thank you for volunteering for the project that i am going to run with the 30
Physics class. I appreciate your interest and willingness to take part in it.
Unfortunately the project is not large enough to accommodate al1 those who
volunteered, and some had to be refused. Consequently I will be unable to use
you in the project.
Thank you again for your consideration.
Yours truly,
Warren Wessel
LETER OF CONSENT TO PARTICIPATE IN A RESEARCH PROJECT
Telep hone:
1, , agree to participate in the research project conducted by Warren Wessel at Campbell Collegiate during February to June of 1993 in Grade XII Physics. I give my permission to use any data that is collected during the study. I understand that the data will be used anonymously in written descriptions of the study, and that the data will be archived for a period of five years and then destroyed.
The nature of the project has been fully explained to me and I am satisfied with this explanation and the requirements of the participants.
I also understand that I am free to withdraw from the project at any time that I may choose and retum to the regular physics class.
Student's signature:
ParentGuardian signature:
Date:
Appendix C
Sample Section of Transcript
IV - 93.03.23 (coding identifyng tape on which the session is located)
04:30:00 (time position on the tape) Tchr' - OK. Just listen, listen. Remember yesterday when we were lifting things.
I decided we could do this with a different type of demonstration which will show you the same type of ideas but instead of using something lime. (The previous day students had worked with a mass supported by two spring scales on strings. They had to measure the angles and calculate components of the measured forces. Students yell out the date for the tape record. This has been the manner in which the date is verbally recorded on the tapes. I am uncoiling a piece of light rope and talking with the kids.)
(I am relating a story of my buming hands on a kite string when helping my kids fly a kite. I have tied the rope through the handle on a large pail. The 25 litre plastic pail contains about 15 kg of sand. A rope is tied to the handle so that the rope can not slide through the handle.
Tchr* - Now, I have here a bucket of sand - a couple of thousand pounds actually - No it's only about a third full because that's al1 the sand that's upstairs. (1 move the desks around at the front of the room) Ryan is pretty macho, so we'll Save him for the hard demonstration here.
04:32:03 (time position on the tape) Tchf - I want you to think about yesterday, what we did. A - Sol this is just bigger then. Tchr* - No, it's the same size, you're just closer. A - It's the same thing? Tchr* - Well, it could be. We'll just take a look here. I need to volunteers here to
get up on the desks. A - I will. (She immediately volunteers; others as well.) Tchf - OK, get up on the desk. Go up on the desk. No, no get up there. It's
Ryan I'm going to make a fool out of by proving that he's not as strong as you two girls.
M* - Anne (1 wave to her to corne up. Marie has volunteered as well) Oh great, another volunteer. (1 move the desks around so that the free standing camera can be used for recording this demonstration.
04:33:00 1 made some camera adjustments and adjusts the desks on which they will stand. I comments that I keeps forgetting to consider camera angle when setting up these demonstrations. This is a management concem that is not in regular classrooms. The camera and recording concems were always with me in the research study. After a time the camera changes and organization became more automatic and did not distract so much from instruction.)
Tchr* - Now, each of you gets a rope.
Voice - You have to get up on the desks. Tchr - No, it will be too hard if you do it that way. Get up there, get up there.
God this is great. Look at this. R - You're pretty tall. Tchr - They are, aren't they. OK now lift. Lift it up. M* - No. (But she and Anne do. and the buckle lifts off the ground) Tchr - Was that hard? A* - No. Tchr - Marie, you weren't even lifting. M* - I was so. Tchr - Oh, you were. OK. Was that hard? Can you lift up. Was that easy? A* - Yeah. Tchr - No problems? M* - No problerns. Tchr - Now get down and move the desks apart. Quite a ways apart. Say 4 or 5
metres. Now get back up on them. They have got to be sort of equally apart from this thing today.
A* - OK. Are they equal? Tchr - Yeah, that's close enough, I think Marie's should be a little further away.
OK. Get up there. OK, now lift it. (Anne and Marie lift the bucket, but not as easily) Is that easier or harder?
M* - (laughing) Harder. A lot harder. Tchr - 1s that a lot harder? A - Not much harder. Tchr - Not much harder? OU move the desks further apart still. Another metre
or so. OK, here we go; get 1 (bucket of sand) half way in between. M* - Is that half way? Tchr - Well I don't know, is it? You're the ones that are supposed to decide. R 8 K - Yeah it is. (AI1 students were engaged in fhis demonstration. This is the
first time 1 have used this demonstration in a class and it appears to be a very good demonstration. I will use it in future classes.)
Tchr - Now, get up there. (Ryan moves up into a front desk on his own.) M' - Ryan, what are you doing? R* - Lift. A - It's harder. M* - I can'i get a grip. Tchr - You're not supposed to lift alone, you've got to lift together, you know, in
unison. You know, it's 1, 2, 3, lift. And you have got to pull towards you. 04:36:00 Tchr - Keep it of the ground .... Use some of that macho strength ... Yeah you can
use both hands ... now lift, pull ... pull. Harder still? (Anne and Marie pull harder to lift the pail. The desks are now each about 3 m from the pail so the rope would be about 35O to 40" to the floor). Put it down, now move them out further. (Some rearranging occurs and the desks are now 4 to 5 metres away from
240
the pail. The students will still be standing on the desks allowing the rope angle to be kept high enough so that they should still be able to lift the pail.)
Tchr - Here they are, here they are, ready to pull. (The camera cannot get the entire scene in the lens. I pan both students and the pail to show the set-up.)
Tchr - OK, do the best you can. Pull hard, pull hard. Lean into it. (Marie and Anne exhibit considerable difficulty as they try to lift the pail. Marie has to lean back and tries to hold on to the door behind her in order to apply a greater pulling force. They do manage to lift the bucket off the floor.) OK, you lifted it. Why is it harder?
A - It9s at a different angle. Tchr - OK, now we're going to make it even more difficult. Ryan. M* - Ryan is going to chuck that rock at me. Tchr - Don'? you throw rocks in this class. (Ryan looks guilty). Ryan and Dean
now get the hard one. 1 want you to lift 1 but don? get on the desks. Same distance, way out here.
R* - Whatever. Tchr - OK. Get down off the desks. Tchr - Don? lift. I want you to pull it at waist level. 04:38:00
(Ryan is pulling very hard. He raises his hands high to about eye level. lntuitively he knows that he must do this to lift the pail of sand.)
Tchr - No, no you're lifting your hands. Why do you want to lift your hands? R* - Because it's easier. Tchr - Why?
(Some additional attempts to lift the pail are made. The pail moves about, but when the force is applied at waist level - an angle of less that 1 O0 - a considerable total force is required to lift the pail.)
Tchr - Why does it rnake a difference? R* - Because the higher it is ... Tchr - Oh, because the higher it is ... what? M* - It's easier to pull. A - The angle is greater. K - It's a smaller angle. Tchr - But, why?
(Many suggestions as to why lifting is harder corne rapidly from the class. None describe accurately the reason for greater force to be applied.)
V's - (replies to garbled to be understood.) Tchr - But why? You automatically know that you want to lift, you want to get
your hands up this high. But why does that make it better? R* - They had it up here. Tchr - They were able to pull it a different angle. R* - Yeah, but that was up here. Tchr - They were at a different angle. But, low angle is hard, eh? Zero angle,
what if Ït were right at zero angle?
K - Then you can't do it. V - It's impossible. Tchr - You can't lift it at zero angle? 04:39:00 Tchr - What about as it goes up? OK now, you can leave the bucket for now. Tchr - Now can you, and this is the whole point, can you describe that sort of ...
why that is the case? Remember there is obviously an angle. You are pulling with a certain force, right. you are pulling with a certain force and yet when the angle goes down you have to pull with a greater force to lift it. That does seem strange because. Does the weight change?
K - No. It's harder to counteract the gravity. Tchr - It's harder to counteract the gravity, so the gravity must be getting bigger,
is that right? K - No. It's staying the same. (camera switch (CS) to Kevin) Tchr - No, it's staying the same .... Why is it gravity doesn't change, the weight of
the bucket doesn't change. What's happening? J* - You're pulling it straight up instead of ... like ... Tchr - I (was) pulling on the rope straight up. J* - You're pulling your force directly opposite to the force that's being pulled on
it, that's pulling it to the side. (Judy shows some signs of knowing her answer is "weak" and not clearly stated. She smiles and shrugs her shoulders.)
Tchr - Well, no, no, no, no, I'm not saying it's wrong. Can you be clearer on that?
J* - No. Tchr - What do you think Todd? T - I'm not thinking ... I don't know. K - Because in order to lift the bucket (CS to Kevin) No. (Hides his head doesn't
like the camera directly on him.) You need a certain amount of ah ... like, weight.
04:41 :O0 R - Get your hand down Kevin. K* - You need a certain amount of like force to lift it up, and when you are
pulling out the force is going out instead of up. Tchr - Then why does it go up at all? K' - What do you mean? Tchr - Well it went up.
(Some comments about the camera and the fact that it makes you feel self conscious. I atternpt to dissuade them of their self-consciousness. The records are required for the project.)
Tchr - Let's get back to the forces though. You know that you are pulling on the rope, but.. .
K - Because the forces are going out instead of up. Tchr - OK, so what? K - So in order lift that up, you need to pull up.
242
C4:42:00 Tchr - OK, let's do it again, talk to the group. In order to pull the bucket up you
have to do what? K* - We have to pull. There has to be a force, I don't know if it's equal ... or
greater than the weight of the bucket going like up. Tchr - OK. K* - And if you are pulling straight frorn the sides, you're not pulling it up.
(Some antics by all. Kevin is rubbing his eyes, etc. Others giggle) Tchr - But if you are then pulling on the ... if you're pulling on the thing at an
angle why does it lift at all? J* - Because you're using a greater force. Tchr - Yeah, you're using a greater force, but the direction of the pull isn't in the
lifting direction, it's off at some angle. In fact, there's two forces off at some angle. (I refer here to the two people pulling wlh a force on the bucket.) Why does it lift at all?
K - Because you are still lifting 1 up a bit. 04:43:00 K - You are lifting it like up and out not just straight out. Tchr - You are lifting it like up and out not just straight out. K - Just use that vector math stuff. Tchr - What do you think Lisa? L* - I think that when you are pulling out at the angle that the gravity's pulling it
straight down, so that the angles are going out like this. but the gravity's pulling straight down, that's why it takes a greater force to pull it that way. (She uses hand motions to indicate directions)
Tchr - So, Î t takes more force, and yet does gravity get stronger? L* - No, it's going straight down. It's not going out to the sides. Tchr - OK, so you're pulling with a tremendous amount of force on that rope
going, I mean, these guys seem to (work hard) K - You need to give an exact opposite force. Tchr - To what? J - Gravity K - To lift it. (CS - still netvous about being on camera) Tchr - But if it's lifting, I don? understand where the force is going then ... I don't
understand where the force is going that you put on the rope because you apply far more force than, I mean, these two guys are straining really hard at a low level to get it to raise it off the ground, yet when you when you stood right over it, it was very simple to rnove it. I mean, Marie and Anne had no trouble at al1 lifting it up. But when you get out to the side you are pulling with a tremendous force and it doesn't lift it off the ground or you're just barely able to get off the ground and you're staggering around. So where's al1 that extra force go?
J* - Part of your force is going from you pulling this way and the other person pulling .
K - It's balancing off.
J* - Part of it is. Yeah, part of i fs balancing off that way, and part of i fs pulling like going opposite gravity
04:45:00 Tchr - So part of it's pulling in what direction? J* - Towards each of the people that are pulling. Tchr - Towards each of the people that are pulling and those two pieces that are
pulling out that way ... J* - Balance off. Tchr - Balance off, and what about the parts that don't? J* - The gravity's pulling down Tchr - Right J' - And the extra force is pulling up. Tchr - So the other piece of the force is pulling up. OK, now is there some thing
that we've talked about in vectors so far that might help us to examine that situation? . . . See, rernember that whenever we're doing these things I want you go back always to the idea of mathematical models. Always go back to the idea that we want to try to explain ... this is a strange phenomenon here .... I mean, it isn't strange because practically you know irnmediately. As soon as Ryan and Marie tried to lift and couldn't at the waist, they Iifted the ropes up, automatically. A very practical application of force, I mean they didn't even hesitate, they just sort of automatically (thought) I'm going to lift it from a higher point. The low angle is impossible. You get down to just at zero angle and you know that it doesn't work at all. OK, but how can we use the idea of vectors to explain that a little more clearly. And that's really what we're trying to do. We're trying to get a better picture and then and maybe calculate how much force is required to pull those things out. And then what we'll do is I have a couple of stands here. We'll set up a couple of smaller systems, a little more laboratory things, because this is hard (the bucket system) because I don't have any springs (scales) that will measure that. But you wanted to Say what about that force? Al1 right, the one that you are pulling on the rope, you guys are saying that it's part this way and part vertical
04:47:00 R' - Well, these ... the ones opposite each other. J' - Yeah. R* - Balance each other out and then Tchr - OK, the ones opposite each other'? J* - But when you're pulling at the angle you're pulling here and you're pulling
here. That's where your force is. Tchr - So what two forces could you pull with that would give you the same
result? Supposing that I put four ropes on that, two horizontal and and two vertical; could you make it do the same thing?
J* Yeah. Tchr - How? R* - As long as the forces were equal.
Tchr - What would you do? J* - Cut the force in half ... half the force going up that way and half the force
going up that way (indicates the directions with her hands). (es to front board. Tchr now tries to draw the situation on the board. He begins with a real worid diagram.)
Tchr* - Here's Our bucket. It has sand in it, probably about 10 kg's worth or something like that. The handles on the bucket like this. And of course when you lift it this way with the two ropes you are al1 able to do that quite easily (direct vertical lift), because it isn't al1 that heavy. Right. It isn't al1 that difficult. But when we put the ropes on this thing out to here (A knock at the door. I go to answer. Some teacher wants to use the fax machine. He is concemed that this may be telecast and that he is interrupting. I tell him that he is not. Generally 1 wanted the project to be part of the school as much as possible. In normal situations 1 would have allowed this interruption. so 1 did so in this case.)
04:49:00 Tchr' - OK, there's Our ropes, and here is somebody up here. R - Marie. Tchr* - Pulling on it (rope) standing on a table. M - I have hair you know. Tchr* - What's that? M - I have hair. Tchr* - Oh, OK. (Puts hair on the diagram.) That's a moustache. (Class laughs)
I shouldn't have done that. (on with the diagram) I have to have different color of hair, because her hairs a different color than yours. isn't it? We'll use blue, it's sort of a nice color. She's happy a smile (on diagram). Another table. These tables have to corne down a bit. (tu the same level as the bucket)
Tchr" - That's al1 we need. Now, you're telling me ... let us, let us suggest that they are pulling ... This is 10 kg, so what force would it take to lift it? Let's Say it's 10 kg. (writes mass on the bucket) So what force will it actually take to lift it? (Tchr wants to calculate 'reasonable" forces for the diagram) How much force will it take to lift it?
T - 10 N, is it wrong? Tchr* - Yeah you're wrong, but why did you pick 10 N? T - Because it's equal to the weight. Tchr" - It is equal to the weight, but not to the mass. 04:s 1 :O0 Tchr* - The force of gravity, Fg is, we've done this before, we need to do it
again. Tchr* - 9.8, but mg, remember. mg and this is 10 kg times 9.8 and you can
either put a couple of set of units but the most, the best one is Nkg, so this is about 100 N. So you need a 100 N to lift it. OK, that's the weight. Are you close - no not really. So let's assume that they are pulling up here with 250 N force. OK. Let's assume that these people are both pulling with 250 N
force and it has to be, and it had to be pretty balanced. Tchr' - It has to be pretty balanced. And that just las, OK so it just lift this .... It just
lifts the thing. (Some concem about the recording of the sound is expressed. The concem is resolved.)
Tchr* - OK, two people lifting. What did you tell me about the forces? 04:53:00
(1 tie the ropes of the bucket to the light fixtures in the room to create a real world situation for the diagram. Students comment about the set up to Tchr, who talks to them establishing rapport.)
Tchr - OK, now, look. We're saying ... you guys know you pull on this rope. And what you people seem to be interpreting is that
04:55:00 Tchr* - OK, wnat you seern to be saying is that on this rope, there's ... you said
there's a force kind of to the side and a forc9 up. Now that's kind of strange, because when I pull on the rope I don't pull to the side and up I just pull in this direction here. I just pull on the rope. So what? (garbled) I'm not saying you're wrong but what are we kind of pretending there? That's a nice word, what are we pretending ?
A* - That we (garbled) Tchr - That we (laughter from class) See, because you've got the right idea, I
don? question that the idea that you've got some force going to the side and some force going up, but how's that possible? What did I talk about a couple of days ago when you saw a force and you drew something at right angles?
Tchr* - (CS) Remember when I was talking about - we have velocity - (draws a V vector) - we and some vector like this perhaps. It could be any vector A - it doesn't have to be ... Then remember I drew sornething that you (garbled) could break it into two forces.
04:57:00 Tchr* - And I think that I called this, Ah, in the horizontal and this, A", in the
vertical. What were those called? What was it called when you do this? (long pause) Are you awake K?
K - Yeah. Tchr* - Yeah, what was it called when we do this? What were these two vectors
called of this vector? (5 second pause) Somewhere in there I mentioned the word. (1 write compon on the board)
K- Components. M - Components. Tchr - Sure they were the components of vector A, but what does that mean? J - They make up vector A. Tchi" - Do they make it up? L - They would. Tchr* - Are they really here? 1s this ... Remember the force is pulled along this
rope. OK, the force is pulled - you see the rope right here. It's pulled in this direction, up. Only up. Are they (the components) really there? Is there really a force down here and one here?
K - Yeah. C - Yes. Tchr* - An interesting question. Then how come I don't run into them and get
sucked up (by the up force in the components)? Or if I step into this one I should get pulled this way. How come that doesn't happen?
J - Because it just doesn't. K - Because they're strange gases. Tchr" - Because they are strange gases. Now that sure answers the question. If
you do it with this diagram over here. (CS)
04:59:00 Tchr* - What we would be saying then is that there is a horizontal force like so,
right, force horizontal, Fh. And a vertical force like so, F,. OK, that's the same, but are those forces really aiere, or is it, and this is kind of important I think ... are the forces really there, or is this another of those mathematical tricks that makes the calculations possible?
Voice - A math trick. Tchi* - You think it's a math trick. K - I think they're actually there. Tchr - Why do you aiink it's a math trick, Judy? J* - Well they're not actually there, but like the 250 N is your total.
Appendix D
Vector Quiz
Chapter 3 Quiz - Vectors
1. A cat and a dog are pulling on opposite ends of a rope. The cat is pulling on one end with a force of 78 N. The dog is pulling on the other end of the rope with a force of 103 N.
a) Draw a "real life" diaaram and vector diagram of the two forces showing the two forces and the resultant sum (or net) of the two forces. This is a one dimensional problem. There should be no triangles in the drawings.
b) What is the resultant (or net) force in this situation? (magnitude and direction)
2. A boat can travel at a velocity of 25 k d h using the power of its engine and travelling on water with no current. The boat is moving upstream (against the current). The current in this river is 8.5 k m .
a) Draw a "real Iife" diagram and a vector diaaram showing the situation in this problem. (This is a one-dimensional problem, there should be no triangles in the diagram).
b) What is the resultant velocity of the boat as it travels uostream?
c) After 30 minutes of travel, what is the displacement of the boa??
3. Vectors have and
Scalars have but no
4. A car is travelling at 45 km/h [SI. Then it turns and goes 75 kmBi [El. Calculate the chanae in velocity that has occurred in this situation. Draw both diagrams here.
5. A jogger runs 2000 m [NI. She then tums and goes 1500 m [S 60" El. Calculate her total dis~iacement. This is the total displacement from her starting point. Remember that displacement is a vector.
Appendix E
Summary of Difficulties Exhibited by the Participants
250
Summary of Difficulties Exhibited by the Participants
The students had to leam how to become active leamers and why this approach to learning would be personaliy valuable. They had to discover the value of a non-threatening cooperative leaming environment.
They had diffîculty understanding the structure of physics; in particular, the roles of theory in physics and mathematics in creating theoretical models. The participants did not distinguish algorithms from theory in physics.
The participants had difficulty in moving beyond employing algorithms to solve problems and seeing the underiying physics principles which provided the pathway to solution of problems.
These students made essentially no attempt to estirnate answers to problems as part of their problem solving process, used formulae without discretion and did not evaluate their solution to see if it were a reasonable answer to the problem.
The participants struggled to represent personal experience of physics phenomena mathematically even when they knew the mathematics that were required.
They had difficulty making connections between physics principles and events in their everyday lives. Laboratory experiences did not appear to provide as much assistance in developing these interconnections as was traditionally beiieved.
The students confused the ternis egual and baianced when required to use them to write vector equations to describe relationships between vector components of forces.
They had difficulty distinguishing vector addition and subtraction frorn algebraic addition and subtraction. On several occasions students added the magnitudes of the vector quantities and ignored the direction of the vectors altogether. Knowledge of vector mathematics developed in separate mathematics classes was not effectively applied to physics problems.
IMAGE EVALUATION TEST TARGET (QA-3)
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