Intuition in Science and Mathematics: An Educational Approach

233 Pages·2002·1.724 MB·English

by  Efraim Fischbein (auth.)

INTUITION IN SCIENCE AND MATHEMATICS
MATHEMATICS EDUCATION LIBRARY
Managing Editor
A.J.Bishop, Cambridge, U.K.
Editorial Board
H. Bauersfeld, Bielefeld, Germany
H. Freudenthal, Utrecht, Holland
J.Kilpatnck, Athens, U.S.A.
Gilah Leder, Melbourne, Australia
T. Varga, Budapest, Hungary
G.Vergnaud, Paris, France
EFRAIM FISCHBEIN
School of Education, Tel Aviv University, Israel
INTUITION IN
SCIENCE AND
MATHEMATICS
An Educational Approach
KLUWER ACADEMIC PUBLISHERS
NEWYORK/BOSTON/DORDRECHT/LONDON/MOSCOW
eBookISBN: 0-306-47237-6
Print ISBN: 9-027-72506-3
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TABLE OF CONTENTS
PREFACE ix
ACKNOWLEDGMENTS xiii
PART I: THE THEORY
CHAPTER 1 / Intuition and the Need for Certitude 3
The Complexity of the Domain 3
The Need for Certitude 7
A PreliminaryDefinition 13
CHAPTER 2 / Intuition and Mathematical Reasoning 15
The Real World and the Mathematical World 15
The Behavioral Meaning of Mathematical Concepts 19
Axiomatic Structures and Intuitive Procedures 22
Intuitive Concepts and the Scientific Community 24
CHAPTER 3 / Investigations in Overconfidence 28
Overconfidence and Intuitive Biases 28
Educational Implications: Overconfidence and Metacognition 36
Summary 41
CHAPTER 4 / General Characteristics of Intuitive Cognitions 43
Self-Evidence 43
Intrinsic Certainty 45
Perseverance 47
Coerciveness 47
Theory Status 50
Extrapolativeness 50
Globality 53
Implicitness 54
Summary 56
CHAPTER 5 / The Classification of Intuitions 57
Piaget’s Classification 57
Classifications of Intuitions Based on Roles and Origins 58
Summary 70
CHAPTER 6 / Inferential Intuitions and Logical Reasoning 72
v
vi TABLE OF CONTENTS
PART II: FACTORS WHICH SHAPE INTUITIONS
CHAPTER 7 / Intuition and Experience 85
The Behavioral Roots of Intuitive Representations 85
Experience and Intuitive Biases 89
Summary 95
CHAPTER 8 / The Practicality of Intuitive Meanings. Analysis of an
Example: The Negative Numbers 97
CHAPTER 9 / Factors of Immediacy 103
Visualization 103
Availability 106
Anchoring 107
Representativeness 108
Summary 109
CHAPTER 10 / Factors of Globality 111
Summary 120
CHAPTER 11 / Intuition and Intuitive Models 121
Classification of Model Types 121
Role of Tacit Intuitive Models 122
Summary 125
CHAPTER 12 / Models and Analogies 127
The Role of Analogy in Model Construction 127
Analogic Models in Mathematics 129
Analogies as Sources of Misconceptions in Mathematics 136
Summary 142
CHAPTER 13 / Paradigmatic Models 143
CHAPTER 14 / Diagrammatic Models 154
Summary 165
CHAPTER 15 / Phenomenological Primitives 167
CHAPTER 16 / Conflicts and Compromises 176
Impetus versus Inertia 176
Striving for Cognitive Compromise 181
Impact of the Impetus Model: Further Experimental Evidence 184
Conciliatory Models 187
Summary 191
CHAPTER 17 / Factors of Perseverance and Closure: The Primacy
Effect 193
Summary and Comments 198
TABLE OF CONTENTS vii
CHAPTER 18 / Summary and Didactical Implications 200
The Role of Intuition: A Summary 200
The Classification of Intuitions 201
Intuitions and Models 202
The Mechanisms of Intuitions 204
Conflicts and Compromises 205
Didactical Implications 206
Concluding Remarks 21 1
BIBLIOGRAPHY 215
INDEX OF NAMES 223
PREFACE
In writing the present book I have had in mind the following objectives:
- To propose a theoretical, comprehensive view of the domain of intuition.
- To identify and organize the experimental findings related to intuition
scattered in a wide variety of research contexts.
- To reveal the educational implications of the idea, developed for science
and mathematics education.
Most of the existing monographs in the field of intuition are mainly
concerned with theoretical debates - definitions, philosophical attitudes,
historical considerations. (See, especially the works of Wild (1938), of Bunge
(1 962) and of Noddings and Shore (1 984).)
A notable exception is the book by Westcott (1968), which combines
theoretical analyses with the author’s own experimental studies.
But, so far, no attempt has been made to identify systematically those
findings, spread throughout the research literature, which could contribute to
the deciphering of the mechanisms of intuition. Very often the relevant
studies do not refer explicitly to intuition. Even when this term is used it
occurs, usually, as a self-evident, common sense term.
The explanation is that intuition is generally seen as a primary phenome-
non which may be described but which is not reducible to more elementary
components. As a matter of fact, intuitively, intuition has the appearance of a
self-evident, self-consistent cognition, like the perception of a color or the
experience of an emotion. The effect is that, generally, no attempt is made by
researchers to use their experimental findings for elucidating the structure of
intuitive phenomena. On the contrary: it is intuition which is used as a
descriptive and explanatory concept. Piaget, for instance, who used the terms
intuition and intuitive thinking very often never made any attempt to refer to
his own findings for obtaining a deeper understanding of the general
structure of intuitive mechanisms as such. With a very few exceptions this is
the situation with most of the researchers. Such an exception is the work of
Andrea DiSessa (1982, 1983a, 1983b) who made an important attempt to
analyse his own experimental findings with the explicit purpose of building a
theory of intuition.
The present book has a similar aim. We expound a theory, we try to get a
richer insight in the mechanisms of intuition using research evidence, and we
try to support and enlarge our conception on the basis of these data.
Our general thesis is that, in principle, cognition fulfils behavioral aims
ix
x PREFACE
and is shaped by behavioral constraints. The same is to be said about
intuition which is a particular form of cognition.
Intuition has its roots in the syncretic type of thinking of the child and of
human beings in the early stages of civilization. But it does not survive in
adults and in highly developed cultures only as a mere residuum. We claim
that intuition expresses a profound necessity of our mental behavior.
During the very course of our reasoning, of our trial-and-error attempts,
we have to rely on representations and ideas which appear, subjectively, as
certain, self-consistent and intrinsically clear. We cannot doubt everything at
every moment. This would be a paralysing attitude. Some representations,
some conceptions have to be taken for granted. They have to appear, subjec-
tively, as autonomous, coherent, totally and directly acceptable cognitions in
order to keep the process of reasoning working fruitfully.
An intuition is, then, such a crystallized - very often prematurely closed
- conception in which incompleteness or vagueness of information is
masked by special mechanisms for producing the feelings of immediacy,
coherence and confidence.
Such mechanisms have been described in the research literature, but very
often without any apparent connection with a theory of intuition. In the
present work an attempt has been made to take advantage of these research
sources. Studies in overconfidence, in subjective probabilities, findings
referring to mental models, to typical errors in naive physics, to misconcep-
tions in mathematics, to the evolution of logical concepts in children etc.
represent, in fact, rich potential sources for a theory of intuition.
I should like to emphasize again this interesting phenomenon: simply
because intuition is tacitly but firmly considered to be a primitive feeling, rich
sources of information, based on experimental findings, have been ignored by
most of the theorists. A primary purpose of this book has been to overcome
this obstacle.
The history of science and mathematics is also an important source for
understanding the dramatic struggle of the scientific mind against intuitive
biases. We have used a number of examples in our work in this respect,
trying to identify the common structure of intuitive difficulties in experts and
novices.
Turning our attention, now, to the educational aspects: Most authors -
experimental researchers and theorists alike - strive to set up recommenda-
tions for avoiding intuitively-based errors in learning and problem solving
and for improving intuitive guesses and evaluations. But, in our opinion, one
can hardly expect such suggestions to be really helpful if they are not based
on a comprehensive theory of intuition. Intuitions are only apparently
autonomous, self-evident cognitions. They are so in order to confer on some
of the individual's ideas the appearance of certitude and intrinsic validity.
But, in fact, these ideas appear very robust as an effect of their being deeply
rooted in the person's basic mental organization. Consequently, in order to
PREFACE xi
eliminate or change or even control an intuitive attitude it would be
necessary to produce a profound, structural transformation in large areas of
mental activity
Therefore, the individual’s self-confidence itself might be endangered if he
learns-that even his deepest beliefs may very often be misleadiag. It follows
that intuitions cannot be treated effectively and positively as mere isolated
symptoms but rather as manifestations of highly articulated and very complex
structures. And for this, a comprehensive theory is necessary, a theory which
would take into acount the behavioral roots and the adaptive functions of
intuitions.
I do not pretend to have solved all these puzzling theoretical problems. It
is probably too early to get definite answers. But, if I have been able to
convince the reader of the importance of this line of thought, if I have been
able to cast some doubts and to produce some new expectations with regard
to his intuition about intuition then my main goal will have been achieved.
The book is divided into two major parts. The first part is concerned with
the theoretical aspects - the theory, the relevance of intuitive forms of
cognition for scientific and mathematical reasoning, the connections between
intuition and other, related, categories of cognition, the genera1 charac-
teristics and the classification of intuitions.
The second part of the book deals mainly with factors which contribute to
shaping intuitions: the role of experience, the role of various types of models
- analogies, paradigms, diagrams, phenomenological primitives - the role of
factors for producing the effects of immediacy and globality.
I believe that the time is now ripe for a formal recognition of intuition as
one of the major components of our cognitive endeavors. A constant
interplay between theory and experiment in this field is nowadays certainly
possible and highly desirable.
It is our conviction that the findings will be beneficial for both cognitive
psychology and educational practice.