Weeding and Sowing: Preface to a Science of Mathematical Education

324 Pages·2004·3.88 MB·English

by  Hans Freudenthal (auth.)

WEEDING AND SOWING
By the same author
Mathematics as an Educational Task
WEEDING
AND SOWING
Preface to a Science of Mathematical Education
HANS FREUDENTHAL
KLUWER ACADEMIC PUBLISHERS
NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 0-306-47234-1
Print ISBN: 90-277-0789-8
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FOREWORD
A title that sounds like poetry, and a subtitle that seems to contradict the
title! But the subtitle is right, and originally it was just the title. A strange
subtitle, isn’t it? Preface to a Science of Mathematical Education. All
sciences – in their prenatal stage – have known this kind of literature: only
the term used was not ‘Preface’, but, for instance, ‘Prolegomena’, which
means the same* though it sounds less provisional. In fact such works were
thicker than the present one, by up to ten times. There is much more that can
be said about a science before it comes into being than after; with the first
results comes modesty.
This is the preface to a book that will never be written: not by me, nor by
anybody else. Once a science of mathematical education exists, it will get the
preface it deserves. Nevertheless this preface – or what for honesty’s sake I
have labelled so – must fulfil a function: the function of accelerating the
birth of a science of mathematical education, which is seriously impeded by
the unfounded view that such already exists. Against this view I have to argue:
it rests on a wrong estimation – both over and under estimation at the same
time – of what is to be considered as science. This explains the first chapter
of this book, ‘What is Science?’, where science is delimited in various direc-
tions, against various sorts of non-science and pseudo-science, against tech-
nology, against faith. All that is expounded in that chapter – arguably out of
a craving for lucidity – extends to many sciences, in particular those in the
area of the social sciences. It is relevant wherever, caught between highly
developed technology and rationally motivated faith, one can scarcely find an
approach to a science. Among these domains education is the most prominent,
and so the title of the second chapter is ‘On Education’, aiming at the role
of education caught between technology and faith.
It is not only a science of mathematical education that we are waiting for.
We need, just as badly, a science of education – with no adjective added –
* In fact, ‘preface’ is derived from ‘praefatio’, the Latin translation of ‘prolegomena’.
VI FOREWORD
which is for the time being even farther away. A science of education is no
precondition of a science of mathematical education. It is just the other way
round, as it has always been in the history of the sciences: mathematics prior
to science, mechanics prior to physics, physics prior to natural science, sci-
ences of languages prior to linguistics. This relation explains why on the way
from the third to the fourth chapter, from ‘On a Science of Education’ to ‘A
Science of Mathematical Education’ the tone gradually changes from criticism
of that which puts on scientific airs, to a search for the silver lining. That is all
and no more: there are not even the first rudiments of a science of mathemat-
ical education here; at most there are indications where to look for such rudi-
ments. I pledge nothing; but I shall redeem all I pledge. Though it is not in
order to make it easier for me to keep my promises that I do not make any,
but in order to prevent anything from being raised to the level of a science
when it is not one.
Someone called my Mathematics as an Educational Task a Summa Contra
Mathematicos. This book could as well be called a Summa Contra Didacticos.
They would be complements of each other, and that is alright. To everyone
his due.
Right from the title onwards this Preface to a non-existent book aims at a
future in the making: since the other book was dedicated to friends of my
own generation, it is fair to dedicate this book to my dear collaborators from
three to thirteen.
IOWO, HANS FREUDENTHAL
University of Utrecht
TABLE OF CONTENTS
FOREWORD V
CHAPTER I/WHAT IS SCIENCE?
Abstract 1
1. Introduction 2
2. Relevance 3
3. Consistency 9
4. Publicity 12
5. The Fringe of Science 15
6. Science and Technology 19
7. Science and Faith 25
8. Values 30
CHAPTER II/ON EDUCATION
Abstract 33
1. What Does ‘Education’ Mean? 34
2. Science and the Picture of Man 41
3. A Case in Point 44
4. Environment and Heredity 47
5. Equal Chances for All 49
6. Education Bottled and Funneled 55
7. The Social Context 59
8. The Heterogeneous Learning Group 60
9. The Strategy of Innovation 64
10. Teacher Training 68
11. Educational Philosophy 75
CHAPTER III/ON A SCIENCE OF EDUCATION
Abstract 77
1. Does it Exist? 78
VIII TABLE OF CONTENTS
2. In Full Bloom 81
3. Atomisation 92
4. The Attainment of Concepts 98
5. Objectives of Instruction 105
5.1. How to Find Them 105
5.2. In a Green Tree 107
5.3. In the Dry Tree 113
5.4. The Distribution of Chestnuts 114
5.5. Searching One’s Own Conscience 116
6. Opinion Polls 119
7. Diagnosis 121
8. Production of the Package 123
9. The Art of Dividing 128
10. Models 130
11. Mathematical Models 136
12. Educationese 138
13. Rituals 141
14. Educational Accountancy 144
15. Educational Research Inc. 157
16. A Sociopsychological View 160
17. The End of the Matter 163
CHAPTER IV/A SCIENCE OF MATHEMATICAL EDUCATION
Abstract 170
1. Introduction 171
2. The Art of Mathematics Teaching 171
3. Team Work as a Source of Research 172
4. The Theoretician in the Team 176
5. The Learning Situation as a Source of Research 178
6. Language as a Vehicle of Research 183
7. Motivation 185
7.1. Through Discontinuities in the Learning Process 185
7.2. Through Goals 187
7.3. Through Make-Up 189
8. Generality by Comprehension and by Apprehension 192
9. Apprehension and Paradigm 201
TABLE OF CONTENTS IX
10. In Vain Quest for the Paradigm 210
11. In Vain Quest for Discontinuities in the Learning Process 214
12. An Apprehending Approach to Algebra 221
13. The Mathematical Background of the
Geometrical Approach to Algebra 225
14. The Algebraic versus the Arithmetical Approach to Algebra 230
15. Levels of Language 233
16. Change of Perspective 242
16.1. Grasping the Context 242
16.2. A Logical Problematic 246
17. The Field of Tension Between Global and Local Perspectives 252
18. The Field of Tension Between
Quantitative and Qualitative Perspectives 257
19. Grasping the Context – Chances 263
20. I See It So 275
21. An Example of Didactical Phenomenology
– Ratio and Proportion 292
21.1. Preparation 292
21.2. Elaboration 296
21.3. Final Remark 304
EPILOGUE 305
BIBLIOGRAPHY OF PUBLICATIONS BY
HANS FREUDENTHAL ON MATHEMATICAL EDUCATION 306
INDEX 311
CHAPTER I
WHAT IS SCIENCE?
ABSTRACT. The query ‘what is science?’ is not answered by a clean cut definition or
by a complete set of characteristics. Science is delimited rather from other domains of
human activity by criteria, some of which are made explicit in this chapter: relevance,
consistency and publicity.
Truth is not mentioned among the criteria, because truth is a property of state-
ments, whereasscience as an activity is not a treasury of truth but a method of asking
questions.
Relevance, as I call it, is a property not only of statements but also of problems
and methods, and as such it is even more crucial than as a property of statements.
Relevance can also be a property of definitions, notations, concepts, classifications,
and, more globally, of problem complexes, theories, domains of knowledge. In this
global sense it meansbeing related to reality rather than floating in empty space.
As a criterion of what science is, consistency looks more like truth, though it seems
to stress its logical component. But this is only the objective aspect of consistency.
Consistency can also be intended as a property of action and of patterns of activity –
as an attitude which faces consequences, asks relevant questions and pursues promising
leads. For logicians the acme of consistency is the logically closed system. This, how-
ever, is an ideal picture, which is realised only in today’s mathematics – even theoretical
physics is a far cry from this ideal. General theory in physics is not a basis from which
deduction takes place but a repository of organising devices. Physics is like a shop of
mini-theories supervised by, though not dervied from, general theory. It is a pity that a
wrong picture of physics, and natural science in general, has served and still is serving
as a model for the social sciences, and the humanities.
Science is a public property, and in spite of the so-called secret sciences, publicity is
one of the characteristics of true science. Nobody can be obliged to submit to initiation
rites before he can study and practise science. Science is publicly accessible to everybody
who agrees to learn its language, and in the long run neither schools nor prophets succeed
in monopolising a domain of science, though sometimes it may be difficult to decide
whether a particular science means more than the language in which it is expressed.
Relevance, consistency, and publicity are criteria by which science contrasts with its
fringe: pseudo-science and non-science. Flying saucers, the mysteries of the Cheops
pyramid, and the paragnosts are no serious problems to science but the Nazi pseudo-
scientific racism was a menace to mankind and new pseudo-sciences may endanger
humanity even more. Pseudo-science often sounds like a protest against public science
as far as publicity means public recognition and is suspected to mean public coercion.
The fringe of science is a social danger worth studying. It may also mean a danger to
serious science. Pseudo-scientific infections may cause a cancer-like growth in serious
science. Language borrowed from serious science may be abused in other sciences; terms
like function, information, model and structure, that originated in mathematics, became
meaningless fashion in many other sciences.