developing critical thinking through probability models

Ben-‐Gurion University of the Negev

Developing Critical Thinking through Probability Models, Intuitive Judgments and Decision-Making

Under Uncertainty

Thesis submitted in partial fulfillment of the requirements for the degree of

“DOCTOR OF PHILOSOPHY”

by

Einav Aizikovitsh-Udi

Submitted to the Senate of Ben-Gurion University of the Negev

October 2010 Beer Sheva

Developing Critical Thinking through Probability

Models, Intuitive Judgments and Decision-Making under

Uncertainty

Thesis submitted in partial fulfillment of the requirements for the degree of

“DOCTOR OF PHILOSOPHY”

by

Einav Aizikovitsh-Udi

under the supervision of Prof. Miriam Amit

Submitted to the Senate of Ben-Gurion University of the Negev

Approved by the advisor ________________________ Date _________

Approved by the Dean of the Kreitman School of Advanced Graduate Studies ____________

September 2010

Beer-Sheva

II

This work was carried out under the supervision of Prof. Miriam Amit

in the Department of Science Education the Faculty of Humanities and Social Sciences.

Acknowledgments

My deepest gratitude is due to Prof. Miriam Amit who fostered my confidence and opened the wonderful world of thinking for me. Many many thanks for the engaging and uncompromising mentorship, for the ENDLESS encouragement, empathy, and great generosity!

I thank the ISEF foundation for the generous financial support that enabled me to carry out this research.

Thanks to Dr. Olga Kuminova for the English editing and the great patience she had for my interminable questions.

Thanks to Mrs. Yeti Varon for the meticulous and extremely professional statistical analyses and the great interest she showed in my work.

Thanks to Mrs. Tirtsa Kauders for her assistance with finding materials and editing my bibliography with the greatest attention and care.

Thanks to Dr. Assaf Marom for your steadfast friendship and the supportive and helpful interest you have taken in my work as a researcher.

Thanks to each of the research participants, teachers and students, for giving their time and their thoughtful participation to this study.

Thanks to all the generous colleagues who read, discussed and gave invaluable feedback on my work.

And finally, I want with all my heart to thank my dearest Ziki, for being a wonderful, unique partner and perfect fellow-‐traveler throughout my long academic journey and our beautiful way. Thank you so much for being you!

II

With love, to our wonderful Yami, Shiri and Tuli

III

Developing Critical Thinking through Probability Models, Intuitive Judgments and Decision-Making under Uncertainty

Table of Contents

Abstract .......................................................................................................................................... i

1. Introduction ................................................................................................................................ i 1.1 Statement of the problem…………………………………………………………………..i

1.2 Rationale and Motivation .................................................................................................... ii 1.3 Uniqueness and Contribution of This Research ................................................................. iii

2. Theoretical Background ............................................................................................................ 5 2.1 Critical Thinking: An Overview .......................................................................................... 5

2.2 Theory of Critical Thinking ................................................................................................ 7 2.3 The Defining Components of the Theoretical Framework ............................................... 17

2.4 Contexts of Critical Thinking ............................................................................................ 20 2.5 The Critical Thinking Movement ...................................................................................... 25

3. Research Method ..................................................................................................................... 27 3.1 The Research Purpose ....................................................................................................... 27

3.2 The Research Questions .................................................................................................... 27 3.3 The Choice of Mixed Methods .......................................................................................... 27

3.4 “Working on the Inside”: The Teacher as a Researcher ................................................... 28 3.5 Stages of the Research ....................................................................................................... 29

3.6 Research Population .......................................................................................................... 30 3.7 Research Instruments ........................................................................................................ 31 3.8 Pilot Study ......................................................................................................................... 36

3.9 Summary of Research Description .................................................................................... 27

4. The Intervention: The Learning Unit “Probability in Daily Life” .......................................... 38

4.1 The Learning Unit "Probability in Daily Life" .................................................................. 39 4.2 Our Intervention ................................................................................................................ 40

5. Dispositions of Critical Thinking ............................................................................................ 52 5.1 The Research Question ...................................................................................................... 52

IV

5.2 Method ............................................................................................................................... 52 5.3 Results of Dispositions ...................................................................................................... 55

5.4 Discussion of Critical Thinking Dispostions ..................................................................... 85

6. Abilities of Critical Thinking .................................................................................................. 71

6.1 Research Question ............................................................................................................. 71 6.2 Method ............................................................................................................................... 71

6.3 Results of Abilities ........................................................................................................... 76 6.4 Discussion of Critical Thinking Abilities .......................................................................... 85

7. Construction of Critical Thinking Skills ................................................................................. 88 7.1 Research Question ............................................................................................................. 88

7.2 Method ............................................................................................................................... 88 7.3 Results ............................................................................................................................... 95

7.4 Qualitative Findings ........................................................................................................ 102

8. General Discussion and Conclusions .................................................................................... 109

8.1 The Research ................................................................................................................... 109 8.2 General Discussion in Light of Research Questions and CT Literature ......................... 109

8.3 Other Points for Discussion Derived from Research Findings ....................................... 110

9. Research Contribution and Implications ............................................................................... 117

9.1 Review of Principal Findings .......................................................................................... 117 9.2 Conclusions ..................................................................................................................... 117

9.3 Recommendations ........................................................................................................... 118 9.4 Limitations ....................................................................................................................... 119

9.5 Research Uniqueness and Contribution .......................................................................... 120 9.6 Recommendations for Future Research and Concluding Remarks ................................. 122

Appendices ................................................................................................................................ 136 Appendix 1: CCTDI A Disposition Inventory

Appendix 2: Abilities Cornell Critical Thinking Test, Level Z

Appendix 3: Critical Thinking Questionnaire: “Probability in Daily Life”

Appendix 4: Mathematics Questionnaire

Appendix 5: The Learning Unit “Probability in Daily Life”


Appendix 7: Sample Problems and Exams from the Course “Probability in Daily Life”

V

List of Tables, Charts and Figures

Table 1: Research population distribution ................................................................................... 31 Table 2: Research population distribution each Test .................................................................. 32 Table 3: Research questionnaires by type group. ........................................................................ 33 Table 4: Stages of the pilot study: goals, tools, population, data collection methods ................. 36 Table 5: Stages of the proposed research.………………………………………………………37 Table 6: Classroom discussion of an article and the infusion of CT skills ................................. 49 Table 7: Scale of critical thinking disposition by Facione .......................................................... 54 Table 8: Number of students each round ..................................................................................... 55 Table 9: Disposition of CT in the “Kidumatica” group .............................................................. 56 Table 10: Disposition towards critical thinking in the HighSchool 1 ......................................... 57 Table 11: CCTDI Total statistical tests results ............................................................................ 58 Table 12: Truth-Seeking sub-scale statistical tests results .......................................................... 59 Table 13: Open-Mindedness sub-scale statistical tests results .................................................... 60 Table 14: Inquisitiveness sub-scale statistical tests results ......................................................... 61 Table 15: Systematicity sub-scale statistical tests results ........................................................... 62 Table 16: Maturity sub-scale: statistical tests results .................................................................. 63 Table 17: Confidence sub-scale statistical tests results ............................................................... 64 Table 18: Analyticity sub-scale statistical tests results ............................................................... 65 Table 19: Classification of items by aspect of thinking in Cornell ............................................. 73 Table 20: Number of ducks according to the different menus .................................................... 75 Table 21: Numbers of students each round ................................................................................. 76 Table 22: Critical Thinking abilities in the “Kidumatica” group ................................................ 77 Table 23: CT abilities in the High School 1 group ...................................................................... 77 Table 24: CTI Total statistical tests results ................................................................................ 78 Table 25: Induction Sub-Scale statistical tests results ................................................................. 79 Table 26: Deduction Sub-Scale statistical tests results ............................................................... 80 Table 27: Observation Sub-Scale statistical tests results ............................................................ 81 Table 28: Assumptions Sub-Scale statistical tests results .......................................................... 82 Table 29: Meaning Sub-Scale statistical tests results .................................................................. 83 Table 30: Construction of Critical Thinking skill ....................................................................... 89 Table 31: Example of Questions and analyses ............................................................................ 93 Table 32: Dispositions toward critical thinking .......................................................................... 93 Table 33: Statistical tests of differences for High School 1 ........................................................ 93 Table 34: Statistical tests of differences for Kidumatica group .................................................. 96 Chart 1: Disposition of CT “Kidumatica”……………………………………………………… 56 Chart 2: Dispositions for CT "High-school"…………………………………………………... 57 Chart 3: CCTDI Total Means………………………………………………………………….. 58 Chart 4: Truth-Seeking sub-scale Means………………………………………………………. 59 Chart 5: Open-Mindedness sub-scale Means…………………………………………………... 60 Chart 6: Inquisitiveness sub-scale Means……………………………………………………… 61 Chart 7: Systematicity sub-scale Means……………………………………………………….. 62

VI

Chart 8: Maturity sub-scale Means…………………………………………………………….. 63 Chart 9: Confidence sub-scale Means………………………………………………………….. 64 Chart 10: Analyticity sub-scale Means………………………………………………………… 65 Chart 11: Total Dispositions during all the Research………………………………………….. 67 Chart 12: Abilities of CT Kidumatica………………………………………………………….. 76 Chart 13: Abilities of CT High School………………………………………………………… 77 Chart 14: CTI Total Means…………………………………………………………………….. 78 Chart 15: Induction Sub-Scale Means………………………………………………………….. 79 Chart 16: Deduction Sub-Scale Means………………………………………………………… 80 Chart 17: Observation Sub-Scale Means………………………………………………………. 81 Chart 18: Assumptions Sub-Scale Means……………………………………………………… 82 Chart 19: Meaning Sub-Scale Means…………………………………………………………... 83 Chart 20: Total Abilities during all the Research………………………………………………. 84 Chart 21: Skill (a) Identifying Variables………………………………………………………. 96 Chart 22: Skill (b) Referring to Sources……………………………………………………….. 97 Chart 23: Skill (c) Identifying conclusions…………………………………………………….. 98 Chart 24: Skill (d,e,f,g) Evaluating , Suspending , Offering…………………………………..100

VII

List of Abbreviations

ACT Abilities toward Critical Thinking CCTDI California Critical Thinking Inventory COT Cornell Test CT Critical Thinking DCT Dispositions toward Critical Thinking H1 High School 1- 1st Round H2 High School 2- 2nd Round HOTS High Order Thinking Skills Iter1 1st Round- preliminary research Iter2 2nd Round- secondary research KD1 Kidumatika Group-1st Round KD2 Kidumatika Group- 2nd Round PIDL Probability in Daily Life StA-Ass Sub-test Assumption StA-dud Sub-test Deduction StA-Ind Sub-test Induction StA-M Sub-test Meaning StA-Ob Sub-test Observation StD -T Sub-test Truth-seeking StD-A Sub-test Analyticity StD-Inq Sub-test Inquisitiveness. StD-Mat Sub-test Maturity StD-O Sub-test Open-mindedness StD-Sc Sub-test Self-confidence

Abstract

i

Abstract In light of the importance of developing critical thinking, and given the scarcity of

research on critical thinking in mathematics education in the broader context of higher-

order thinking skills, we have carried out a research that examined how teaching

strategies oriented towards developing higher-order thinking skills influenced the

students’ critical thinking abilities. The guiding rationale of the work was that such

teaching can foster the students’ skills of and dispositions towards critical thinking. In

this research, a primary attempt has been made to examine the relations between

education for critical thinking and mathematics education through examining teaching

and learning critical thinking according to the infusion approach, which combines critical

thinking and mathematical content, in this case, “Probability in Daily Life.”

The purpose of this research was to examine how and to what extent it is possible to

develop critical thinking by means of the learning unit “Probability in Daily Life” using

the infusion approach. The research questions were:

(1) To what extent does the study of “Probability in Daily Life” in the infusion approach

contribute to the development of critical thinking dispositions?

(2) To what extent does the study of “Probability in Daily Life” in the infusion approach

contribute to the development of critical thinking abilities?

(3) What are the processes of construction of critical thinking skills (e.g., identifying

variables, postponing judgment, referring to sources, searching for alternatives) during

the study of the “Probability in Daily Life” learning unit in the infusion approach?

The present research involved nine groups of gifted and high-achieving mathematics

students in eleventh grade from all the social groups and strata of Israeli society. The

students studied the learning unit “Probability in Daily Life” modified by the researchers

to include critical thinking teaching in the infusion approach. The research combines

quantitative and qualitative methods: on the one hand, the students took two critical

thinking tests, CCTDI and the Cornell Critical Thinking Test, the results of which were

statistically analyzed. On the other hand, the students were selectively interviewed, with

subsequent qualitative analysis of the interviews and lesson transcripts. Mixed method

was chosen in order to achieve deeper insight into the data and strengthen the validity of

the results. The research findings can be summed up in the following categories:

Abstract

ii

(i) In all three rounds of experimental teaching, a moderate improvement was detected in

the critical thinking dispositions of all experimental groups. (ii) Throughout these rounds,

a moderate improvement was also detected in the students' critical thinking abilities. (iii)

Teaching critical thinking contributed to the construction and use of these skills in the

framework of mathematics. Thus, when teachers consistently emphasize critical thinking

skills, the students are more likely to succeed in the subject of mathematics. (iv) This

research did not detect a clear-cut distinction between the critical thinking abilities and

dispositions of excellent and average mathematics students. That is, no direct correlation

has been found between the development of mathematical knowledge and the

development of critical thinking.

The main contribution of this work is the connection it elucidates between critical

thinking and the study of mathematics and new insights it provides into the mechanisms

of critical thinking development, and its place and importance in mathematics education,

in spite of “the transfer problem” relating to the students' ability to apply critical thinking

skills learned in one discipline to other academic subjects and areas of life. This research

uncovers a potential for strengthening the status of mathematics studies in imparting

higher-order thinking skills in various frameworks, in parallel with and beyond the

formal program of studies. To conclude, critical thinking within the framework of

mathematics education does not develop spontaneously but requires effort. It is not

algorithmic, i.e. its patterns of thinking and action are not clear or predefined. Critical

thinking skills rely on self-regulation of the thinking processes, construction of meaning,

and detection of patterns in supposedly disorganized structures. Critical thinking tends to

be complex and often terminates in multiple solutions that have advantages and

disadvantages, rather than a single clear solution. It requires the use of multiple,

sometimes mutually contradictory criteria, and frequently concludes with uncertainty.

Conventional teaching is not appropriate for the changing and challenging world we live

in, which demands critical/evaluative thinking based on rational thinking processes and

decisions. In this research we find that combining different instruction strategies, such as

asking questions, independent investigation of phenomena, or experimenting in the

framework of open discussion and drawing conclusions considerably improves the

students' critical thinking abilities and dispositions. These findings are consistent with

those of earlier studies showing that critical thinking relies on cognitive activity directed

at focused, inquisitive interpretation of relevant information.

Introduction

i

1. Introduction “What makes a child gifted and talented may not always be good grades in school, but a different way of looking at the world and

learning” (Senator Chuck Grassley)

1.1 Statement of the problem

It definitely seems that in the last decade, there has been a rapidly growing awareness of

the importance of promoting the development of thinking skills in the Israeli educational

system, and the system has been making considerable progress towards integrating the

curriculum learning materials that contribute to the development of higher-order thinking

skills1. In 1994, the Ministry of Education recognized thinking skills as a distinct subject

of studies. This recognition lead to the establishment of a Subject Committee for

Thinking Skills, which is in charge of consolidating appropriate didactic materials, as is

the case with the rest of the academic subjects in the school system. The complex and ceaselessly changing contemporary reality, which requires independent

decision-making on a daily basis, makes it extremely important to impart to students the

ability to think critically. Critical thinking is needed in every field of activity, as it allows

the individual to deal with reality in a reasonable, mature and independent way

(Lipmann, 1991). The need for developing critical thinking in different disciplines is

anchored in the ideals of education for democracy, as our freedom to think about and

criticize the reality and society in which we live is a form of expression of our autonomy

as individuals. Today this idea is even more vital, because of the growing need to be

capable of engaging in inquiry and evaluation based on rational considerations regarding

the various messages we are exposed to in different areas of life (Feuerstein, 2002,

Perkins, 1992, Swartz, 1992).

In the field of education, mathematics has traditionally been considered a branch of

knowledge particularly suited to the teaching and learning of higher-order thinking skills,

such as critical thinking. Mathematics curricula all over the world, including Israel,

identify the acquisition of these skills as one of their goals. The idea that mathematics is a

discipline suited to teaching critical thinking also appears in the research literature2.

However, in spite of this assumption, very few empirical studies to date have engaged

with the question of whether the study of mathematics indeed develops or even requires

1 Critical, deductive, creative, inventive and other types of thinking: on the interconnections between different types of thinking. 2 A number of articles on critical thinking in mathematics use the term in other contexts and deal with imparting technical tools such as making an estimation, comparison or inference, verifying a result, evaluating an exercise, application and interpretation, solution strategies etc.

Introduction

ii

this mode of thinking. The answer to this question is far from being clear. The present

research tackles precisely this basic question, “Is it possible to develop critical thinking in

the framework of mathematics studies?”

1.2 Rationale and Motivation

In light of the great change that has recently taken place in the status of knowledge, and

for the sake of fulfilling the school’s true purpose, many researchers emphasize education

for thinking. Perkins (1992) emphasizes the need for fostering thinking as a means of

understanding the acquired knowledge. He claims that many students graduate from

school with what he terms “Fragile Knowledge Syndrome3”, having acquired knowledge

that they cannot make sense of or apply, and the reason for this is that the students are not

involved in thinking about the topics.

New developments in science, technology, economy, society and culture require far-

reaching changes in the educational process. These changes should find expression in

occasional reconsideration of school curricula and paying adequate attention to the

development of the human mind. The development and improvement of the mind,

through imparting higher-order thinking and learning skills that are not acquired routinely

or develop on their own, is especially important in the education system. There is a

consensus today among researchers and educators about the importance of not only

imparting information, but also taking the students’ thinking to the level of mastering

various modes of higher-order thinking. In the last two decades, the need for changing the

old traditional methods of teaching has received international acceptance. Textbook-

based, rote learning came to be considered less valuable than exposing pupils to varied

experiences and allowing them to actively construct their knowledge. In Israel,

developing a range of cognitive skills is considered to be of the utmost importance, and

school curricula increasingly incorporate tasks requiring higher-order thinking modes,

such as critical, deductive, creative and inventive thinking.

One excellent example of mind development is fostering critical thinking. Critical

thinking is a crucial area in educating the next generation and, in our opinion, comprises

an integral part of general school studies4. Development of critical thinking is gradually

becoming an agreed-upon educational goal, which is assigned the highest importance in 3 Perkins, Smart Schools: Better Thinking and Learning for Every Child (1992). 4 See researches assigning a great importance to imparting “critical thinking” in various fields in Chapter 2.

Introduction

iii

general school studies, and particularly in the study of mathematics. In the real world, we

constantly need to make personal decisions based on complex situations. In our modern

age, rational judgment is vital in order to process information we face on a daily basis

(Feuerstein, 2002, Perkins 1992, Swartz, 1992). Hence it is extremely important to instill

in our students the ability to think critically. Critical thinking is used in every profession,

and it allows people to deal with reality in a reasonable and independent manner

(Lipman, 1991). It is also indispensable for educating critically thinking citizens in a

democratic society. To sum up, teaching critical thinking is crucial if we want to prepare

our students for life and not merely for their matriculation exams, if we seriously intend

to help them learn how to transform their knowledge and abilities into positive,

responsible actions (Zoller, 1999, 2001). In education, mathematics has been considered

a field of thought which is particularly suitable for promoting higher-order thinking

skills, including critical thinking. There is a strong claim that the fields of mathematics

and science are perfectly suited to teaching critical thinking in high school. However, in

order to develop and foster critical thinking, we need first to define it and to understand

the mental processes it involves.

Previous research has investigated the ways in which students acquire technical tools

such as evaluation, verifying results, assessing problems, making comparisons and

conclusions, choosing solution strategies etc.. Our study, by contrast, focuses on the more

general and universal aspects of critical thinking, investigating the ways in which

students develop abilities, such as induction, deduction, value judging, observation,

checking the sources’ reliability, identifying assumptions, and extracting meaning, and

dispositions, such as truth-seeking, open-mindedness and inquisitiveness, according to

the taxonomy of Ennis that we will elaborate on later. The purpose of our research is to

determine whether critical thinking abilities and dispositions can be developed through

the study of probability.

1.3 Uniqueness and Contribution of This Research

Educators in Israel, who wonder, like their colleagues worldwide, about the goals of the

education system that could guide the different educational frameworks, may find in this

research an idea that can unify different topics and study programs, in order to prepare

learners for life in a changing society, and develop their ability to think in a systematic

and independent way. More generally, this research is expected to contribute to the public

discourse of the mathematical education community on this issue.

Introduction

iv

It raises the public awareness of the need to develop critical thinking in the framework of

mathematical education, which may enable future examination and promotion of the

development of critical thinking through mathematics teaching in a fuller and more

informed way.

Drawing on infusion-approach study of "Probability in Daily Life," the present research

establishes points of reference to critical thinking dispositions and abilities among

students learning mathematics in different environments (high school and the

Kidumatica mathematics club). This combination has not been examined so far by the

literature in the field. This research has identified and measured differences between

dispositions, abilities, and construction of skills characteristic of critical thinking in

mathematics, and completes other researchers conducted in other environments. The

combination of the Cornell test and the CCTDI test in the evaluation of critical thinking

abilities and dispositions is unique to this research; it has not been performed in previous

studies.

To conclude, the main contribution of this research lies in revealing the connection

between critical thinking and the teaching of mathematics. Despite the problem of

transfer discussed earlier, the scientific contribution of this research lies in the new

insights it provides into critical thinking, its place and importance in teaching

mathematics. Thus it will be possible to strengthen the status of the study of mathematics

in imparting higher-order thinking skills, both in parallel with and beyond the formal

education program.

Theoretical Background

5

2. Theoretical Background “Learning without thinking is a wasted blessing" (Confucius)

The present research examines assumptions about the connection between mathematical

studies (specifically, the “Probability in Daily Life” learning unit) and the development of

critical thinking. In this chapter, while discussing research literature, I will focus on two

central questions: what is critical thinking, and what is the place of teaching mathematics

(specifically, “Probability in Daily Life”) in developing this form of thinking. Remarkably,

in spite of the discussion that has been going on since antiquity, about the contribution of

mathematics to the development of critical thinking, and about teaching critical thinking

through mathematical practice, almost no empirical studies have been carried out on this

issue so far5.

2.1 Critical Thinking: An Overview Critical thinking is a topic that has interested humans since ancient times. Development

of critical thinking has been defined as one of the most important goals of education since

the Middle Ages until today. The ancient ‘fathers’ of the idea of critical thinking are

considered to be the sophists6 (740-399 BC) and Socrates (5th c. BC). Socrates, who was

and still remains an extremely influential philosophical figure, dealt mainly with the

theory of ethics and the issues of governing society and the state. Walking the streets of

Athens, he approached people with questions about the nature of the world. In order to

understand their opinion on a certain issue, he first had to clarify their definition of that

issue and whether that definition was true. He was a person who thought independently,

and taught others to think for themselves. Therefore, if one wanted to be a disciple of

Socrates, one would have to think independently, and if necessary, to be able to detach

oneself from previously known and generally accepted ideas and definitions (Bryan,

1987). Socrates is known to have resisted the greatest cultural innovation of his time – the

writing of books. He claimed that writing on parchment does not allow open argument

and contestation, which are crucial for thinking (Regev, 1997). Socrates used a technique 5 In fact, aside from two researches I will mention later, I have found no works that would attempt to examine the connection between the study of mathematics and learning critical thinking. One of the possible reasons is the use of the term “critical thinking” without a clear definition, and several factors that make it difficult to check this connection, e.g. the “problem of transition” that I discuss in Appendix 10.3.3. 6 The sophists developed the theory of rhetoric, the basis of non-formal logic, which later became an important component in education for critical thinking.


6

called elenchus (ελενχος), a mixture of questions somewhat like a cross-examination,

which later became known as the “Socratic method” or “Socratic debate,” and in which

Socrates refrained from openly introducing his own opinions. Socrates makes all his

conclusions from the answers of his opponent, which served him later in the debate to

defeat the latter’s opinions; thus in his constant striving for the absolute knowledge he

created a method of critical thinking, and posed an ideal model of critical thinking for his

successors. What made him such a model was that he investigated questions, was the first

to raise the problem of definition, sought after the meaning of things, sought to find self-

evident arguments and proofs, used inductive arguments and did not grant axiomatic

validity to definitions (Bryan, 1987). Socrates was sentenced to death on the charge of

treason and “corruption of youth.” Only later did Plato’s writings, “the Socratic

dialogues,” defend the good name of Socrates and prove that he was wrongfully

convicted (Bryan, 1987). The pedagogy of questioning and thinking, according to

Aristotle, begins with wondering, with the primary question. The ability to ask questions

is crucial for a human being, and in Jean Paul Sartre’s terms, it is essential to be able “to

see what is lacking – about facts, reasons, explanations that we lack – to explain what is

present and is experienced as lacking” (Harpaz & Adam, 2000).

2.1.1 The Educational and Social Importance of Critical Thinking

As in the distant past, the need for developing critical thinking today is anchored in the

ideals of education for democracy, which postulate our freedom to think about and

criticize reality and society in which we live, as an expression of our being autonomous

individuals. Today this idea becomes even more vital, because of the increasing need to

be able to investigate and evaluate various messages presented to us in different fields, on

the basis of rational considerations. In this sense, to develop a critical approach and

attitude towards various issues means to “be aware” (Feuerstein, 2002). Matthew Lipman

in his article “A Functional Definition of Critical Thinking” points at the traditional

distinction between ‘knowledge’ and ‘wisdom’. ‘Knowledge’ refers to the sum of

information and ‘truths’ passed from generation to generation, while ‘wisdom’ refers to

the person’s ability to make sensible decisions in complicated and unclear situations.

Wisdom is highly prized, because previous knowledge is not sufficient in order to know

what the best way to act is. According to Lipman, in periods of transition and change,

when the reservoir of traditional knowledge becomes insufficient to deal with reality,


7

wisdom, which is characterized by intellectual flexibility and originality, is highly

esteemed. In our days, according to Lipman, the term ‘wisdom’ came to be replaced with

the term ‘critical thinking’. The principles of critical thinking, according to Lipman, are

the ability to exercise judgment (judgment is activated when we use our knowledge to

arrive at practical decisions), use of criteria in decision-making (when we have several

options that we critically compare to each other in order to choose the one that appears

best), sensitivity to context (when we take into account the specific conditions of context

and choose a suitable way to act, instead of acting as we are accustomed to, without

regard for the specific situation), and finally, self-correcting thinking (when we encounter

a problem that springs from our course of action, we are prepared to make a re-evaluation

and to correct that course). The contemporary reality is complex and ceaselessly

changing. It constantly demands arriving at independent decisions, therefore it is

extremely important to instill in the students at school the ability to think critically,

according to the above principles. Critical thinking is necessary in any field of

occupation, since it allows the individual to deal with reality in a reasonable and

independent way.

2.2 Theory of Critical Thinking "Critical thinking is that mode of thinking - about any subject, content or problem - in which the

thinker improves the quality of his or her thinking by skillfully taking charge of the structures

inherent in thinking and imposing intellectual standards upon them" (NCECT)

2.2.1 Definitions of Critical Thinking A historical survey over several decades shows that the existing plethora of definitions of

‘critical thinking’, vagueness and lack of true understanding surrounding the term have

led to a structural disagreement about the nature of this phenomenon among researchers,

psychologists, informal logicians, philosophers, educators and theorists (Ennis,

1985,1987; Lipman, 1991; McPeck, 1981; Passmore,1980; Paul, 1993; Siegel, 1998;

Johnson & Blair, 1994). The situation of this term is similar to that of the term

‘environment protection’. Everyone agrees about the importance of the activity and its

goals, but the lack of clarity about the exact nature of the goals and the means of

achieving them prevents necessary action in many cases. Some see critical thinking


8

simply as “everyday, informal reasoning” (Galotti 1989), whereas others feel differently.

Shafersman (1991) proposes that a critical thinker is one who asks questions, offers

alternative answers and questions traditional beliefs. He believes that such people, who

seem to be challenging society, are not welcome, and for this reason critical thinking is

not encouraged. Lipman considers it to be different from ordinary thinking because it is

both more precise and more rigorous, and furthermore, it is also self-correcting (1991). It

has also been described by Halpern (1998) as being “purposeful, reasoned, and goal-

directed. “Since there exist in literature dozens of widely varying definitions of ‘critical

thinking’7, I have no way of posing one definition, as is conventionally done in

dissertations. I will try to dispel this vagueness by presenting different definitions and the

disagreements between different specialists concerning these definitions.

On the basis of extensive reading in the field, it seems that critical thinking is a thinking

that establishes criteria for examining beliefs, opinions and truths, in order to give a

rationally based preference to certain beliefs, opinions and truths over others – and to be

prepared to doubt even these8. Thus, critical thinking is a thinking that criticizes

phenomena, ideas and products on the basis of rational and emotional criteria. Ennis

(1987) defines critical thinking as “reasonable and reflective thinking focused on

deciding what to believe or do.” This definition replaced a narrower definition Ennis

proposed in 1962, as “correct evaluation of statements,” which encountered much

criticism and opposition, because it was based on logical skills alone. Ennis, who is

known as one of the most important writers on critical thinking, presents in his article “A

Taxonomy of Critical Thinking Dispositions and Abilities” (1987) the taxonomy of

critical thinking, which includes fourteen dispositions and twelve abilities, subdivided

into sub-abilities. Some of the dispositions and abilities essential for a critical thinker will

be described below: dispositions such as searching for a question, making an argument,

taking care to be well-informed, using reliable sources, searching for alternatives, taking

a stand, and abilities such as clarity, grounding of claims, inference, and interconnection.

7 In fact, each definition relates to a certain area in the field of education and includes several important aspects. 8 The concept of ‘critical thinking’ raises a wealth of associations. Some will imagine critical thinking as doubting whatever is said, others as a kind of protest, provocation or an inclination to argue. With regard to the term ‘doubt’ I will adhere to the positive meaning of the word: not discarding an old idea and seeking to replace it with a new idea, but recognizing the value of the old idea while creating and posing a new one alongside with the old. Without doubting and testing ideas, it would be impossible to develop new and better ideas. Doubt is, without a doubt, a crucial force in learning and development.


9

Siegel (1988) discusses Ennis’ taxonomy very favorably and refers to the inclusive set of

dispositions, characteristics and abilities proposed by Ennis as a “regulative ideal”9. He

claims that critical thinking guides our judgments and provides us with criteria of

excellence on which evaluation of educational activities can be based – therefore it is

called a regulative ideal. Siegel in his article deals with the question, “Who is a critical

thinker?” He claims that a critical thinker has to be an individual with a certain type of

personality, dispositions, traits of character and thinking habits. The critical thinker has to

know how to evaluate statements, and to be prepared to match judgment and action to a

principle, to demand justification and to question ungrounded claims.

McPeck (1981) defines critical thinking as correct use of reflexive skepticism in a given

field, and sees the essence of critical thinking in the behavioral aspect of doubting, or

“postponement of judgment.” McPeck’s behavioral approach differs from Ennis’

definition (according to Ennis, the essence of critical thinking is the logical-analytic

activity of analyzing statements). In order to deeper understand both McPeck’s and

Ennis’ definitions, I will review and compare the various approaches again. McPeck sees

the essence of critical thinking in its behavioral aspect, while Ennis completely ignores

this aspect. McPeck opposes those who see critical thinking as primarily evaluation of

statements, because evaluation of statements concentrates on questions of validity and not

on checking the reliability of information sources. According to McPeck, we do not

routinely analyze conclusions, but rather evaluate data, information and facts. In order to

do this, one needs to be well acquainted with the field that the evaluated information

belongs to. Therefore, according to McPeck, “acquisition of specific skills is neither a

necessary nor a sufficient condition for critical thinking”; a more detailed definition can

be found on the official website of NCECT10: Critical thinking is that mode of thinking –

about any subject, content, or problem – in which the thinker improves the quality of his

or her thinking by skillfully analyzing, assessing, and reconstructing it. Critical thinking

is a self-directed, self-disciplined, self-monitored, and self-corrective thinking. It

presupposes assent to rigorous standards of excellence and mindful command of their

use. It entails effective communication and problem-solving abilities, as well as a

commitment to overcome our native egocentrism and sociocentrism.

9 Siegel defines regulative criteria for excellence, the ability to choose between methods, kinds of policy, and educational acts. 10 The National Council for Excellence in Critical Thinking, http://www.criticalthinking.org/about/nationalCouncil


10

Watson and Watson & Glaser (1980) claim that critical thinking is: (1) an investigative

approach that involves an ability to recognize and accept the general need of proving

whatever is assumed to be true; (2) knowledge of the nature of valid conclusions, and of

abstractions and generalizations in which the measure of validity of different kinds of

evidence is established in a logical way; (3) skills of applying the above knowledge and

approaches. Critical thinking is also defined as result-based, rational, logical and

reflective evaluative thinking in terms of what to reject or accept and what to believe,

following which thinking a decision is made what to do (or not to do), and then to act

accordingly, taking responsibility for the decisions that were made and for their

implications. Elsewhere, critical thinking is defined as the ability and readiness to

evaluate claims in an objective manner, based on solid arguments (Wade & Tavris,

1993). From all of the above definitions it can be concluded that critical thinking is

characterized both by behavioral components, such as doubting, postponement of

judgment and inquisitiveness, and by cognitive components, such as the process of

investigation and drawing conclusions. We would like to focus on three specific

definitions that deal with both abilities and dispositions. McPeck defines critical thinking

as “skills and dispositions to appropriately use reflective skepticism” (McPeck, 1981).

Lipman claims that critical thinking is “thinking which enables judgment, is based on

criteria, corrects itself, and is context-sensitive” (Lipman, 1991). The third definition is

the one we have based our research. Ennis (1962) defines critical thinking as “a correct

evaluation of statements". Over twenty years later, Ennis broadened his definition to

include a mental element, defining it as “reasonable reflective thinking focused on

deciding what to believe or do” (Ennis, 1985).

2.2.2 Critical Thinking Taxonomy The abilities related to critical thinking are divided into two categories: skills, which

include the ability to analyze, evaluate, and draw conclusions, and dispositions, such as

the motivation, inclination and urge of the student to apply critical thinking to discussing

issues, making decisions, and/or solving problems. In addition to critical thinking skills, it

is also important to evaluate the students’ dispositions towards critical thinking, since


11

they may point at the learner’s inclination to practically apply critical thinking in various

contexts.

2.2.3 Critical Thinking Abilities According to Ennis’ Taxonomy11 Critical thinking depends on a number of skills, such as identifying the source of

information, assessing the source’s reliability, evaluating the extent of the new

information’s consistency with previous knowledge, and making a conclusion on the

basis of all these mental acts. In the literature, critical thinking skills are considered

necessary for encouraging meta-cognitive understanding. According to Ennis

(1963,1987,1991,2002) critical thinking is a reflective activity (in which the person

examines his/her own thinking activity) and at the same time a practical activity, the goal

of which is a rational belief or action. There are five key concepts here: practical,

reflective, rational, belief, and action. In light of these, Ennis upgraded his taxonomy of

critical thinking12 and divided it into a system of dispositions and abilities presented

below. The principal areas of the critical thinking ability are clarity, grounding, inference,

and interrelatedness. The critical thinking abilities are: focusing on the question;

analyzing statements; asking questions; evaluating the reliability of the source; deduction;

value-judging; defining terms; identifying assumptions; making decisions about action;

interrelatedness with others. It is important to note that the principal areas presented in

Fig. 2.1.3 have an intuitive dimension: we want to be clear about what is happening; we

want to have an acceptable grounding for our judgments; we want our inferences to be

logical; we want our interrelations with others to be sensitive, and we want that the

dispositions and abilities for critical thinking should be active (Harpaz, 2002).

2.2.4 Critical Thinking Dispositions according to Facione Critical thinking has been investigated largely in terms of thinking skills that involve the

cognitive domain. For decades, the promotion of students’ thinking has been the focus of

11. Ennis emphasizes that the dispositions and abilities in his taxonomy relate to general critical thinking. In order to infuse them into the general curriculum, it will be necessary to teach them several times, at different levels of difficulty and in the framework of different study subjects (see Fig. 2) 12 In his first article from 1962, Ennis defined critical thinking as “correct evaluation of statements. In 1987, he replaced this definition with a new one: “Critical thinking is a reflexive rational thinking focusing on the decision what to believe or to do”. Ennis upgraded his taxonomy, which he published 25 years earlier, according to his new definition. The first taxonomy only included abilities and skills, while the present one, a “taxonomy of dispositions and abilities for critical thinking,” also includes dispositions (14 dispositions and 12 abilities).


12

educational studies and programs (Boddy, Watson, & Aubusson, 2003; De Bono, 1976;

Ennis, 1985; Kuhn, 1999). Each of these programs has its own definition of thinking

and/or of skills. Some use the phrase ‘cognitive skills’ (Leou et al., 2006; Zoller et al.,

2000) and others refer to ‘thinking skills’ (Aizikovitsh & Amit 2008, 2009, 2010; Boix-

Mansilla & Gardner, 1998; De Bono, 1990; Egan, 1997; Resnick, 1987; Zohar & Dori,

2003; Zohar, 2004), but they all distinguish between higher- and lower-order skills.

Resnick (1987) maintained that thinking skills resist precise forms of definition; yet,

higher order thinking skills can be recognized when they occur. Our ever-changing and

challenging world requires students, our future citizens, to go beyond the building of their

knowledge; they need to develop their higher-order thinking skills, such as system critical

thinking, decision making, and problem solving (Zohar, 1999; 2000, Zoller, 2002; 2007).

There have been significant changes in the past decades in the field of education.

Whereas earlier the teacher was at the center and the emphasis was put on what to teach,

today’s education involves teaching how to think, and in particular, how to be a critical

thinker. Critical thinking is necessary in every profession, and it allows one to deal with

reality in a reasonable and independent manner (Lipman, 1991; McPeck, 1994; Paul,

1993). There seems to be no clear consensus as to what exactly critical thinking is. Some

see it as simply being “everyday, informal reasoning” (Johnson & Blair, 1994), whereas

others feel differently. Yet, it seems evident at this point that the ability to think critically

is not something that we are born with, and it is widely accepted that it is in fact a learned

ability that we need to teach.

There are taxonomies that set out a list of reasoning skills involved in critical thinking

(12 skills according to Ennis’ taxonomy of 1962 or 15, according to Dick, 1991). Many

of these approaches assume that when these skills are taught and used properly, the

students will become better thinkers. Other approaches see dispositions as playing a vital

part in the process of critical thinking. Beyer (1987) describes dispositions for critical

thinking as involving "an alertness to the need to evaluate information, a willingness to

test opinions, and a desire to consider all viewpoints."

Halpern (1996) emphasizes the importance of the students’ dispositions, since skills are

useless unless put into practice. In addition to successfully using the appropriate skill in a

given context, critical thinking implies also the disposition to recognize the need for


13

using a particular skill in a certain situation, and the willingness to make the effort of

applying it.

Facione and Facione (1994, 2000) describe dispositions towards critical thinking as

containing elements of intellectual maturity, searching for truth, open-mindedness,

systematicity, self-confidence in critical thinking, analyticity, and inquisitiveness.

They developed the California Critical Thinking Disposition Inventory (CCTDI), which

was originally meant to be used to assess critical thinking dispositions in college students,

but has been successfully adapted also for use in high school. There are seven scales on

the CCTDI. Each describes an aspect of the overall disposition toward using one's critical

thinking to form judgments about what to believe or what to do. People may be

positively, ambivalently, or negatively disposed on each of seven aspects of the overall

disposition toward critical thinking.

The CCTDI also provides a total score which gives equal weight to each of the seven:

Truthseeking, Open-mindedness, Analyticity, Systematicity, Critical Thinking, Self-

Confidence, Inquisitiveness, Maturity of Judgment. Truth seeking is the habit of always

desiring the best possible understanding of any given situation; it is following reasons

and evidence where ever they may lead, even if they lead one to question cherished

beliefs. Truth-seekers ask hard, sometimes even frightening questions; they do not ignore

relevant details; they strive not to let bias or preconception color their search for

knowledge and truth. The opposite of Truthseeking is bias which ignores good reasons

and relevant evidence in order not to have to face difficult ideas.

Open-mindedness is the tendency to allow others to voice views with which one may not

agree. Open-minded people act with tolerance toward the opinions of others, knowing

that often we all hold beliefs which make sense only from our own perspectives. Open-

mindedness, as used here, is important for harmony in a pluralistic and complex society

where people approach issues from different religious, political, social, family, cultural,

and personal backgrounds. The opposite of open-mindedness is closed-mindedness and

intolerance for the ideas of others. Analyticity is the tendency to be alert to what happens

next. This is the habit of striving to anticipate both the good and the bad potential

consequences or outcomes of situations, choices, proposals, and plans. The opposite of

analyticity is being heedless of consequences, not attending to what happens next when

one makes choices or accepts ideas uncritically. Systematicity is the tendency or habit of


14

striving to approach problems in a disciplined, orderly, and systematic way. The habit of

being disorganized is the opposite characteristic to systematicity.

The person who is strong in systematicity may or may not actually know or use a given

strategy or any particular pattern in problem solving, but they have the mental desire and

tendency to approach questions and issues in such an organized way. Critical Thinking

Self-Confidence: the tendency to trust the use of reason and reflective thinking to solve

problems is reasoning self-confidence.

This habit can apply to individuals or to groups; as can the other dispositional

characteristics measured by the CCTDI. We as a family, team, office, community, or

society can have the habit of being trustful of reasoned judgment as the means of solving

our problems and reaching our goals. The opposite is the tendency to be mistrustful of

reason, to consistently devalue or be hostile to the use of careful reason and reflection as

a means to solving problems or discovering what to do or what to believe. Inquisitiveness

is intellectual curiosity. It is the tendency to want to know things, even if they are not

immediately or obviously useful at the moment. It is being curious and eager to acquire

new knowledge and to learn the explanations for things even when the applications of

that new learning is not immediately apparent. The opposite of inquisitiveness is

indifference. Maturity of Judgment: Cognitive maturity is the tendency to see problems

as complex, rather than black and white. It is the habit of making a judgment in a timely

way, not prematurely, and not with undue delay. It is the tendency of standing firm in

one's judgment when there is reason to do so, but changing one's mind when that is the

appropriate thing to do.

It is prudence in making, suspending, or revising judgment. It is being aware that

multiple solutions may be acceptable while appreciating the need to reach closure in

certain circumstances even in the absence of complete knowledge. The opposite,

cognitive immaturity, is characterized by being imprudent, black-and-white thinking,

failing to come to a closure in a timely way, stubbornly refusing to change one's mind

when reasons and evidence indicate one is mistaken, or revising one's opinions without a

substantial reason for doing so.

Ennis (1985, 1987, 1989) presents 14 dispositions, the first 13 of which are defined as

necessary for critical thinking, while the last one, “being sensitive,” is not exactly a basic

disposition yet nevertheless has to be present in the totality of dispositions. The


15

dispositions are: seeking for clarification of a thesis or question; searching for arguments;

trying to be well-informed; using reliable sources; taking into account the general

situation; trying to stay relevant to the central issue; consistently remembering what the

original or basic issue is; searching for alternatives; seriously considering different points

of view; postponing judgment; taking a stand; seeking a high degree of precision; dealing

with the components of the whole in an organized way; sensitivity.

Figure 1: The Taxonomy of Critical Thinking

2.2.5 Development and Learning of Critical Thinking There is an ongoing discussion in the field of education regarding the ways in which

critical thinking skills can be developed. Some researchers believe that there is a need to

plan specific critical thinking courses. Others claim that developing these skills can be

accomplished in the framework of regular courses (Ennis, 1989; McPeck, 1981; Resnick,

1987; Weinberger,1992). There is a debate as to whether these skills are completely

general or specific to subject matter and concepts. Most agree that critical thinking has

both general and specific attributes. Feuerstein's study (2002) showed that after teachers

were provided with theoretical and pedagogical knowledge, they were able to foster

critical thinking in their students. Zohar and Tamir (1993) found that critical thinking

does not develop on its own. Based upon this conclusion and upon the small amount of

Abilities Informational Basis: Accepted, Previously Achieved Conclusions

Deduction Induction Value-

judging

Decision Regarding Belief or Action

Interrelatedness

Ennis’ Taxonomy

Dispositions


16

existing research in the field of critical thinking in mathematics, this study examines the

affinity between education for critical thinking and mathematical studies. Many

researchers, beginning with the philosopher Passmore (1980), hesitate regarding many

questions related to critical thinking, such as, what does ‘being critical’ mean? Is it

possible to educate for critical thinking, and what does this mean? How can we know that

we have succeeded in this task? In his article “Teaching to Be Critical,” Passmore

discusses the meaning of education for critical thinking and raises the question, “Is being

critical a matter of habitual behavior acquired through experience, that is, a habit?”

In addition, Passmore relates to the confusion between mere grumbling and critical

thinking. According to Passmore, it should be clear that a critical person is a person who

has imagination. In the same way as imagination should be distinguished from delusion,

being critical has to be distinguished from a grumbling expression of discontent, or from

mere slander. They are as easy to confuse as imagination and delusion. According to

Schafersman (1991), critical thinking is a learned ability that should not be left to develop

of its own accord, nor should it be taught by an untrained teacher. Both training and

knowledge are necessary to promote critical thinking abilities in students. Moreover,

Schafersman suggests that because society does not welcome people who challenge

authority, critical thinking is not often encouraged. Thus, in his opinion, "most people do

not think critically."

Resnick (1987) corroborates Schafersman’s point of view, arguing that despite the fact

that developing critical thinking has been one of the most important goals of education

for centuries, problems that demand critical thinking are often dealt with ineffectively.

Expertise in any field can only be achieved with critical thinking (Wagner 1997), and it is

therefore necessary to help students understand how valuable it is and how they can

achieve it. In Zohar and Tamir's (1993) study as well, the researchers concluded that

critical thinking does not develop on its own and efforts are required in order to develop

it. As Barak, Ben-Chaim and Zoller (2002, 2007) summarize, previous research has

shown a need for improving critical thinking skills among students, since most students

do not use sophisticated thinking even at the higher education level. In general, there is a

consensus that the ability to think critically is becoming increasingly important for being

successful in contemporary life, because of the ever-increasing pace of changes and the

complexity and interconnectedness of various phenomena we encounter. People today are


17

not expected to ‘know their place’, but rather to establish and reinvent their position in

the world. As the world is advancing, more and more people are required to make

rational decisions based on evaluative/critical thinking, instead of accepting others’

authority. Thus, students need to be ready to examine truth values, to raise doubts, to

investigate situations and to search for alternatives in the context of school and everyday

life. In accordance with the above, De Bono (1976) had proposed a long time ago that it

is hard to teach thinking skills by means of a formal logical process, using principles and

axioms. He developed a number of approaches to teaching thinking, and showed that

students who received lessons in thinking produced a greater number of solutions for

problems, compared to students who did not receive such lessons. Our research is based

on three key elements: a critical thinking taxonomy that includes skills and dispositions

(Ennis, 1987); the learning unit "Probability in Daily Life" (Lieberman & Tversky,

1996,2001); and the infusion approach of integrating subject matter with thinking skills

(Swartz, 1992).

2.3 The Defining Components of the Theoretical Framework13 This section presents the three fundamental components that this research is based on:

Ennis taxonomy, the Probability in Daily Life learning unit and the infusion approach.

2.3.1 The Choice of Ennis’ Taxonomy Ennis claims that critical thinking is a reflective and practical activity aiming for a

moderate action or belief. There are five key concepts and characteristics defining critical

thinking: practical, reflective, moderate, belief and action. In accordance with the

categories this definition employs, Ennis developed a taxonomy of critical thinking skills

that include an intellectual as well as a behavioral aspect. In addition to skills, Ennis’

taxonomy also includes dispositions and abilities. In this study, we focus on students’

abilities rather than their dispositions. We have chosen to use Ennis’ definition and

taxonomy of critical thinking because it distinguishes between abilities and disposition,

and because teaching thinking skills according to a taxonomy suits the hierarchical

structure of our learning unit in probability studies.

19 See illustration on p. 22.


18

2.3.2 The Learning Unit "Probability in Daily Life" (Lieberman & Tversky, 2001) This unit in probability studies is part of the formal high school curriculum of the Israeli

Ministry of Education. It was chosen because its rationale is to make the students to

"study issues relevant to everyday life, which include elements of critical thinking”

(Lieberman & Tversky 2001, Introduction p.3). In this unit, students must analyze

problems using statistical instruments, as well as raising questions and thinking critically

about the data, its collection, and its results. Students learn to examine data qualitatively

as well as quantitatively. They must also use their intuitions to estimate probabilities and

examine the logical premises of these intuitions, along with misjudgments of their

application. The unit is unique because it explores probability in relation to everyday

problems. This involves critical thinking elements such as tangible examples from

everyday life, confronting credible information, accepting and dismissing generalizations,

rechecking data, doubting, comparing new knowledge with the existing knowledge. This

unit is characterized by questions such as “Define the term ‘critical thinking’,” “Give

examples of a problem while using a controlled experiment,” “Give examples of failures

and misleading commercials,” and “Give examples of a scientific truth that was

dismissed.” While studying the subject, the connection is checked between statistical

judgment and intuitive judgment, and intuitive mechanisms that produce wrong

judgments are explored. While studying the subject, students are expected to acquire the

tools for critical thinking. In the beginning, students learn the mathematical tools

necessary for performing calculations, and later on they use the probability part: causal

connection, and mechanisms of intuitive judgment, which are considered more of a

psychological projection (Gilovich, Griffin & Kahneman, 2002; Kahneman et. al, 1996)

2.3.3 The Infusion Approach (Swartz, 1992) In light of the evidence that has accumulated in the field of teaching thinking, the

question arises whether thinking skills are general or content-dependent (Perkins &

Salomon, 1988,1989; Perkins, 1992). Out of this question there developed four major

approaches: the general approach, the infusion approach, the immersion approach, and

the mixed approach. The general approach teaches thinking skills as a range of general

skills detached from other study subjects, as a separate course in the curriculum. In the


19

infusion approach the skills are taught in the framework of a specific study subject, and

thinking turns into an integral part of teaching specific materials, while general principles

and terminology of thinking are explicitly emphasized. In the immersion approach, the

study material is taught in a thought-provoking way and the students are “immersed” in

the topic of study, without explicit reference to the principles of thinking. The mixed

approach combines the general and the infusion approaches.

The present research employs the infusion approach, where thinking is taught and

learned in the context of the learning unit “Probability in Daily Life.” It is important to

elaborate here on the distinctions between the general and the infusion approach.

The field of education has recognized for decades the need to concentrate on the

promotion of critical thinking skills. The question is how this can be best accomplished.

Some educators feel that the best path is to design specific courses aimed at teaching

critical thinking, which is called the general skills approach. Integrating the teaching of

these skills in regular courses in the curriculum is a different approach known as the

infusion approach.

The question at the heart of the argument is, whether critical thinking skills are general or

depend on content and on the system of concepts specific to that particular content.

According to Swartz and Parks (1994), the infusion approach aims at teaching specific

critical thinking skills along with different study subjects, and instilling critical thinking

skills through teaching the set learning material.

According to this approach, such lessons are expected to improve the students’ thinking

and help them to learn the contents in different study subjects. Swartz also emphasizes

that the students should not only employ critical thinking skills in class, but also be able

to activate them in real-life situations and to recognize situations when these skills should

be used. For this, an appropriate motivation should be fostered; otherwise these skills will

remain passive.

In this study, conducted according to the latter approach, we have combined the

mathematical content of the "Probability in Daily Life” learning unit with critical

thinking skills according to Ennis' taxonomy, restructured the curriculum, tested different

learning units and evaluated the participants’ critical thinking skills, to examine whether

the learning unit “Probability in Daily Life,” by using the infusion approach, does indeed

develop critical thinking.


20

_______________________________________________________________________________________________________

Figure 2: The Infusion Approach According to Swartz

2.4 Contexts of Critical Thinking The following section addresses the different aspects of critical thinking: research, learning and teaching.

2.4.1 Studies Dealing with Critical Thinking in Mathematics An extensive literature review conducted in this research has shown that a number of

works have been published on the topic of “critical thinking in mathematics,” yet very

few of them proceed from the same context or ‘spirit’ as the present study, namely, that

of seeking for a general definition of “critical thinking” and giving this definition a

scientific grounding (Akbari-Zarin & Gray,1990; Avital & Barbeau, 1991; Battista et al.,

1989; Becker, 1984; Boucher, 1998; Cherkas, 1992; Coon & Birken, 1988; Dion,

1990; Dubinsky, 1989, 1986; Fridlander, 1997; Garofalo, 1986, 1987; Gray & St.

Ours, 1992; Innabi & Sheikh, 2007; Johnson, 1994; Kaplan, 1992; Kaur & Oon, 1992;

Kloosterman & Stage, 1992; LeGere, 1991; McCoy, 1990; Movshovitz-Hadar, 1993;

Olson .& Olson, 1997; Lawrenz & Orton, 1989).

As pointed out before in the “Theoretical Background” section, critical thinking has been

defined in many different ways, on the basis of various theories. In science, and in

Learning unit “Probability in Daily

Life” (Lieberman & Tversky, 2001)

The use of a critical-thinking promoting learning unit in the context of a specific

subject (mathematics)

Infusion Approach (Swartz, 1992)

Rewriting the lesson plans in various subjects for

direct teaching of critical thinking

Ennis’ Taxonomy

(1987)

Direct teaching of critical thinking out of specific subject context

The research’s scientific basis - Ennis’ taxonomy of critical thinking skills and

dispositions


21

particular in mathematics, none of the classical definitions cited in the “Theoretical

Background” section have been presented. Researchers who do relate to critical thinking

in the field of mathematics use this term in other contexts, and in fact deal with imparting

technical tools such as performing an assessment, checking the correctness of results,

evaluating a certain exercise, comparison, inference, application and interpretation,

solution strategies, etc. Reviewing these articles, we have searched for the term “critical

thinking” used in the sense relevant to the present study. Strategies for critical thinking in

learning: define your purpose, what it is you want to study; clarify questions and answers

with your teachers or other specialists in the subject. The purposes of study can be

formulated in simple phrases: “Plumbing regulations in suburban neighborhoods,”

“Structure and terms in the human skeleton.” Think about what is already known to you

on the subject: what do you already know that may help you in your study? What are

your preconceptions on the subject? What means do you have for carrying out the study,

and what is your timetable? Gather information; keep your thinking open so as not to

exclude opportunities, Ask questions; what are the preconceptions of the sources’

authors? Organize the information you have collected into structures that make sense to

you and ask questions again and again.

2.4.2 Studies of Critical Thinking in Different Disciplines14

Critical thinking is a field in philosophy and psychology15 dealing with tools and methods

for seeking and grounding knowledge (of any kind and in any field). As we have already

shown, similar thinking processes occur in different fields of human knowledge and

action. Whether we are reading a newspaper item, interrogate a witness in court, diagnose

an illness or carry out a scientific experiment in a lab, in a certain sense we are playing

the same kind of a game based on investigation, clarity, and addressing questions to a

relevant source of information. In this chapter we will review the place of critical

thinking development in the field of education in different areas: media, Jewish scriptural

exegesis (Midrash), art, electronics, and sciences (specifically, chemistry and biology).

14 This section does not bear directly on the topic of the present research, yet it points at the importance of developing general critical thinking. 15 Cognitive psychology, in particular, attributes a great importance to education for critical thinking.


22

2.4.2.1 Critical Thinking in Natural Sciences Zielinski (2004) proposes in her article a way of helping students to develop higher-order

thinking skills, in particular, critical thinking, through chemistry studies. Zohar and

Tamir (1993), in their research “Developing Critical Thinking through Teaching

Biology,” claim that critical thinking does not develop on its own, and purposeful efforts

are needed in order to develop it. The purpose of the researchers is to improve critical

thinking through the study of biology, by means of teaching directed in a clear and

explicit way towards the acquisition of critical thinking by the students. In Zohar and

Tamir’s research there participated 77 ninth-grade students that were divided into four

groups. Two of the groups studied according to the regular biology curriculum and

comprised the control group. The teaching in the other two groups was carried out by the

researchers themselves. All the groups studies for exactly the same number of hours,

using the same textbooks, while the two experimental groups were also exposed to the

critical thinking teaching project. The goal of this project, titled “Critical Thinking in

Biology,” was to develop a range of activities integrated into the regular biology

curriculum, without taking up any additional class time. The activities should be fully

integrated into the existing biology curriculum, rather than comprising an alternative

curriculum. To enhance the effectiveness of thinking skills teaching, specific thinking

skills were repeatedly integrated into a range of different contents. All the activities

started with posing a specific problem arising from the biology topic studied in class.

Towards the end of the activity, after the pupils had tried themselves in the specific

thinking skills while solving a problem, a discussion on the meta-cognitive level was

conducted, in which the thinking skills and the use that the students had made of them

was discussed. Only a limited time was devoted to these activities, in order that the other

teaching goals should not be impeded. On the basis of the research findings, the

researchers concluded that it is possible to integrate activities for development of critical

thinking into the regular biology curriculum. This integration does not involve additional

class time and does not detract from the students’ level of knowledge in biology,

moreover, improves this level. In addition, the researchers claim that this project makes a

significant contribution by improving the students’ ability to perform tasks in biology that

require critical thinking.


23

2.4.2.2 Critical Thinking in Media Studies A research by Feuerstein (2002) examines the connection between teaching media studies

and development of critical thinking and shows that the theoretical and pedagogical

components of the curriculum develop the students’ critical thinking abilities. Feuerstein

checked whether the regular undergraduate program of media studies is sufficient to

educate the students for critical consumption of media (based on the development of

critical thinking), or whether there is a need for integrating in this program specific

education for critical thinking. Feuerstein’s research is the first attempt to examine the

affinity between education for critical thinking and media studies, in two aspects: first,

teaching and learning critical thinking in an integrated mode, along with imparting

subject-specific contents, and secondly, the potential of various elements in the media

studies curriculum for developing critical thinking. The research showed that the students

demonstrated a critical approach to media and a clear inclination to cast doubt about

information and messages presented by the media. We attribute this critical attitude on

the students’ part to the explicit messages of the study program and its pedagogical

approach based on the constructivist theory of learning that emphasizes active learning

and construction of knowledge on the student’s part, and systematic critical analysis of

the media contents. Feuerstein concludes that one should not rest content with the goals

and messages of the media studies program, but it is essential to engage in teaching and

learning directed specifically at the development of thinking and of the students’

awareness of its importance. In other words, to the experience of exercising critical

thinking about media it is important to add the definitions derived from the theory of

thinking, and from the pedagogy of media studies, on the meta-cognitive level.

2.4.2.3 Critical Thinking in the Midrash Studies Searching for the meaning of a term as common and popular as “critical thinking”

increases the wondering and lack of understanding of the term’s meaning. When we

cannot recover the literal meaning of the term, we check its etymology and search for its

appearance in the earliest sources. Thus, we have discovered an additional context of

critical thinking in the Jewish Midrash literature, one of the earliest sources of this

concept and practice. Moreover, this kind of thinking contributed to a social uprising


24

against the existing order, since it allowed all the social layers of the nation to participate

indentify with the highly prized moral stock of the Torah students. This mode of thinking

is capable of examining and giving different interpretations to the scriptural text. As we

will see further, according to Ennis’ (1987) definition and taxonomy of critical thinking,

Midrashic thinking can be classified as critical.

2.4.2.4 Critical Thinking in Art In Nevo’s (2005) article, “What is the Contribution of Art Studies to the Educational

System,”16 she claims that appropriately planned art lessons can become lessons in

critical thinking. Nevo treats the educational system as a system that must aspire to give

the students thinking tools that will enable them to confront the complex and changing

reality in a rational, critical and intelligent way. According to Nevo, education should

allow the student to explore him- or herself, his or her talents, abilities and areas of

interest, and to express him- or herself in a creative way. In Nevo’s words:

“As someone who studies and wants to continue studying art, I know that I must examine

my own work in a critical way, that is, to ask myself critical questions during the process

of work, and to receive critical appraisals of my work. I must formulate for myself in a

critical way what I want to convey in my work, by what means it can be conveyed, and

finally, whether the external public will be capable to ‘communicate’ with the work, that

is, to understand the contents arising from it. It is also important for me to ask why it is

important for me to express these specific contents in the work, and whether the work

indeed succeeds in expressing these contents. In other words, I must apply rational and

critical deliberation and to learn to look at my work ‘from the outside’. The same

procedures also apply when looking at another artist’s work. I make an effort to ask

myself not only whether I ‘like’ or am ‘attracted’ to the work in an intuitive and

emotional way, but also what contents arise from the work, what does the work connect

to, and what artistic means did the artist use in order to convey to the spectator the

specific experience or content” .

16 http://cms.education.gov.il/EducationCMS/Units/Mazkirut_Pedagogit/Omanut/MaagareyMeida/Maamarim


25

2.5 The Critical Thinking Movement At the time of unprecedented confusion regarding the appropriate goals of education, the

educational Critical Thinking Movement, with headquarters in the United States,

proposes a profound discussion of one educational goal rooted far back in the antiquity.

This movement has a considerable influence in the U.S. and is increasingly influential

also in other countries, including Israel. The movement’s thinkers – philosophers,

psychologists and educators – claim that critical thinking is a worthy educational goal

that suits the spirit of the present time and answers its challenges.

The movement is a sub-current of a larger and older, internationally renowned movement

called “Education for Thinking,” which began 30 years ago in the United States in

response to the failure of school education to realize its goals. “Education for Thinking”

set out to propose an educational ideal that should guide all the educational institutions.

In the framework of the “Critical Thinking” movement, its title concept was given

different and even contradictory theoretical and didactic definitions.

Many educators from different disciplines are trying to define the field of critical thinking

and create a common concept. Yet, in spite of the variety of definitions and the

disagreements on the meaning of the movement’s central idea, the purpose of the

movement is to educate young people for critical thinking and personality, prepared and

able to examine the accepted beliefs. David Perkins, one of the movement’s most notable

thinkers, emphasizes the need for fostering critical thinking as a tool for understanding

knowledge, and not as a goal for its own sake (Harpaz, 1996,1997).

The Critical Thinking Movement seeks to encourage students to cast intelligent doubt

about what the authorities – teachers, specialists, textbooks, books, newspapers,

television – tell them. It seeks to bring up critical pupils who ask questions such as, on

what grounds does a certain text or person claim what they claim? From what point of

view are they claiming this? Why prefer their claim over other, contradicting or different

claims? The idea of educating for critical thinking, as well as the idea of educating for

creative thinking, has far-reaching consequences for school education. At present, it is

mostly an idea, rather than action, but one can already see practical attempts to realize

this idea in the educational field. One of the ways to realize the idea of educating for

critical thinking is to “translate” it into a range of skills.


26

Thus, for instance, the devoted promoters of this idea developed a new field called

“informal logic,” which helps to locate, criticize and construct propositions in natural

language. Other supporters of education for critical thinking developed a classification of

skills, such as the skill of examining reliability of information sources, the skill of

uncovering basic assumptions, the skill of identifying biases, etc. Other supporters

developed study programs based on conflicts between different worldviews, standpoints

and versions. Still others composed programs of critical reading, critical watching and

critical “consumption” of media (Harpaz & Adam, 2000).

Research Method

27

3. Research Method “As soon as a question of will or decision or reason or choice arises, human science is at a loss." (Noam Chomsky)

As stated above, the present research examines the influence of the learning unit

“Probability in Daily Life” on the development of critical thinking among high-school

students in various educational frameworks. To achieve the research purpose most

effectively, we have combined between qualitative and quantitative approaches to

mutually validate the findings of each.

3.1 The Research Purpose The purpose of this research is to examine how and to what extent it is possible to

develop critical thinking by means of the learning unit “Probability in Daily Life” using

the infusion approach.

3.2 The Research Questions What, if any, are the influences of implementing the “Probability in Daily Life” learning

unit in the infusion approach on:

3.2.1 the development of critical thinking dispositions, according to the taxonomies of

Ennis (1987) and Facione (1992)?

3.2.2 the development of critical thinking abilities, according to the taxonomy of Ennis

(1987)?

3.2.3 the processes of construction of critical thinking skills (e.g., identifying variables,

postponing judgment, referring to sources, searching for alternatives) during the study of

the “Probability in Daily Life” learning unit in the infusion approach?

3.3 The Choice of Mixed Methods Due to the pragmatic nature of the knowledge claims for this consequence-oriented study,

a mixed methods approach was used in the design of the methodology. Drawing on the

work of Cherryholmes (1992), and his own interpretation, Creswell (2003,2009) makes

the following statement regarding pragmatically-based knowledge claims: “Pragmatism

is not committed to any one system of philosophy and reality. This applies to mixed

Research Method

28

methods research in that inquirers draw liberally from both quantitative and qualitative

assumptions when they engage in their research. Individual researchers have a freedom of

choice. They are ‘free’ to choose the methods, techniques, and procedures of research

that can best meet their needs and purposes” (p. 13). This paradigm provided me with the

liberty to select multiple methods, draw on different worldviews and assumptions and to

make use of different forms of data collection and analysis (Creswell, 2003; Tashakkori

& Teddlie, 2003).

In this particular study, the conceptualization, method and inference stages of the

research process all drew on what would be traditionally classified as both qualitative and

quantitative approaches (Tashakkori & Teddlie, 2003). Exploratory (qualitative) as well

as confirmatory (quantitative) questions were asked, and both quantitative and qualitative

data were collected to answer these. Subsequently, in keeping with pragmatic foundations

outlined by Pierce (see section 3.3.1), both inductive (qualitative) as well as deductive

(quantitative) investigations of analysis into the inquiry were utilized in the inference

stage to form a meta-inference at the end. This type of methodology is referred to by

Tashakkori and Teddlie (2003) as a "fully integrated mixed model design."

The importance of such combination was emphasized by Sabar Ben Yehoshua (2000) as

reinforcing the results’ validity. Supplementing quantitative tools and results with

interviews gives a deeper and more reliable ‘picture’ of the results. Quoting what the

research participants say reinforces and clarifies the findings. On the other hand, Shkedi

(2006) claims that interviews in qualitative research comprise a crucial but not the only

source of information – in other words, qualitative research in its turn can be

strengthened by being combined with quantitative tools. In the present research,

interviews were the main tool used to answer the third research question.

The remaining sections of this chapter outline the site, sampling and data collection and

analysis stages in the study to shed further light on this process, and samples of an

experimental lesson and test questions are quoted and analyzed.

3.4 “Working on the Inside”: The Teacher as a Researcher Another aspect of the methodological approach that this research adopts is termed

“working on the inside.” Researching “on the inside” relates to using the researcher’s

Research Method

29

workplace in the classroom as a site in which s/he researches teaching and learning (Ball,

2000); it usually refers to research that teachers conduct on their ways of working in the

classroom, and the ways in which their pupils learn (Feldman & Minstrel, 2000).

Initially, the main purpose of research on the inside was not to produce new knowledge

but to improve and change what is being done in the classroom, and in this way to

develop the teacher’s/researcher’s ability of self-exploration and reflexive and critical

thinking (Feldman 1996). The teacher was supposed to learn from experience and go

through a continuous process of constructive evaluation in order to create change.

Research in which the researchers use their teaching work as a basis for academic work is

a relatively new field, to which the present research belongs. Such research “in the first

person” introduces into the academy the teacher’s voice and point of view (Ball, 2000). A

number of definitions of this concept can be found in research literature, e.g. as

“systematic research aiming at changing and improving educational work of the groups

of participants by means of understanding the work’s significance and personal self-

reflection regarding the results of these actions” (Ball, 2000); “systematic study of

attempts to change and improve educational functioning by groups of participants,

through their practical action and self-reflection on the influence of this action” (Ball,

2000).

3.5 Stages of the Research Our purpose in this research was to develop, use and evaluate a model of teaching and

learning critical thinking in mathematics. This work had three partially overlapping

principal stages:

1. Developing a learning unit based on the syllabus of “Probability in Daily Life.”

2. Carefully controlled application of the unit, including full documenting17.

3. Researching the unit’s influence on the development of critical thinking.

The purpose of the first stage was to develop teaching materials and methods that

promote critical thinking through the study of probability in daily life. This purpose was

achieved by constructing a very precisely thought-out learning unit (see Chapter 4,

“Intervention,” for the account of how the unit was designed, and the Appendix for

17 See chapter 4, “Intervention”

Research Method

30

specific samples of materials from the unit) that integrated critical thinking skills with

various activities.

The second stage involved an experimental application of the learning unit and its

recurrent evaluation after each lesson throughout the process (see Chapter 4,

“Intervention”).

In the third stage, the impact of the learning unit was examined and researched. We have

investigated how and to what extent the study of “Probability in Daily Life” contributed

to the development of general and field-specific critical thinking18; established levels and

criteria of critical thinking (according to Ennis) and described a number of crucial factors

one has to confront when attempting to teach critical thinking while using the learning

unit “Probability in Daily Life.” Finally, we have checked whether there is a connection

between the specific mathematical knowledge (the student’s level of advancement in

mathematics) and their level of critical thinking.

3.6 Research Population This research was carried out in three rounds. The first round was the pilot study,

described in detail in section 3.8. In the second round, the unit was taught to two

Kidumatica math club groups and a regular high school in central Israel (High School 1).

The first round included only experimental groups and was taught by the researcher

alone. In the second round, the research was extended to another high school (High

School 2) and two additional experimental groups were added, taught by another teacher

and not the researcher herself; also a control group was added in this round. The students

in the experimental groups took the specially developed intervention learning unit (see

Chapter 4) while the control group students took a standard course in probability (i.e.

were not exposed to the intervention). Both groups studied for the same number of hours.

The sample was chosen out of tenth-grade students and consisted of 11 groups. It was our

intention for the groups to represent, as far as possible, the multicultural society of Israel:

city dwellers, kibbutz adolescents, and adolescents from the religious and Arab sector.

Five groups were taught by the researcher in the framework of the Kidumatica program at

Ben-Gurion University and in a high school in central Israel, and the rest were taught by

another teacher.

18 Mathematics – Probability in Daily Life.

Research Method

31

Table 1: Research Population Distribution 3.7 Research Instruments In this research, the following data collection and research instruments were used:

research questionnaires, personal interviews, observations, documentation of lessons and

analysis of lesson plans and records. The first and second research questions were tackled

with quantitative tools, while the third research question was answered by means of

qualitative tools described in sections 3.7.3 and 3.7.4. We have used triangulation

between the different research tools to increase the validity of the findings.

Qualitative data collection for the third research question was also, in turn, triangulated

by means of the following tools:

Unit Taught by Research Framework Number of Students First round Group type

the researcher

“Kidumatica” project Ben-Gurion University of the Negev, in the framework of

mathematics studies

41

2 groups

“Kidumatica”

the researcher

A regular high school, in the

framework of formal mathematics studies

28-30

one group (experimental )

“High school”

Unit Taught by Research Framework Number of Students Second round Group type

the researcher

“Kidumatica” project

25

2 groups

Kidumatica

another teacher

A regular high school, formal

mathematics studies

46

2 groups (experimental and control)

High School 1

the researcher.

Regular high school, formal

mathematics studies

42

2 groups (experimental and control)

High School 2

another teacher

Regular high school, formal

mathematics studies

24

A class of excellent students parallel to

“Kidumatica”

Research Method

32

1) Randomly conducted personal interviews: five students were interviewed at the end of

a session and one week after. The personal interviews were conducted in order to reveal

change in the students' attitudes during the academic year.

2) Collecting the students' products: exams, in-class papers and homework.

3) Recording, transcription and analysis of all sessions. The teacher kept a journal (log)

on every session. These data were processed by means of qualitative methods, which

enabled to follow the students' patterns of thinking and interpretation with regards to the

material in different contexts.

3.7.1 Questionnaires In order to answer the research questions and to evaluate the change in the students’

knowledge, pre and post questionnaires were used according to the Counter Balance

method19 (Birnbaum, 1993). To achieve these goals, we have passed the following

questionnaires: (1) the CCTDI (Facione, 1992) Likert test testing critical thinking

dispositions to answer the first research question (for details see section 5.3); (2) the

Cornell Critical Thinking Test, version “Z” (Ennis, 1985), testing critical thinking

abilities to answer the second research question (for details, see section 6.3); (3) a

questionnaire testing critical thinking in a specific field of knowledge, “Probability in

Daily Life”, to answer the third research question. Table 2.presents the numbers of

students taking each test in each of the two rounds.

Disposition Test Ability Test

Group School First Round Second Round First Round Second Round

Experiment Kidumatica 41 17 41 25

High School 1 30 22 28 29 High School 2 31,19 25,17

Control High School 1 29 25

High School 2 32,27 21 Total 71 177 69 142

Table 2: Research Population Distribution each Test

19 Birnbaum’s counter-balance method is a move directed at minimizing external diversities’ effect on research results, through such techniques as dividing the experimental group into subgroups to which different versions of the questionnaire are given (see Table 4.3.1 and the section on “Control Groups”).

Research Method

33

Table 3 describes the way in which each questionnaire was delivered to the different

populations, following the principle of counter balance.

Table 3: Research questionnaires by type and the manner in which they were passed to each group.

3.7.2 Personal Interviews20 Semi-structured interviews (47 interviews in all) were conducted during the first and

second year of research. The purpose of the interviews was to hear from the students how 20 Fontana & Frey 2003

The research question that the

questionnaire answers

Post -test for the two

groups

Pre -test for the two

groups

Type of

questionnaire

Groups

H (1, 2)

K (1, 2)

H (1, 2)

K (1, 2)

To what extent did the study of “Probability in Daily Life” contribute to the development of dispositions toward general critical thinking (according to Ennis’ taxonomy, 1985) in the framework of mathematics studies?

1-35 Items

(1)

36-75 Items

(1)

36-75 Items

(1)

1-35 Items

(1)

CCTDI

36-75 Items

(2)

1-35 Items

(2)

1-35 Items

(2)

36-75 Items

(2)

To what extent did the study of “Probability in Daily Life” contribute to the development of general critical thinking abilities (according to Ennis’ taxonomy, 1985) in the framework of mathematics studies?

A+B+C Sections

(1)

D+E

Sections (1)

D+E

Sections (1)

A+B+C Sections

(1)

Cornell

D+E

Sections (2)

A+B+C Sections

(2)

A+B+C sections

(2)

D+E

Sections (2)

What are the processes of construction of critical thinking skills (e.g. doubting, suspension of judgment, referring to sources) during the study of “Probability in Daily Life” in the infusion approach?

During the Unit

Questionnaire in critical thinking in a specific field of knowledge, “Probability in Daily Life”

Research Method

34

they perceive what is taking place in class during mathematics lessons, to recognize and

distinguish between different strategies of teaching that can improve critical thinking

skills and throw light on the ways in which the students understand the concept of critical

thinking. Each interview lasted for approximately 30 minutes. The interviews were

recorded at the interviewees’ permission, in addition to note-taking. The interviews were

read twice and then analyzed into coded segments. For interviewing, 7-10 students were

selected from each group. The choice of students was made on the basis of their answers

to the questionnaires (particularly interesting answers, unclear answers, etc.) The

interviews were conducted at the end of the period of class observation and at the end of

analyzing the post-questionnaires.

3.7.3 Observations Systematic observations were conducted in the mathematics lessons in all the groups

participating in the research. The purpose of observation was to provide data regarding

the character of teaching in the class and the way the learning unit is taught, as well as to

become acquainted with the teacher’s teaching style and strategies, the research

participants, and the cultural climate in the class (observation sheet, “critical” events,

problematic points). The observations were conducted alternately during the second year

of the study. The data were written down in a journal that was analyzed and interpreted

by the researchers. The researcher sat among the students at the back of the class and

documented the ways in which the teacher presented new topics to the class, her use of

the different teaching methods, and her interaction with the students, while focusing on

the strategies related to critical thinking. The observations were content-analyzed by one

researcher and divided into categories. Three expert researchers reviewed the categories

in order to evaluate the findings’ reliability.

3.7.4 Statistical Tools The participant groups were tested twice, both Pre and Post the experiment. For each

session (Pre, Post) the groups were tested on both questionnaires: Dispositions of Critical

Thinking and Abilities of Critical Thinking. Each of the questionnaires has sub-scores

and a total-score.

Research Method

35

For the First Round (since there was no control group) a Paired t-test was conducted in

order to compare the difference between the Post-scores and the Pre-scores for the two

questionnaires.

For the Second Round we used the following statistical tools:

1. T-tests between the Experimental group and the Control group for the Pre-scores

and Post-scores for the two questionnaires.

2. Paired t-tests for each of the groups (Experimental, Control) to compare the

difference (delta) between the Post-scores and the Pre-scores for the two

questionnaires.

3. For comparison between the groups’ Post-scores we used the ANCOVA F test.

When comparing between the groups' Post-scores the ANCOVA F test controls

for initial differences (if they exist) between the Pre-scores. When there are no

initial differences (on the Pre score) between the two groups ANCOVA is not

necessary. ANCOVA F test can only be performed when the regression slopes (of

Post-scores on Pre-scores) for the two groups (Experimental, ,Control) are not

significantly different. When the regression slopes are significantly different

ANCOVA test is not-allowed. When the ANCOVA test is significant, the

conclusion can be that the difference between the Post-scores of the groups is due

to the treatment it received.

3.7.5 Intervention: The Uniqueness of the “Probability in Daily Life” Learning Unit21 The main characteristic of the learning unit "Development of Critical Thinking by Means

of Probability in Daily Life" in the infusion approach (Swartz, 1992) is the innovative,

‘different’ teaching method that lies at the basis of this unit. The intervention learning

unit is based on reworking and enriching the syllabus of the standard “Probability in

Daily Life” learning unit22. The length of the unit is 15-16 double lessons (30-32

academic hours), and it has a fixed lesson structure. For details on the intervention see

Chapter 4.

26 For elaboration on the new learning unit see Chapter 4. 22 Includes the following topics: rules of probability, conditional probability and Bayes theorem, statistical connection and causal

connection, judgment by representativeness, regression, and failures in perception of regression.

Research Method

36

3.8 Pilot Study

Before beginning the present research, a pilot research was conducted in order to check

its feasibility and advisability. A corresponding purpose of the pilot research was to test

the tools chosen and developed for the research.

3.8.1 Pilot Study Description

Performing a pilot study helped to test the research tools and the learning unit. This pilot

study had been conducted between October 2006 and June 2007 by the researcher. The

study was carried out as an experimental application of the learning unit during the

school year 2006-7 in the framework of "Negishut – Access to Higher Education"

program that involved 40 participants. The duration of the research was 8 months

approximately. In the course of the research, all three questionnaires were answered by

the participants. The researcher taught the learning unit while developing, changing and

designing it in parallel with the teaching. At the end of every lesson the researcher wrote

a lesson report and field notes (see Appendices).

Table 4: Stages of the Pilot Study: Goals, Tools, Population, Data Collection Methods

Data collection and

processing

Populatio

n

Method

Purpose

Time of research

Stage of research

Documenting the course by means of teacher’s journals, students’ work and field notes. Questionnaires accompanied by personal interviews. Qualitative and quantitative processing of data.

Negishut 41 students

Designing, applying and evaluating the learning unit. Shaping and correcting it according to need. Unit duration: 30 academic hour

Developing critical thinking skills

Spring and Fall semesters 2006-7

Pilot

Research Method

37

3.9 Summary of Research Description Table 5 presents in detail the entire process of research in chronological order, showing

what stages of intervention, data collection and processing methods, and research

populations were involved in each stage.

Table 5: Stages of the Proposed Research: Goals, Instruments, Population, and Data Collection Methods

Data collection and processing methods

Research population

Method Research purpose

Time of research

Stage of research

Documenting the course in teacher's journals, students' works and field notes. Questionnaires accompanied by personal interviews. Qualitative and quantitative processing of the data.

Negishut

41 students

44 Students

60 Students

Designing, applying and evaluating the learning unit. Shaping and correcting it according to need. Teaching of the learning unit, by the researcher Teaching of the learning unit, by the researcher and an additional teacher

Developing critical thinking skills Development of critical thinking skills

Spring and Fall semesters 2006-7 Fall Semester, Spring Semester 2007-2008 Fall Semester, Spring Semester 2008-2009 2009-2010 2009-2010

Pilot Applying the learning unit model Round number one Applying the learning unit model Round number two Evaluation Carrying out the research

The Intervention: The Learning Unit “Probability in Daily Life”

38

4. The Intervention23: The Learning Unit “Probability in Daily Life” The business of education is less about what the teachers do and more about what they make the students to do.

(David Perkins)

The purpose of the intervention (teaching the present learning unit) is for the students to

develop critical thinking skills and dispositions while thoroughly studying the unit's

contents. We created a new experimental version of the learning unit (described in

section 4.2 below) by adding new questions to the problems presented in the unit, which

call for exercise of specific critical thinking skills, as well as adding several entire new

problems to the original unit (described in section 4.1 below). The teaching of critical

thinking skills in this research is integrated with the contents of the learning unit and

oriented to foster transfer, conservation and improvement of these skills by means of

thinking dispositions and abilities, which include the motivation to apply the skills in

specific subjects. This method of teaching does not require changes in the curriculum and

integrates well into its present structure.

The construction of the learning unit based on a taxonomy of skills for developing critical

thinking was expected to enable the students to thoroughly think through problems in this

specific field of knowledge, use their prior acquaintance with the field, apply logical

patterns to the problems arising in the field, make inferences based on mathematical

models, develop skills of mathematical-logical thinking (inference, generalization,

analysis, proposing, testing and proving hypotheses), and examine the resulting answers

in an informed and critical way.

Section 4.2 reviews the infusion-approach teaching of critical thinking within the specific

topic and learning unit, “Probability in Daily Life.” In the study of this topic, which

appears objectively scientific, there is much room for critical learning and teaching, and

asking questions such as why, how, for what reason a certain phenomenon takes place in

certain situations and not in others. Are the results contingent or representative? Is there

any regularity? Studying our version of “Probability in Daily Life” foregrounds critical

thinking abilities while not requiring an extensive prior knowledge of mathematics,

introduces a different discourse into the class, in addition to the mathematical discourse.

In the framework of this other discourse, the students discover the connection between

23 see appendix 6: Unit description.


39

mathematics and life. Thus, this learning unit will be analyzed here as a jumping-board

for critical thinking.

One excellent way of learning to reflect on one’s own thinking, provided by the unit, is

examining the intuitive errors in probability judgments. Kahneman, Slovic & Tversky

(1982) claimed that intuitive errors proceed from using certain heuristic principles that

often lead to erroneous probability judgments24. For instance, according to the principle

of representativeness, people assess the probability of an event according to the extent to

which this event’s description reflects the way they perceive the set of its most likely

consequences, or alternatively, the process that produces the event25. In a number of cases

when adult respondents were asked to compare the probability of two different events,

they displayed a tendency to assess them as equiprobable, while in reality one event was

more probable than the other. For instance, the chance of getting a 6:6 pair in throwing

two dice was assessed as equal to the chance of getting a 5:6 pair. It was found that this

tendency was not affected by the respondent’s age, acquaintance with the theory of

probability, or experience in games of chance. This erroneous judgment proceeded,

according to the researchers, from activating a cognitive model according to which

random events are perceived as equally probable by nature. In other words, different

aspects of psychological mechanisms of probability intuitions have been examined so far,

but the researchers did not examine whether probability intuitions might not be connected

to what Piaget (1978) called “the operational (logical and analytic) capacities of the

individual.”

4.1 The Learning Unit "Probability in Daily Life" (Lieberman & Tversky, 2002) This unit in probability studies is part of the formal high school curriculum of the Israeli

Ministry of Education. It was chosen because its rationale is to make the students to

"study issues relevant to everyday life, which include elements of critical thinking”

(Lieberman & Tversky 2002, Introduction p.3). In this unit, students must analyze

problems using statistical instruments, as well as raising questions and thinking critically

about the data, their collection, and their results. Students learn to examine data 24 The errors are divided into the following types: 1. Representativeness. 2. Gambler’s Fallacy. 3. Conjunction Fallacy. 4. Availability. 5. The Falk Phenomenon. 6. Insensitivity to the sample size. 7. Simple and Compound Events – Equiprobability Bias. 25 For instance, in the game of lottery, the sequence of 1, 16, 13, 20, 6 will be perceived as more probable than the sequence 1, 2, 3, 4, since the former better represents the randomness of lottery results (Kahneman & Tversky, 1972).


40

qualitatively as well as quantitatively. They must also use their intuitions to estimate

probabilities and examine the logical premises of these intuitions, along with

misjudgments of their application. The original unit is unique in its own right because it

explores probability in relation to everyday problems. It involves elements of critical

thinking such as tangible examples from everyday life, checking the credibility of

information, accepting and dismissing generalizations, rechecking data, doubting,

comparing new knowledge with the existing knowledge. This unit is characterized by

questions such as “Define the term ‘critical thinking’,” “Give examples of a problem

where a controlled experiment can be used,” “Give examples of failures and misleading

commercials,” and “Give examples of a scientific truth that was dismissed.” While

studying the subject, the connection is checked between statistical judgment and intuitive

judgment, and intuitive mechanisms that produce wrong judgments are explored. While

studying the subject, students are expected to acquire the tools for critical thinking. In the

beginning, students learn the mathematical tools necessary for performing calculations,

and later on they use the probability part: causal connection and mechanisms of intuitive

judgment, which are considered more of a psychological projection.

4.2 Our Intervention: The Uniqueness of the “Probability in Daily Life” Learning Unit Modified for Infusion Approach Teaching26 The main characteristic that distinguishes the learning unit "Development of Critical

Thinking by Means of Probability in Daily Life" modified by the researcher for infusion

approach teaching (Swartz, 1992) from the original one is the innovative, ‘different’

teaching method that lies at the basis of this unit. Teaching in the infusion approach

necessitates a different structure of teacher-student interaction in class that diverges from

the traditional one used in most ‘traditional’ math lessons. This method is known in

literature as "dialogical teaching" (Ron, 1993) or "negotiating knowledge" and is

26We are dealing with a "negotiating knowledge" (Ron, 1993) and is characterized by classroom discussions between the teacher and the students and among the students themselves during the lessons, so that the teacher won't be the only person speaking in class. The teacher's role is to encourage the students to make changes in their systems of perceptions and concepts, convince them that such changes should be made, and help the students to make these changes (in particular, by allowing the students to talk the topic over among themselves and discuss it together in small groups, in a less ‘threatening’ form than doing it in the larger forum of the whole class). To fulfill this function, the teacher can use various teaching methods at his/her disposal: oral explanation, texts, experiments, demonstrations, video films, computer programs, the students' own work, group discussions, etc. The method of negotiating knowledge in the classroom emphasizes that the use of any teaching methods or tools must be accompanied by a dialog between the teacher and the class, and among the students themselves, i.e. by classroom and group discussions. Such discussions may have different purposes (according to specific situations).


41

characterized by classroom discussions between the teacher and the students and among

the students themselves during the lessons, so that the teacher won't be the only person

speaking in class. The teacher's role is to encourage the students to make changes in their

systems of perceptions and concepts, to convince them that such changes should be

made, and to help the students to make these changes (in particular, by allowing the

students to talk the topic over among themselves and discuss it together in small groups,

in a less ‘threatening’ form than doing it in the larger forum of the whole class). To fulfill

this function, the teacher can use various teaching methods at his/her disposal: oral

explanation, texts, experiments, demonstrations, videos, computer programs, the

students' own work, group discussions, etc. The method of negotiating knowledge in the

classroom emphasizes that the use of any teaching method or tool must be accompanied

by a dialog between the teacher and the class, and among the students themselves, i.e. by

classroom and group discussions. Such discussions may have different purposes

(according to specific situations). During such discussions the students may, for example,

express their opinions about the topic currently studied, present the insights they acquired

as a result of different learning and teaching strategies, ask questions, make comments,

argue about interpretations, and so on. It is important to emphasize that the main

characteristic of these sessions is a meaningful, authentic dialog where the students feel

free to express their original thoughts instead of the ideas the teacher expects them to

learn.

The “Probability in Daily Life" learning unit has been included since 2005 in the

Ministry of Education’s official curriculum for students who take five or four learning

units. This topic was added to the curriculum because we are daily required to make

decisions under the conditions of uncertainty. Our decisions in all aspects of life are

made after collecting data, processing them and making a judgment, all of which stages

already require application of critical thinking dispositions, abilities and skills. Making a

judgment consists of two parts: statistical judgment, based on numerical data, and

intuitive judgment, which is a personal evaluation of the situation. “Probability in Daily

Life" includes the following topics: rules of probability, conditional probability and

Bayes theorem, statistical connection and causal connection, judgment by

representativeness, regression and failures in perception of regression.


42

In addition to the skills found in the "Probability in Daily Life" learning unit, we have

added some problems where a student is required to use analysis, problems that raise

questions and make the student apply critical thinking about data and information27. As a

result, the modified version of the unit is unique in providing an opportunity to study in a

regular mathematics course topics that are both interesting, relevant to everyday life, and

include elements of critical thinking, such as checking the reliability of information,

supporting or refuting generalizations, re-checking quantitative data, doubting,

comparing the newly presented knowledge with prior knowledge. Yet, even after the

modifications were introduced, the length of the unit remained 15-16 double lessons (30-

32 academic hours), same as the original unit’s length. The main purpose is achieved by

means of the teaching method on which the modified unit is based, the "dialogical

teaching" by means of classroom negotiation of knowledge. This system, when used

optimally, enables the students to discuss their own and their friends’ perceptions, as well

as those proposed by the teacher or the textbook, in a critical and skeptical way. Students

are encouraged not to accept an idea or an explanation unless they understand its

meaning and agree that it describes the state of things in the best possible way.

Classroom or group dialogic discussion is the tool recommended for clarifying ideas,

asking questions, doubting, and reaching a shared meaning.

This unit is characterized by questions of the following kind: define the concept "critical

thinking," give an example of a problem solved by means of a controlled experiment,

give examples of failures and misleading advertisement, give an example of a scientific

truth that has been refuted. Studying probability, we examined the connection between

statistical and intuitive judgment, raised the psychological points of failure that lead to

wrong judgments. In other words, while studying this topic, the students also learned

how to think critically. In the beginning they learned mathematical tools for performing

calculations, and then learned how to use the professional terminology: causal

connection, inversion of a relation and intuitive judgment mechanisms, which can be

defined as psychological implications of dealing with probability. 27 Problems of the type described here are complex not only because they deal with a single event, but also because they do not always have a single straightforward answer. As noted in the theoretical background section, the purpose of this learning unit is to teach the students not to be satisfied with a numerical answer but to check the data and their validity, and in those cases when there is no single numerical answer, to know how to ask the appropriate questions and analyze the problem qualitatively and not only quantitatively. Along with imparting statistical tools, the unit also introduces intuitive mechanisms used by people for evaluating probability in everyday situations, and examines the biases and errors that intuitive judgment often involves, through contrasting the intuitive judgment with the probability calculation that the discipline of probability requires.


43

The new probability unit included additional questions taken from daily life situations,

newspapers and surveys, and combined with critical thinking skills. Each of the fifteen

lessons that comprised the unit had a fixed structure. The lesson began with a short article

or text that was presented to the class by the teacher. A generic question relating to the

text was then written on the blackboard. Then an open discussion of the question took

place in small groups of four students. Ten minutes were allotted for the discussion, and

there was no intervention by the teacher. Each group offered their initial suggestions

about how the question could be resolved, and included practicing the critical thinking

skills. An open class discussion then followed. During the discussion, the teacher asked

the students different questions to foster the students’ thinking skills and curiosity and to

encourage them to ask their own questions. The students presented their different

suggestions and tried to reach a consensus. The teacher related to the questions raised by

the students and encouraged critical thinking, while instilling new mathematical

knowledge. Thus, the intervention combined teaching critical thinking skills (abilities and

dispositions) and mathematical knowledge (probability) using the infusion approach.

Using Ennis’ clear distinction between abilities and dispositions, this study focuses only

on the development of the abilities. A future study will deal with developing the

dispositions. The mathematical topics taught during the fifteen lessons included

introduction to set theory, probability rules, building a 3D table, conditional probability

and Bayes theorem, statistical and causal connection, Simpson's paradox, and judgment

of representativeness. With regard to critical thinking, the following skills were

incorporated in all fifteen lessons: a clear search for a hypothesis or question, evaluation

of the sources’ reliability, identifying variables, “thinking out of the box,” and a search

for alternatives (Aizikovitsh & Amit, 2009, 2010). Below we cite two examples of

lessons from the “Probability in Daily Life” learning unit according to the infusion

(intervention) approach. First, in order to illustrate the structure of a lesson in the unit, we

have included here a detailed description of one lesson called “The Aspirin Case.”

Following the description, we analyze the lesson according to the following techniques:

referring to information sources, raising questions, identifying variables, and suggesting

alternatives and inferences. The lesson’s topic was conditional probability. The critical

thinking skills practiced in the lesson were evaluating the reliability of the source,

identifying variables, suggesting alternatives, and inference.


44

Example 1: “Shoes and Mathematics”

Avi: “There is a connection between the size of shoes and the level of mathematical knowledge” Beni: “Can’t be” Avi: “Go to the school next door and see for yourself” Beni: “You are right, the kids who wear bigger shoes really know math better!” Why is this phenomenon true? What do you think about the conclusion?

Example 2: “Calcium and Vitamin D”Read the following text, “Calcium and Vitamin D

Contribute to Dental Health,” from Yediot Ahronot, and answer the following questions.

Calcium and Vitamin D Contribute to Dental Health

Taking calcium and vitamin D as food supplements can help to keep one’s teeth healthy. This connection

arises from a research conducted in the Boston University School of Dental Medicine, which was published

in The American Journal of Medicine.

The study involved 145 participants aged above 65. Part of them took 500 mg calcium and 700 UE vitamin

D daily, and the rest took placebo. In the control group, 27% of participants lost at least one tooth in the


45

course of the three years of research, as opposed to only 13% in the experimental group. The researchers

performed an additional check several years after the end of the experimental period, and found that 40%

of the experimental group lost at least one tooth since the end of the experiment, as opposed to 59% of the

control group.

What connection does the news item discuss? Is it possible to provide a logical explanation for

this connection? Propose at least two factors that can mediate the connection described in the

news item.

4.2.1 Case Study I: The Aspirin Case In the first phase of the lesson, a copy of the following text was handed out to students:

Your brother woke up in the middle of the night, crying and complaining he has a stomachache.

Your parents are not at home and you don’t know what to do. You give your brother aspirin, but

an hour later he wakes up again, suffering from bad nausea and vomiting. The doctor that

regularly takes care of your brother is out of town and you consider whether to take your brother

to the hospital, which is far from your home. You read from a book about children’s diseases and

find out that there are children who suffer from a deficiency in a certain type of enzyme and as a

result, 25% of them develop a bad reaction to aspirin, which could lead to paralysis or even

death. Thus, giving aspirin to these children is forbidden. On the other hand, the general

percentage of cases in which bad reactions such as these occur after taking aspirin is 75%. 3% of

children lack this enzyme.

(Probability Thinking, p. 30, with slight revisions made by the researchers)

The second phase was to divide the class into small groups (up to five students) and to

present them with the following questions that they had to discuss: should you take your

brother to the emergency room? What should you do? Can aspirin consumption be lethal?

The next phase of the lesson was a continuation of the discussion in the framework of the

whole class, under the teacher’s direction. The generic questions on the blackboard were:

Should you take your brother to the emergency room? What should you do?

The following is the transcription of the discussion that took place in the classroom. Teacher: What do you think?* Student 1: Where is the information taken from? Can we see the article for ourselves?* S2: Is the source reliable? How can we check it?* S3: Where is the article taken from? What is its source?


46

S1: Should I answer the identification of the sources question? T: Not yet. We are focusing on searching for questions. Please think of other questions. S3: What connection does the article discuss? S2: first we need to identify the variables!!! T: Right. First, we ask what the variables are. S4: You can infer it from the title that suggests that a connection exists between aspirin and death. T: According to the data from the article, Can we find a statistical connection? (the student already know this subject) S2: I know! We can ask: suggest at least 2 other factors that might cause the described effect. S5: The question is what causes what? S6: Can aspirin consumption be lethal? T: What do you think? T: How can you be sure? S6: Umm… S3: Are there other factors, such as genetics!? T: Very good. What did student 3 just do? S1: He suggested an alternative!! T: How can we check it? Do you have any suggestions? Can you make a connection between this problem and the material we have learned in the past few lessons? Can you offer an experiment that would solve the problem? S3: Of course. An observational experiment.

The fourth phase of the lesson focused on encouraging critical thinking and instilling

new mathematical knowledge (Bayes theorem) and statistical connections by referring to

students’ questions and further discussion. A teacher-led discussion focused on methods

of analysis using such critical thinking skills as source identification (Medical manual);

evaluating the source reliability (high); identifying variables (A – enzyme deficiency, D –

adverse reaction to aspirin). The mathematical knowledge the students had to use was

Bayes theorem: Data: P(D/A)=0.25 P(D)=0.75 P(A)=0.03, To prove: P(A/D)=? Using

Bayes theorem (or a two-dimensional matrix), the result is that only 1% of the children

without the enzyme develop an adverse reaction to aspirin, thus there is no need to go to

the hospital. Even so, is it worth taking the risk? What do you think? (question to the

class).

4.2.2 Case study II: Calcium, Vitamin D, and Dental Health The following detailed description of a session will be analysed with respect to the

following skills: evaluating reliability of information sources, raising questions,

identifying variables, suggesting alternatives and inference. The session subject was


47

statistical connection and causal connection. The session's aim was to teach the students

to determine the presence of causal connection. The following mathematical concepts

were used in the session: determining how a third factor can affect a statistical connection

between two existing factors, including Simpson's paradox (the combination of A and B

seeming to cause reversal of “success”).

4.2.3 Session plan Phase A. At the beginning of the session, the teacher presented a short article about a

research that indicates a connection between calcium and vitamin D intake and dental

health. The research is taken from a daily Israeli newspaper that translated an article from

The American Journal of Medicine. The teacher writes a question on the blackboard. The

students are requested to address the question. Phase B involved discussion in small

groups about the article and the question. Phase C consisted of open class discussion.

During the discussion the teacher asked the students various questions to foster the

students’ thinking skills and curiosity and to encourage them to ask their own questions.

In Phase D, the teacher referred to the questions raised by the students and encouraged

critical thinking while instilling new mathematical knowledge – the identification of and

finding a causal connection by a third factor and finding a statistical connection between

C, and A and B, as well as Simpson's paradox.

The discussion conducted in class The practiced skills

The article presented to the class was "Calcium and vitamin D contribute to dental health" and claimed that the consumption of calcium and vitamin D nutritional supplements can help protect the teeth. The data were taken from a research conducted in a dentistry school at a university in Boston and published in The American Journal of Medicine. In this research there participated one hundred and forty five people aged thirty five and above. Part of them took calcium and vitamin D and the rest of them took placebo. Of the placebo group, 27% lost at least 1 tooth during the experimental period, in comparison to 13% of the calcium and vitamin D group.

The generic question on the blackboard was:

Is calcium good for your teeth?

1 Teacher: Last week I visited a friend who is

In paragraph 1 we encounter skills such as "searching for the question"- a fundamental skill. First there is a need to clarify the starting point for the interaction with the student. We also need to clarify to ourselves what is the thesis and what is the main question before we approach decision-making. The paragraph also demonstrates relevance to daily life.

In paragraph 2 the students are taking a step back, we refer to "identifying information source and evaluating the source's reliability" skill. This step is crucial, as it helps us to assess the quality and the


48

a dentist. When we sat to the table she served a variety of cheeses and told me she read in the newspaper calcium was good for our teeth and presented me with the article. What should I check before I decide whether I should increase the amount of calcium I consume? Should I eat more calcium or not? What do you think?*

2 Student 1: Where is the article taken from? Can we see the article for ourselves?*

3 S2: Is the article's source reliable? How can we check it?*

4 S3: Where is the article taken from? What is its source?

5 S1: Should I answer the identification of the sources question?

6 T: Not yet. We are focusing on searching for questions. Please think of other questions.

7 S3: What relation does the article discuss?

8 T: A very good question. Before you look for the relation, what do you need to do?

9 S2: To identify the variables!!!

10 T: Right. First, we ask what the variables are. Then we refer to the relation between them.

11 S3: Do you mean a statistical connection?

12 S4: What a silly question. It's obvious.

13 S3: What’s so obvious?

14 S4: The connection is obvious – statistical relation between the vitamin and healthy teeth.

15 S3: How do you know?

16 T: There are no silly ideas or silly questions in this class. In fact, student 3's question is excellent. Student 4, please try and think why student 3's question is a good one. Try to follow student 3's line of thought, remembering our discussion last week.

17 S4: If there is a connection, then it must be a statistic al relation, right?

18 T: Did you calculate the existence of P(A/B) ≠P(A/B)?

19 S4: You can infer it from the title that suggests that a relation exists between taking vitamins and healthy teeth.

20 S3: According to the data from the article, you can find a statistical relation (the student

validity of the article discussed. This skill was practiced in past lessons. See the paragraph that summarizes the article.

In paragraph 6 we encounter "searching for the question" skill again. We will continue searching for the main question through practicing the "variables identification" skill.

Raising the search for alternatives. Posing questions enables the practice of this skill.

P(A) , P(B), N(S)

Paragraph 10 deals with identifying the variables and understanding them by a 2D table and a conditional probability formula

( )( / )( )

P A BP A BP B∩

= ⇒

The mathematical part P(A/B)≠ P(A/B).

Calculations according to sets and supplementary sets.

In paragraph 16 the teacher builds the students' self esteem by encouraging them to express their ideas and opinions (even if they are not always correct or relevant). She prevents any intolerance of other students. The method of instruction that aims at fostering the confidence and the trust of the students in their critical thinking abilities and skills is, according to Ennis, "relating to other peoples points of view" and "being sensitive towards other peoples' feelings.”


49

specifies the calculation).SF

21 T: Very good. An excellent inference. I want you to keep thinking of other questions.

22 S4: Can you give a reasonable explanation for the relation we found?

23 S2: I know! We can ask: suggest at least 2 other factors that might cause the described effect.

24 S5: The question is what causes what?

25 S6: Does vitamin D contribute to healthy teeth?

26 T: What do you think?

27 S6: Vitamins contribute to healthy teeth.

28 T: How can you be sure?

29 S6: Umm…

30 S4: Does deficiency in vitamin D cause damage to the teeth?

31 S3: Are there other factors, such as genetics!?

32 T: Very good. What did student 3 just do?

33 S1: He suggested an alternative!!

34 T: How can we check it? Do you have any suggestions? Can you make a connection between this problem and the material we have learned in the past few lessons? Can you offer an experiment that would solve the problem?

35 S3: Of course. An observational experiment.

In paragraph 23 the student is referring to other sets and finding the connection between them.

Paragraph 31 demonstrates the skill of "Searching for alternatives".

Paragraph 35 refers to a controlled experiment or an observational experiment.

An additional grouping and finding the connection between the variables by Bayes theorem or a 2 dimensional table.

Table 6: Classroom discussion of an article and the infusion of CT skills

4.2.3.1 Analysis According to Critical Thinking Skills According to the infusion approach, students practice28 their critical thinking while

acquiring technical probability skills. In this session, the following five skills are

exercised. (i) Raising questions – asking question about the article and probing on the

main question about the connection between calcium and vitamin D contribute to dental

28During such discussions the students may, for example, express their opinions about the topic currently studied, present the insights they acquired as a result of different learning and teaching strategies, ask questions, make comments, argue about interpretations, and so on. It is important to emphasize that the main characteristic of these sessions is a meaningful, authentic dialog where the students feel free to express their original thoughts instead of the ideas the teacher expects them to learn.


50

health (see paragraph 1). (ii) Referring to information sources and evaluating the source's

reliability - the article went through a number of interpretations. It was published in an

Israeli newspaper, which translated it from an American journal, which, in turn,

published a research that had been conducted in a dentistry school in a university located

in Boston but not mentioned by name. All of the above raised the students’ scepticism

(paragraph 2) (iii) Identification of variables – students identified the variables: calcium,

vitamin D, dental health (paragraph 6) (iv) Following these skills, another skill, searching

for alternatives (paragraph 31), was used. In class we spoke about suggesting alternatives,

not taking things for granted but examining what has been said and suggesting other

explanations. At this stage, we combined the mathematical aspect of the session – the

connection reversal (a third factor that reverses the conclusion made beforehand). We

found the connection between the tree events (A, B and C) (v) Another skill that was

practiced is inference, in light of the alternatives suggested. Thus, the skills practiced in

the described session are: raising questions, evaluating the source's reliability, identifying

variables, suggesting alternatives and inference.

In order to understand and monitor the student's attitudes toward critical thinking as

manifested by the skills specified above, interviews were conducted after the above

session. In these interviews, the students expressed their acknowledgement regarding the

importance of critical thinking. Moreover, students were aware of the infusion of

instructional strategies that advances critical thinking skills. For instance, Student 4 said,

while defining critical thinking:

"I think critical thinking is important when you study mathematics, when you study other

topics and when you read a newspaper, but it is most important when you deal with real

life situations, and you need the right instruments in order to do so (deal with these

situations)."

When student 2 was asked about important components during the last few classes and

the present class, she answered:

"Now I understand 'variables identification' and it helps in everyday life. The issue of

"intermediate factor" and the meaning of "reversing the connection" is also very


51

important. Besides,” she added with a grin, “now I’m more sceptical about what they

write in the paper."

To conclude, this unit is unique because it provides an opportunity to study, in a regular

mathematics course, topics that are both interesting, relevant to everyday life, and

include elements of critical thinking, comparing the newly presented knowledge with

prior knowledge. In other words, while studying this topic, the students also learned how

to think critically. In the beginning they learned mathematical tools for performing

calculations, and then learned how to use the professional terminology: causal

connection, inversion of a relation and intuitive judgment mechanisms, which can be

defined as a psychological implication of dealing with probability Figure 3 presents all

the stages of the intervention according to the times when each questionnaire was passed.

Figure 2: Description of the Research Process

Intervention Probability in daily life

Pre- CCTDI and Cornell test

Statistical Connection

Causal Connection

Judgment of Representativeness

Post-CCTDI and Cornell test

Dispositions of Critical Thinking

52

5. Dispositions of Critical Thinking: Methods, Results29 and Discussion “The most important thing is not to stop asking” (Albert Einstein)

This chapter presents the methods and results regarding the dispositions toward critical

thinking according to Ennis’ taxonomy and Facion’s theory and discusses them. The

major tool through which the results were obtained is the CCTDI thinking test.

5.1 The Research Question

To what extent does the study of “Probability in Daily Life” in the infusion approach

contribute to the development of critical thinking dispositions?

5.2 Method

5.2.1 The Instrument: CCTDI Critical Thinking Test CCTDI was used in order to evaluate the students’ dispositions toward critical thinking.

This tool, based on the seven positive aspects of the disposition for critical thinking, was

designed to measure general dispositions profile of the students. CCTDI is divided into

seven sub-tests: truth-seeking (sub-test T), intellectual openness (sub-test O), analyticity

(sub-test A), systematicity (sub-test S), self-confidence in critical thinking (sub-test C),

inquisitiveness (sub-test I) and maturity (sub-test M). The following descriptions of the

sub-questionnaires are based on Facione et al. (1995) and Facione (1992). The sub-test T

deals with the inclination to investigate in order to arrive at the fullest and most adequate

information possible in a given context, raising questions in a courageous way, as well as

honesty and objectivity in searching for information, even if this information goes against

the investigator’s personal interests or opinions. For instance, those who incline for truth-

seeking will not agree with the following claim: “All people, including myself, always

present claims that follow from their personal interest,” or, “If there are four reasons ‘pro’

and one ‘contra’, I will be in favor of the four.” The sub-test O deals with the inclination

for intellectual openness and tolerance towards other opinions, while remaining aware of

the fact that one’s own opinions are different. People who are not tolerant towards views

29 The results were analyzed both with and without the high-achieving Kidumatica group, in order to maximize the high school teachers' gain from and application of the results. It was important to show that improvement was achieved not only in high-achieving groups but in regular high school groups as well, with research population similar to those students the teachers are regularly working with. In practical terms, the Kidumatika group was excluded from the total research population in order for the teachers to be able confidently apply the research recommendations.


53

different from their own will agree with the claim “It’s important for me to understand

what other people think on various issues.” The sub-test A focuses on application of

cause and proof, awareness of problematic situations and an inclination to predict

outcomes. For instance, students are asked to answer “agree” or “disagree” to the

following statement: “People need reasons for disagreeing with someone else’s

opinions.” The sub-test S checks organization, order, focus, and commitment to

investigation, and uses test statements of the following kind: “My opinion on

controversial issues depends mostly on the last person I have had a conversation with.”

The sub-test C measures the person’s confidence in his/her own thinking process, and

uses test statements such as “I do better in exams that demand thinking, not only

memorizing,” or “I take pride in my ability to understand other people’s opinions.” The

sub-test I measures the keenness to acquire knowledge and find explanations, even when

this knowledge does not seem immediately applicable. Representative test-statements are:

“No matter what the topic, I am keen to learn more about it,” or “Learn all you can, you

never know when you may need to use this knowledge.” Finally, the sub-test M measures

the person’s inclination to be critical about his/her own decision-making. A mature

person who thinks critically can be defined as a person who approaches problems

inquisitively and makes decisions while knowing that some problems are inherently

poorly constructed, and others have multiple solutions. The CCTDI total score is a

measure that estimates one's overall disposition toward critical thinking. A person may be

positively and strongly disposed toward seeking to solve problems and address questions

using reflective judgment, that is critical thinking; or ambivalent toward that, or even

negatively disposed and hostile toward that approach. The total score is based on all 75

items. Facione et al. report correlations that support the simultaneous validity between

scores in CCTDI sub-tests and psychological tests.

Example of Statements (Selected)30

6. It disturbs me when people rely on weak claims to defend good ideas.

8. It disturbs me that I may be under influences that I am not aware of.

15. Most topics studied in school are not interesting and not worth participation.

16. Exams that demand thinking and not only memorizing are better for me.

22. It is easy for me to organize my thoughts.

30 For elaboration see appendix.


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24. There is a limit to openness when we get to the question of what is right and what is wrong. 31. I must have a basis for my beliefs. 39. It is very hard not to be biased when discussing my own opinions.

Scale Code No. of items

Description Sample Item Reliability

Truth-Seeking

T 12 The inclination to investigate in order to arrive at the fullest and most adequate information possible in a given context, raising questions in a courageous way, as well as honesty and objectivity in searching for information, even if this information goes against the investigator’s personal interests or opinions.

- If there are four reasons ‘pro’ and one ‘contra’, I will be in favor of the four.

- (negative item) All people, including myself, always present claims that follow from their personal interest

.61

Open-Mindedness

O 12 The inclination for intellectual openness and tolerance towards other opinions, while remaining aware of the fact that one’s own opinions are different.

(negative item) It’s important for me to understand what other people think on various issues

.64

Inquisitiveness I 9 Measures the keenness to acquire knowledge and find explanations, even when this knowledge does not seem immediately applicable.

- No matter what the topic, I am keen to learn more about it

- Learn all you can, you never know when you may need to use this knowledge

.75

Systematicity S 11 Checks organization, order, focus, and commitment to investigation.

My opinion on controversial issues depends mostly on the last person I have had a conversation with.

.70

Maturity M 11 Measures the person’s inclination to be critical about his/her own decision-making. A mature person who thinks critically can be defined as a person who approaches problems inquisitively and makes decisions while knowing that some problems are inherently poorly constructed, and others have multiple solutions.

.71

Self-Confidence C 10 Measures the person’s confidence in his/her own thinking process.

- I do better in exams that demand thinking, not only memorizing.

- I take pride in my ability to understand other people’s opinions.

.79

Analyticity A 10 The application of cause and proof, awareness of problematic situations and an inclination to predict outcomes.

People need reasons for disagreeing with someone else’s opinions

.71

Table 7: Scale of Critical thinking Disposition by Facione


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In addition, they report a Cronbach’s alpha of 0.60 to 0.78 for sub-tests and of 0.90 for

the questionnaire as a whole among college students (N=1019). The CCTDI includes 75

items of six levels each. Each of the seven sub-tests is composed of 9-12 items dispersed

over the entire questionnaire. The CCTDI was translated into Hebrew, and the content of

most items remained identical with the original, with a few minimal adjustments to the

Israeli context. In a similar way, the questionnaire was translated word for word into

Italian, with the exception of a few minor adjustments. The Italian version was translated

back into English to make sure that the meaning of the items was preserved. A pilot test

of the questionnaire was conducted, which revealed that the language and the meaning of

the items is clear to test subjects in Israel. It took one lesson for the test subjects to

answer all 75 items in the questionnaire.

5.2.2 The research population Table 8 summarizes population sizes of all groups

Disposition Test

Group School First Round Second Round

Experiment Kidumatica 41 17

High School 1 30 22

High School 2 31,19

Control High School 1 29

High School 2 32,27

Total 71 177

Table 8: Number of students each round 5.3 Results of Dispositions The first part describes the findings of the first round, the second part describes the

second round, and the third part describes the difference between the rounds in critical

thinking dispositions.

5.3.1 Results of the First Round (n=71) The first round of teaching the unit was conducted only in two groups: the Kidumatica

group and a regular high school group.


56

5.3.1.1 The “Kidumatica” group (n=41) This round is the first round in the Kidumatica class. Chart 1 schematically describes the

Post vs. Pre average CCTDI test sub-scale scores for “Kidumatica” (the t-test values are

presented in full in Table 9). Points on the diagonal represent cases where the pre- and

post-scores were equal. Consequently, points above the diagonal show an improvement.

Only the Analyticity Scale showed a significant change, i.e. the Pre-test was significantly

higher then the Post-test.

Post vs. Pre for Kidumatica

20

24

28

32

36

40

20 24 28 32 36 40 Pre

Post

Truth-SeekingOpen-mindednessInquisitivenessSystematicityMaturityConfidenceAnalyticity

Chart 1: Disposition of CT “Kidumatica”

Table 9: Disposition of CT in the “Kidumatica” group

5.3.1.2 The “High School 1” Group (n=30) Chart 2 schematically represents the Post vs. Pre average CCTDI test sub-scale scores for

“High School 1” (the exact t-test values are presented in full in Table 2). Both the chart

and the table reveal that in the first round there was a significant improvement in the sub-

scales of Systematicity, Maturity and Analyticity, whereas there was no improvement in

the other sub-scales.

t value Post-test Pre-test Sub-scale SD Mean SD Mean N=41

0.73 5.97 39.32 6.74 38.59 Truth-seeking 0.26 5.63 33.27 5.10 33.05 Open-mindedness 0.58 8.55 24.68 8.07 23.78 Inquisitiveness -0.52 8.04 31.27 7.26 31.88 Systematic 0.68 9.29 33.12 8.85 32.17 Maturity -1.32 9.08 24.81 8.32 26.42 Confidence

-2.14(*) 6.85 28.29 6.48 30.44 Analyticity 0.26 5.72 30.68 4.58 30.90 CCTDI Total

(*)= difference significant at the .05 level


57

Post vs. Pre for HighSchool-1

20

24

28

32

36

40

20 24 28 32 36 40 Pre

Post Truth-SeekingOpen-mindednessInquisitivenessSystematicityMaturityConfidenceAnalyticity

Chart 2: Dispositions for CT "High-school"

Table 10: Disposition towards critical thinking in the “HighSchool 1” group: Means and standard deviations by sub-tests, pre and post questionnaires.

5.3.2. Results of the Second Round (n=177) In the second academic year, i.e. the second round, there were 177 students, 88 of them in

the control group and 89 in the experimental group. The Kidumatica class had 17

students.

5.3.2.1. Results for CCTDI Total Chart 3 schematically presents the mean values of the pre and post CCTDI Total score

for the students in the experiment and control groups. Table 11 presents the complete

results of all the statistical tests conducted on the data relevant to the CCTDI Total score.


0.71 4.36 35.20 6.10 34.30 Truth-seeking 1.08 4.77 29.97 5.68 28.77 Open-mindedness 0.33 4.71 27.73 7.31 27.27 Inquisitiveness

2.47(*) 5.14 37.10 7.15 33.23 Systematic 2.43(*) 5.83 33.10 7.10 28.70 Maturity -1.51 5.62 26.40 7.42 28.57 Confidence 2.72(*) 3.14 30.83 5.35 28.70 Analyticity 1.90 1.81 31.48 4.28 29.93 CCTDI Total

(*)=difference significant at the .05 level


58

Graph for CCTDI Total

20

25

30

35

40

exp pre exp post control pre control post

EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL

CONTROL-ALL

Chart 3: CCTDI Total Means

CCTDI Total

Pre-test Post-test Difference ANCOVA

Mean S.D.

t value for comparison

Exp with control

Mean S.D.


Exp with control

Mean S.D. t value for comparing Post to Pre


Exp with control

F value for comparison of

Post-score Exp with control

Control 32.49 4.46 34.25 3.04 1.75 4.78 3.44(***)

Exp 30.64 3.76 -2.99(**) 36.16 2.68 4.45(***) 5.52 5.30 9.82(***) 4.96(***) not allowed(a)

Exp without KD

30.64 4.01 -2.74(**) 36.24 2.54 4.43(***) 5.60 5.43 8.75(***) 4.77(***) not allowed(a)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 (a) significant difference between slopes of regression lines for the 2 groups (Experiment vs. Control) Table 11: CCTDI Total Statistical tests Results There was an initial (in the pre-test) significant difference (p<0.01) between the experiment

and the control group. Therefore we tried to used ANCOVA analysis to compare between

those groups. This analysis couldn't be performed for this score. The results show that the

experiment group improved by about six points whereas the control by only about two

points. The improvement in the experimental group was at least threefold compared to that of

the control group. This difference can be attributed to the learning process, as will be further

discussed. As discussed in previous chapters, the CCTDI test contains seven different sub-

scales. In the following sections each of the sub-scales will be analyzed. To unmask the

possible implications of the learning unit, one must decompose the CCTDI test to its

components, thus allowing incipient improvements. Following are the seven components of

the CCTDI test.


59

5.3.2.2. Results for sub-scale Truth-Seeking

Chart 4 schematically presents the means of the pre and posts CCTDI Truth-Seeking sub-

scale score for the students in the experiment and control groups. Table 12 contains the

complete results of all the statistical tests conducted on the data relevant to the CCTDI

Truth-Seeking sub-scale score.

Graph for Truth-Seeking

20

25

30

35

40



CONTROL-ALL

Chart 4: Truth-Seeking sub-scale Means

Truth Seeking


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control



Control 35.00 8.37 36.77 5.93 1.77 9.32 1.78 0.01 Exp 37.80 6.58 2.47(*) 37.35 6.25 0.63 -0.45 7.92 -0.54 -1.71 - Exp without KD 37.75 6.53 2.33(*) 37.28 6.26 0.52 -0.47 7.72 -0.52 -1.63 0.001

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 Table 12: Truth-Seeking sub-scale Statistical tests Results There was an initial (in the pre-test) significant difference (p<0.05) between the

experiment and the control group. Therefore we used ANCOVA analysis to compare

between those groups. The ANCOVA analysis reveals that the two groups were not

different in the post-test on this sub-scale. The control group improved by about two

points, but this improvement is not statistically different from zero and is not different

from the change that occurred in the experiment group. The results for the experiment


60

group coincide with literature reports of truth seeking in mathematics educations, as there

was no significant improvement. It is possible that the natural process of the development

of the truth seeking should not interfere and should not be included in the goals of the

learning unit.

5.3.2.3. Results for sub-scale Open-Mindedness

Chart 5 schematically presents the mean values of the pre and post CCTDI Open-

Mindedness sub-scale score for the students in the experiment and control groups. Table

13 presents the complete results of all the statistical tests conducted on the data relevant

to the CCTDI Open-Mindedness sub-scale score.

Graph for Open-mindedness

20

25

30

35

40



CONTROL-ALL

Chart 5: Open-Mindedness sub-scale Means

Open Mindedness


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control

F value for comparison

of Post-score Exp

with control Control 31.74 5.28 33.77 5.67 2.03 8.01 2.38(*)

Exp 31.20 5.93 -0.64 33.66 5.40 -0.13 2.46 7.72 3.01(**) 0.36 0.02 Exp without KD 30.83 5.98 -1.02 33.93 5.50 0.18 3.10 7.85 3.35(**) 0.84 0.03

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 Table 13: Open-Mindedness sub-scale Statistical tests Results

Chart 5 shows the means of the pre- and post-CCTDI - Open-mindedness sub-scale scores

of the students in the experiment and control groups. There was no difference between


61

the two groups for the Pre test and for the Post test scores. It is clear that there was a

significant improvement in the CCTDI Open-mindedness score (mean improvement of

approximately 2-3 points) for both groups. As can be drawn from comparison of the

graphs, the improvement in the experiment group was very similar to that of the control.

This lack of difference can perhaps be attributed to the developmental, psychological and

cognitive processes of young teenagers as well as to the learning process. Elaboration on

this topic follows in the discussion.

5.3.2.4. Results for sub-scale Inquisitiveness Charts 6 schematically presents the means of the pre and post CCTDI Inquisitiveness

sub-scale score for the students in the experiment and control groups. Table 14 contains

the full results of statistical tests conducted on the data relevant to the CCTDI

Inquisitiveness sub-scale score.

Graph for Inquisitiveness

20

25

30

35

40


EXPERIMENT-WITHOUT-KIDUMATICA

EXPERIMENT-ALL

CONTROL-ALL

Chart 6: Inquisitiveness sub-scale Means

Inquisitiveness


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control



Control 28.73 10.57 33.93 8.49 5.21 11.31 4.32(***)

Exp 24.84 8.71 -2.67(**) 39.34 6.41 4.78(***) 14.49 12.36 11.06(***) 5.22(***) not allowed(a)

Exp without KD 24.96 8.79 -2.42(*) 39.18 6.16 4.39(***) 14.22 12.17 9.92(***) 4.85(***) not allowed(a) (*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 (a) significant difference between slopes of regression lines for the 2 groups (Experiment vs. Control) Table 14: Inquisitiveness sub-scale Statistical tests Results


62

There was an initial (in the pre-test) significant difference (p<0.01) between the


between those groups. The ANCOVA analysis reveals that the two groups don't have

equal slopes, therefore ANCOVA can't be used. The results show that the Experiment

group improved by about 15 points whereas the control only by about five points. The

improvement in the experimental group was at least threefold compared to that of the

control group. This difference can be attributed to the learning process, as will be

discussed further.

5.3.2.5. Results for sub-scale Systematicity

Charts 7 schematically presents the means of the pre and post CCTDI Systematicity sub-


full results of statistical tests conducted on the data relevant to the CCTDI Systematicity

sub-scale score.

Graph for Systematicity

20

25

30

35

40



CONTROL-ALL

Chart 7: Systematicity sub-scale Means

Systematicity


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


of Post-score Exp

with control Control 34.66 6.17 35.30 6.33 0.64 8.37 0.71

Exp 30.79 6.21 -4.16(***) 34.69 4.91 -0.72 3.90 8.27 4.45(***) 2.61(**) 0.38 Exp without KD 30.89 6.62 -3.72(***) 34.88 4.96 -0.46 3.99 8.53 3.96(***) 2.50(*) 0.10

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 Table 15: Systematicity sub-scale Statistical tests Results


63



between those groups. The ANCOVA analysis reveals that the two groups were not

different in the post-test on this sub-scale. The experiment group improved by about four

points whereas the control group improved by close to one point. This difference can be

attributed to the learning process in the learning unit, as will be further explained.

5.3.2.6. Results for sub-scale Maturity Charts 8 schematically presents the means of the pre and post CCTDI Maturity sub-scale

score for the students in the experiment and control groups. Table 16 contains the full

results of statistical tests conducted on the data relevant to the CCTDI Maturity sub-scale

score.

Graph for Maturity

20

25

30

35

40



CONTROL-ALL

Chart 8: Maturity sub-scale Means

Maturity


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


of Post-score Exp

with control Control 32.56 7.73 38.50 5.66 5.94 9.14 6.1(***)

Exp 32.79 7.39 0.20 37.62 6.77 -0.94 4.83 9.44 4.83(***) -0.80 0.93 Exp without KD 32.53 7.40 -0.02 38.06 7.00 -0.44 5.53 9.74 4.81(***) -0.28 0.20

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 Table 16: Maturity sub-scale: Statistical tests results


64

On the Maturity sub-scale, there was no difference between the two groups for the Pre

test and for the Post test scores. There was a significant improvement in both groups,

experiment as well as control, from the Pre test to the Post test (mean improvement of

approximately 5-6 points for both groups). As can be drawn from comparison of the

graphs, there was practically no difference between the improvement in the experiment

group and that of the control. This lack of difference can perhaps be attributed to the

developmental, psychological and cognitive processes in teenagers as well as to the

learning process. Elaboration on this topic follows in the discussion.

5.3.2.7. Results for sub-scale Confidence Chart 9 schematically presents the means of the pre and post CCTDI Confidence sub-


full results of statistical tests conducted on the data relevant to the CCTDI Confidence

sub-scale score.

Graph for Confidence

20

25

30

35

40



CONTROL-ALL

Chart 9: Confidence sub-scale Means

Confidence


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


of Post-score Exp

with control Control 32.30 7.73 33.90 7.35 1.60 10.81 1.39

Exp 30.57 8.49 -1.41 36.43 7.36 2.29(*) 5.85 10.19 5.42(***) 2.69(**) 5.71(*) Exp without KD 30.78 8.84 -1.16 36.47 7.11 2.24(*) 5.69 9.80 4.93(***) 2.48(*) 5.56(*)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 Table 17: Confidence sub-scale Statistical tests Results


65

There was no initial (on the pre-test) difference between the experiment and the control

group. On the post-test the experiment group performed significantly better. The

experiment group improved by six points, whereas the control group by only about two

points. The improvement in the experiment group was at least threefold when compared

to that of the control. This difference can be attributed to the learning process, as will be

further discussed.

5.3.2.8. Results for sub-scale Analyticity Charts 10 schematically presents the means of the pre and post CCTDI Analyticity sub-


complete results of statistical tests conducted on the data relevant to the CCTDI

Analyticity sub-scale score.

Graph for Analyticity

20

25

30

35

40



CONTROL-ALL

Chart 10: Analyticity sub-scale Means

Analyticity


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


of Post-score Exp

with control

Control 32.48 8.91 27.56 6.62 -4.92 10.05 -4.59(***)

Exp 26.52 8.89 -4.45(***) 34.07 8.01 5.89(***) 7.55 12.19 5.84(***) 7.42(***) 34.17(***) Exp without KD 26.72 8.91 -4.06(***) 33.89 7.30 5.75(***) 7.17 11.76 5.17(***) 7.01(***) 33.32(***)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 Table 18: Analyticity sub-scale Statistical tests Results


66



between those groups. The ANCOVA analysis reveals that the two groups were

significantly different in the post-test on this sub-scale. The analyticity score results were

especially interesting, in light of the fact that for the control group the mean post-test

score was significantly lower than the mean pre-test score, and at the same time there was

a significant improvement by about seven points in the experiment group. While the

differences in the experiment group may stem from the advantages of the learning unit, it

is somewhat difficult to explain the results of the control group. This case will receive its

proper attention in the discussion, however, additional data may have further implications

on deciphering the control results.

5.3.3 To Summarize of Disposition of Critical Thinking To summarize, the preliminary round resulted in improvement in one subtest (Maturity)

for the Kidumatica students and three subtests (Systematicity, Maturity and Open

mindedness) for the High-school group. The results of the second round were even more

interesting: those can categorized into three groups – in some of the tests (Systematicity,

Inquiness, Confidence, Analityicity), there was a notable improvement is the CCTDI

score when compared to the improvement of the control group. In a second group

(Maturity, Open mindedness) the improvement rate was similar in both groups and may

be attributed to cognitive and psychosocial processed in teenage life, as will later be

discuss. A third group of results was especially intriguing (Analyticity and truth seeking)

because in those parameters not only an improvement was not observed, the post- CCTDI

score was lower in the experiment group while the control was significantly higher.

These peculiar results may be explained along the following lines: it is possible that we

had here a kind of “Hypercorrection” which will be discussed more fully in the

concluding discussion in Chapter 8.


67

Total Disposition

25

30

35

40

Pre CCTDI Post CCTDI

Preliminary-Highschool

Preliminary-Kidumatika

Secondary-Highschool

Secondary-Kidumatika

Compare Preliminary to Secondary onD=Post-Pre,For Highschool:t=3.62,p<0.001For Kidumatika:t=3.54,p<0.001

Chart 11: Total Dispositions during all the Research As can be inferred from Figure 4, there are differences between the rounds. The second

one was more successful, and this observation can be attributed to the improvement of

the learning unit between the rounds. Comparison of the Kidumatica and High School

group reveals a similar improvement in the second round.


68

5.4 Discussion of Dispositions31 Findings indicate that in most CCTDI sub-tests, excluding the category of critical

thinking maturity, the experimental group has scored high success levels in the post test

in the two rounds (see figure 3). A possible explanation for this is that a consistent effort

to encourage higher-order thinking skills does not only promote student level of critical

thinking during the course period, but also has a long-term effect of becoming an integral

part of these students’ thinking habits.

The statistic comparison of averages in CCTDI tests focused on the relative rate of

improvement. A series of t-tests has shown that the students of the Experimental group

have considerably improved their critical thinking dispositions relative to the Control

group. These findings raise the question, what is a "thinking disposition," and how can

we differentiate between the different types of thinking dispositions? According to

Harpaz (2000), one can reflect on the origin of thinking dispositions from two

perspectives: one perspective suggests that thinking dispositions (and dispositions of

character in general) originate ”from below,” i.e. from unconscious sources - primal

drives, suppressed feelings, and other mechanisms that shape the individual psyche,

including its cognitive ‘tip of the iceberg’.

According to the second perspective, thinking dispositions originate “from above” – from

opinions, standpoints, values, decisions, etc., which the individual has formed or chosen

after a certain amount of deliberation. Apparently, dispositions are formed from both

"below" and "above," and from the intricate connections between these sources.

Education for thinking, however, tends to draw mainly on the second source, reinforcing

the aspects of conscious choice, informed preference and reasonable standpoint. Thus,

one could define "thinking disposition" as a rational impulse toward a particular thinking

pattern or thinking quality (openness, depth, systematicity, etc.) imbued with motivation

“from above.”

This research indicates that parts of the students’ thinking dispositions were developed

because they were stimulated from "above." While teaching the learning unit and

conducting this research, we have restricted our consideration of thinking dispositions in

31 31 This section discusses dispositions solely. Chapter 8 presents an integral discussion.


69

two dimensions. The first dimension is depth: thinking dispositions do not apply to

personality in general; they are not traits of personality or character. According to this

assumption, intellectual dispositions are related to traits of character in an intricate

manner; these are not necessarily consistent with each other. A person can be

intellectually courageous, yet timid in other areas of life (for example, one may construct

audaciously innovative theories, or write hair-raising adventure stories and still be afraid

of leaving one’s own home).

The second limitation we put on thinking dispositions has to do with breadth: thinking

dispositions in a specific area do not necessarily extend to thinking patterns in general.

For example, one may tend to deep and nuanced thinking in one’s area of specialization

but exhibit superficial thinking on political matters. Thinking dispositions are context-

dependent.

The dispositions approach originated as a criticism of the skills approach, in two stages.

In the first stage, which we will call the dependence stage, thinking dispositions were

considered "energy suppliers" for thinking skills, that is, a link that connected thinking

skills with action. This link is vital because the individual who possesses the appropriate

thinking skills may still lack the drive, the will or inclination to act upon them. At this

dependence stage, thinking dispositions were derived from thinking skills: every skill was

associated with a corresponding disposition. Thus, the skill of "searching for alternatives

to an idea" was associated with the disposition to search for alternatives (cf. Ennis, 1996).

In the second stage, which we shall call "the independent stage," the dispositions

demanded "self-definition", that is, to be considered as a "[primary] unit of analysis of

cognitive behavior" (Perkins et al., 2000a, p.72). In the independent stage, a "thinking

disposition" came to be perceived as the primary foundational and explanatory factor of

"good thinking". As a result, thinking dispositions came to be derived from the general

cultural image of a "good thinker" (wise, intelligent, rational, etc.) rather than from

specific skills.

Thinking dispositions were now valued independently of the actions they were meant to

carry out. Although thinking dispositions are far less numerous than thinking skills

(according to Sternberg, there are "close to a thousand" thinking skills, and according to

Lipmann, "the list is infinite"), it is worth differentiating between two types of thinking

dispositions: thinking disposition and disposition towards thinking (Sternberg 1987, p.


70

251; Lipmann 2003, p. 162). This distinction is by no means clear-cut - thinking

dispositions include and encourage disposition to thinking - but it has theoretical and

practical justifications. A thinking disposition, as we define it, is a rational (“from

above”) impulse toward a particular thinking pattern or thinking quality, which

encourages becoming actively involved in the process of thinking, investing oneself in

thinking. Dewey believes that the disposition toward thinking is the most significant

quality of good, “reflective” thinking.

For Dewey, "reflective thinking" consists in turning the topic over in one’s mind and

giving it a serious and constructive consideration (Dewey, 1933/1998, p. 3). A

disposition to thinking is an act of devotion to thinking, of withdrawal from common or

"public" opinion; it is associated with practical objectives and "personal" or "authentic"

considerations of the very nature of these objectives and the thought process in and of

itself. Unfortunately, school does not traditionally provide room for the type of thinking

which involves intellectual awakening; it is even adversary to it. Baber describes the

following dialogue between a teacher and his student: "Teacher: What are you doing?

Student: I'm thinking. Teacher: Then stop thinking and start working!" (Baber, 1997

p.180). Only the school that devotes time to this type of thinking and encourages students

to "stop and think" (i.e. promotes students' disposition to thinking) deserves to be

considered a school. Such a school, as I later argue, is a fundamentally different

institution than the commonly found present-day school.

To conclude, teaching “Probability in Daily Life” unit in the infusion approach has

greatly developed the students' critical thinking. These findings support the assumption

that one of the fundamental elements of good critical thinking is the development of the

dispositions this research addresses.

Abilities of Critical Thinking: Methods, Results and Discussion

71

6. Abilities of Critical Thinking: Methods, Results and Discussion "Learning without thinking is a wasted blessing" (Confucius)

This chapter presents the methods and results regarding the abilities towards critical

thinking according to Ennis’ taxonomy and Facion’s theory and discusses them. The

major tool through which the results were obtained is the Cornell Critical Thinking Test.

6.1 Research Question To what extent does the study of “Probability in Daily Life” in the infusion approach contribute to the development of critical thinking abilities? The first part describes the findings of the first round, the second part describes the second round, and the third part describes the difference between the rounds critical thinking abilities. 6.2 Method

6.2.1 The Instrument: Cornell Critical Thinking Test Level Z In order to check the development of the students’ critical thinking abilities according to

the taxonomy of Ennis, we decided to use the Cornell Test. This test was developed by

Ennis and his colleagues (Ennis and Millman, 1985). The Cornell Critical Thinking Test,

Level Z was chosen by the researchers as it was more suited to the advanced high-school

students in the group. The test includes general content with which most of the students

would be familiar and it assesses various aspects of critical thinking. It is a multiple-

choice test with three choices and one correct answer. Although the test is meant to be

taken within a fifty-minute period, we predicted that the students in the group would be

unable to complete it within that time limit. For this reason we decided to give them

eighty minutes in which to take the test. The test includes five sub-tests and evaluates

different aspects of critical thinking including induction, deduction, value judging,

observation, credibility, assumptions and meaning. The process of critical thinking,

however, involves an overlap of these aspects as they are all dependent on each other. In

the test, this inter-dependence is evident in the fact that frequently an item is assigned to

several different aspects. It is important to note that both observation and credibility are

evaluated according to the same items in the test (items twenty two – twenty-five).


72

According to Ennis et al. (2004), this test is a general ability test, involving many aspects

of critical thinking. This test is part of an ongoing research on critical thinking and was

developed in the 1980s by Ennis and his colleagues32. The test has two versions, X and Z,

where level X is designed for younger students aged between 4-14, and level Z is

designed for high-school students and adults. This is a multi-aspect test that includes four

sub-tests: inductive inference, credibility of information sources and observations,

deduction, and recognition of implicit assumptions. Level Z includes 52 multiple-choice

items. A general test on critical thinking ability may include induction, deduction, value

judging, observation, credibility (of other people’s statements), identifying assumptions

and meaning (including definition, sensitivity to meaning, and ability to handle

ambiguity). By contrast, such a test would not include the attitudes and dispositions of a

critical thinker, such as intellectual openness, caution, and assigning high value to being

well-informed, all of which are attitudes that are no tested (Ennis et al., 2004, p.2).

Aspects of general ability for critical thinking chosen for inclusion in this test are

presented in Table 1 (ibid.), along with the numbers of items that test each of the aspects.

Even though these aspects are presented separately here, there is a significant overlap and

interdependence between them in the process of critical thinking. This interdependence is

reflected in the tests, as one can see from the overlaps in Table 19, where some of the

tasks are related to more than one aspect of critical thinking. For instance, items related to

recognition of assumptions also appear under the rubric of deduction, since deduction is

used in recognizing possible assumptions inherent in a specific line of thought. Making a

prediction in order to test a hypothesis also necessitates deduction, at least in a loose

mode. Therefore, prediction items of level Z appear under rubrics of both induction and

deduction (ibid.).

32 Ennis and his colleagues (Ennis, Millman, & Tomko, 1985) point out in the introduction that as far as they know there exist no tests on critical thinking in specific fields of knowledge that would be offered on the commercial market. They make a number of suggestions designed to help those who wish to compose a new test that would suit their needs better than the tests currently offered on the market. However, it is important to remember that composing a valid and reliable test on thinking skills requires a high degree of competence. Thus Norris and Ennis (1985) recommend that in addition to composing new tests, it is advisable to pass also the existing tests, since they were developed with great care and effort and thus, in their opinion, will serve the user better.


73

Table 19 summarizes items of Cornell test:

Items in Level Z Aspects of Thinking in the Test

17, 26-42 Induction

,1-10 39-52 Deduction

22-25 Observation

22-25 Credibility

43-52 Assumptions

,11-21 43-46 Meaning

Table 19: Classification of Items by Aspect of Thinking in Co. test Level Z

It is possible to claim that deduction (even in its loose mode) is often involved in

inductive thinking, which explains the overlap between the items classified under

induction and those classified under deduction rubric (Ennis et al., 2004, p.3). It is true if

one assumes that deduction usually involves induction dependent on the best-explanation

hypothesis. It is also possible to claim that observation and credibility judgments

necessitate application of principles, a deductive process, and thus they also have to be

listed under the rubric of deduction. In addition, one can claim that since basic deduction

is in fact equivalent to knowing the meaning of words and statements, everything listed

under the rubric of deduction can be also listed under that of meaning. Ennis et al.

conclude that basic deduction and ability to deal with meaning exist, in the final account,

in every aspect of critical thinking, which causes the theoretical difficulty in grading each

part of the exam separately, and introduces strong independent factors in a factor

analysis. And yet, “critical thinking is not a unidimensional concept either, making it

difficult to obtain high internal consistency reliability estimates”. An additional overlap

occurs between the categories of observation and credibility (items 22-25 in level Z).

Observation statements made by another person, which comprise many of the items in the

test, are subject to both the criteria of credibility and those of observation statements

(ibid.). The rest of these items are pure credibility items, so that in the final account, all

the items are justly listed under the category of credibility. Since there is hard to clearly

distinguish outside of context between observation and inference, there are some items

listed under both observation and credibility, which might also be listed under credibility

alone. As a result of this complication, and because observation statements made by other

people are subject to criteria of credibility, there arises again the theoretical difficulty of


74

grading different parts of the exam separately and the presence of strong independent

factors in a factor analysis (ibid.). Ennis et al. discuss the above issues in order to provide

the user of the test with general knowledge about the field covered by the test, and also to

engage the question of the test’s validity and as a preface to justifying the answers key for

each item. These topics will be discussed further in the dissertation in more detail. To

conclude, the Cornell Critical Test Level Z is general and not dependent on a specific

field of knowledge. In the present research we have chosen the fourth edition of the Z-

level exam (Ennis, Millman & Tomko, 1985, 2005), since the age of the research

participants was 16 and older. In this test there are 52 multiple-choice items that the

student is allotted 50 minutes to answer, but the test can be taken in two parts by students

who have been recognized to need extra time. For each question, three answers are

offered, one of which is correct. In the introduction there is a detailed definition of the

concept ‘critical thinking’ and what it means in the context of the present test; there are

also instructions on conducting and grading the test, a discussion of the test’s reliability

and validity, analysis of selected items and an explanation of the answers key.

6.2.1.1 Sample Question33 from the Cornell Critical Thinking Test34

An experiment was performed by Drs. E.E. Brown and M.R. Kolter in the veterinary

laboratory of the British Ministry of Agriculture and Fisheries. The doctors were

interested in what happens to ducklings that eat cabbage worms. Several cases had been

reported to them in which ducklings had “mysteriously” died after being in a cabbage

patches containing cabbage worms. Three types of ducklings were secured (Mallards,

Pintails, and Canvasbacks), two broods of each. Each brood was then split into two equal

groups as much alike as possible. For a one-week period they were provided an approved

diet for ducklings. All had this diet, except that half of each brood were provided

something more: two cabbage worms daily per duckling. The condition of the ducklings

at the end of the week was observed and is reported in the following table:

33 An additional sample question can be found in the introduction to the test. 34 The Method of Analyzing the Questionnaire:The analysis of the test was performed using the tools provided by the authors of the test.


75

Table 20: Number of ducks according to the different menus

The doctors drew the following conclusion: CABBAGE WORMS ARE POISONOUS TO DUCKLINGS.

The experiment attracted a great deal of attention. Many statements were made about the experiment and

about the protection of ducklings.

Items 22 through 25 each contain a pair of statements (A & B), which are underlined. Read both, then

decide which, if either is more believable. Mark items 22 through 25 according to the following system:

If you think the first is more believable, mark A.

If you think the second is more believable, mark B.

If neither statement is more believable than the other, mark C.

22. A. Cabbage worms are poisonous to ducklings (said by Dr. Kolter).

B. Six Canvasbacks died during the week of the experiment (said by Dr. Kolter).

C. Neither statement is more believable.

23. A. Six Pintails were healthy at the end of the experiment (said by Dr. Kolter).

B. Four worm-fed ducklings were ill at the end of the experiment (said by Dr. Brown).

C. Neither statement is more believable. 35

6.2.2 Interviews with Students In order to understand and monitor the students’ attitudes toward critical thinking as it

was manifested by the skills specified above, interviews were conducted with five

randomly-selected students after the aforementioned lesson. The interviews were

conducted by the teacher with the student and lasted fifty minutes. In these interviews,

the students acknowledged the importance of critical thinking. Moreover, students were

aware of the infusion of instructional strategies that advance critical thinking skills.

35 Robert H. Ennis, Jason Millman (2005). Cornell Critical Thinking Test, Level Z. Fifth edition.. pp. 7-8.

Regular Diet plus worms

Regular Diet Original number in brood

Type of Ducking

Dead Ill Healthy Dead Ill Healthy 2 2 1 3 8

Mallard 3 3 6 3 1 2 6

Pintail 3 1 1 3 8 3 1 4 8

Canvasback

3 1 1 3 8

17 4 1 1 3 18 44 Totals


76

6.2.3 Research population Table 21 summarizes population sizes of all groups

Ability Test Group School First Round First Round

Experiment Kidumatica 41 25 High School 1 28 29 High School 2 25,17

Control High School 1 25 High School 2 21

Total 69 142 Table 21: Numbers of students each round 6.3 Results of Abilities the First Round (n=69) The first round of the teaching unit was conducted only in two groups: the Kidumatica

group and a regular high school group.

6.3.1 The “Kidumatica” group (n=41) This round is the first round in Kidumatica classes

Chart 12 schematically describes the Post vs. Pre average Cornell test sub-scale scores for

“Kidumatica” (exact t-test values are presented in table 22). In all the sub-scales there was

a significant improvement from pre to post tests (p<0.001).

Post vs. Pre for Kidumatica

0

20

40

60

80

100

0 20 40 60 80 100

Pre

Post

InductionDeductionObservationAssumptionsCredibility

Chart 12: Abilities of CT Kidumatica


77

Table 22: Critical Thinking abilities in the “Kidumatica” group

6.3.2 The “High School 1” Group (n=28) Chart 13 schematically describes the Post vs. Pre average Cornell test sub-scale scores for

High School 1 (exact t-test values are presented in table 23).

Post vs. Pre for High School 1

0

20

40

60

80

100

0 20 40 60 80 100

Pre

Post

InductionDeductionObservationAssumptionsCredibility

Chart 13: Abilities of CT High School

Table 23: CT abilities in the High School 1 group (N= 28) (*)= difference significant at the .05 level

t value Post-test Pre-test Sub-scale SD Mean SD Mean N=41 6.97(***) 17 69.8 22 41.1 Induction 4.66(***) 13 52.1 14 45.4 Deduction 9.66(***) 22 80.1 24 25.7 Observation 4.76(***) 21 48.3 21 35.9 Assumptions 4.04(***) 12 36.5 11 30.8 Meaning

10.68(***) 10 56.7 9 39.3 CTI Total


-1.31 12 50.4 15 56.2 Induction -2.51(*) 15 43.9 10 53.3 Deduction 2.59(*) 34 64.3 30 45.5 Observation 0.38 18 47.5 19 45.7 Assumptions 1.97 11 43.1 10 38.1 Meaning -1.32 8 46.6 .9 50.1 CTI Total

(***)= difference significant at the .001 level


78

6.3.2. Describing the Results of the Second Round In the second academic year, i.e. the second round, there were 142 students. 46 of them

were in the control group and 96 in the experimental group. The Kidumatica class had 25

students.

6.3.2.1. Results for CTI Total Chart 14 schematically presents the means of the pre and post CTI Total score for the

students in the experiment and control groups. Following the chart Table 24 contains the

full results of statistical tests conducted on the data relevant to the CTI Total score.

Graph for CTI Total

20

30

40

50

60

70



CONTROL-ALL

Chart 14: CTI Total Means

CTI Total

Pre-test Post-test Difference ANCOVA(a)

Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control

F value for comparison of Post-score Exp

with control Control 33.57 8.46 33.07 11.64 -0.50 14.48 -0.23

Exp 32.17 14.97 -0.71 42.33 8.29 4.84(***) 10.16 11.57 8.60(***) 4.72(***) not allowed(b) Exp without KD 36.11 11.79 1.35 43.20 8.12 5.15(***) 7.10 8.21 7.29(***) 3.24(**) not allowed(b)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 (a) Not necessary since there is no initial difference, on the Pre score, between the 2 groups (Experiment, Control) (b) significant difference between slopes of regression lines for the 2 groups (Experiment vs. Control) Table 24: CTI Total Statistical tests Results It was found that the improvement among students in the experimental group was greater

than the improvement among students in the control group. While the control group


79

scores have not changed, the improvement in achievement of the experiment group is

unequivocal. The Post-test of the experiment group is significantly higher then the Post-

test of the control group. As mentioned in previous chapters, the CTI test contains five

different sub-scales.

In the following sections each of the sub-scales will be analyzed.

6.3.2.2 The Induction Sub-Scale Chart 15 schematically presents the means of the pre and post Induction Sub-Scale score

for the students in the experiment and control groups. Table 25 contains the full results of

statistical tests conducted on the data relevant to the Induction Sub-Scale score.

Graph for Induction

20

30

40

50

60

70



CONTROL-ALL

Chart 15: Induction Sub-Scale Means

Induction


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


with control

Control 28.62 19.46 30.56 21.24 1.93 28.72 0.46

Exp 34.61 21.00 1.63 41.96 14.12 3.31(**) 7.35 14.65 4.91(***) 1.21 not allowed(a) Exp without KD 39.28 18.93 2.94(**) 43.04 14.17 3.51(***) 3.76 11.68 2.71(**) 0.41 not allowed(a)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 (a) significant difference between slopes of regression lines for the 2 groups (Experiment vs. Control) Table 25: Induction Sub-Scale Statistical tests Results While the control group scores have not changed, the improvement in achievement for

the experiment group is quite noticeable . The Post-test of the experiment group is

significantly higher then the Post-test of the control group. This observation is reinforced


80

by the fact that the experiment group consists of two subsets of students, where one is

inclusive of the Kidumatica students and one is not. The two subsets share the same

tendency.

6.3.2.3. The Deduction Sub-Scale Chart 16 schematically presents the means of the pre and post Deduction Sub-Scale score


statistical tests conducted on the data relevant to the Deduction Sub-Scale score.

Graph for Deduction

20

30

40

50

60

70



CONTROL-ALL

Chart 16: Deduction Sub-Scale Means

Deduction


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


with control

Control 40.31 12.67 37.68 14.49 -2.63 20.79 -0.86

Exp 33.51 16.84 -2.68(**) 44.92 12.00 3.14(**) 11.41 13.07 8.56(***) 4.20(***) not allowed(a) Exp without KD 36.44 14.97 -1.45 45.19 11.67 3.09(**) 8.74 9.69 7.61(***) 3.47(***) not allowed(a)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 (a) significant difference between slopes of regression lines for the 2 groups (Experiment vs. Control) Table 26: Deduction Sub-Scale Statistical tests Results

While the control group scores have not changed significantly, the improvement in

achievements for the experiment group is obvious. The Post-test of the experiment

group is significantly higher then the Post-test of the control group. This observation


81

is reinforced by the fact that the experiment group consists of two subsets of students,

one of which is inclusive of the Kidumatica students and one is not.

6.3.2.4. The Observation Sub-Scale Chart 17 schematically presents the means of the pre and post Observation Sub-Scale

score for the students in the experiment and control groups. Table 27 contains the full

results of statistical tests conducted on the data relevant to the Observation Sub-Scale

score.

Graph for Observation

20

30

40

50

60

70



EXPERIMENT-ALL

CONTROL-ALL

Chart 17: Observation Sub-Scale Means

Observation


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


with control

Control 22.28 20.57 22.83 26.78 0.54 34.76 0.11

Exp 28.91 23.88 1.62 58.85 25.39 7.77(***) 29.95 31.54 9.30(***) 5.03(***) 56.96(***) Exp without KD 33.45 21.93 2.76(**) 61.27 22.67 8.34(***) 27.82 29.14 8.04(***) 4.58(***) 62.31(***)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 Table 27: Observation Sub-Scale Statistical tests Results While the control group scores have practically not changed, the improvement in

achievements for the experiment group is quite significant. The Post-test of the

experiment group is significantly higher then the Post-test of the control group. The

ANCOVA F is also highly significant. This observation is reinforced by the fact that


82

the experiment group consists of two subsets of students, one is inclusive of the

Kidumatica students and one is not. The two subsets share the same tendency.

6.3.2.5. The Assumptions Sub-Scale Chart 18 presents the means of the pre and post Assumptions Sub-Scale score for the

students in the experiment and control groups. Following the chart Table 28 contains

results of statistical tests conducted on the data relevant to the Assumptions Sub-Scale

score.

Graph for Assumptions

20

30

40

50

60

70



EXPERIMENT-ALL

CONTROL-ALL

Chart 18: Assumptions Sub-Scale Means

Assumptions


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


with control

Control 28.04 20.18 28.91 19.46 0.87 28.58 0.21

Exp 28.75 23.68 0.17 46.56 16.66 5.59(***) 17.81 17.90 9.75(***) 3.69(***) not allowed(b) Exp without KD 30.00 22.80 0.47 46.90 17.04 5.27(***) 16.90 16.53 8.62(***) 3.45(***) not allowed(b)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 (a) Not necessary since there is no initial difference, on the Pre score, between the 2 groups (Experiment, Control) (b) significant difference between slopes of regression lines for the 2 groups (Experiment vs. Control) Table 28: Assumptions Sub-Scale Statistical tests Results While the change in control group scores is negligible, the improvement in

achievements for the experiment group is quite noticeable. The Post-test of the

experiment group is significantly higher then the Post-test of the control group. This

observation is reinforced by the fact that the experiment group consists of two subsets


83

of students, one of which is inclusive of the Kidumatica students while the other is

not. The two subsets share the same tendency.

6.3.2.6. The Meaning Sub-Scale Chart 19 schematically presents the means of the pre and post Meaning Sub-Scale score


statistical tests conducted on the data relevant to the Meaning Sub-Scale score.

Graph for Meaning

20

30

40

50

60

70



EXPERIMENT-ALL

CONTROL-ALL

Chart 19: Meaning Sub-Scale Means

Meaning


Mean S.D.


Exp with control

Mean S.D.


Exp with control



Exp with control


with control

Control 29.42 11.03 28.84 10.89 -0.58 14.73 -0.27

Exp 26.11 14.79 -1.49 34.72 11.36 2.93(**) 8.61 12.87 6.56(***) 3.80(***) not allowed(b) Exp without KD 30.05 12.59 0.28 34.74 11.49 2.77(**) 4.69 7.68 5.15(***) 2.24(**) not allowed(b)

(*) 0.05>p>0.01, (**) 0.01>p>0.001, (***) p<0.001 (a) Not necessary since there is no initial difference, on the Pre score, between the 2 groups (Experiment, Control) (b) significant difference between slopes of regression lines for the 2 groups (Experiment vs. Control) Table 29: Meaning Sub-Scale Statistical tests Results While the control group scores have not changed, the improvement in achievements

for the experiment group is quite notable. The Post-test of the experiment group is

significantly higher then the Post-test of the control group. This observation is

reinforced by the fact that the experiment group consists of two subsets of students,


84

one of which is inclusive of the Kidumatica students while the other is not. The two

subsets share the same tendency.

6.3.3 Summarizing the results of Abilities The preliminary round demonstrated an improvement in all sub-subscales for the

Kidumatica students and in one sub-scale (observation) for the high-school 1 group. For

the second round, all the sub-scales of the CTI test show the same tendency. In other

words, the results support a significant improvement of the experiment group in all the

Cornell test parameters.

Total Ability

20

25

30

35

40

45

50

55

60

pre CT post CT

Preliminary-Highschool

Preliminary-Kidumatika

Secondary-Highschool

Secondary-Kidumatika

Compare Preliminary to Secondary onD=Post-Pre,For Highschool:t=3.78,p<0.001For Kidumatika:N.S.

Chart 20: Total Abilities during all the Research Once again, differences between the iterations can be demonstrated, and the pattern of

improvement in the second round is retained. Previous explanations for this observations

can be reinforced by the fact that the teaching improved in the second round, by the

changes that were aimed to improve the learning unit.


85

6.4 Discussion of Critical Thinking Abilities36 The research findings indicate that in most of the Cornell sub-tests, excluding first round

induction and deduction, the experimental group reached higher scores in the post test. It

seems that here too, as in the case of critical thinking dispositions, consistent effort to

encourage high order thinking skills contributed to the development of the students’

ability to think critically (not only during the experimental learning period but also in the

long run, as these skills become an integral part of the students' thinking habits). Statistic

comparison of these students' averages in the Cornell tests focused on the relative rate of

improvement.

A series of t-tests showed that the students of the experimental group have considerably

improved their critical thinking abilities. One may argue that deduction (even if only in

its free form) involves a considerable amount of induction, which explains the overlap

between induction and deduction items on the test (assuming that free deduction usually

involves induction based on the best possible explanation hypothesis).

One can also argue that observation and credibility judgments require implementation of

principles, a deductive process, and therefore should be also listed under deduction. In

addition, since basic deduction actually entails knowing the meaning of words and

statements, one can say that anything classified under deduction can be also classified

under meaning.

Conclude, the above factors indicate that basic deduction skills and the ability to deal

with meaning are part of every aspect of critical thinking. The subsequent theoretical

difficulties in grading each test separately are strong independent factors of analysis.

Nevertheless, critical thinking is not a one-dimensional term, which makes it difficult to

achieve high reliability or validity. Another overlap can be found between observation

and credibility/reliability (items 22-25 of level Z).

Statements of observation by another person, a description that applies to many items on

the test, is subject to credibility criteria as well as criteria for statements of observation.

The rest of the items directly relate to credibility. Thus, most of the above items appear

under the rubric of "credibility," while most of them also appear under "observation"

Since there is no clear connection outside of context between observation and drawing

36 *** This section discusses abilities solely. Chapter 8 presents an integral discussion.


86

conclusions, there are items classified under both observation and credibility, but might

be classified under credibility alone by other researchers. For this reason, and because

statements of observations by others are subject to credibility criteria, there arises again

the theoretical problem with grading each test separately and the absence of strong

independent factors in factor analysis (Ennis and Millman, 2005).

In the ‘beginning’, the historical and ideological beginning of general critical thinking

abilities there arose the abilities approach. The skills approach was posed against the

traditional or "old" education, as Dewey terms it. This type of education focused on

passing on knowledge. Changes in the way knowledge functions and is perceived

undermined the principles of traditional education and paved the way for the skills

approach.

On the basis of these changes, the skills approach claims that in the current era, when

knowledge “explodes” (is doubled in short time periods), becomes outdated (new

findings constantly provide grounds for new theories and vice versa), and is generally

accessible (by means of the Internet and other media), it no longer makes sense to

concentrate on transmitting knowledge to students. Moreover, in our (postmodern) era,

when knowledge is seen as relative – affected by interests, perspectives, and frames of

references devoid of objective justifications – it makes no sense to sanctify knowledge

and commit to it the following generations.

It follows then that instead of imparting to students bodies of knowledge, it now makes

more sense to impart the skills necessary to locate, process, criticize and create

knowledge, in other words, to think well. Thinking well means using thinking skills.

These arguments have greatly affected the discourse of education, and the "educational

market" (particularly in the United States) was flooded with thinking skills of various

kinds – termed critical, creative, and productive thinking (Harpaz, 2005).

As with thinking dispositions, also here there arises the question, what is a "thinking

skill," and how one can differentiate between the various types of these skills. While the

term "thinking skills" is most frequently employed in the discourse of ‘education for

thinking’, it is afflicted with the greatest ambiguity precisely in this discourse. In the

context of ‘education for thinking’, this term has two basic meanings that we will term

“internal” (or subjective) and “external” (or objective) meanings. In the latter sense,

"thinking skills" are different instruments of thinking (strategies, heuristics, algorithms,


87

routines, frames, tools, etc.) that make the process of thinking more efficient. In the

‘internal’ sense, however, a "thinking skill" is equivalent to making good use of thinking

instruments, that is, using them in a quick and precise way with minimal investment of

‘mental energy’. In the literature of ‘education for thinking’, the term "thinking skills" is

used occasionally it its ‘internal’ meaning, at other times in its ‘external’ meaning and

sometimes in both, which greatly contributes to the sense of ambiguity about this term (in

other fields, the thinking ‘instruments’ are not considered as an integral part of the

abilities).

Combining the two meanings of "thinking skills," it seems that good thinking is skilled

thinking, and that skilled thinking is a type of thinking that operates different instruments

of thought in a quick, precise manner (adapted to the particular circumstances or

problem).

To conclude, studying the learning unit "Probability in Daily Life" in the infusion

approach has contributed to the development of the students' thinking skills. If so, it is

possible to say that one of the foundational elements of good critical thinking is the

development of the abilities this research is dealing with.

Construction of Critical Thinking Skills: Methods, Results and Discussion

88

7. Construction of Critical Thinking Skills: Methods, Results and Discussion “As soon as a question of will or decision or reason or choice arises, human science is at a loss" (Noam Chomsky)

The purpose of the intervention (teaching the present learning unit) was for the students

to develop critical thinking skills and dispositions while thoroughly studying the unit's

contents. The concept of critical thinking skills in this research is linked to the contents

of the learning unit and points to the need of fostering transfer, conservation and

improvement of these skills by means of thinking dispositions and abilities, which

include the motivation to apply the skills in specific subjects. This system does not

require change in the program of studies and integrates well into the present structure of

the curriculum.

7.1 Research Question What are the processes of construction of critical thinking skills (such as identifying

variables, suspending judgment, referring to sources, searching for alternatives) during

studying the topic “Probability in Daily Life” in the infusion approach?

The third research question examined the processes of construction of critical thinking

skills (e.g. identifying variables, suspension of judgment, referring to sources, searching

for alternatives) during the study of "Probability in Daily Life" learning unit in the

infusion approach. In order to answer this research question, we have closely reviewed

the contents of the learning unit for connection to any relevant thinking skills. The skills

that were found relevant were: (a) identifying variables; (b) referring to sources; (c)

identifying assumptions; (d) evaluation of statements; (e) suspending judgment; (f)

offering alternatives. At the end of each stage, the students completed a questionnaire

testing acquisition and conservation of thinking skills, according to the topics studied at

each stage, as presented in Table 10.

7.2 Method The distinction between general thinking skills and critical thinking skills is not clear-cut,

as it is difficult to distinguish between simpler and more complex skills. On the one hand,

the more complex skills require mastery of simpler ones, but on the other hand, also

simpler skills require mastery of more complex ones (such as decision-making necessary

for the purposes of comparison). The learning unit was divided into three parts, according


89

to the increasing number and complexity of critical thinking skills that one needs to use

for solving the problems in each part. Table 19 details the successive addition of more

complex thinking skills, in the second and third parts of the course, to the first three basic

skills taught in the first part of the course: identifying variables, referring to sources, and

evaluating/analyzing statements. The order in which critical thinking skills were

introduced corresponds to the mathematical topics studied in each part, where the latter

were also ordered hierarchically, from the simpler to the more complex. We argue that

studying the mathematical topics below, when taught in the infusion approach, both

promotes and is promoted by the development of the relevant thinking skills in the

students. For example, teaching Simpson’s paradox both requires and reinforces the skills

of suspending judgment and proposing alternatives.

Table 30: Construction of Critical Thinking Skill

Process of

construction

The specific skills practiced

(Ennis, 1987)

Topics studied in "Probability

in Daily Life" (Lieberman &

Tversky, 2001)

Stages of the learning unit

a+b+c

a. Identifying variables b. Referring to sources

c. Evaluating/ analyzing statements

Ø Proportion

Ø Conditional proportion

Ø Bayes formula

Stage A

Statistical connection

a+b+c

d+e+f+g


c. Evaluating/ analyzing statements

d. Evaluating the source's reliability e. Suspending judgment

f. Proposing alternatives

Ø Connection reversal

phenomenon

Ø Simpson's paradox

Ø Observation research

Ø Controlled experiment

Stage B

Cause-effect connection

a+b+c

d+e+f+g+h+i+j


c. Evaluating/analyzing statements

d. Evaluating the source's reliability

e. Suspending judgment

f. Proposing alternatives

g. Willingness to investigate

h. Statements of cause

i. Interpreting the author's intention

j. Renewed investigation

Stage C

Judgment

of representativeness


90

Figure 4 graphically represents the processes of construction of critical thinking skills

through and the interconnections between some study topics and various dispositions,

abilities and skills presented in Table 19, with more emphasis on the skill.

Figure 4: Processes of construction of critical thinking during the learning unit

D

ispo

sitio

ns

Judgment of Representative

Statistical Connection

b c e f a

a

b

c

Causal Connection

A

bilities

Skills as: (a) Identifying variables (b) Referring to sources (c) Identifying conclusions (d) Evaluating (e) Suspending judgment (f) Offering alternatives

Processes of construction of critical thinking during studying the topic “Probability in Daily Life” in the

infusion approach


91

7.2.1 Questionnaire on critical thinking in a specific field of knowledge37, “Probability Life in Daily”

For the purposes of this research, a questionnaire was built following the model of the

textbook “Probability in Daily Life” and the suggestions of its workbook (Lieberman &

Tversky, 2001). The questions of the pilot study (see below and Appendix 6 and 7) were

built as follows: authentic fragments containing elements of drawing conclusions were

selected from the press. The questions were constructed after the same model as the

questions used by Ennis to illustrate his taxonomy and questions from “Probability in

Daily Life,” and passed internal validation and a validation by a pilot study (by 80

students). After some more consolidation and fine-tuning, the questionnaire underwent an

additional round of validation by experts in statistical methodologies. In the process of

compiling the questionnaire, after an initial processing of the selected texts, the following

kinds of questions were asked: Is there a connection between factors? What does this

connection mean? Why is this finding not plausible? Give an example of a problem that

can be solved by means of a controlled experiment; give examples of failures and

misleading advertisement, give an example of a scientific truth that was refuted, etc. The

problems of the kind described here are complex not only because they have to do with

more than a single event, but also because they do not always have a single simple

answer. As noted in the theoretical background section, the purpose of the learning unit is

to teach the students not to be satisfied with a numerical answer, but to examine the data

and their validity, and in cases where there is no clear numerical answer, to be able to ask

the appropriate questions and analyze the problem in a qualitative way and not only by

calculation; the questionnaire gauges the extent to which the students have learned these

skills and attitudes.

7.2.2 Examples of Questions from the Questionnaire on Critical Thinking in the Specific Field of Knowledge

The following examples demonstrate the questions used to evaluate the acquisition of critical

thinking skills at the end of each of the three parts of the course. The tests were evaluated

according to a standardized baseline of answers, exemplified below in Table 14 and in full in 37 The “Probability in Daily Life” questionnaire was written by the researcher with explanations and emphasis on the principles of critical thinking; the questionnaire was validated by specialists and passed pilot and statistical testing.


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Appendix 6, which was specially developed by the researcher in order to evaluate thinking skills

construction on the basis of test answers. If the student’s answer contained the basic elements

detailed in the table, the skill was considered acquired/conserved.

Example one: The Aspirin Case Your brother woke up in the middle of the night, crying and complaining he has a

stomachache. Your parents are not at home and you don’t know what to do. You give

your brother aspirin, but an hour later he wakes up again, suffering from bad nausea

and vomiting. The doctor that regularly takes care of your brother is out of town and you

consider whether to take your brother to the hospital, which is far from your home. You

read from a book about children’s diseases and find out that there are children who

suffer from a deficiency in a certain type of enzyme and as a result, 25% of them develop

a bad reaction to aspirin, which could lead to paralysis or even death. Thus, giving

aspirin to these children is forbidden. On the other hand, the general percentage of cases

in which bad reactions such as these occur after taking aspirin is 75%. 3% of children

lack this enzyme.

(Probability Thinking, p. 30, with slight revisions made by the researcher)

Example two: Ronald Fisher Case


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The statistician Ronald Fisher was a heavy smoker. In the middle of the nineteen-fifties, the first

connections between smoking and the increased risk of lung cancer were being discovered.

Fisher’s students approached him and asked him to try and smoke less, for the sake of his lungs.

They gave the recent findings in support of their request. Fisher refused, stating that the

correlation itself does not prove that a causes b. He said it was possible that cancer in its early

stages caused a need for nicotine, resulting in the patient smoking, and only afterwards did the

tumors begin to develop. Fisher died in 1962. It was only in the seventies that scientists proved

that the increased need for nicotine did indeed cause an increase in the risk of becoming ill with

lung cancer. Some people may say that Fisher behaved foolishly , while others will say that

Fisher was perfectly correct. What do you think? Was Fisher right or wrong?

One of the students answered as follows: "I don't know enough about this topic and

therefore cannot answer this question ". The answer shows that the student understands

that there is a statistical connection but that s/he does not know enough about the causal

connection in order to give a definitive yes or no answer.

Table 31 presents the thinking skills each of the two questions elicits, and shows what

answers are yielded by exercising each of the skills.

Table 31: Example of Questions and analyses

The skill practiced Questions in intermediary questionnaires

Identifying variables

Referring to sources

Identifying conclusions

Evaluating the source’s reliability (profession-alism, absence of conflicts of interest)

Suspending judgment (when evidence and arguments are not sufficient, looking for new and contradictory evidence)

Proposing alternatives (looking for alternative explanations)

Readiness to research (proposing plans of experiments, including plans for controlling the variables)

Claims (regarding people’s beliefs and positions)

Making value judgments (apparent application of accepted principles)

Aspirin

Enzyme deficiency Pathological response to aspirin

Medical manual

False alarm (one should go to the emergency room)

Ronald Fisher

Smoking Cancer

Real story Not possible to know. Fisher was right

Narrative source

No connection between the cause and the effect, other factors possible

In order for the patient to feel better, he has to receive nicotine

Controlled experiment


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7.2.3 The Method of Analyzing the Questionnaires by Dispositions and Abilities Table 14 presents the complete scheme of evaluation for all of the skills taught in the

course. The skills evaluated in each task of the three intermediary tests are coded with

two letters and a number: for example, [Aa1] means that skill (a) is tested by task #1 in

the test concluding stage A of the course. One can see from the table that with each stage

the number of skills tested increases, why the skills taught earlier are also retained as

tested items along with the newly added ones. Thus it was possible to evaluate not only

acquisition of new but also conservation of the already acquired skills.

Table 32: Dispositions toward critical thinking

Stages Stage A

Stage B

Stage C

Critical thinking Skills practiced



and Causal

Connection

Statistical connection,

causal connection, and judgment of

representativeness

Number of item

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

a) Identifying variables [Aa1] [Aa2] [Aa3] [Aa4] [Aa5] [Ab1] [Ab2] [Ab3] [Ab4] [Ab5] [Ac1] [Ac2] [Ac3] [Ac4] [Ac5]

[Ba6] [Ba7] [Ba8] [Ba9] [Ba10] [Bb6] [Bb7] [Bb8] [Bb9] [Bb10] [Bc6] [Bc7] [Bc8] [Bc9] [Bc10] [Bd6] [Bd7] [Bd8] [Bd9] [Bd10] [Be6] [Be7] [Be8] [Be9] [Be10] [Bf6] [Bf7] [Bf8] [Bf9] [Bf10] [Bg6] [Bg7] [Bg8] [Bg9] [Bg10]

[Ca11] [Ca12] [Ca13] [Ca14] [Ca15] [Cb11] [Cb12] [Cb13] [Cb14] [Cb15] [Cc11] [Cc12] [Cc13] [Cc14] [Cc15] [Cd11] [Cd12] [Cd13] [Cd14] [Cd15] [Ce11] [Ce12] [Ce13] [Ce14] [Ce15] [Cf11] [Cf12] [Cf13] [Cf14] [Cf15] [Cg11] [Cg12] [Cg13] [Cg14] [Cg15] [Ch11] [Ch12] [Ch13] [Ch14] [Ch15]

b) Referring to sources

c) Identifying conclusions

d) Evaluating the source’s reliability ( professionalism, absence of conflict of interests)

Not Relevant

e) Suspending judgment (when evidence and arguments are insufficient, searching for evidence and counterevidence)

f) Offering alternatives (searching for alternative explanations) g) Readiness for investigation


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7.3 Results The results in this chapter are presented in two parts: first the quantitative and then the

qualitative results. The quantitative part was collected from the students' open answers to

the questionnaires we passed to them (see the Methodology chapter). We have isolated,

coded, collected and statistically analyzed various elements in the students' answers that

corresponded to the different skills we were looking for.

The qualitative part is based on the students' interviews and written assignments.

7.3.1 Quantitative Findings38 Acquisition of skills was examined as reported in Table 33 and 34. The group's results

show that, although in the learning process the students were occupied most of the time

with investigation, discovery and solving probability problems, while the time devoted to

direct practice of thinking skills was significantly reduced, the students still conserved

most of the skills they had acquired at a high level. Table 33 and table 34 presents the

qualitative findings on acquisition of skills in High-school group (N=20) and Kidumatica

group (N=18) .

Skills of Critical Thinking Stages Mean Std Dev t-Value

Identifying variable

A→B 0.01 0.079 0.57 A→C 0.12 0.386 1.39 B→C 0.11 0.334 1.47

Referring to sources A→B 0.02 0.194 0.46 A→C 0.24 0.56 1.92 B→C 0.26 0.528 2.2(*)

Identifying conclusions A→B 0.1 0.189 2.36(*) A→C 0.16 0.359 1.99 B→C 0.06 0.421 0.64

Evaluating the source’s reliability B→C 0.16 0.376 1.9

Suspending judgment B→C 0.07 0.578 0.54

Offering alternatives B→C 0.48 0.575 3.74(**)

Readiness for investigation B→C 0.25 0.576 1.94

Table 33: Statistical tests of differences for High School 1 group * 0.05>p>0.01, ** 0.01>p>0.001

38 For each of the seven skills, the statistical correlations between the learning stages (A, B and C) are described in a table summarizing the results (tables X and Y), both for a two-stage process (A→B, B→C) and a one-stage process (A→C). Also, the mean, standard deviation and t-value for each permutation were calculated.


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Skills of Critical Thinking Stages Mean Std Dev t-Value

Identifying variable

A→B 0 0.168 0 A→C 0.156 0.483 1.37 B→C 0.156 0.397 1.66

Referring to sources A→B 0.011 0.211 0.22 A→C 0.322 0.575 2.38(*) B→C 0.333 0.54 2.62(*)

Identifying conclusions A→B 0.078 0.239 1.38 A→C 0.022 0.417 0.23 B→C 0.1 0.401 1.06

Evaluating the source’s reliability B→C 0.356 0.493 3.06(**)

Suspending judgment B→C 0 0.434 0

Offering alternatives B→C 0.089 0.496 0.76

Readiness for investigation B→C 0.356 0.551 2.74(*)

Table 32: Statistical tests of differences for Kidumatica group (N=18) * 0.05>p>0.01, ** 0.01>p>0.001

7.3.1.1 Analyzing the Findings by Specific Skill

Skill (a): Identifying Variables

Identifying variables is a fundamental skill in critical thinking. As we can see in Chart 21,

this skill was examined in three sequential grades: when statistical connection was taught,

when causal connection was added, and when judgment of representativeness was added

to the learning unit. The correlation between each two pairs was calculated and

construction of the skill was defined as retaining the skill throughout the stages. The bold

lines pertain to cases of such nature: both High School 1and Kidumatica groups gained

the skill, whether this was a one-stage or two-stage process.

Identifying Variables

0.7

0.8

0.9

1

Statistical Connection Statistical Connection and CausalConnection

Statistical Connection and CausalConnection and Judgment of

Representativeness

High SchoolKidumatica

Chart 21: Skill (a) Identifying Variables


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Skill (a), identifying variables, was defined here as a basic skill. In Chart 21 one can see

that this skill has improved between the "Statistical Connection" and "Cause-Effect

Connection" sections of the learning unit and was conserved in both groups when they

moved on to the "Judgments of Representativeness" section.

Skill (b): Referring to Sources

Among the skills of critical thinking, referring to sources is more difficult to construct as

Chart 22 shows. Between the first two stages only the Kidumatica group gained this skill,

but when it was examined in the third part of the study, only the high-school students had

the skill. This may be due to the effect of the difference in time spent with the teacher

(1.5 hours a week for the Kidumatika group and 4.5 hours a week for the high-school

group). As will be discussed later, this may imply that referring to sources is a skill that it

is time-consuming to develop.


0.4

0.6

0.8

1



Representativeness


Chart 22: Skill (b) Referring to Sources

Skill (b), referring to sources, was defined here as a basic skill. In Chart 22 one can see that

this skill has improved between the "Statistical Connection" and "Cause-Effect

Connection" sections of the learning unit and was conserved in both groups when they

moved on to the "Judgments of Representativeness" section.

Skill (c): Identifying conclusions

Skill (c) Identifying conclusions was defined here as a complex, non-basic skill. In Chart 23

can see that this skill has generally improved. It deteriorated slightly in both groups

between the "Statistical Connection" and "Cause-Effect Connection" sections of the


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learning unit, but improved in both groups when they moved on to the "Judgments of

Representativeness" section. This finding will be analyzed in the "Discussion" section.

Identifying Conclusions

0.7

0.8

0.9

1



Representativeness


Chart 23: Skill (c) Identifying conclusions

Skills: (d) Evaluating the sources, (e) Suspending judgment, (f) Offering alternatives and

(g) Readiness for Investigation (among High-School 1 Group)

As Chart 24 shows, the skills of Suspending judgment and Evaluating the sources

improved in both groups, while the skill of Offering alternatives did not improve and

even deteriorated, and the skill of Readiness for Investigation was retained. These effects

will be analyzed in the discussion section.

Skills over time class High School

0

0.2

0.4

0.6

0.8

1

Statistical Connection and Causal Connection Statistical Connection and Causal Connectionand Judgment of Representativeness

Evaluationg the Source'sReliabilitySuspendeing Judgment

Offering Alternatives

Readiness for Investigations

Chart 24: Skill (d,e,f,g) Evaluating the sources, Suspending judgment, Offering alternatives and Readiness for Investigation over high school group.


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Skills: (d) Evaluating the sources, (e) Suspending judgment, (f) Offering alternatives and (g) Readiness for Investigation (among Kidumatica Group) As one can see in Chart 25, skills (d), (e) and (f) have been retained by the students,

while skill (g) deteriorated significantly. When progressing from stage B to stage C,

retention of the skills (d)-(f) supports progress towards constructions.

Skills over time class Kidumatica

0

0.2

0.4

0.6

0.8

1

Statistical Connection and CausalConnection


Representativeness

Evaluationg the Source's Reliability

Suspendeing Judgment



Chart 25: : Skill (d,e,f,g) Evaluating the sources, Suspending judgment, Offering alternatives and Readiness for Investigation


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7.3.3 Summarizing the Quantitative Results To sum up the findings of the third research question, the present teaching method is

based on thinking activities that transform implicit thinking processes taking place in the

student's consciousness into explicit thinking processes open to observation, hearing,

emulation and internalization by the student's partners in the learning process. The

transformation of implicit into explicit thinking processes is the foundational strategy of

this method. When thinking processes in the classroom become explicit, the students

think about their thinking processes and improve them. When the students talk, write

about and schematically draw their thinking processes concerning a certain idea or

problem, they perfect these processes and deepen their understanding of the

idea/problem. If so, the learning unit has accomplished its goals in most of the cases: for

the Kidumatica group, skills (a), (b), (c), (e), and (f) were constructed and for the High-

school group skills (a), (c), (d), (e), and (g) were constructed (see chart 27,28). These

important results are illustrated in the following charts, and a possible explanation as to

why the other skills were not constructed will be provided in the discussion (see chart

26).

In addition, a control group of students who have not undergone the teaching process had

virtually none of the skills, as demonstrated in Figure 6.

Compare Experiment to Control on Skills

0

0.2

0.4

0.6

0.8

1

Iden

tifyi

ngV

aria

bles

Ref

ferin

g to

Sou

rces

Iden

tifyi

ngC

oncl

usio

ns

Eva

luat

iong

the

Sou

rce'

s R

elia

bilit

y

Sus

pend

eing

Judg

men

t

Offe

ring

Alte

rnat

ives

Rea

dine

ss fo

rIn

vest

igat

ions

Control (n=24)

Experiment (n=38)

Chart 26: Experiment and control group


101

Skills over time class Kidumatica

0.4

0.5

0.6

0.7

0.8

0.9

1

Statistical Connection Statistical Connection andCausal Connection

Statistical Connection andCausal Connection and

Judgment ofRepresentativeness


Reffering to Sources






Chart 27: All the skills over time Kidumatica class

Skills over time class High School

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S tatis tic al C onnec tion S tatis tic al C onnec tion and C aus alC onnec tion

S tatis tic al C onnec tion and C aus alC onnec tion and J udgment o f

R epres entativenes s


Reffering to Sources






Chart 28: All the skills over time High school class


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7.4 Qualitative Findings As noted above, this comparative research is based on a combination between

quantitative and qualitative methodology, in order to present several perspectives of

observation and interpretation of the students’ critical thinking abilities with regard to

mathematics. It was important to take into account the context of the educational reality

in each of the frameworks and the socio-cultural context of the research participants,

while entering into a direct dialog with them (Sabar Ben-Yehoshua, 2000). The mixed

method of research allows to describe and explain the researched reality in its multiple

aspects and to reveal the knowledge that the research participants have on thinking

processes in the context of mathematics study. The choice of mixed method also helps to

achieve insights about the specific qualities of the learning unit the perceptions, thinking

skills and reflective ability of the participants, to have a deeper sense of what they

experienced in the learning process aimed at thinking development, and to evaluate the

significance of this experience for them. Twenty-seven interviews related to critical thinking were conducted with the students

towards the end of our course, in order to closely examine their personal attitude towards

mathematics, critical thinking and the development of thinking, and to reveal the

students’ thinking patterns in their interaction with “Probability in Daily Life” and

mathematics. The interviews allowed to create a direct, open and flexible dialog with the

students, which provided an additional source of information for evaluating their critical

thinking abilities. An additional body of findings is derived from the group discussions

aroused by the learning unit, which shows the centrality of critical thinking in everyday

life. With this set of findings, as with the others, the purpose of the analysis was to

examine the students’ patterns of critical thinking in the mathematical, social and cultural

context.

In the course of teaching the unit, we have interviewed a number of students and asked

them a number of questions concerning critical thinking. During the interviews we have

identified a number of recurrent elements presented below. The interviews were of two

kinds: closed/ structured interviews, where questions were composed in advance, and

open/ semi-structured interviews, where questions were also composed in advance but

selected and/or modified according to the interviewee’s answers. In all of the interviews,


103

three main elements recurred throughout the students’ answers: the usefulness of critical

thinking as an instrument for life and study; the importance of critical thinking as a more

empowered attitude towards authoritative sources of information and opinion; and

finally, the role of critical thinking in promoting the students’ general understanding of

the world.

As mentioned in the theoretical background section, there is no single agreed-upon

definition of critical thinking, and the specialists are divided as for its meaning. There are

a number of definitions in the literature but none of them embraces all the aspects of the

phenomenon. Fig. 6 presents the categories we found recurring in the interviews, which

are in good agreement with the research literature reviewed above.

Figure 6: The three main elements that appeared in the interviews

7.4.1 The Findings of the Structured Interviews To the question, “What is critical thinking?” or the prompt “Critical thinking is…,” the

students gave the following answers, which define critical thinking in three main

dimensions: as a tool they can use in life and studies, as an attitude towards authority and

sources of information, and as a way to improve their general understanding of the world.

The answers that define critical thinking as a useful tool would say, for example: “It’s

something for which you need to use your brain properly. Something about critique. For

instance: an ad in a newspaper that is not true” [B536]; “To know how to check

findings, opinions, reliability; to research, to doubt” [R505]; “Not to trust everything

[you hear], to check before you decide. Not to believe any odd survey [right away]. To

Instrument

Empowering Attitude

Understanding of the world

Three Main Aspects of Critical Thinking

Dispositions Abilities


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think about every thing” [A847]. The latter definition brings forward a strong aspect of

critical attitude towards authoritative sources of information, as does the following one:

“In my opinion, the importance of critical thinking is that you don’t take everything they

tell you for granted, but check whether it’s true and whether it’s possible that the person

who is explaining is wrong, and you accept mistakes. “If we didn’t have critical thinking

we wouldn’t be able to understand well” [Y318].

Yet another extensive definition focuses on critical attitude towards information sources,

but also puts a lot of emphasis on the role of critical thinking in learning to understand

the world better: “Often I used to see only the external aspect of things and wouldn’t

really see what they are about. All of a sudden things become explicit, something lights

on me, and it has to do with understanding. When I understand something, it also helps

me to understand myself better. I have a greater power. When we studied investigation, I

felt that my voice was becoming strong. [I could ask,] Who is doing the research, how

many people, what are the purposes? I got power out of understanding, to understand

more things better” [E886]. The aspect of empowerment acquired by mastering critical

thinking should be noted here as well. Also the following definition, “Every time I study,

I discover new things, things are becoming clearer to me” [A427], focuses on the aspect

of improved general understanding that critical thinking provides. Finally, one student’s

definition of critical thinking as “A way of life” [E886], while it eludes going into a

detailed analysis, does captures the all-encompassing influence of acquisition of critical

thinking on the students’ lives and perception of the world. To sum up, the main elements

in the students’ definitions of critical thinking are as follows:

1. openness to a variety of opinions and ideas;

2. serious consideration of other points of view;

3. suspension of judgment when evidence and arguments are insufficient;

4. consolidating or changing an opinion when evidence supports doing so;

5. looking for precision in information, searching for reasons and arguments, examining

all the possibilities.

Answers to the question “Who is a critical thinker?” are closely related to the definitions

of critical thinking itself, but also add an important dimension of personal wisdom and

intelligence as traits closely associated with critical thinking, as, e.g. in the following

answers: “A critical thinker knows how to examine things, put things into question, to go


105

deep and think about what s/he sees” [R505]; “A critical thinker for me is an intelligent

person, with a lot of world knowledge and life wisdom, which they can draw on when

they are thinking critically about what they read or what they get. They also need

mathematical thinking” [S210]. To sum up, the main elements of the students’ definition of the critical thinker are as

follows: (i) someone who tends to shape, correct and change their beliefs in light of

convincing arguments (ii) someone who is capable of understanding at least two

opposing, well-defined points of view on the same subject while maintaining one’s own

standpoint regarding the subject.

In answering the question “In what ways can critical thinking be developed?” the

students emphasized learning from other people, reading, and the importance of patience

and perseverance. One student said it can be “learned from reading books, criticisms,

articles, listening to other people’s opinions. In researches they discuss methods that one

can examine” [R505]. Another student emphasized the challenge that learning and

practicing critical thinking poses, and the importance of insistently pursuing it: “What’s

interesting about critical thinking is that at first everything is very difficult and

complicated, and then, when you peel off leaf after leaf, you discover some little treasure;

at first it seems very complex, so you need to remember that all the time you need to keep

exploring, because as long as we go on it becomes more and more beautiful” [E886]. In their answers to the question about the ways of developing critical thinking, the

students named three main abilities that need to be developed:

1. The ability to distinguish between opinion and fact: the difficulty of distinguishing

between utterances expressing the position of the speaker/writer on a certain reality, and

the expressions of facts/events comprising this reality.

2. The ability to identify information intended to influence the reader emotionally, such

as using emotional manipulation as a means for presenting an argument and persuading

the reader.

3. The ability to recognize stereotypes and avoid using them: it is difficult to identify

overgeneralization that leads to stereotyping and is likely to create bias and acceptance of

a stereotype as a scientific fact.


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7.4 Discussion Acquisition and construction of higher-order thinking skills by students in general and

mathematics students in particular has become one of the main targets of the education

system widely accepted by educators around the world. The acquisition of these skills

will enable the student to function as an active and productive citizen, and the challenge

at present is to find ways of teaching and developing this approach not only in the

excellent students but in the total population of students in schools.

Higher-order thinking involves applying many different criteria that frequently contradict

each other, as well as self-regulation of thinking processes (independence of others at

every stage of thinking). Earlier in this dissertation (see section 2.2.2), we have referred

to the general skills that characterize critical thinkers in the context of the learning unit

we use. With the qualitative methodology we have chosen for this research, it was

possible to examine these different aspects from several perspectives that enabled

observation and interpretation of the educational reality in each group and conducting

direct dialogue with the research participants (Sabar Ben Yehoshua, 2000). Thus it

became possible to point out several tendencies that became apparent during the research.

Analyzing the findings, we have arrived at the following generalizations regarding the

process of critical thinking skills construction and teaching:

(1) It seems that critical thinking skills do not develop spontaneously and that even good

students acquire them by means of explicit instruction. This finding is in direct opposition

to Rohwer's claim (1971) that learning skills and learning strategies develop in the

student spontaneously, without direct instruction.

(2) To a large extent, the construction and teaching of critical thinking skills are

determined by specific contents and tasks the teacher uses. In this research, the skills

were chosen with respect to the contents and the increasing difficulty level of the learning

unit. This finding corresponds with other researches, such as Bransford, Sherwood,

Rieser, and Vye (1986), and Glaser (1984, 1985).

(3) It is possible to significantly influence and change the mathematical discourse in the

classroom and the students' language of critical thinking, by providing appropriate

conditions and using appropriate instruction methods. In the literature, this finding

applies not only to older and/or more successful students, but also to younger and/or


107

underachieving ones (Weinstine and Meier, 1986; Feuerstein, Rand, Hoffman, and

Miller, 1980).

(4) Excellent students (the Kidumatica group) were capable of operating a greater

number of skills automatically, quickly, utilizing a minimal degree of conscious effort.

However, this automatic application of thinking has only been acquired after much

practice and exposure to different learning contexts. What is more, even expert learners

are likely to return to a much slower and more conscious way of learning when

confronted with unfamiliar tasks or connections.

Thus, the learning process of expert learners is often characterized by cycles of higher

and lower automatism (Bransford, Sherwood, Rieser, & Vye, 1986; Lesgold, 1986).

Resnick (1987) argues that it is difficult to define higher-order thinking skills, but easy to

recognize them when used by someone. She believes that higher-order thinking is not

algorithmic, and that thinking patterns are not clearly defined in advance. This type of

thinking often concludes with multiple solutions, each of which has its advantages and

disadvantages, but does not yield a single clear solution.

High-level thinking has to do with skills in solving problems, asking questions, thinking

critically, making decisions and taking responsibility (Zoller, 1993, 2000; Zoller & Ben-

Chaim, 1998). Decision is an essential part solving a problem that involves a gap between

an initial situation and a final goal and there is no easy, well-known way of finding a

solution. Based on the findings of this research, it seems that significant learning of

thinking skills in the context of mathematics enables the students to develop basic critical

thinking skills (in this research, skills a, b, c) by way of solving probability problems.

This type of learning emphasizes the development of skills in the process of solving

mathematical problems.

7.4.2 Discussion of Correlation between Mathematical Knowledge and Critical Thinking Skills According to various statistical analyses (including the t-test), there is no clear correlation

between the development of critical thinking and the development of mathematical

knowledge (P>0.05 in both rounds). A possible explanation for this is the "ceiling effect”:

since the experimental group in both rounds consisted of students at the highest level of


108

mathematics studies (5 learning units), the quantitative tools were not sensitive enough to

the small differences at the high level of grades.

The following findings arose concerning the first round: according to the statistical

analyses (t-test), it is possible to say that the interaction between the development of

critical thinking and development of mathematical knowledge is not significant (P>0.05).

It should be noted that in the first round there was an improvement in the students’

achievements in mathematics, but it was impossible to determine whether this

improvement took place as a result of the proposed learning unit or vice versa. We have

assumed that no differences are expected between grades, because the skills tested by the

Cornell test and the CCTDI test are different from mathematical skills learned in the

framework of regular mathematics studies.

This assumption was confirmed, which may suggest that the dispositions towards critical

thinking or skills of critical thinking do not depend on previous mathematical knowledge

or the course of studies. Also in the second round, both in the experimental and the

control group, no connection was found between the development of critical thinking and

the development of mathematical knowledge. It is possible that in both rounds there was

a “ceiling effect,” since the research population studied mathematics at a high level (4-5

learning units), and therefore the quantitative tools were not sensitive to the small

differences in the high-level grades. This finding will be discussed further.

ther.

General Discussion and Conclusions

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8. General Discussion and Conclusions "It is not subject matter that makes some learning more valuable than others, but the spirit in which the work is done"

(John Holt) This chapter consists of three sections. The first section provides a general discussion

relating to all three research questions and the broader implications of the findings (e.g.

the acquisition of the language of critical thinking), as well as clarifying the relation of

my research findings to the literature. The second section, general conclusions, evaluates

the results with regard to the research purposes, presents the limitations of the present

research, outlines the practical implications of the findings for the teaching practice, and

proposes directions for further research. The final section describes the contribution of

the present research to the field of general and science education

8.1 The Research

The earlier chapters presented in detail my research findings (see chapter 5). The

following is a short outline of its four goals:

1. An examination of the contribution of infusion-approach study of “Probability in Daily

Life” to the development of CT dispositions.

2. An examination of the contribution of infusion-approach study of “Probability in Daily

Life” to the development of CT abilities.

3. An examination of the processes of construction of critical thinking skills (e.g. putting

a statements into question, postponing judgment, referring to sources) during the

infusion-approach study of the “Probability in Daily Life” unit.

8.2 General Discussion in Light of Research Questions and CT Literature

The results presented above provide a clear indication that consistency in instruction that

promotes critical thinking, as this was performed by both the researcher and teacher,

contributes to the development of several significant factors related to critical thinking:

truth-seeking, open-mindedness, self-confidence in critical thinking, maturity of

judgment, and reasoned decision-making. Moreover, the results show that a study that

improves the students’ thinking abilities also improves their ability to evaluate

information and to identify and prove that information in order to make deductions. The

above results support the research hypothesis regarding critical thinking skills, which also


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highlights the potential inherent in identifying specific instruction strategies in a cause

and effect relationship. Weinberger’s research (1998) indicates the implications of

experimenting with systematic learning that uses a program designed to promote critical

thinking. Her research shows that the thinking skills of science teaching students have

developed as a result of a systematic learning during the course on teaching thinking with

science. Feuerstein's (2002) research demonstrated a connection between the curriculum

(of a communication course) and the development of critical thinking, which shows that

the theoretical and pedagogical components of the curriculum strongly develop the

students’ thinking abilities. Zohar and Tamir's (1993) study also shows that critical

thinking does not evolve naturally: rather, it takes a deliberate effort to promote and

develop. This effort was made possible by including activities that encouraged critical

thinking in the regular high-school biology curriculum, in a project called Haviv. This

integration did not require extra teaching time and did not come at the expense of the

student's knowledge of biology; on the contrary, it has improved this knowledge. Zohar

and Tamir believe that this project greatly contributes to its participants' ability to

perform tasks involving critical thinking in the field of biology.

8.3 Other Points for Discussion Derived from Research Findings

I would like to consider two interesting points that arose in this research, without

connection to the research questions: first, the development of the language of critical

thinking, and secondly, the teachers' view of the specific teaching strategies used in this

research, and their definition of critical thinking. These points do not directly relate to

any of the research questions, but are nonetheless significant.

8.3.1 The Development of the Language of Critical Thinking in the Classroom

The findings of the qualitative part of my research indicate that a new language has been

developed in the classroom. In his book Smart Schools, Perkins (1992) focuses on how to

change our teaching and our schools to enable children to learn more meaningful

information. One of his suggestions is that we should have a "thoughtful school," which

means that teachers should teach by using a language of thinking. Language is a central

component of mathematics and of mathematical education (Fuson & Twon, 1999); it both


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supports the thinking processes and is a medium of teaching mathematics and of shaping

knowledge.

The main use of language in mathematics has to do with definition of principles and

terms, expression of mathematical ideas in the form of formulas and speech, solving

mathematical problems in general and textual and geometrical problems in particular

(Cummins, 1991; Geary et al., 1996; MacGregor & Price, 1991; Tishmann, 2000).

Since the learner's thinking development is determined by language, it is impossible to

separate language from learning (Vygotsky, 1962), since thinking to a large extent

consists of language that comprises a metaphor for thinking (Sfard & Linchevsky, 1994).

The student develops language out of experience, and acquires a variety of words as

symbols for different concepts, thus, the meaning of a word is a combination of thought,

experience, and communication (Vygotsky, 1962). Many researchers, particularly in the

last decade, have articulated the significance of language for the instruction of

mathematics. The NTCM (National Council of Teachers of Mathematics) and the ASA

(American Mathematical Association) both emphasize the significance of language for

the study of mathematics and warns about the possible difficulties students may

encounter due to language barriers. These difficulties may be the result of dealing with a

new vocabulary on the one hand, and misunderstanding the exact meaning of words on

the other hand. In this research we have seen that students began to speak in a new

language. Here are some examples: - “First we should check the information source’s reliability”

- “The conclusion is not valid because we don’t have all the data.”

- “Despite all the numerical data, I don’t accept the researcher’s conclusion“

- ”We may have found a statistical connection, but we didn’t find a causal connection between the

factors, so we can’t determine the direction of the connection” - “I think critical thinking is important when you study mathematics, when you study other

- topics and when you read a newspaper, but it is most important when you deal with real-life situations,

and you need the right instruments in order to do so”

Students experienced a process of reflection and communication. They conducted

mathematic conversations and developed the language of critical thinking. These findings

correspond with those of Panama et al. showing that while the children's level of

understanding and problem solving ability have improved their basic skills remained


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unaffected (Aizikovitsh & Amit, 2009; Panama, Carpenter, Frankie, Levi, Jacobs, &

Empton, 1994).

Development of the language of critical thinking can also be well demonstrated by

comparing the actual student answer sheets. Those obtained from students before the new

learning process demonstrate that the very basic component of learning mathematics had

already existed. The students use the regular methods of approaching a question in

probability, using the two-dimensional table and the Bayes formula for conditional

probability. Similar answer sheets from students that have undergone the learning unit

demonstrate the verbal explanation added to the two dimensional table, the richness of

the language and the additive value of critical thinking.

Figure 3: Before the Learning Unit


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Figure 4: After the Learning Unit


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8.3.2 The Teachers' Perspective on the Instruction Strategies and Definition of Critical Thinking

Observations carried out throughout the second year of research confirmed the fact that

teachers integrate critical thinking skills in their teaching. Thus, for example, both

teachers encouraged making connections between the learned material and daily life.

They integrated inquisitive learning and raised open questions that stimulate the students'

thinking. They also encouraged the students to ask questions and created a new language

in the classroom.

An interview with teachers was conducted, in which they were asked the following

questions: What is critical thinking? When should it be used, and when should it be

avoided? What is "critical thinking" in its strong sense? What factors enable and what

factors block critical thinking? What promotes it? How did you feel while teaching the

activities and strategies fostering critical thinking, particularly, identification of

assumptions, mapping and evaluating claims according to context and such criteria as:

clarity, explicitness, relevance, acceptability and sufficiency?

These semi-structured interviews were conducted with both teachers of the experimental

groups and focused on their teaching strategies and their definition of the term "critical

thinking." M. is a relatively young teacher, with six years of teaching experience. She has

an M.A in mathematics instruction. Occasionally she participates in after school

workshops for teachers, and is fairly aware of the novel strategies of learning and

teaching. She enjoys teaching mathematics, and on the basis of our class observations it

seems that her students like her. L. is an exceptionally skilled teacher with 27 years of

teaching experience. She has a B.Sc in mathematics instruction. L. is the coordinator of

her school's mathematics curriculum, and a member of the administration. She is a senior

instructor of the Ministry of Education, takes part in continued education programs for

mathematics teachers, and actively participates in committees in charge of the

development and implementation of innovative study programs in mathematics.

Both teachers were asked to provide examples of specific methods to encourage high

order thinking skills among their students, and how they understand the term "critical

thinking". In the interviews, both teachers expressed a similar ‘global’ or one could say


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‘holistic’ world outlook. That is, they stress the significance of the whole, and its

reciprocal dependence on all its parts. Thus, they consider mathematics as highly relevant

to daily, real-life matters. According to them, their teaching strategies include presenting

problems from real life. M. believes her teaching strategies promote critical thinking: "I

encourage students to ask questions in the classroom, to interrogate phenomena and make

assumptions… I teach them new terms in the context of daily life. It is crucial not to

remain on the level of transmitting information: you must teach them to think as well."

M. also believes it is important to create a connection between mathematical terms and

the students' daily experiences: One way to connect the student to "the world of

mathematics" is to teach him/her that mathematics is everywhere and in everything… and

that mathematics (in this case probability) has the capacity to explain many phenomena

familiar from everyday experiences. It is a great challenge for me to create these

connections. Often I refer to other physical and biological phenomena outside the field of

this study unit. For example, when I teach the topic of "exponential functions," I discuss

it in relation to the reproduction of microorganisms or radioactive radiation… I don't

stick to the limits of the discipline. M's interview gives the impression that she

deliberately blurs the boundaries of the discipline. When she was asked to define the term

"critical thinking," she replied:

I believe that "critical thinking" is a method of organizing thinking, basing it on logic in a

systematic way. I expect my students to use critical thinking when solving problems in a

systematic way. I expect them to be able to make assumptions and draw conclusions

based on previous knowledge and using the tools they have acquired in class. M's

definition of "critical thinking" is close to Ennis’s (1985), who defines the term as

reflexive, logical thinking. When M. was asked about the importance of critical thinking,

she said: I believe that critical thinking is important when one studies science or

mathematics. It is significant for other disciplines as well, but it is most crucial when it is

related to real-life situations, when the right tools are necessary to deal with these

situations. L. adheres to similar notions.

She believes that "when it comes to science, it is important to teach not only 'facts and

numbers', but also how to think critically and creatively." She found it difficult to explain

what "critical thinking" is, but when she did, her definition seemed close to that of Zohar

(1993): "I find it very difficult to define what thinking is… I think it is important for


116

students to reflect on their thoughts and to understand the profound meaning of things…

to help them make a decision." Strengthening the students' motivation was one of M's

goals in teaching, and L. similarly aims at cultivating her students' positive reaction to

and experience of mathematics: "I want my students to ‘live’ mathematics… I want my

teaching to influence what they feel about this discipline." When asked about the

significance she attributes to critical thinking, L., like M., stated that she believes critical

thinking is extremely important for all the future citizens in our society.

In light of the above, it should be considered what the role of the teacher is, in the

framework of this approach that aims to promote critical thinking among students. The

role of the teacher is to encourage students to perform the expected changes in their

terminology and perception, to persuade them that such changes are necessary, and to

help them make these changes.

Among other things, the teacher should be able to talk with the students about this and

reflect on the matter in smaller groups, less intimidating than the classroom) Instruction

by way of class negotiation highlights the principle that the use of each of these tools

needs to be accompanied by dialogue between the teacher and the students, and between

the students themselves, i.e., class and group discussions. In these discussions, the

students can express their opinion about the learning material, present their insights, ask

questions, make comments, argue about different interpretations, etc. Importantly, the

purpose of class discussions is to provide room for genuine dialogue where meanings are

clarified and in which students feel free to express their actual thoughts rather than what

the teacher expects them to learn and say. Thus, a class that studies according to the

‘class negotiation’ system, will be characterized by relatively extended class discussions

in which the teacher and the students discuss the studied topic and its meanings

Contribution and Implications

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9. Research Contribution and Implications "Kids in school are simply too busy to think." (Kohn)

But are they?

In this chapter I will briefly review my findings, and then discuss the research

conclusions, recommendations, limitations and scientific contribution, and provide

suggestions for further examination.

9.1 Review of Principal Findings This research has focused on a learning unit aimed to promote higher-order thinking

skills and more specifically, critical thinking. (i) In all three rounds, a moderate

improvement has been detected in the critical thinking dispositions of all experimental

groups. This improvement may be attributed to maturation and accumulating life

experience as well as learning proper. All of these are significant factors affecting the

development of the students' critical thinking, particularly within the framework of

probability. (ii) Throughout these rounds, a moderate improvement was also detected in

the students' critical thinking abilities.

As in the case of dispositions, this improvement can also be ascribed to maturation,

accumulating life experience, knowledge in other mathematical fields (e.g. geometry

contributes to the development of deductive skills), and learning proper. (iii) Teaching

critical thinking also contributes to the construction of these skills in the framework of

mathematics. Thus, when teachers consistently emphasize critical thinking skills, the

students are more likely to succeed. (iv) This research did not detect a clear-cut

distinction between the critical thinking abilities and dispositions of excellent and

average mathematics students. That is, no direct correlation has been found between the

development of mathematical knowledge and the development of critical thinking.

9.2 Conclusions Within the framework of mathematics studies, critical thinking does not develop

spontaneously but requires an effort. Critical thinking skills rely on self-regulation of the

thinking processes, construction of meaning, and detection of patterns in supposedly

disorganized structures. A considerable mental work is involved in the processes and


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judgments it requires. Critical thinking is not algorithmic, i.e. its patterns of thinking and

action are not clear or predefined. Critical thinking tends to be complex. It often

terminates in multiple solutions that have advantages and disadvantages, rather than a

single clear solution.

It requires the use of multiple, sometimes mutually contradictory criteria, and frequently

concludes with uncertainty. The latter conclusion corresponds with Zohar's research

(1996, p.21). (i) Conventional teaching is not appropriate for the changing and

challenging world we live in, which demands critical/evaluative thinking based on

rational decisions and dispositions. In this research we find that combining different

instruction strategies (such as asking questions, independent investigation of phenomena,

or experimenting in the framework of open discussion and drawing conclusions

considerably improves the students' critical thinking abilities and dispositions. These

findings correspond with those of earlier researches (Facione, 2002) showing that critical

thinking relies on cognitive activity directed at focused, inquisitive interpretation of

relevant information, and constant reference to the student's dispositions. (ii) (Partial)

transfer between disciplines is possible.

One of the main goals of teaching higher-order thinking skills, such as critical thinking, is

the transfer of these skills to all disciplines and fields. However, transfer within and

between disciplines is difficult to put into practice (Bransford et al., 1999). In this

research the instruction of higher-order thinking skills was used within the framework of

mathematics studies, but the students' success in critical thinking tests indicates their

ability to transfer their critical thinking skills to other fields, since these tests are based on

generic questions that are not confined to specific disciplines. I will elaborate on this

conclusion further in the “Research Limitations” section.

9.3 Recommendations We live in a period of non-stop dynamic changes in all the areas of life. The amount of

knowledge accumulated by the different research disciplines is immense and ever-

growing, which makes it impossible to endow students with all the information they may

need in the future. Thus, the education system needs to adapt itself to the world of

tomorrow. Along with imparting basic knowledge, education needs to impart skills


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needed for independent confrontation with new information and with the challenges of

the 21st century.

We believe that a graduate of the education system should be capable of critical thinking

(which is a central empowering mental tool necessary to the citizen of the modern

democratic society as a learner, consumer, professional, and more) and adopt critical

thinking as a way of life. The term "critical thinking" refers to the individual's ability to

adequately evaluate claims by means of logical-analytic skills, to use reflective thinking

that raises question questions regarding inclinations, beliefs, perceptions and ways of

action (Facion, 2002; Ennis, 1989). These research findings have major educational

implications concerning the training of teachers for taking part in programs designed to

promote critical thinking.

The empirical results convincingly show that conscious, consistent instruction of critical

thinking in mathematics increases the students' chances of success. This conclusion is

extremely important for the process of changing teachers' beliefs and instruction

strategies in the discipline. We propose that the programs of professional development be

designed in such a way as to help teachers better understand what is higher-order thinking

and have a more coherent sense of what is critical thinking. We also propose to

encourage teachers to employ a wider range of teaching strategies, as presented in this

and other researches, in order to help their students fulfill tasks that require higher-order

thinking in general and critical thinking in particular.

9.4 Limitations This research has a number of limitations, both with regard to teachers and students and

to the contents of teaching material.

9.4.1 Limitations Concerning Teachers and Students Population

This research did not include a thorough examination of all the teachers' level of

involvement in the teaching process, or their attitude to teaching as a means of

developing critical thinking in mathematics; the teachers’ thinking functions in learning

and teaching have not been evaluated. Other teacher-related factors that can influence the

process, such as the teachers’ level of education, motivation and attrition, have not been


120

sufficiently examined. As pointed out in the Results Chapter, this research only examined

students who study mathematics at a high level (5 learning units) but not an inclusive

sample of the total population of students who study mathematics.

9.4.2 Content Limitations—the Problem of Transfer39. Educators around the world disagree on the best way of promoting thinking in general

and critical thinking in particular. There are two central questions in dispute: (a) Are

thinking skills general, or do they depend on specific content and system of concepts?

(McPeck, 1981) (b) To what extent and under what circumstances can critical thinking be

"transferred" from one discipline to another? I will attempt to provide an answer to the

second question, since it has greater implications for teaching strategies. The

characterization of the transfer procedure is highly controversial, since this procedure

depends to a large extent on the specific context and criteria40. Bransford et al. (1999)

argue that the transfer takes place when information acquired in a certain context is

applied in another context. This procedure is central for our understanding of how human

beings develop different skills that are open for diverse interpretations (Bransford et al.,

1999). In this research the "transfer problem" has not been thoroughly examined. We

know that transfer has taken place in this research, yet do not know to what extent or

under what circumstances it took place.

9.5 Research Uniqueness and Contribution This research was designed as a continuation of a large-scale pilot study, conducted by

the Ben-Gurion University of the Negev and a general high school (located at the center

of Israel) in 2007. The purpose of the pilot study was to examine the students' critical

thinking abilities in different environments drawing on infusion-approach study of

"Probability in Daily Life".

40 This topic appears in the research literature of critical thinking instruction and is largely connected with the larger question of the place of mathematics instruction in general.


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(i)The present research establishes points of reference to critical thinking dispositions

among students learning mathematics in different environments (high school and a

mathematics club).

This element has not been examined so far by the literature in the field (ii) This research

has identified and measured differences between dispositions, abilities, and construction

of skills characteristic of critical thinking in mathematics, and completes other

researchers conducted in other environments (iii)The combination of the Cornell test and

the CCTDI test in the evaluation of critical thinking abilities and dispositions is unique to

this research; it has not been performed in previous studies.

9.5.1 Research Contribution Educators in Israel, who wonder, like their colleagues in the West, about the goals of the

education system that could guide the different educational frameworks, may find in this

research an idea that can unify different topics and study programs, in order to prepare

the learners for life in a changing society, and develop their ability to think in a

systematic and independent way.

In much of the literature, critical thinking development is referred to as an important goal

of the educational system. This research may contribute to the public discourse of the

mathematical education community on this issue. It also raises the public awareness of

the need to develop critical thinking in the framework of mathematical education, which

may enable future examination and promotion of critical thinking development through

mathematics teaching in a fuller and more informed way.

To conclude, the main contribution of this research lies in revealing the connection

between critical thinking and the teaching of mathematics. Despite the problem of

transfer discussed earlier, the scientific contribution of this research lies in the new

insights it provides into critical thinking, its place and importance in teaching

mathematics. In this manner, it will be possible to strengthen the status of the study of

mathematics in imparting higher-order thinking skills, both in parallel with and beyond

the formal education program.


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9.5.2 Implications for the Formal School Curriculum Current mathematics teaching approaches espouse the conceptual understanding of

mathematics and stress the significance of problem solution, mathematical literacy and

mathematical discourse. According to this approach, teachers function as mediators

between the students and the information they need to acquire by asking questions,

posing challenges, and research. Thus teachers help students better comprehend

mathematical terminology, ideas and associations. This method of teaching is extremely

challenging for both students and teachers: it necessitates the teacher's profound

understanding of mathematics, intellectual effort and creativity, and the student's

confrontation with unfamiliar situations and contents.

The implications of this research for the education curriculum were designed on the basis

of previous studies41. Feuerstein (2002), Zohar and Tamir (1993), as well as Weinberger

(1998) point out the importance for the students to experience learning that develops

critical thinking by means of diverse study programs with special characteristics. The

findings of the present research may assist in developing curricula and instruction

methods for different ages and learning levels in mathematics, on the basis of the

connection between critical thinking and the study of mathematics through the learning

unit "Probability in Daily Life."

In light of the above, the implications of this research for the formal school curriculum is

in the opportunity it provides to expand the implementation of programs for critical

thinking development and their infusion into mathematics curricula, according to the

requirements specified by the formal education program 42.

9.6 Recommendations for Future Research and Concluding Remarks From this research’s findings and discussion, there arise the following research

recommendations: A more comprehensive examination of the processes of critical

thinking: to what extent could the students describe, orally and in writing, the processes

of thinking, activate them and apply the thinking skills they studied on the procedural and

meta-cognitive level? Did they make an informed use of terms and strategies of higher- 41 See appendix for the studies on which this research implications are based 42 I.e., "the student knows how to draw conclusions from mathematical models," "the student will develop logical mathematical thinking skills, such as drawing conclusions, making generalizations, analysis, making and supporting assumptions, self-criticism.”


123

order thinking, including critical thinking? On the basis of the former, it should be

examined what use the research participants make of the “language of thinking,” or, in

the words of Costa and Marzano, “do they speak thought?” (Costa & Marzano, in

Harpaz, 1997; Costa, 1991). Developing such a language involves, on the part of the

teacher, such skills as using precise vocabulary, presenting critical questions, presenting

data rather than answers, aspiring for exactness, giving directions, and developing meta-

cognition. Examination of the attitudes and perceptions of education students in colleges

for teacher training, practicing teachers and researchers of mathematical education with

regard to teaching that develops critical thinking in mathematics; evaluation of these

students’ and professionals’ critical thinking functions in teaching, learning, and research.

Teaching “Probability in Daily Life” and conducting the same research among all the

strata of the students’ population and not only among those who study mathematics at the

higher level. Examining the gender, age, and ethnicity aspects of critical thinking

development.

Concluding remark:

There is neither consensus nor coherence in contemporary approaches to

education for critical thinking. The research reported in this thesis has

demonstrated the viability of integrating the purposeful promotion of critical

thinking with the teaching of conventional mathematics content. It is hoped

that the findings of this study will contribute to our understanding of the

nature of critical thinking and to the further development of instructional

approaches relevant to its promotion.

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Appendices

136

Appendices

Appendix 1: CCTDI A Disposition Inventory

Appendix 2: Abilities Cornell Critical Thinking Test, Level Z


Appendix 4: Mathematics Questionnaire

Appendix 5: The Learning Unit “Probability in Daily Life” Appendix 6: Critical Thinking Questionnaire: “Probability in Daily Life”

Appendix 7: Sample Problems and Exams from the Course “Probability in Daily Life”

Appendices

137

Appendix 1: CCTDI A Disposition Inventory Instructions:

1. Separate the last page (answers page) from the back of the exam.

2. Mark with X in the appropriate slots on the answers page to what extent you agree

with each of the proposed statements.

3. Begin the mark regarding statement #1 and proceed through statement #75.

Statements (Selected)

6. It disturbs me when people rely on weak claims to defend good ideas.

8. It disturbs me that I may be under influences that I am not aware of.

15. Most topics studied in school are not interesting and not worth participation.

16. Exams that demand thinking and not only memorizing are better for me.

22. It is easy for me to organize my thoughts.

24. There is a limit to openness when we get to the question of what is right and what is

wrong.

31. I must have a basis for my beliefs.

39. It is very hard not to be biased when discussing my own opinions.

41. Frankly, I am trying to be less critical.

44. It is not very important to continue trying to solve difficult problems.

48. Others expect me to pose reasonable standards with regard to decisions.

50. I look for facts that confirm my opinions, not those that contradict them.

55. I really enjoy figuring out how things work.

60. There is no way of knowing whether one solution is better than another.

63. I am known as someone who approaches complex problems in an orderly way.

67. Things are as they appear.

69. Others expect me to decide when a problem reaches a solution.

73. Others are entitled to have their opinions, but do not have to listen to them.

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138

Appendix 2: Abilities Cornell Critical Thinking Test, Level Z (An Example) Section IA:

In the first five items, two men are debating about voting by eighteen-year-olds. Mr. Pinder is the

speaker in the first three items, Mr. Wilstings in the last two. Each item presents a set of

statements and a conclusion. In each item, the conclusion is underlined. Do not be concerned with

whether or not the conclusions or statements are true.

Mark items 1 through 5 according to the following system:

If the conclusion follows necessarily from the statements given, mark A.

If the conclusion contradicts the statements given, mark B.

If the conclusion neither follows necessarily nor contradicts the statements, mark C.

If a conclusion follows necessarily, a person who accepts the statements is unavoidably

committed to accepting the conclusion. When two things are contradictory, they cannot both be

correct.

CONSIDER EACH ITEM INDEPENDENT OF THE OTHERS.

1. “Mr. Wilstings says that eighteen-year-olds haven’t faced the problems of the world, and

that anyone who hasn’t faced these problems should not be able to vote. What he says is

correct, but eighteen-year-olds should be able to vote. They’re mature human beings,

aren’t they?”

2. “Furthermore, eighteen-year-olds should be allowed to vote because anyone who will

suffer or gain from a decision made by the voters ought to be permitted to vote. It is clear

that eighteen-year-olds will suffer or gain from the decisions of the voters.

3. “Many eighteen-year-olds are serving their country. Now there can be no doubt that

many people serving their country ought to be allowed to vote. From this you can see that

many eighteen-year-olds ought to be allowed to vote.”

4. “I agree with Mr. Pinder that anyone who will suffer or gain from a decision ought to be

permitted to vote. And it is true that eighteen-year-olds will suffer or gain from these

decisions. But so will ten-year-olds. Therefore, eighteen-year-olds shouldn’t be allowed

to vote.”

Appendices

139


Intermediary Questionnaires: Statistical Connection, Causal Connection, Judgment by Representativeness

Part 1: Statistical connection Question 1

In a certain school, a poll was conducted about introducing the "zero hour." Half of the respondents were teachers and the other half students. 60% of all the respondents supported the "zero hour" and 20% of the students who responded supported it.

a) What is the share of students among those who support "the zero hour"?

b) In the school newspaper, a heading appeared that said "Poll found teacher respondents support introducing 'zero hour." Show that this heading is correct.

(Source: matriculation exam 2004, modified by the researcher.)

Question 2

In order to enroll in an army music band, one has to provide a recommendation letter from school. Five hundred high school graduates applied for acceptance. Out of them, 200 had good recommendations. Out of the latter, 50% were accepted. In total, 350 candidates were accepted.

a) Anat was accepted to the army music band. What is the probability that she had a good recommendation from school?

b) It was claimed that the chances of being accepted to the band are higher for those who do not get a good recommendation from school. What is this claim based on?

(Source: pre-matriculation exam circular to teachers, modified by the researcher)

Question 3 In a certain research group it was found that out of 200 smokers, heart diseases occurred in 40, while out of 300 non-smokers heart diseases occurred in only 24. Is there a connection between smoking and heart disease in this group? Explain.

(Source: "Probability thinking," p. 30, modified by the researcher)

Appendices

140

Question 5 In a survey conducted last year participated 600 people, out of them 200 men. Among the men participants, 50 supported the law about equal opportunities, which allows men to take paternity leave. 1/5 of the supporters of this law in the survey were men.

a) One participant was chosen at random and turned out to be a woman. What is the probability that she supports the law?

b) One participant was chosen at random; what is the probability that this is a woman who does not support the law?

c) One male participant was chosen randomly, and he can be described as follows: "Roee is a young married student. He takes an active part in housework, loves to cook and takes care of his home most of the time to help his wife succeed in her studies." How, in your opinion, will your friend answer the question: "What is more probable – that Roee supports the law or doesn't support it?"

(Source: matriculation exam 2009, modified by the researcher)

Part 2: Causal Connection

Question 6 A daily newspaper published results of a survey that examined the connection between the level of education and income among 2000 wage-earners in Israel aged between 35-40, having at least 5 years of working experience. Two thirds of the workers who earned higher than average wages had an academic degree. 20% of the workers with an academic degree do not earn higher than average wages. 62.5% of the participants had an academic degree.

a) How many of the participants earned higher than average wages? b) One participant was chosen randomly and it turned out that his wages are lower than average. What is the probability that he had an academic degree? c) The researcher who published the survey claims that based on the findings it is possible to conclude that there is a higher percentage of workers with wages higher than average among those who have an academic degree. What findings is the researcher basing this statement on? d) Why did the researcher point out that all the participants were aged between 35-40 and had at least 5 years of working experience? e) A reader who responded to the survey found out that most of the participants who had no academic degree were women. He claims that this finding puts into question the researcher's conclusions. Explain how this finding can put into question the survey's findings and conclusions.

(Source: Edit Cohen, Marianne Rosenfeld, problem database in mathematics, modified by the researcher)

Question 7

Appendices

141

The statistician Ronald Fisher was a heavy smoker. In the middle of the nineteen-fifties, the first connections between smoking and the increased risk of lung cancer were being discovered. Fisher’s students approached him and asked him to try and smoke less, for the sake of his lungs. They gave the recent findings in support of their request. Fisher refused, stating that the correlation itself does not prove that a causes b. He said it was possible that cancer in its early stages caused a need for nicotine, resulting in the patient smoking, and only afterwards did the tumors begin to develop. Fisher died in 1962. It was only in the seventies that scientists proved that the increased need for nicotine did indeed cause an increase in the risk of becoming ill with lung cancer. Some people may say that Fisher behaved foolishly, while others will say that Fisher was perfectly correct. What do you think? Was Fisher right or wrong? Question 8 Tall students usually make fewer spelling mistakes than shorter students. What do you think? (From "Probability Thinking," p.30, modified by the researcher) Question 9 Consider the following publication in a newspaper: Research: Avocado Prevents Ulcer In addition to lowering the cholesterol levels and improving male sexual potency By: Dvora Namir, Yediot Ahronot correspondent. A good news came yesterday from an international conference of avocado growers that opened in Tel-Aviv: eating avocado protects the stomach mucous membrane and prevents development of ulcer. Prof. Moshe Hashmonai, Head of the Surgery Ward in the Rambam Medical Center, Haifa, told the participants, hundreds of growers and marketers of avocado from all over the world, about his last research that shows how important it is to eat avocado. The research was carried out by a group of doctors from the Rambam hospital in cooperation with the Technion Medical School and a group of doctors from the Lund Medical Center in Sweden. The research was carried out on two groups of rats. In both groups an acute inflammation of the stomach mucous membrane was provoked by introducing alcohol into the stomach. The experimental group received a single dose of avocado mixed with saline solution; the control group receive only saline solution. The experiment showed that the erosion of mucous membrane was much lower in the experimental group than in the rats that did not receive avocado. According to the researchers, this finding shows that eating avocado contributes to preventing damage to the stomach mucous membrane and therefore helps prevent

Appendices

142

stomach ulcer. If the ulcer has already developed, eating avocado, which contains phospholipids, can contribute to healing the wound. Until now, it was recommended to eat avocado for lowering cholesterol levels and also for improving sexual potency in men. Now it is the turn of the ulcer." Express your opinion about this publication. (Source: Yediot Ahronot, 1995) Question 10 In order to check whether there is a connection between success in the psychometric test and taking a preparation course for this test, the grades of 400 students who took such a course and 400 students who didn't were examined. 75% of those who took the course succeeded in the exam. 60% of those who succeeded in the exam took the course.

a) Among the students who didn't take the course, what is the proportion of those who succeeded in the exam?

b) Is it possible to establish on the basis of these data whether there is a statistical connection between taking the preparatory course and success at the psychometric test? Explain your answer.

c) In addition, the grades were analyzed according to the region of the student's residence: those who live in central Israel as opposed to those who live in other regions. The data are presented below: For students from central Israel:

Took the course Did not take the

course Succeeded in the test

280 80

Did not succeed in the test

70 20

For students from other regions:

Took the course Did not take the

course Succeeded in the test

20 120

Did not succeed in the test

30 180

Is it possible to conclude, on the basis of the additional data, that taking the preparatory course is a cause of success in the psychometric test? Explain your answer. (Source: matriculation exam, 2004). Part 3: Judgment by representativeness

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143

Question 11 A newspaper article titled "The Height Does Matter, Even in a Baby": see Appendix 5, "Sample Questions with Solutions" Question 12 A stock exchange consultant foresees a drop in the price of a certain share and recommends the client holding this share to sell it. According to the accumulated data, it was found that for this share drops were registered on 30% of trading days. It is also known that the advisor is right in 75% of the cases, whether he predicts a drop or a rise: in 75 % of cases when there was a rise he predicted a rise, and in 75% cases of actual drop he predicted a drop.

a) What is the advisor's opinion about the option that the share will drop? What is this opinion based on?

b) What is your subjective estimate of the chances that the client's share will drop? c) Make a conclusion about the advisor's level of predictive ability.


Question 13 The authors of the book Critical Thinking believe that by means of teaching this subject it is possible to improve the quality of the students' thinking and their ability to analyze information. In order to verify this, they have passed a test checking the students' ability to analyze and evaluate information. A large group of students participated in the test, while half had previously taken the course "Critical Thinking" and the other half had not. The analysis of the results showed that out of those who had taken the course 70% passed the exam successfully.

a) Express your opinion on the following statement in a well-argued way: "The 70% rate of success is a high rate , therefore it seems that the course indeed seems to improve the students' abilities of analysis and thinking."

b) Here are additional data: out of those who succeeded, 87.5% had taken the course "Critical Thinking."

- According to all of the above data, is there a statistical connection between success in the test and taking the course?

- According to the above data, is there a causal connection, in other words, does taking the course bring about success in the exam? If yes, explain; if not, explain and propose a way of checking such a causal connection.

- Is it possible that some two features have a causal connection but have no statistical connection?

c) One student was randomly chosen, and it turned out that he failed the test. What is

the probability that he had taken the course "Critical Thinking"?

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(Source: matriculation exam, Winter 2007)

Question 14 Inspectors of the Israeli Nature and Parks Authority released ostriches in the Negev in order to observe their behavior for the purposes of research. They have attached a radio transmitter on the body of each bird. One of the ostriches crossed the border and approached a post of Egyptian soldiers who detected a radio transmitter on its body.

a) Which of the following reactions of the Egyptian soldiers is the more likely: 1). The soldiers let the ostrich go. 2). The soldiers shot the ostrich. 3). The soldiers laughed at the ostrich with an antenna. 4). The soldiers laughed at the ostrich and let it go. 5) The soldiers called the commander and laughed.

b) A figure is approaching an Egyptian border post from the Israeli territory. In which of the cases is it more likely that the figure belongs to a spy? 1). If the figure carries an antenna on its body. 2). If the figure is an ostrich carrying an antenna on its body.

c) Suppose that it is known that 3% of the figures approaching the border post are

spies. The soldiers recognize as spies 25% of the spies that approach their post. When the ostrich approached, there were 20 soldiers at the post; 15 of them thought that the ostrich is a spy and should be shot. One of the soldiers shot at the ostrich and killed it. What is the probability, in light of these data, that the ostrich was indeed an intelligence tool?


Question 15 Research: Teachers Discriminate Outstanding Students By: Rali Saar

It appears from a research currently conducted by the Ministry of Education on about 120,000 matriculation exams from recent years…….

Appendices

145

Appendix 4: Questionnaire in Mathematical Knowledge

1) Solve the system of equations:

2 1

5 4 6x y x y x+ −

− = −

22 4x y− =

(Checks the students’ knowledge of systems of equations with two variables of the first

degree).

_______________________________________________________________

1) A truck left from Tel-Aviv, stopped in two army bases on the way, and came back to Tel-Aviv.

Distance

60

40

20

Time 8 7 6 5 4 3 2 1 0 (Hours) Referring to the graph, answer the following questions:

a) For how long did the truck stop at the first army base, and for how long at the

second?

b) What is the distance between the first and the second base?

c) What was the truck’s velocity in the first hour of its trip?

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146

d) What was the truck’s velocity on the way back from the second base to Tel-Aviv?

(Checks the students’ ability to read data from a graph.)

_______________________________________________________________ 3) A formula is given:

v = F – 3P 4

a) Express F by means of P and V.

b) Express P by means of F and V.

c) Given P=3, v= -1, calculate F.

(Checks the students’ ability to change the subject of an equation.)

_______________________________________________________________

4) The initial salary of a worker was 3500 shekels per month. Every month his salary was

raised by 40 shekels.

a) What was the worker’s salary in the 12th month of his work?

b) How much has the worker earned in the first 12 months of his work?

(Checks the students’ understanding of progressions from a verbal question.)

_______________________________________________________________ 5) The price of a product is M shekels. The formula for calculating the price N of the

product without VAT (in shekels) is

100M = N

117

a) Write a formula for calculating price M including VAT when N without VAT is

given.

b) Write a formula for calculating the VAT in shekels, T, when N is given.

c) Give an example of a problem from everyday life that would use this formula.

(Checks the students’ ability for bi-directional inference and applying the formula to

everyday life.)

_______________________________________________________________ 6) The figure below shows a graph of the function y = x ² – 4x + 3 y

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147

.

x

a) Find the points of intersection of the graph with each axis.

b) What is the lowest value that the function receives, and at which point is this

value received?

c) For what values of x does the function decrease?

(Checks the students knowledge of square function.)

____________________________________________________________

7) A sportsman walks for 7 hours non-stop. Each hour the distance he covers equals 45

of

the distance he covered in the previous hour. In the third hour he covered 4000 meters.

a) Calculate the distance he covered in the first hour.

b) Calculate the total distance the sportsman covered in 7 hours.

(Checks the students’ ability to understand a verbal question describing a progression.)

____________________________________________________________ 8) The following graph was published in one of the evening newspapers in 1998.

The graph describes the change in the index of shares between one Monday to the next.

Index

(points)

190 188 186 184

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148

M S Th W T M Consider the graph and answer the questions a – d.

a) On what day of the week was the index at its highest?

b) On what day of the week was the index at its lowest?

c) By how many points did the index drop between Wednesday and Sunday?

d) On what days was the index 185 points?

(Checks the student’s ability to read graphs from everyday life.)

______________________________________________________________

9) Substitute numbers in the following exercise: even numbers for each letter in the word

in such a way as to receive a correct ,”פרט“ odd numbers for each letter of the word ,"זוג"

equation. It is given that the letters of the word ” פרט ” comprise an arithmetical

progression. What is the value of “פרט”?

ז ו ג

+ א ו

____

ט פ ר

(Checks the students’ ability to understand a question of an unfamiliar type, which

requires thinking differently.)

_______________________________________________________________

10) Try to receive, in three different ways, the number 2000 by means of the numbers

1,2,3,4,5, using all of these numbers, each of them only once, and the four rules of

arithmetic.

(Checks the students’ ability to understand a question of an unfamiliar type, which

requires thinking differently.)

GOOD LUCK!

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149

Appendix 5: The Learning Unit “Probability in Daily Life”

5.1: Sample Problems and Exams from the Course “Probability in Daily Life”

Sample Problems and Exams from the Course “Probability in Daily Life”

3.1 Final Exam in the Course “Critical Thinking and Probability in Daily Life”

Part 1 (33 points)

Choose three concepts out of the following six. Explain each concept and give

detailed examples.

- the phenomenon of reversal relation

- mediating factor

- conditional probability

- diagnosticity

- controlled experiment

- observational research

Part 2 (67 points)

Choose 2 questions out of the following 3.

Question 1.

An eleventh grade at a certain school took the first part of the mathematics matriculation

exam in winter. 7% of the class got “excellent” in the exam. The class will take the

second part of the exam in summer.

Yosi is a student in this class. The teacher defines Yosi as an excellent student. It is

known from previous experience that 90% of the students who receive “excellent” in the

exam are defined by the teachers as excellent student. Also, 90% of the students who did

not receive “excellent” in the matriculation exams are defined by the teachers as less than

excellent.

a. What is the chance that Yosi will indeed receive “excellent” in the exam?

b. According to what you have learned in “Probability Thinking,” what is the answer that

people who didn’t study “Probability Thinking” will tend to give?

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150

c. What percent of the students have to receive “excellent” in the next exam so that the

probability that Yosi will receive “excellent” in the same exam will be 90%, if the

teacher’s evaluation of him does not change?

Question 2.

A certain daily newspaper published the results of a survey that checked the connection

between the level of education and the level of income among 2000 wage-earners in

Israel aged between 35-40, with 5 years of working experience at least. 2/3 of the

workers who receive higher-than-average wages have academic degrees. 20% of the

workers with an academic degree do not receive higher-than-average wages. 62.5% of the

wage-earners participating in the research have an academic degree.

a. How many workers participating in the survey receive higher-than-average wages?

b. One participant was chosen at random from the sample, and it turned out that he

received lower-than-average wages. What is the probability that he has an academic

degree?

c. The researcher who published the survey claimed that on the basis of the findings it

can be inferred that among workers with an academic degree, a higher percentage of

workers receives higher-than-average wages than among workers without higher

education. On what findings does the researcher based this conclusion?

d. Why did the researcher point out that the workers researched were aged between 35-40

and had at least 5 years of working experience?

e. A reader responded to the survey results found that most of the participants without an

academic degree are women. According to the reader, this finding puts into question the

researcher’s conclusions. Explain how the fact pointed out by the reader can put into

question the survey’s results.

Question 3.

In one of the high schools, the teachers began to have an impression that many students

do not prepare their homework, and decided to check the issue out. On one of the school

days, all the teachers checked the students’ homework in the first class. Upon

examination it was found that the number of students who didn’t do their homework is

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151

greater by 2.25 than of those who did. It was found that 55% of those who prepared

homework were girls. 9/20 of the students who did not prepare the homework were boys.

a. Calculate the percentage of the students who did not do their homework for the check-

up day.

b. Calculate the proportion of the male students who did do their homework.

c. Is there a statistical connection between gender and the inclination not to prepare

homework?

Sample Questions with Solutions

Question 11.

A newspaper item titled “The Height Does Matter, Even in a Baby” describes a research

conducted by scientists from Finland and U.K. The researchers arrived at a conclusion

that the height of the baby in the first 12 months of his life determines the kind of work

he will be doing in his adult life and therefore also the level of his wages. They found that

babies who were taller than average on their first birthday earned more at the age of 50

than their counterparts who were shorter than average as babies, and this connection held

irrespectively of the participants’ family background. The research examined 4500 men

aged about 50.

Let us assume that 2/3 of the participants were taller than average as babies. 375 men

earned higher-than-average wages at the age of 50 but were shorter than average as

babies. 20% of the men who earned higher-than-average wages at age 50 were shorter

than average as babies.

Mark the group of men who were taller than average at the age of 12 months as A, and

the group of those who were shorter as B.

a. A man was chosen randomly from the sample and turned out to have been taller

than average at the age of 12 months. In light of the research data, what is the

probability that his wages today are higher than average?

b. Is there a statistical connection between the level of the man’s wages at age 50

and his height at age 12 months?

c. Do the findings support the researchers’ claim?

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152

d. 1) Why, in your opinion, was the research conducted on men only?

2) Why did the researchers point out the connection held “irrespectively of the

participants’ family background”?

e. What is the diagnosticity of identifying men who will earn more than average

according to their height as babies at 12 months?

f. In a large factory there work 200 men aged about 50. 60 of them were taller than

average as babies. If we choose randomly one of the fifty-year-old men whose

wages are higher than average, what is more probable: that he was a taller-than-

average or a shorter-than-average baby? Explain and support your answer with

appropriate calculations.

Solution:

Given: S is the sample of men who participated in the research

a) P (B/A)

According to the formula for calculating conditional probability, we will receive:

P (Ā/B) =

Using a table

A Ā

B 1/3 1/12 5/12

1/3 1/4 7/12

2/3 1/3 1

we will receive:

P (B/A) =

b). P (B/A) =

therefore, according to these data, there is a statistical connection between the wages of a

man at age 50 and his height at age 12 months.

c). We have found that P (B/A) =

That is, the percentage of men whose wages are higher than average is higher among

those who were taller-than-average babies at the age of 12 months. This finding

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153

corresponds to the claim that taller babies are more likely to earn higher wages at the age

of 50.

d). 1) The researchers assumed that apparently there is a connection between the level of

wages at age 50 and the worker’s gender.

2) The researchers suspected that these factors influence the examined variables and

therefore neutralized them.

e). P (B/A) =

f). Given: N (S) = 200, N (A) = 60 … and we will receive

R =

P (B/A) =

Therefore there exists a higher probability that he was a shorter-than-average baby at the

age of 12 months. The result seems surprising and it stems from the high percentage of

men who were shorter-than-average babies (70%) in the sample.

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154

Table 1: Skills and Topics: Intermediary Questionnaires.

The skill practiced Questions in intermediary questionnaires

Identifying variables


Identifying conclusions

Evaluating the source’s reliability (profession-alism, absence of conflicts of interest

Suspending judgment (when evidence and arguments are not sufficient, looking for new and contradictory evidence)

Proposing alternatives (looking for alternative explanations)

Readiness to research (proposing plans of experiments, including plans for controlling the variables)

Claims (regarding people’s beliefs and positions)

Making value judgments (apparent application of accepted principles)

1. “The Zero Hour”

Teachers/ students, for/ against

Survey at school

The meaning of 100% or 0%

2. “The Army Band”

Coming with / without recommend-ations

Not relevant

Relating to the claim in paragraph 3

3. “Heart Diseases”

Smoking/ not smoking Suffering/ not suffering from heart disease

Not relevant

The meaning of the connection between smoking and heart diseases

4. “Aspirin” Enzyme deficiency Pathological response to aspirin

Medical manual

False alarm (one should go to the emergency room)

5. “The Law of Equal Opportunities”

Men/ Women Support/ do not support the law

Survey/ sample

Roee supports the law

6. “Education”

Possessing/ not possessing an academic degree Highly paid/ low-paid

Daily newspaper

People with academic degrees earn higher-than-average wages

Daily newspaper, not a scientific journal

Neutraliza-tion of inter-mediate factors age and work experience

The worker’s gender Women are usually paid less than men carrying out the same work

If more women with an academic degree took part in the study, their highest wage rate wouldn’t be higher than that of workers without an academic degree

7. “Ronald Fisher”

Smoking Cancer

Real story Not possible to know. Fisher was right

Narrative source

No connection between the cause and the effect, other factors possible

In order for the patient to feel better, he has to receive nicotine

Controlled experiment

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155

8. “Spelling Mistakes”

Spelling mistakes

Necessary Explanation of the existing connection

Necessary Where and by whom was the survey conducted?

Looking for evidence

Other factors such as age, school class

Separate observation research in different classes and age groups

9. “Avocado” Eating avocado Ulcer prevention

“Yediot Achronot”

A connection was found between eating avocado and ulcer healing, but the meaning of the connection is still unclear.

Daily newspaper from 1985, citing a research by a group of physicians from Rambam hospital, Technion, and a group of Swedish physicians. Presented at an international conference of avocado growers.

The rats were hungry.

It is possible that avocado helps to heal the ulcer.

Controlled experiment.

10. “The Psychometric Test”

Studying in prep courses Success in the exam

Some survey

Taking prep courses is not necessarily the reason for success in the exam.

Who conducted the survey? Who ordered the survey?

The evidence is insufficient.

Analyzing the grades by area of residence.

Looking for a clearer reason, since causality was not confirmed.

11. “The Baby’s Height”

Men taller than average at 12 months Men paid higher than average

There is a statistical connection between the wages of men at age 50 and their height at age 12 months.

Research conducted in Finland and U.K. Source unknown, even to the newspaper presenting the research.

The finding corresponds to the claim that taller babies will be higher paid when arriving at age 50.

12. “Equities” Trade days when the equity dropped. The set of days on which a drop was predicted.

Story

13. “Critical Thinking”

Some survey

14. “The Israel Nature and Parks Authority”

The set of cases in which the silhouette belongs to a

Story

Appendices

156


Question 1

In a research published in Gynecology and Obstetrics journal, 2006, doctors from

Columbia University of New York and the Hadassah Hospital in Jerusalem examined the

influence of the father’s age on the incidence of miscarriage. The research was based on

data collected from women who gave birth in Jerusalem between 1964-1976 and on their

medical gynecological history (pregnancies and births) as reported by the women.

The research analyzed data of 13,865 women, out of which 1,506 had had miscarriages

and the other 12,359 had not. Data such as the mother’s age, presence of diabetes,

smoking habits, earlier history of miscarriages, marital status and time between

pregnancies and miscarriages were checked. The research found that where the father

was aged over 35, the rate of miscarriages for the mother was about three times as high as

for fathers aged below 25. The statistical dependency on the age of the father was found

significant after neutralizing the influence of the mother’s age and lifestyle.

The rate of miscarriages when the father’s age was below 25 was about half of that from

fathers aged between 25-29. The rate of miscarriages from fathers aged between 30-34

was about 1.5 times higher than that from 25-29 year old fathers, and the rate of

miscarriages from fathers aged between 35-39 was about twice as high as that from 25-29

year old fathers.

The researchers claim that the increase in the father’s age is an important variable in the

incidence of miscarriages and is independent of the age, health or other characteristics of

the mother. The research shows that in the same way as women’s fertility decreases with

spy. The set of cases when a soldier notices a spy.

15. “Do Teachers Discriminate?”

Daily newspaper

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157

age, so does the quality of the male semen, causing an increase in the rate of miscarriages

and probably other genetic defects.

Is it possible to say that the researchers found a close statistical connection between

miscarriages and the age of the father?

Is it possible to say that the researchers did not find a connection between

miscarriages and environment factors?

Is it possible to conclude that the research findings indicate that the higher is the age

of the father, the higher is the probability of miscarriage in the mother?

Question 2

Avi: “There is a connection between the size of shoes and knowledge in math.”

Beni: “Can’t be.”

Avi: “Go to the school next door and see for yourself.”

Beni: “You are right, kids with bigger numbers of shoes know more math.”

Express your opinion about this conclusion.

Question 3

In a certain school there are 40 teachers. Orna is one of the teachers. It is known that

Orna likes hiking and is a member of a hiking club. Several teachers went on the school

yearly tour. Which one of the following sentences sounds more plausible to you:

2) Orna is a mathematics teacher.

3) Orna is a mathematics teacher and she went on the yearly tour.

Question 4

Is it true that in larger and better equipped hospitals the rates of mortality are higher than

in small and less well equipped hospitals?

Explain your answer.

Question 5

Read the attached passage from Yediot Ahronot titled “Calcium and Vitamin D

Contribute to Dental Health,” and answer the following questions.

Calcium and Vitamin D Contribute to Dental Health

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158

Taking calcium and vitamin D as food supplements can help to keep one’s teeth healthy.

This connection arises from a research conducted in the Boston University School of

Dental Medicine, which was published in The American Journal of Medicine.

The study involved 145 participants aged above 65. Part of them took 500 mg calcium

and 700 UE vitamin D daily, and the rest took placebo. In the control group, 27% of

participants lost at least one tooth in the course of the three years of research, as opposed

to only 13% in the experimental group. The researchers performed an additional check

several years after the end of the experimental period, and found that 40% of the

experimental group lost at least one tooth since the end of the experiment, as opposed to

59% of the control group.

What connection does the news item discuss? Is it possible to provide a logical

explanation for this connection? Propose at least two factors that can mediate the

connection described in the news item.

Question 6

The culture of consumerism offers a hierarchy of values in which the person is measured

by his/her possessions, and not by his/her actions or ideas. Money is the ideal, and every

means of obtaining it is acceptable. This culture is referred to by various names, such as

“Americanization,” “the golden calf,” materialism, advertisement culture, or rating

culture. It also exacts a heavy price.

The factors that encourage the culture of consumerism are:

Lack of time: when we work more, in an attempt to buy more, we have less time to

devote to the spouse, friends, and things that matter to us. Society deteriorates, and we

have ‘no life’.

Destruction of the family: parents spend less time with their children. Parents try to

compensate lacking attention to children with presents and money. The long hours at

work do not contribute to the well-being of the couple, and no holiday tour can

compensate for the daily damage done in the race after the money to afford it.

Materialistic children: from very young age they are a target for advertisement. While

parents spend more time at work and the education system is crashing, the children

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159

undergo an incessant brainwashing. The advertisers acquire them as consumers for life,

and we lose them.

Is it possible to conclude that if we earn less and buy less, we will live happier?

Is it possible to conclude that money “destroys families”?

Is it possible to conclude that advertisement is harmful?

Question 7

It is a question that does not concern people much, but probably should concern them

more so that they are more careful: what is the chance for an Israeli to be hurt in a road

accident? The answer is 0.76%, in other words, in 1999, out of every 1000 residents 7.6

were injured. The table below presents data for 1999 (according to the Central Bureau of

Statistics).

Percent of each type of injury

Total injured

Degree of injury Type of participation in traffic

Lightly injured

Severely injured

Killed

8.36 3,803 2,721 915 167 Pedestrians 38.57 17,549 16,576 844 129 Vehicle

passengers 43.56 19,822 18,870 830 122 Vehicle

drivers 7.50 3,413 3,027 363 23 Cyclists 1.30 593 440 122 31 Motorbike

drivers 0.61 278 240 35 3 Motorbike

passengers 0.10 45 39 5 1 Other

(unknown) 100.00 45,503 41,913 3,114 476 Total

Is it possible to conclude that motorbike drivers stand the highest chance to be

killed? What data in the table confirm your conclusion?

A newspaper published a warning to parents: “Do not allow your children to drive a

bicycle without a helmet.” Express your opinion about this warning.

Question 8

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160

On the basis of data of the Central Bureau of Statistics for 1999, the statistics of road

accidents were examined for vehicles from Arab and Jewish settlements of over 10,000

residents within the 1967 borders (77,223 vehicles from Arab settlements, 1,361,295

from Jewish settlements). The comparison of the two groups showed the following

results:

- in fatal accidents (1.8% of all accidents), the involvement of Jewish vehicles was

44% more per vehicle than that of Arab ones;

- in severe accidents (10.5%) there were 13% less accidents per Jewish vehicle than

per Arab one;

- in light accidents (87.7%) there were 109% more accidents per Jewish than Arab

vehicle.

- Altogether (100% accidents) Jewish vehicles are involved in 95% more accidents

than Arab vehicles.

What problems are there with the data of this research?

Question 9

The graph below presents statistics of vehicle theft in different countries. Israel is in the

second highest place, with 450.9 thefts per 100,000 residents, after Switzerland (838.1

thefts per 100,000).

Claim: Switzerland has most vehicle thieves.

Express your opinion about this claim.

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161

Relate to the number for Holland. In Holland there are fewer thieves, because most

people ride bicycles. What do you think?

Question 10

The statistician Ronald Fisher was a heavy smoker. In the middle of the nineteen-fifties,

the first connections between smoking and the increased risk of lung cancer were being

discovered. Fisher’s students approached him and asked him to try and smoke less, for

the sake of his lungs. They gave the recent findings in support of their request. Fisher

refused, stating that the correlation itself does not prove that a causes b. He said it was

possible that cancer in its early stages caused a need for nicotine, resulting in the patient

smoking, and only afterwards did the tumors begin to develop. Fisher died in 1962. It was

only in the seventies that scientists proved that the increased need for nicotine did indeed

cause an increase in the risk of becoming ill with lung cancer. Some people may say that

Fisher behaved foolishly, while others will say that Fisher was perfectly correct.

What do you think? Was Fisher right or wrong?

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162

Appendix 7: Sample Problems and Exams from the Course “Probability in Daily Life” 6.1 Course Description

Course title: Critical and Probability Thinking in Daily Life.

Purpose of the course: development of critical and probability thinking in daily life.

Central topics of the course:

1. Historical background and early problems in probability: back in the tunnel of

time to the 16th century, following the correspondence between great

mathematicians (from Pascal to Bayes) on calculating probability in various

games of chance.

2. Grounding of mathematical and probability thinking: where, if anywhere, can

statistics and probability be applied? Characteristics of probability function,

probability laws, conditional probability and Bayes theorem.

3. Statistical connection and causal connection: decision making based on numerical

data and subjective evaluation of the situation, measures of central tendency and

dispersion. Mechanisms of intuitive judgment: psychological mechanisms people

employ to arrive at intuitive judgments. Critical evaluation of surveys.

4. Regression toward the mean and failures in perception of the phenomenon of

regression. Predicting values of one variable if values of the other variable are

known and if the connection between the variables is linear (optional).

Teaching methods: lecture; mathematical discourse.

Appendices

163

Introduction and Contents of the Course Textbook (First Round; Written by the

Researcher)

Developing and Fostering Critical Thinking through the Learning Unit “Probability

Thinking in Daily Life

Lecture Notes

In the Framework of “Access to Higher Education” Program

Ben-Gurion University of the Negev

Course design and teaching: Einav Aizikovitsch

Table of Contents

1. Introduction ………………………………………………………………… 1

2. Rationale ……………………………………………………………………. 2

3. Topic distribution by hour…. ……………………………………………… 3

4. Topic distribution by lecture ………………………………………………. 4

5. Lesson plans by chapter ……………………………………………………. 7

6. Bibliography ………………………………………………………………… 100

1. Introduction

In the learning unit “Probability in Daily Life,” the student is required to analyze

problems, raise questions and think critically about data and information. This program

discusses the concept of probability in the context of everyday problems. The uniqueness

of this program is in allowing the student to study interesting and relevant topics from

daily life, involving elements of critical thinking, in the framework of mathematical

studies (Lieberman, 2002). Problems of this kind are complex not only because they deal

with a single event, but also because they do not always have a simple single solution.

Appendices

164

The purpose of the unit is to go beyond the numerical answer, to check the data and their

validity, and in cases where there is no unambiguous numerical answer, to know how to

ask the appropriate questions and analyze the problem qualitatively, not only by

calculation. Aside from imparting statistical tools, the course also presents intuitive

mechanisms used by humans when they need to estimate probability in daily life and

examines inclinations and errors that often accompany these estimations by juxtaposing

intuitive mechanisms with probability calculation dictated by probability laws.

2. Rationale

The Rationale for Designing and Teaching the Learning Unit

The topic of “Probability Thinking in Everyday Life” has been introduced into the

curriculum because we are daily required to make decisions under conditions of

uncertainty. Our decisions in all areas of life are made after collecting data, processing

them and arriving at a judgment, which has two components: statistical judgment, based

on numerical data, and intuitive judgment, based on subjective evaluation of the situation.

In this course we will examine the connection between statistical and intuitive judgment

and will study the psychological failures that bring about erroneous judgment. In other

words, while studying probability, we will also acquire the bonus of learning correct

procedures of critical thinking.

First we will study the mathematical tools necessary for performing calculations, and

then the topics of causal connection and mechanisms of intuitive judgment, which can be

defined as psychological implications of probability judgment.

3. Outline of Topics by Hour

Recommended

Number of Hours*

Topic Studied

5

Part 1: Introduction to the Theory of Probability

Historical background, early probability problems in the history of

Appendices

165

* According to the recommendation of Dr. Varda Lieberman.

mathematics.

5

Part 2: Introduction to Set Theory

10

Part 3: Basic Concepts in Probability

Characteristic of the probability function, the complementarity

principle, the inclusion-exclusion principle, the unification principle,

conditional probability, Bayes formula and its applications, statistical

connection and lack of dependence.

6

Part 4: Judgment by Representativeness

The mechanism of representativeness, combinatorial failure, base rate

influence, biased evidence.

2

Part 5: Prevalence, Probability, and the Degree of Belief

The meaning of the concept of probability and the connection between

prevalence approach and subjective approach.

3

Part 6: Statistical Connection and Causal Connection

What is the relation between statistical and causal connection;

controlled experiment, biased choice, relation reversal (Simpson’s

paradox).

2

Part 7: Basic Concepts in Descriptive Statistics

Measures of central tendency (mean, median, percentile), measures of

dispersion (divergence and standard deviation, range).

7

Part 8: Regression toward the Mean and Non-Regressive Judgment

The phenomenon of regression toward the mean, calculating the line of

regression to predict Y by X and X by Y; failures in perception of the

phenomenon of regression in daily life.

Appendices

166

4. Course Curriculum by Date

Number

of hours Page Lesson Topic and Contents Date

2.5

4

Opening lesson:

Defining the concepts of critical thinking and probability

thinking.

הקלאסית.

10.11

2.5

6

• History of the theory of probability.

1) About mathematicians.

2) The problem out of which probability theory first

developed.

• Interesting questions from the past.

17.11

2.5

8

Basic concepts.

Defining the concept of “probability.”

Introduction to set theory.

תרגול ברמה בסיסית בתורת ההסתברות הקלאסית.

רקע היסטורי בתורת ההסתברות.

שאלות מעניינות מההיסטוריה המשך -

24.11

2.5 12 Introduction to set theory + basic exercises.

1.12

2.5

15 Introduction to set theory, classical probability + basic exercises.

8.12

Appendices

167

2.5

17

Basic concepts in probability:

Characteristics of probability functions, complementarity

principle, inclusion-exclusion principle, unification principle,

conditional probability, conditional probability functions, Bayes

formula and its applications, statistical connection and lack of

dependence.

15.12

2.5

18






dependence.

29.12

2.5 19






dependence.

5.01

2.5






dependence.

12.01

Appendices

168

2.5

Judgment by representativeness

The mechanism of representativeness, combinatorial failure,

base rate influence, biased evidence.

26.01

2.5




2.02

2.5




9.02

2.5

Prevalence, Probability, and the Degree of Belief

The meaning of the concept of probability and the connection

between prevalence approach and subjective approach.

16.02

2.5

Statistical Connection and Causal Connection


controlled experiment, biased choice, relation reversal

(Simpson’s paradox).

23.02

2.5

Statistical Connection and Causal Connection


controlled experiment, biased choice, relation reversal

(Simpson’s paradox).

9.03

Appendices

169

2.5

Basic Concepts in Descriptive Statistics

Measures of central tendency (mean, median, percentile),

measures of dispersion (divergence and standard deviation,

range)

16.03

2.5

Regression toward the Mean and Non-Regressive Judgment

The phenomenon of regression toward the mean, calculating the

line of regression for predicting Y by X and X by Y; failures in

perception of the phenomenon of regression in daily life.

23.03

2.5



line of regression for prediction of Y by X and X by Y; failures

in perception of the phenomenon of regression in daily life.

2.5



line of regression for prediction of Y by X and X by Y; failures

in perception of the phenomenon of regression in daily life.

Changes and Remarks

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developing critical thinking through probability models

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