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 6: Critical Thinking Questionnaire: “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.
Theoretical Background
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,
Theoretical Background
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
Theoretical Background
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.
Theoretical Background
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
Theoretical Background
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
Theoretical Background
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).
Theoretical Background
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
Theoretical Background
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
Theoretical Background
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
Theoretical Background
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
Theoretical Background
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
Theoretical Background
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.
Theoretical Background
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
Theoretical Background
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.
Theoretical Background
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
Theoretical Background
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.
Theoretical Background
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.
Theoretical Background
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
Theoretical Background
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
Theoretical Background
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.
Theoretical Background
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.
The Intervention: The Learning Unit “Probability in Daily Life”
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).
The Intervention: The Learning Unit “Probability in Daily Life”
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).
The Intervention: The Learning Unit “Probability in Daily Life”
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.
The Intervention: The Learning Unit “Probability in Daily Life”
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.
The Intervention: The Learning Unit “Probability in Daily Life”
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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.
The Intervention: The Learning Unit “Probability in Daily Life”
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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
The Intervention: The Learning Unit “Probability in Daily Life”
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?
The Intervention: The Learning Unit “Probability in Daily Life”
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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
The Intervention: The Learning Unit “Probability in Daily Life”
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
The Intervention: The Learning Unit “Probability in Daily Life”
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.”
The Intervention: The Learning Unit “Probability in Daily Life”
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.
The Intervention: The Learning Unit “Probability in Daily Life”
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
The Intervention: The Learning Unit “Probability in Daily Life”
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.
Dispositions of Critical Thinking
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.
Dispositions of Critical Thinking
54
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
Dispositions of Critical Thinking
55
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.
Dispositions of Critical Thinking
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
Dispositions of Critical Thinking
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.
t value Post-test Pre-test Sub-scale SD Mean SD Mean N=30
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
Dispositions of Critical Thinking
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.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
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.
Dispositions of Critical Thinking
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 4: Truth-Seeking sub-scale Means
Truth Seeking
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison of
Post-score 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
Dispositions of Critical Thinking
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 5: Open-Mindedness sub-scale Means
Open Mindedness
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
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
Dispositions of Critical Thinking
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICA
EXPERIMENT-ALL
CONTROL-ALL
Chart 6: Inquisitiveness sub-scale Means
Inquisitiveness
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison of
Post-score 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
Dispositions of Critical Thinking
62
There was an initial (in the pre-test) significant difference (p<0.01) 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 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-
scale score for the students in the experiment and control groups. Table 15 contains the
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 7: Systematicity sub-scale Means
Systematicity
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison
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
Dispositions of Critical Thinking
63
There was an initial (in the pre-test) significant difference (p<0.001) 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 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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 8: Maturity sub-scale Means
Maturity
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison
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
Dispositions of Critical Thinking
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-
scale score for the students in the experiment and control groups. Table 17 contains the
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 9: Confidence sub-scale Means
Confidence
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison
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
Dispositions of Critical Thinking
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-
scale score for the students in the experiment and control groups. Table 18 contains the
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 10: Analyticity sub-scale Means
Analyticity
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison
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
Dispositions of Critical Thinking
66
There was an initial (in the pre-test) significant difference (p<0.001) 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
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.
Dispositions of Critical Thinking
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.
Dispositions of Critical Thinking
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.
Dispositions of Critical Thinking
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.
Dispositions of Critical Thinking
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).
Abilities of Critical Thinking: Methods, Results and Discussion
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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.
Abilities of Critical Thinking: Methods, Results and Discussion
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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
Abilities of Critical Thinking: Methods, Results and Discussion
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.
Abilities of Critical Thinking: Methods, Results and Discussion
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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
Abilities of Critical Thinking: Methods, Results and Discussion
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
Abilities of Critical Thinking: Methods, Results and Discussion
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
t value Post-test Pre-test Sub-scale SD Mean SD Mean N=28
-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
Abilities of Critical Thinking: Methods, Results and Discussion
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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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 14: CTI Total Means
CTI Total
Pre-test Post-test Difference ANCOVA(a)
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
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
Abilities of Critical Thinking: Methods, Results and Discussion
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 15: Induction Sub-Scale Means
Induction
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison of Post-score Exp
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
Abilities of Critical Thinking: Methods, Results and Discussion
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
for the students in the experiment and control groups. Table 26 contains the full results of
statistical tests conducted on the data relevant to the Deduction Sub-Scale score.
Graph for Deduction
20
30
40
50
60
70
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICAEXPERIMENT-ALL
CONTROL-ALL
Chart 16: Deduction Sub-Scale Means
Deduction
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison of Post-score Exp
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
Abilities of Critical Thinking: Methods, Results and Discussion
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICA
EXPERIMENT-ALL
CONTROL-ALL
Chart 17: Observation Sub-Scale Means
Observation
Pre-test Post-test Difference ANCOVA
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison of Post-score Exp
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
Abilities of Critical Thinking: Methods, Results and Discussion
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
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICA
EXPERIMENT-ALL
CONTROL-ALL
Chart 18: Assumptions Sub-Scale Means
Assumptions
Pre-test Post-test Difference ANCOVA(a)
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison of Post-score Exp
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
Abilities of Critical Thinking: Methods, Results and Discussion
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
for the students in the experiment and control groups. Table 29 contains the full results of
statistical tests conducted on the data relevant to the Meaning Sub-Scale score.
Graph for Meaning
20
30
40
50
60
70
exp pre exp post control pre control post
EXPERIMENT-WITHOUT-KIDUMATICA
EXPERIMENT-ALL
CONTROL-ALL
Chart 19: Meaning Sub-Scale Means
Meaning
Pre-test Post-test Difference ANCOVA(a)
Mean S.D.
t value for comparison
Exp with control
Mean S.D.
t value for comparison
Exp with control
Mean S.D. t value for comparing Post to Pre
t value for comparison
Exp with control
F value for comparison of Post-score Exp
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,
Abilities of Critical Thinking: Methods, Results and Discussion
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.
Abilities of Critical Thinking: Methods, Results and Discussion
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.
Abilities of Critical Thinking: Methods, Results and 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,
Abilities of Critical Thinking: Methods, Results and Discussion
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
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
a. Identifying variables b. Referring to sources
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
a. Identifying variables b. Referring to sources
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
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.
Construction of Critical Thinking Skills: Methods, Results and Discussion
92
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
93
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
94
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
Statistical connection
Statistical connection
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
95
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.
Construction of Critical Thinking Skills: Methods, Results and Discussion
96
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
97
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.
Referring to sources
0.4
0.6
0.8
1
Statistical Connection Statistical Connection and CausalConnection
Statistical Connection and CausalConnection and Judgment of
Representativeness
High SchoolKidumatica
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
98
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
Statistical Connection Statistical Connection and CausalConnection
Statistical Connection and CausalConnection and Judgment of
Representativeness
High SchoolKidumatica
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.
Construction of Critical Thinking Skills: Methods, Results and Discussion
99
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
Statistical Connection and CausalConnection and Judgment of
Representativeness
Evaluationg the Source's Reliability
Suspendeing Judgment
Offering Alternatives
Readiness for Investigations
Chart 25: : Skill (d,e,f,g) Evaluating the sources, Suspending judgment, Offering alternatives and Readiness for Investigation
Construction of Critical Thinking Skills: Methods, Results and Discussion
100
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
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
Identifying Variables
Reffering to Sources
Identifying Conclusions
Evaluationg the Source's Reliability
Suspendeing Judgment
Offering Alternatives
Readiness for Investigations
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
Identifying Variables
Reffering to Sources
Identifying Conclusions
Evaluationg the Source's Reliability
Suspendeing Judgment
Offering Alternatives
Readiness for Investigations
Chart 28: All the skills over time High school class
Construction of Critical Thinking Skills: Methods, Results and Discussion
102
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,
Construction of Critical Thinking Skills: Methods, Results and Discussion
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
104
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
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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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106
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
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
Construction of Critical Thinking Skills: Methods, Results and Discussion
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.
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109
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
General Discussion and Conclusions
110
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
General Discussion and Conclusions
111
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
General Discussion and Conclusions
112
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
General Discussion and Conclusions
113
Figure 4: After the Learning Unit
General Discussion and Conclusions
114
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
General Discussion and Conclusions
115
‘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
General Discussion and Conclusions
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
117
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
Contribution and Implications
118
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
Contribution and Implications
119
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
Contribution and Implications
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.
Contribution and Implications
121
(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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122
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.”
Contribution and Implications
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
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Appendices
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 6: Critical Thinking Questionnaire: “Probability in Daily Life”
Appendix 7: Sample Problems and Exams from the Course “Probability in Daily Life”
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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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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.”
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Appendix 3: Critical Thinking Questionnaire: “Probability in Daily Life”
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)
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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
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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? 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
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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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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.
(Source: Edit Cohen, Marianne Rosenfeld, problem database in mathematics, modified by the researcher)
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?
(Source: Edit Cohen, Marianne Rosenfeld, problem database in mathematics, modified by the researcher)
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…….
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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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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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.
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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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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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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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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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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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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Table 1: Skills and Topics: Intermediary Questionnaires.
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)
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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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
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Appendix 6: Critical Thinking Questionnaire: “Probability in Daily Life”
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?
Appendices
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.
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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
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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
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.
29.12
2.5 19
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.
5.01
2.5
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.
12.01
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168
2.5
Judgment by representativeness
The mechanism of representativeness, combinatorial failure,
base rate influence, biased evidence.
26.01
2.5
Judgment by representativeness
The mechanism of representativeness, combinatorial failure,
base rate influence, biased evidence.
2.02
2.5
Judgment by representativeness
The mechanism of representativeness, combinatorial failure,
base rate influence, biased evidence.
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
What is the relation between statistical and causal connection;
controlled experiment, biased choice, relation reversal
(Simpson’s paradox).
23.02
2.5
Statistical Connection and Causal Connection
What is the relation between statistical 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
Regression toward the Mean and Non-Regressive Judgment
The phenomenon of regression toward the mean, calculating the
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
Regression toward the Mean and Non-Regressive Judgment
The phenomenon of regression toward the mean, calculating the
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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