HANSFREUDENTHAL MAJOR PROBLEMS OF MATHEMATICS EDUCATION 1 0 Forgive me, it was not me whose chose the theme, though when it was first suggested to me, I experienced it as a challenge. A challenge, indeed, but to be sure not as one to emulate Hilbert, who at the Paris International Congress of Mathematicians in 1900 pronounced his celebrated 23 mathematical problems, which were to profoundly influence, nay presage, the course of mathematics for almost a century. If it were not modesty that prevents me from even trying it, it should be the obvious fact that problems, problem solving, problem solvers mean different things in mathematics education from what they mean in mathematics. But first let us look at the other noun in the title: education. It can mean roughly three things: the educational process taking place in the family, at school, on the street, and everywhere, an administrative establishment, a theoretical activity, called educational research. The major problems I am going to summarise, are ofthefirst kind, partly related to the second, while I will disregard the third kind. What I am interested in is problems in mathematics education as a social activity rather than problems as an entrance to educational research. However, at the end I will cast a glance at educational research as one of the major problems of mathematics education. Yet let me come back to what I announced as a difference between problems, problem solving, and problem solvers in mathematics on the one hand and mathematics education on the other. Mathematical problems are problems within a science arising for a large part from this science itself or from other sciences. Education problems are problems of life arising from changing needs, moods and whims of a changing society. Hilbert's problems have been seminal for a century. The address I deliver today may be forgotten never to be remembered ten years from now. From olden times mathematicians have posed problems for each other, both major and minor ones: here is the problem, solve it; if you can, tell me; Educational Studies in Mathematics 12 (1981) 133-150. 0013-1954/81/0122-0133 $01.80 Copyright 1981 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.
H A N S F R E U D E N T H A L
M A J O R P R O B L E M S OF M A T H E M A T I C S E D U C A T I O N 1
Forgive me, it was not me whose chose the theme, though when it was first
suggested to me, I experienced it as a challenge. A challenge, indeed, but to be
sure not as one to emulate Hilbert, who at the Paris International Congress
of Mathematicians in 1900 pronounced his celebrated 23 mathematical
problems, which were to profoundly influence, nay presage, the course of
mathematics for almost a century. If it were not modesty that prevents me
from even trying it, it should be the obvious fact that
�9 problem solving,
�9 problem solvers
mean different things in mathematics education from what they mean in
But first let us look at the other noun in the title: education. It can mean
roughly three things:
�9 the educational process taking place in the family, at school, on the
street, and everywhere,
�9 an administrative establishment,
�9 a theoretical activity, called educational research.
The major problems I am going to summarise, are of thef irs t kind, partly related
to the second, while I will disregard the third kind. What I am interested in is
problems in mathematics education as a social activity rather than problems as an
entrance to educational research. However, at the end I will cast a glance at
educational research as one of the major problems of mathematics education.
Yet let me come back to what I announced as a difference between problems, problem solving, and problem solvers in mathematics on the one hand and mathematics education on the other.
Mathematical problems are problems within a science arising for a large part from this science itself or from other sciences. Education problems are problems of life arising from changing needs, moods and whims of a changing society. Hilbert's problems have been seminal for a century. The address I deliver today may be forgotten never to be remembered ten years from now.
From olden times mathematicians have posed problems for each other, both major and minor ones: here is the problem, solve it; if you can, tell me;
Educational Studies in Mathematics 12 (1981) 133-150. 0013-1954/81/0122-0133 $01.80 Copyright �9 1981 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.
134 HANS FREUDENTHAL
I will listen to check whether it is a solution. In education, problem solving is not a discourse but an educational process. In mathematics the problem solvers are mathematicians. In education, problems are properly solved by
the participants in the educational process, by those who educate and by those who are being educated.
Moreover in mathematics you can choose one major problem, say from Hilbert's catalogue, solve it, and disregard the remainder. In education all major problems, and in particular those I am going to speak about, are strongly interdependent. As a matter of fact major problems of education are charac- terised by the fact that none can properly be isolated from the others. The best
you can do at a given moment is focus on one of them without disregarding the others, and this is in fact what I am going to do here.
In a sense the title of my address is wrong: all major problems of mathematics education are problems of education as such. In another sense it is exactly
right: if you look for major problems the best paradigm of cognitive education is mathematics.
In order to terminate this introduction, I should add two points. First,
there is no authority in mathematics education comparable to those in
mathematics. The problems which I think are major ones have been chosen
according to my philosophy of learning and teaching mathematics, which I
will not recapitulate here since it will be implicit in the problems and in the
way they are submitted. Second, although the problems have been inspired
by my own experience and philosophy, I do not claim to have invented them.
Originality too, has another meaning in mathematics education from that
which it has in mathematics. My ideas have been anticipated not once but many times in the past. This then is the reason why I will refrain from quoting
and citing, whether a particular problem I mention has been successfully tackled before or not.
Allow me to start with the most earthly problem I can think of. Among the
major ones it is the most urgent. What is even a problem is how to formulate it correctly and unmistakably. Let us try a preformulation. It runs:
Why can Johnny not do arithmetic?
Does it sound sexist? I would not change it into
Why can Mary do arithmetic?,
lest it may sound even more sexist, suggesting that girls are less able than boys.
MAJOR PROBLEMS OF MATHEMATICS EDUCATION 135
As a matter of fact both are wrong. My problem is not John Roe and
Mary Doe. It is a problem, indeed, why many children do not learn arithmetic
as they should, and it is a major one because more than anything else, failure
in arithmetic may mean failure at school and in life. My concern, however,
is not, or not primarily, what is wrong in classrooms and textbooks- today
- that creates a host of underachievers. Let me change the question. I now ask:
�9 Why can Jennifer not do arithmetic? Rather than an abstraction like John and Mary, Jennifer is a living child
(though I have changed her name) whom I can describe in all detail. The two details that matter here, are that she was eight years old and could not do
arithmetic. Meanwhile the question
Why can Jennifer not do arithmetic?
is no question any more, because today Jennifer is eleven and excels in arithmetic. Yet when she was eight, somebody observing her stumbling with numbers succeeded in answering the question and after ten minutes of remedial
teaching, the problem Jennifer had had ceased to exist. Was it a miracle? Not
at all. It was just an easy case. But there are so many. Noticed and unnoticed,
cared for and uncared for. But what about the less easy cases? They have grown out of those easy ones that remained unnoticed and uncared for.
Diagnosis and prescription are terms borrowed from medicine by education-
alists who pretend to emulate medical doctors. What they do emulate is medicine
of a forgone period, which is the quackery of today. Medical diagnosis in
former times aimed at stating what is wrong, as do the so-called diagnostic tests
in education. True diagnosis tells you why something went wrong. The only way to know this is by observing the child's failure and trying to understand it.
An expensive way. Would it be cheaper by computer? No, because in fact
observing and understanding the individual child is not expensive, what is
really expensive is wasting the vast resources of human experience.
Let me explain what I mean. By chance I know why Jennifer failed and I know about quite a number of other children because there were people who
observed their failures and analysed them. All were different, and nevertheless they were an infinitesimal minority of those who need help. On the other hand I am sure that my own experience is only an infinitesimal part of a vast amount of knowledge, which has never been recorded nor even made conscious. Would we be better off if this bulk of knowledge had actually been recorded and reported? Not at all. Useful educational theory does not arise from blind generalising. What we need are paradigmatic cases, paradigms of diagnosis and prescription, for the benefit of practitioners and as bricks for theory builders.
136 HANS FREUDENTHAL
Let me add two examples, two paradigmatic cases though not enough for
To test underachievers it is useful to ask them what is 2 + 9. Rather than
at the result look at how the task is performed. Is it done by counting nine
forward from 2, that is: 3, 4 . . . . ? Whether you call it a diagnosis or not, if it
happens this way, you can be sure about at least two sources of failure: no
awareness of commutativity nor of the use of the positional system.
Another example: a twelve years old girl whom I taught understanding
fractions simplified 16124 to 3/8 - an unexpected failure. She explained it by
16 = 2 • 8, 24 = 3 • 8, thus 3/8. When I dug more deeply, the source appeared
to be a failure of short term memory, that is, an error in storing or retrieving intermediate results.
This experience led me to better understand her failures and those of other
children which in the past I had wrongly interpreted, for instance attributed
to a lack of concentration. I started remedial work to improve short term
memory, which proved successful, even in transfer: factorising mentally whole
numbers below 100 with at least three prime factors. This exercise draws
heavily on short memory, indeed. Say 48 = 6 x 8, 6 = 2 x 3, 8 = 2 x 4, and
now you can imagine what happens, the badly stored 2 x 3 cannot be retrieved.
A fortnight after this exercise the girl performed the same task with no
difficulty at all, and short term memory, in general, had greatly improved.
In former times medical diagnosis started with anamnesis. Anamnesis of class-
room experiences can be a ticklish affair. In most cases of arithmetic disability
I came across, the case history was clear by circumstantial evidence, such as in
the example of ignorance of commutativity: the subjects had never received
any instruction, or at least any decent instruction, in arithmetic - I do not
blame their teachers, who obviously had never learned what and how to teach.
Studying illness is easier than studying health. Human biology indeed, started
with medicine and medicine grew from cure to prevention. This would suggest
my second major problem, as it were an extension of the first:
How should children learn?,
in particular mathematics, which I immediately change into �9 How do people learn?
which is the proper question, and the way to answer it would be:
By observing learning l~ocesses, analysing them and reporting paradigms -
MAJOR PROBLEMS OF MATHEMATICS EDUCATION 137
learning processes within the total educational system, learning processes of pupils, groups, classes, teachers, school teams, councillors, teacher students, teacher trainers, and of the observer
�9 Learning to observe learning processes, this is my second major problem in mathematics education. In mathematics
education because I believe that learning to observe mathematical learning processes is the easiest approach to the problem of learning to observe learning processes in general. Observing involves analysing, by which I don't mean averaging or applying other statistical procedures nor fitting the observational data into preconceived patterns of developmental psychology. Grasping
�9 how people do learn would be a first step towards solving the every day problems of practitioners.
�9 how to teach learning and towards
�9 building a learning theory, which should be based on evidence, rather than on preconceived ideas.
In the preceding I stressed
observing learning processes against
testing learning products.
Amongst the learners I mentioned I forgot one, the biggest one. Mankind too is a learner. Observing its learning process is what w.e call history. How can the individual learner profit from knowledge about the big learning process of mankind? Rather than from its detail he can profit from the fact as such. Each stage in the growth of mathematics meant:
�9 knowledge acquired by insight transformed by
�9 schematising and memorising (or call it, codifying) into
�9 skills and insight of a higher order. Let me explain by a few examples what I mean by schematising. A problem
of long standing:
A farm with chickens and rabbits, 20 heads and 56 legs; how many chickens and how many rabbits?
I am sure you will solve it by insight. But as early as Babylonian antiquity one knew the schematism of a pair of linear equations with two unknowns to solve
this kind of problem, and in more recent times this schematic insight has been anew schematised by the rule: Put the unknown numbers x andy, write down the connecting algebraic relations, and solve them by algebra. The most modern schematisation of this idea is vector space.
Clumsy algebra by insight was schematised by Vieta and Descartes' formal
methods, and this process of schematising still continues in modern algebra.
Calculation of areas, volumes, gravity centres, and moments, which once required the genius of an Archimedes - and even harder problems - are today
within the grasp of our freshmen, thanks to Newton and Leibniz' schematising
infinitesimal methods in what is known as Calculus.
But let us explain it in an even more elementary way. As early as written
sources can remember, counting was schematised by introducing higher
level units such as 5, 10, 100, 1000, and as early as reckoning was invented,
it was schematised by a positional idea: higher level units materialised by
counters on the abacus. Schematising arithmetic proceeded one step further by transforming
sets of counters into digits,
the hardware abacus into a written one, written into sand or on paper, from which by a progression of schematising our present
schematism of columns arithmetic arose.
Let me illustrate this process by a modem didactical version, a few sheets that need hardly any explanation.
The schematisation of packing units,
the positional arrangement transformed into the positional abacus - the schematisation that makes each bead worth its position,
MAJOR PROBLEMS OF MATHEMATICS EDUCATION 139
z 7 6 2 7 6
the schematism of changing tens of units into higher ones,
I 3 / 2 I / 0 2 2. 0 2
the schematism of the drawn and written abacus and of addition,
again schematised by the position card,
[ l l l ~
the last step before the final schematisation of the usual column addition.
History of mathematics has been a learning process of progressive schema- tising. Youngsters need not repeat the history of mankind but they should not be expected either to start at the very point where the preceding generation stopped. In a sense youngsters should repeat history though not the one that actually took place but the one that would have taken place if our ancestors had known what we are fortunate enough to know.
Schematising should be seen as a psychological rather than a historical progression. I think that in the mental arithmetic of whole numbers we can
fairly well describe schematising as a psychological progression, or rather as
a network of possible progressions, where each learner chooses his own path or all are conducted along the same way. Quite a few textbooks witness efforts
to teach learning the traditional algorithms of column arithmetics of whole
numbers by a progression of schematising steps, though I am not sure whether
their ideas are supported by actual learning and teaching. In teaching fractions,
decimal numbers, algebra, calculus, I see little if any attempts at progressive
schematising. The idea that mathematical language can and should be learned
in such a way - by progressive formalising - seems even entirely absent in the whole didactical literature.
This then is my third major problem of mathematics education:
�9 how to use progressive schematisation and formalisation in teaching any mathematical subject whatever?
A cherished antagonism in teaching and learning mathematics is putting on one side of a deep gorge such noble ideas as
�9 insight, understanding, thinking,
and on the other side such base things as �9 rote, routines, drill, memorising, algorithms.
If I were malicious, I would add another pair of opposites �9 theory versus practice,
suggesting that learning by insight and understanding be a noble theory while the base practice is learning by rote and memorising. However, it is not that simple, and it has never been so. Even in our computerised age children memorise tables of addition and multiplication and acquire certain skills by rote, though one might argue that by the rise of the computer the balance has
shifted in favour of the nobler activities. It is not that simple, firstly because the option is not one between both
sides of a gorge but rather bridging it by the learning process I called schema- tising. Secondly, I do believe that at any time more mathematics has been
MAJOR PROBLEMS OF MATHEMATICS EDUCATION 141
taught from the viewpoint of insight and more has been learned by insight than
we are aware of. All agree and textbook writers witness that elementary arithmetic cannot be learned in any other way than by insight whether it is
taught that way or not. But it is also true that as things go on, as teaching proceeds to higher grades, to column addition and multiplication, to long division, to fractions - ordinary and decimal - to algebra, to learning math- ematical language, the part played by insight changes. Not just by diminishing, but its character changes. There is a tendency that the learner's insight is
superseded by the teacher's, the textbook writer's, and finally by that of the adult mathematician. And the same holds on the long winding road, starting with concretely understood word problems and leading to highly formalised and badly understood applied mathematics.
This is why people who advocate learning by insight, disagree about what is insight. The wrong perspective of the so-called New Math was that of replac-
ing the learner's by the adult mathematician's insight. Yet this is not my main point. I have still to explain why we are not aware
of how much is nevertheless being learned by insight. It is a most natural
thing that once an idea has been learned, the learner forgets about his learning process, once a goal has been reached, the trail is blotted out. Skills acquired
by insight are exercised and perfected by - intentional and unintentional
- training. This is a good thing. What is bad, is �9 sources of insight clogged by acquired routines, never to be reopened,
and this is what usually happens. It explains why teachers at higher grades
so often complain about teaching habits at lower grades. If it is restricted to the first acquisition of some idea, learning by insight does not deserve this
name. What is crucial, is �9 retention of insight,
which is gravely endangered by
�9 premature training, �9 too much training, �9 training as such.
This then is my fourth major problem in mathematics education: �9 How to keep open the sources of insight during the training process, �9 how to stimulate retention of insight, in particular in the process of
How can this goal be pursued? The solution I propose is �9 having the learner reflect on his learning processes.
To a large degree, mathematics is reflecting on one's own and others' physical
142 HANS FREUDENTHAL
mental and mathematical activity. The origin of proving theorems is arguing
what looks obvious. Nobody tries to prove a thing unless he knows it is true.
This he knows by intuition, and the way to prove it is by reflecting on one's
intuition. Successful learning processes, if observed, should be made conscious
to the learner in order to be reinforced and in order to be recalled if needed.
This, however, is not what usually happens. Let us illustrate this abstract exposition by an example:
Many children and adults can tell you that in order to multiply by 100 you
have 'to add two zeros' (which just holds for whole numbers) and most of them cannot explain why. It is even worse: most of them do not even understand that you can argue it and why you should do so. Did they learn such rules by rote? I do not believe it. I have observed to o many children applying such rules intuitively before they were verbalized and formally taught at school. Rather
than being taught the rules, they should have been taught to argue their intuitions, reflecting on what looks obvious. But this requires more patience than teachers can afford.
Let me add a personal experience. It happened with a sixth grade girl, a serious underachiever. When I started teaching her, she did not know anything about fractions, decimal fractions, percentage, the metric system. Fortunately
- I should add. (The only thing she knew was: area equals twice length plus width.) So it was a fresh start. As I taught her mathematics, she taught me patience. By patience I don't mean, not getting angry, but rather abstaining
from teaching her any rules. She learned by insight and schematising, and she now performs fairly well without knowing any rules or schematisms. At one
moment at school when they noticed she had made progress, they tried to
teach her roles for fractions. It was a catastrophe. I needed weeks to restore what had been destroyed. Now one year later she performs quite decently in
mathematics. She does not master rules, she heavily depends on insight and half conscious schematisms. I have not yet succeeded in getting her to reflect on her mental activities. Is it too early? I am afraid it is too late. I guess that
such an attitude should be acquired early. This then is my fifth major problem of mathematics education: �9 How to stimulate reflecting on one's own physical, mental and math-
Reflecting on one's own physical, mental and mathematical activities is an important component of what is called
�9 a mathematical attitude.
MAJOR PROBLEMS OF M A T H E M A T I C S E D U C A T I O N 143
I have hesitated whether I should include in the list of my major problems
�9 how to develop a mathematical attitude?
My own problem is that while we fairly well know what we understand under
a mathematical attitude, we can describe it only by a long catalogue of
examples and counter-examples - too many to be dealt with in one hour or
even in one week. I once tried to identify a few components of mathematical attitude. I will repeat this list here, while omitting examples:
�9 developing language above the demonstrative and linguistically relative
level, in particular on the level of conventional variables,
�9 change of perspective, a complex field of strategies with the common
feature that the position of data and unknowns in a problem or field of
knowledge is - partially - interchanged; including the recognition of wrong
changes of perspective,
�9 grasping the degree o f precision which is adequate to a given problem,
�9 identifying the mathematical structure within a context if there is any,
and keeping off mathematics where it does not apply,
�9 dealing with one's own mathematical activity as a subject matter in order to reach a higher level.
Without examples this is a meaningless catalogue. The reason why I mentioned
mathematical attitude is that you would have rightly objected if I had not,
and the reason for showing a few components of mathematical attitude was to
fight the usual mistake: testing a mathematical attitude by asking questions about an attitude towards mathematics.
Let us consider one feature of progressive schematising that I did not yet
consider; I mean that in such a progression not all learners progress at the same
pace and reach the same goals. Progressive schematising is a way to account
for natural differences in aptitudes and abilities. Differentiation is a general
problem of education. In spite of the wide variability of linguistic mastery, it
is a fact that people can communicate with each other in their mother tongue
on a broad scale of subjects. Why is mathematics different, and should it really be different?
There are reasons - social reasons - why in spite of their diverging learning
abilities learners should learn together as they are expected to work together
in the society. Cooperation involves levels of work. Cooperative learning presupposes levels of learning. It is a fact that mathematics lends itself, as
no other subject does, to distinguishing levels, in mathematics and in learning mathematics.
144 HANS FREUDENTHAL
My seventh major problem of mathematics education is �9 How is mathematical learning structured according to levels and can this
structure be used in attempts at differentiation?
Perhaps you will complain that up to now I have paid almost no attention to
subject matter and its didactics. If by subject matter you mean some chapter of a textbook, you will be disappointed. These are no major problems. But I agree that teaching is always teaching something. Something rather than
anything. Something worth being taught. But what is worth being taught? In order to be taught it should be applicable, in some sense, in any sense,
in any sense whatever. What does this mean? Teaching as much mathematics
as the science teachers pretend they need? Or after a block of compulsory
algebra and calculus a few choice subjects like probability, numerical methods,
linear programming, or mechanics? Everybody knows that it does not work. From an educational point of view,
application is a wrong perspective cherished by old math and even more by
New Math. The right perspective is primarily from environment towards mathematics rather than the other way round. Not: first mathematics and then
back to the real world but first the real world and then mathematising. The real world - what does it mean? Forgive this careless expression. In
teaching mathematising 'the real world' is represented by a meaningful context
involving a mathematical problem. 'Meaningful' of course means: meaningful
to the learners. Mathematics should be taught within contexts, and I would like the most abstract mathematics taught within the most concrete context.
Let me tell a little story about what context can mean, not only for learning mathematising but even for learning mathematical skills. In a context of sharing and leasing gardens, fourth graders had to figure out, among other things, the rent of a plot like
MAJOR PROBLEMS OF MATHEMATICS EDUCATION 145
where each square pays five florins. All children who stayed within the context
of squares of land and of florins to be paid got the correct answer of 22�89 florins, whereas all the others who prematurely divorced the problem from
its context and schematised it as the numerical multiplication problem 4�89 x 5 got the wrong solution 20�89
What is this little story to suggest? That in the learning process mathema- tising a situation deserves priority over solving word problems by schematisms.
After this interruption let me formulate my eighth major problem of
mathematics education: �9 How to create suitable contexts in order to teach mathematising?
Environment involves space, objects in space and happenings in space. The
mathematised spatial environment is geometry, the most neglected subject of mathematics teaching today. For centuries geometry in the English terminology was synonymous with Euclid. But in history geometry started long before
Euclid, and in children's life it starts even before kindergarten. Or shouldn't it? In any case, what starts, is the grasp of space and the relations in space, the
grasp of space by seeing, by listening, by moving in space. When can this rightly be called geometry? The traditional answer is: when it can be verbalised
in terms of definitions, theorems, and proofs. This then meant that geometry
education started at an age when children were pronounced able to speak the conventional language of geometry, either Euclid's or the textbook's. Unfortunately you cannot learn the language in which a subject matter is
expressed if you have not experienced the subject matter itself. How do you learn the subject matter itself?. The way is by becoming
conscious about one's intuitive grasp of space. Let us take an example. Everybody knows that the diagonals of a parallelo-
gram bisect each other. How do you know it? Well, you can prove it, by
Euclid, by congruent triangles, and so on. But you knew it long before you learned formal geometry. How did you know it then? You do not remember? Perhaps you did not care. Why didn't you? Because your teachers did not care. But why should teachers not care?
Ask a child: how do you know that these lines in that figure bisect each other? Every answer is welcome, even a wrong one, if it shows the child reflecting on his spatial intuitions.
This is my ninth problem of mathematics education:
�9 Can you teach geometry by having the learner reflect on his spatial intuitions?
I am obliged to say something about calculators and computers. You would
protest if I did not. I could refuse because I can prove I am incompetent. I know almost nothing about calculators and computers. It is a lack of knowledge that prevents me from tackling any minor problem of calculators
and computers in mathematics education. It does not prevent me from indi-
cating what in my view is a major problem.
Technology influences education. The ballpoint, Xerox, and the over-
head projector have fundamentally changed instruction. But this is as it
were unintentionally educational technology. Programmed instruction, teaching machines, language laboratories, which were intentional educational technology, founded on big theory, did not fare as well, to say the least
Calculators are being used at school, and they will be used even more in the future. Computer science is taught and will be taught even more. How to
do it - these are minor questions. Computer assisted instruction has still a long way to go even in the few cases where it looks feasible.
What I seek is neither calculators and computers as educational technology nor as technological education but as a powerful tool to arouse and increase mathematical understanding.
Let me illustrate what I mean by a few examples. If you find they are trivial, or silly, please look for better ones.
John and Mary are playing with their calculators. John starts at 0, Mary starts at 100. Alternately John adds 2 while Mary subtracts 3. Where will they meet?
Or another: John starts at 0 and Mary at 100. Alternately John adds 3
while Mary adds 2. Where will John catch up? Or still another: John and Mary are asked to share 100 (say marbles) in the
ratio 2 : 3. They will do it by alternating subtractions of 2 and 3 or multiples
of them, while using their calculators. I hope you understand what I mean: discovering the laws governing ratio
by numerical experiments, facilitated by calculators. It would be marvellous, indeed, if calculators, which know neither ratios nor fractions, could be helpful or could be even a key to understanding these fundamental mathematical concepts.
With the provisos I have made before I would formulate my tenth major
problem of mathematics education as: �9 How can calculators and computers be used to arouse and increase math-
MAJOR PROBLEMS OF MATHEMATICS EDUCATION 147
Is this not a marvellous display of problems of mathematics education? Show
me the wonderful world where these problems might be solved, would be solved! Rather than in the armchair or in the laboratory, educational problems are solved in the educational process - I claimed half an hour ago. If this is
true, then in our real world solving them will be a slow process, a social
process, a long learning process of the society. Can it be steered, can it be
guided? Can there possibly be a strategy of change?
I said change, rather than innovation. I do not like the term innovation.
Innovation means newness. It suggests news as a necessary and sufficient
condition for higher quality. In the sixties people believed in curriculum
development as a strategy for change: curriculum prescribed by governmental
decree or dropped as a new subject matter condensed in colorful textbooks
on schools and classrooms, sold as easily in highly industrialised as in develop-
ing countries where for the majority of youth school life may end at the
age of ten. Though I know almost nothing about developing countries,
there are few things that have shocked me more than the curricula sold to
Curriculum development viewed as a strategy for change is a wrong perspec-
tive. My own view, now shared by many people, is educational development. This means an educationally integrated activity, aiming totally at education,
rather than at details. Totally means:
�9 simultaneously at all levels,
�9 and viewing every subject area in its larger context.
Isn't it an illusion, aiming at total education rather than at a number of aspects? Isn't it too many irons in the fire? No, aiming at education totally
just means not overestimating one's modest forces, not dissipating one's
feeble energy. Only with a view of the whole can you discern the salient points, the nerve-fibres to influence education.
My eleventh major problem of mathematics education is: �9 How to design educational development as a strategy for change?
Where can you find the nerve-fibres to influence education? I will choose two
extremes, the most conservative and the most progressive medium, the most
powerful determinant of present and the most powerful of future education, that is
148 HANS FREUDENTHAL
�9 textbooks, and �9 teacher training.
Let us start with textbooks. Teachers, most often, heavily depend on text- books. I do not blame them for this lack of self-reliance. After three or four years of inadequate training, textbooks might be their last, their only resource.
You have been confronted here with ten problems of mathematics edu- cation that I thought of as major ones, and with the eleventh, of how to solve them. But from the start onwards I have warned you that problem soMng in
education is not a job for theoreticians but for the participants in the edu- cational process. Textbook authors are participants, who in turn depend on the presumptive users of their production. Should I ask textbook authors to solve my problems? Of course not, even not to try it. The least and the most I can expect them to do is to ponder my problems. Perhaps they already did - indeed it was not me who invented my problems. If they did, and if in some
respect they succeeded, let it be known, not by guidelines and teacher manuals, which are often belied by the textbook, but by the textbook itself, by its
built-in features. For instance, progressively schematised subject matter would be a good case, provided it is leading the teacher and learner, not along firmly
preprogrammed paths but along reflection and retention of insight. Math-
ematising the environment is a hard thing to be taught by textbooks but just
for this reason worth being tried. And so I could comment on quite a few
among my problems.
I now turn to teacher training. Should I duplicate what I said about text- books and ask teacher trainers as participants in the educational process to
contribute to solving my major problems? Yes, I will, but that is not enough.
Teacher Students learning mathematics are expected to learn it as a didactical feature. In teacher training each of my major problems of mathematics edu-
cation has its didactical counterpart, from
why can Jennifer not teach arithmetic?
reflecting on one's own teaching,
developing a didactical attitude,
how can calculators and computers be used to arouse didactical understanding?
Things are even more involved. Teacher students, in general, belong to the large group of adults where the sources of what they once learned by insight, have been clogged by the knowledge and skills they acquired meanwhile. To
MAJOR PROBLEMS OF MATHEMATICS EDUCATION 149
say it more concretely: they care neither about why multiplication by 100 is
carried out by 'adding two zeros' nor about the fact that you can or why you should argue such a piece of knowledge. So they have to undergo remediation: relearning such facts while teaching children and observing their learning processes. The higher the level of learning, the more paradoxical this con- clusion may sound. Knowing a piece of mathematics too well may be a serious. impediment to teaching it decently as long as the teacher is unconscious about the learning process that produced his excellence.
So he needs relearning by observing learning processes of less skilled people,
of children. But now we are faced with one of the big problems in teacher training. Whereas in the school environment one can easily arrange for observ-
ing short term learning processes, it is impractible and hence impossible to do the like for long term learning processes. Thought experiments, such as under- taken by textbook authors, cannot f'dl this gap if undertaken by unskilled
people. Lack of experience in long term learning processes is the proper cause of the teacher student's future dependence on textbooks as their only sources
for long term learning processes. How to solve this dilemma is a problem worth
studying. Let me restrict myself to this one. Teacher training as a whole should
be rethought and reshaped.
Educational development includes, amongst other things, educational research.
Though I will not deal with any problems of educational research I promised you I would tackle educational research itself as a major problem of math- emafics education.
Perhaps you know that in the past I have severely criticised, if not castigated, educational research because of its irrelevance, and as a danger for math- ematics education. What is called educational research represents an enormous production, still expanding both as to volume and variety. I admitted, and I still admit, that my knowledge, even if I include the most superficial kind, is restricted to a diminutive part of this field. Although amongst the work I do know, there is little of good quality, it is still so large an amount - and I should add that during the last few years it has increased by one order of magnitude - that by extrapolating to the enormous quantity I am not acquainted with, I would guess that there ought to be a large amount of high quality educational research. But how to retrieve it, buried as it is under mountains of irrelevant and worthless production? Both in mathematical and educational research, production is high. The difference is that the retrieval of good and relevant research is easy in mathematics and almost impossible in education.
150 H A N S F R E U D E N T H A L
I will not repeat any details of my well-known criticism of actual educational
research, but for honesty's sake I feel obliged to say that since I first uttered
it, the quantity of relevant research has sensibly increased, at least with respect
to mathematics. Good research is, however, still in danger of being suffocated
by the mass production of worthless, irrelevant educational research, which in
addition is a danger to education itself.
There are people who claim physics is the curse o f nature and biology is the
curse of life. I do not believe they are right. Science is a good thing, for edu-
cation, too, provided we see to it that educational research does not become
the curse of education. I trust we will succeed. My hope is set on educational
research as part of educational development.
Not: research of education. But: research in education.
z Address to the Plenary Session of ICME 4 at Berkeley on August 10, 1980. - I am indebted to J. Adda, H. Bauersfeld, A. J. Bishop, F. Goffree, A. Z. Krygowska, A. Treffers for critical remarks they made on the first version of this address even in those cases where I did not follow their advice.