What should be the output of mathematical education?
by Ronald Brown
Introduction
I had hoped to write an overall view of output as suggested by
the title, but this turned out too ambitious an undertaking.
I am also conscious that a good deal of the readership for this
article will be people who have had experience of teaching in
schools, of teaching the teachers over many years, and who are
well aware of the work that has been done in mathematical
education. It seems advisable therefore to talk about the areas
in which I have had experience, and leave the readers to debate
the possible analogies and relevance is to wider issues.
The use of the word "should" in the title is normative, and
might suggest that I think I know all the answers. To the
contrary, what I hope this article will do is direct attention
to some interesting questions.
The question of "output" is not a bad place to start in
considering an activity. The quality of mathematics education is
an important issue for the future social and economic strengths
of our countries. We are also all interested, for example as
taxpayers, in the output of activities which are publicly
funded. How should we judge this? Can we give reasonable
expectations for research in mathematical education if we do not
know what we can reasonably want from mathematical education
itself? Otherwise, it might even be suggested that an aim of a
conference on mathematical education is to improve the status of
the research and the researchers.
In discussing "output" I am going to start at the "top" end,
namely that of research in mathematics. This may at first seem
foolish, since any comprehension of what goes on in research in
mathematics is generally thought to be way beyond the needs and
comprehension of the vast majority of those who go through the
educational process.
Against this I would set the argument that it would be
desirable for there to be a clearer view in mathematical
education as the nature of the mathematical beast, in order to
know how to deal with it. If it is unclear as to what is
mathematics, and what are its main achievements, what
constitutes performance in it, what is the hope of teaching it
in a clear way, of coming up with new practical hints as to how
it should be taught more effectively? I wondered after attending
ICMI94 whether this itself was not the crucial unasked question,
which required debate. Indeed, one speaker did refer to the
paucity of practical hints which were available. It is more
difficult to give practical hints for teaching an activity not
itself well defined. Not all the resources of psychology,
linguistics, or philosophy, will help education and training in
the sport of, let us invent one, say, Yarangoo, if the rules of
Yarangoo, and the method by which one wins a game of Yarangoo,
are not understood in the first place. For this, it does not
matter whether the aim is that of a beginner or a first division
player.
I find some support for this view from articles I recently came
across in Mathematics Teaching, December, 1986, "Special Issue:
The roots of mathematical activity". Raffaella Borasi writes
that: "Several recent research studies investigating the
difficulties that many students encounter in learning
mathematics have suggested that what students believe to be the
nature of mathematics may influence considerably their
possibilities for success in the subject." Phil Boorman explains
how his outlook on teaching mathematics is shaped by his view of
mathematics as the study of pattern and structure: "Now, if I am
right in defining mathematics in this way, everyone is very,
very good at `doing' mathematics - if they were not they would be
dead- for all thought is mathematical."
Here is another example of how a view of mathematics can affect
teaching and assessment. A colleague in London, say Fred, tells
the story of how his son, say Bob, was given as an investigation
the problem of determining the number of diagonals of a regular
polygon. Bob and two friends came in one day when Fred was
watching television and started discussing the problem. They
tried a few low dimensional cases, came up with a general
formula, tested it out for pentagons, saw they had got it wrong
and needed to divide by two, and so arrived at the correct
formula. Fred was delighted with their progress. The teacher was
not so delighted. One of the boys just wrote down the answer
and got 0. It appeared they were supposed to write a nine page
project, testing special cases, drawing graphs, and so on. As
the teacher explained: "They had to learn: there is more to
mathematics than thinking." None of this had been explained
before they started the project.
Since the ICMI94 was broad in the areas of study which were
used to try and illuminate the question of what should be
research in mathematical education, it is right that mathematics
itself, what it advances towards, and the way it advances,
should have some pride of place. Otherwise, we might have Hamlet
without the prince.
A further point is the psychological truism that the behaviour
of people is more similar if they are compared at the limits of
their ability. The struggles that we have to understand and
master advanced new ideas could give some sympathy for those who
are also struggling at a much lower level of performance, and
perhaps some help in improvement.
An advantage of the research viewpoint is that it counters the
assumption that there is always a "right" mathematics, and that
the problem is necessarily to get people to behave in the way
assumed by this mathematics. To the contrary, from the research
point of view, we are especially interested in those parts of
mathematics where we feel uncomfortable, since this feeling
could be a pointer to a different approach being required. It is
easy to forget the way in which was once research becomes, maybe
over centuries, part of everyday mathematics, and so a part of
mathematics teaching. Sometimes mathematics teaching is forced
into the latest view of "modern" mathematics, to its detriment.
Views even on basic mathematics are changing, partly in response
to research needs and the developing language to describe new
kinds of structures, and the ways in which structures interact,
but also in response to new understandings of what mathematics
is about and how it works.
David Tall in the abstract of his ICMI lecture at the ICM
Zurich, 1994, writes: "There is thus a significant difference
between the flexible thinking of the mathematician and the form
in which the product of that thought is eventually
communicated." One of the ways mathematics progresses is in the
closing of this gap, by finding forms, structures and languages
in which the way we think can be more properly, more accurately,
and more understandably expressed, and even, more importantly,
to yield new ways of thinking about a topic. These new forms
then allow for more elaborate patterns of thought, calculation,
and deduction. For example, mathematics was held up for
centuries for lack of the "trivial" concept of zero. (I owe this
point in this context to A. Grothendieck, the twentieth century
master of the development of concepts.) Were children once
beaten for not being able to add properly in Roman numerals?
As an example, a striking change in basic thinking in
mathematics is the existence of a move away from set theory as a
"foundation" for mathematics, and a realisation that a more
flexible and intuitive approach is needed. The tools for this
move are provided by category theory. In this method, emphasis
is placed not on set theory and not on the "elements" of a set,
since these are somewhat counterintuitive when one comes to
large sets, such as the real numbers. Instead, emphasis is on
the functions, and so on the relations between structures,
through the homomorphisms between them. The constructions that
can be made on objects are defined by the relations of these
constructions to all other objects. In this view, functions
return somewhat to their more intuitive role, rather than the
passive role as a set of ordered pairs, a logical device as part
of a thrust to give mathematics a "safe" foundation.
Thus the difficulties which pupils, students, and professors
have, and will always have, in learning concepts, procedures,
and skills, in mathematics is one aspect of a process with two
variables, the learner and the mathematics. The processes of
teaching, and so of trying to understand in order to explain to
others, has often been a stimulus to the development of
mathematics. It would be fascinating if philosophy and the
social sciences of education, linguistics, and psychology, which
figured so much in the discussions at ICMI94, can help in this
process.
The output of research in mathematics
It could be surprising to you that there might be any question
about this. However the public and Government are in general
unclear as to what mathematicians produce and how valuable what
they produce is. Mathematics undergraduates, even very able
ones, are often unclear that any research goes on in
mathematics. This is not surprising since they are usually not
required to look at original papers, nor to acquire any
understanding of current work in progress.
Whereas astronomers, physicists, biologists, engineers,
chemists, and so on, have worked hard to convince the public and
Government as to what they are doing, and what constitutes a
significant advance, it is less clear that this has happened in
mathematics. The biggest splash recently has been for the
possible solution of Fermat's last theorem. Thus the solution of
famous problems is often advertised as the main success of
mathematics, and this view is encouraged by mathematicians.
Certainly the solution of such problems will bring fame within
mathematics to the solver. What will be the effect on science,
technology and the general public?
The history of mathematics shows that the contribution of
mathematics to science and technology has been to provide a
precise and developing language for the invention,
representation, and discussion of certain concepts and
relationships, together with a mode of deduction, calculation
with and exploration of these. Mathematics has been able to say,
in places where it can be applied: This is true, and that is
false. This has enabled mathematics to reveal astonishing
elaboration of patterns and structure, to provide tools for
applying these structures, and to show new problems. The fact
that mathematics is, if you like, the science, craft, and art,
of pattern and structure, explains why it underlies so many
other scientific and technical areas, which are themselves
seeking to understand the patterns and structures in nature. We
should avoid, though, the idea that the interest of mathematics
rests on these applications, rather than the joy of the
investigation itself. Indeed, some of the structures which have
been forced on mathematicians by the logic of their arguments,
and which have seemed weird and strange, have later, sometimes
many years later, found their true place in applications. A good
example is the theory of fractals, and of chaos.
Within mathematics there has long been a need to decide what is
good mathematics. This occurs most crucially at the sharp end of
publication. Authors have to decide for which journal their work
is good enough. Referees and editors of journals have to decide
whether or not a submitted paper should be accepted, and while
competition for space in the top journals continues to increase,
this decision has to be made on the basis of which are the
"best" papers. Yet the question of what is good mathematics is
little debated at the professional level, and an understanding
of this is not explicitly part of a qualification for an
undergraduate degree or even postgraduate. By contrast, students
of, say, design, or musical performance, are introduced to this
question as a basic object of their studies.
To give some focus to these questions, I would like to give
some account of what I have been doing mathematically for the
last thirty years. In any case, I like explaining this
background, and the way this research has gone has influenced
my views on mathematics as a whole. There has been a "reach for
the stars" aspect of this research, and it has been satisfying
that some basic foundations of this route have now been firmly
laid down.
After my PhD in topology, not so sure of the direction I wished
to go, I embarked, as a kind of displacement activity, on
writing a topology text. Its principal aim was to explain the
main basic results in algebraic topology, particularly the
notion of a cell complex as a way of representing a space as
constructed out of "nice" and comprehensible bits, namely the
"cells" or "balls", and the use of the so called fundamental
group as a topological invariant.
A crucial part of this theory is the calculation of the
fundamental group, in which a main tool is what is known as the
Van Kampen theorem, first found in the mid 1930's. It shows how
the fundamental group of a big space can be obtained if the
space is the union of two "nice" parts, such that the
fundamental groups of these parts, and of their intersection, is
known. However, to use this theorem, the intersection of the
parts has to be connected, to have only one "piece", and this
prevented the theorem from being used to calculate the
fundamental group of a basic example of a space, the circle,
where the result is the additive group of integers. This example
had to be determined by another method, the use of the
exponential function from the real line to the circle, which
wraps the line around and around the circle, like a rope around
a bollard. Of course, this wrapping method is a nice, intuitive,
and important method, but still I found the diversion
unaesthetic.
In 1965 I came across a 1964 paper of Philip Higgins on the
applications of groupoids to group theory. The notion of
groupoid was introduced by Brandt in 1926, as a tool for
extending important work of Gauss on the composition of
quadratic forms, from the case of two variables to that of four
variables. A groupoid should be thought of as a group in which
the multiplication is not everywhere defined, so that many
identities are allowed. Intuitively, the notion of groupoid
corresponds to that of travelling between many points, so that a
journey from London to New York can be composed with one from
New York to Tokyo, but not with one from Washington to Montreal.
By contrast, with a group, you always return to the starting
point. (The word groupoid is, unfortunately, also used for the
quite different notion of a set with a binary operation.)
The notion of groupoid was found in topology as the fundamental
groupoid. It was recognised in the 1950s that groupoids gave a
nice account of one aspect of the theory, that of change of base
point. Perhaps this aesthetic feature should earlier have been
taken as a clue that something potentially important was going
on. However, groupoids were regarded as something of a
curiosity, since the real interest was felt to be in the widely
used notion of abstract group, recognised as one of the central
concepts of mathematics.
To my surprise, I found in 1965 that the notion of groupoid
solved my expository problem, since the Van Kampen theorem
extended neatly to the fundamental groupoid, and so yielded an
elegant determination of the fundamental group of the circle.
Indeed, one obtained a simpler proof of a more powerful theorem,
which, as one might say, can't be all bad. Later, in 1967, a
famous analyst, George W Mackey, told me how he had been using
groupoids for years in ergodic theory. All this convinced me
that the thrust of my book should be on groupoids, and that this
area of basic theory was most naturally expressed using that
term and that language. This became the pattern of the book,
published in 1968.
The excitement of this extension from groups to groupoids was
the very wide and important uses of groups in mathematics and
science, particularly in applications of symmetry. The obvious
question raised by this was to what extent parts of mathematics
and science which used groups could be better served by using
groupoids.
I was brought up in homotopy theory, which studies higher
dimensional versions of the fundamental group, called higher
homotopy groups, basically by replacing what might be thought of
as loops of strings, by their higher dimensional versions, the
n-spheres, which for n =2 are just the surfaces of balls. Could
there be higher homotopy groupoids and a higher dimensional
version of the Van Kampen Theorem? By now I was very familiar
with the proof of this theorem, since I had written it out say
five times in various versions. It seemed quite clear that if
one had the right language of higher homotopy groupoids, then
the proof of the Van Kampen theorem would generalise, at least
to dimension two rather than one, and probably even further. So
I had an outline proof in search of a theorem. Unfortunately,
the theorem itself could not be formulated because the statement
required several concepts which had not been defined, and which
indeed were unclear.
This started me out on a long road. Every so often I would say
"Tonight's the night!", and start to write the basic paper in
the area, giving the basic definitions and propositions, only to
find it drifting into the sands, and getting thoroughly stuck.
The questions, though, always seemed to recur, and the drawing
of a few diagrams kept on strengthening the conviction that they
must represent some real mathematics. There was a bit of the old
adage: "If a fool will but persist in his folly, he will become
wise!"
Gradually, with clues and methods from here and there, through
collaborations with Chris Spencer in 1971 and 1972, with Philip
Higgins for fifteen years from 1974, with Jean-Louis Loday in
1981 to 1987, with research students who inputted key ideas and
results, and contributions from others, a large theory and
method took shape. It really is true that a higher dimensional
group(oid) theory exists, which is not available for groups
alone, and that a number of phenomena in topology, and even in
group theory itself, can be better understood, and new results
found and proved, from this viewpoint. The higher dimensional
Van Kampen theorems also express, and have their roots in, some
long standing traditions in topology and group theory, such as
the notion of a cycle, namely ``something'' with no boundary,
like the surface of a sphere, and also with methods of gluing
pieces together to make larger pieces. I have used these vague
words deliberately, since the early literature is unclear as to
what is a cycle, and how one "adds" pieces. The later
clarifications, in terms of chains and homology, use a trick of
working with formal sums which has a considerable success, but
does not fully represent the intuitive idea. The intuitive idea
is that of gluing bits a and b together, rather than writing a
"formal sum" a + b . Finding some mathematics which represents
this idea ends up by yielding a statement and proof of the above
theorem, and so allowing for new understanding and new
calculations in homotopy theory not currently possible by other
methods. These calculations are not the main aim of the theory,
but they are satisfactory as a test that the ideas are working,
and that they do something new.
Chris Spencer remarked that a strange feature of work in this
area is that you can get very confused, until suddenly you
realise that you have not been doing the "natural" thing. When
you do so, it all works out beautifully. This gives a comforting
feel about the area. On the other hand, the theory in dimensions
greater than 2 was much more technical to find and write down
than I had naively expected, and it would not have been done
without the high level and varied knowledge and skills of those
who came to work in the area.
This theory has not solved any really famous problem. Rather,
it has solved problems not previously formulated, and suggested
a new set of problems, and areas of investigation. Also, there
is a strong intuitive pull. Draw a square, then divide it into
smaller squares, and you can easily convince yourself that there
has to be a theory which expresses the way the big square is
built up. The algebra which expresses this should be of general
importance, since the method of building a complicated object
from small standard pieces is quite widespread. It is
interesting to see the conceptual and technical advances that
are required to express these ideas, even in the simple format
used so far.
History shows that, in the long run, new methods win out over
new theorems, and over the solution of famous problems. The
latter can often be more in the nature of a test of a new method
rather than an indication of what the method will eventually
achieve.
The aim of the theory of higher dimensional group(oid)s is that
it should come to bear to ordinary group theory a similar
relation to that of many variable to one variable calculus. This
may seem a case of overweening ambition. On the other hand, it
is difficult to resist the voice which whispers: "It is
ambitious, but what if....? How do you know it can't be true?
Which of the prospects, true or false, would be most fun? What
needs to be done to make it work?"
It is by this stage, after almost thirty years of thinking, off
and on, about this area, difficult for me to see what might be
the obstruction to this happening. It will be interesting to see
what actually happens.
These experiences have led me to emphasise the conceptual mode
of progress in mathematics. This allows for the view that good
mathematics can be easy. On the other hand, it should also be
said that the technical requirements to set up these concepts
and make sure they work are considerable, and not what I had
expected to happen. One aim of mathematics is to set up
machinery of which you do not need to know or test all the parts
before you use it, just as you can drive a car without knowing
the workings of the internal combustion engine. This is the
function of lemmas and theorems.
These points are relevant to a recent debate, sparked off by an
October, 1993, Scientific American article on "The death of
proof". A point that I think has not so far been made in this
debate is that mathematics increases certainty by the
development of new concepts, and the formalisation of ways of
thinking, so that the framework of an assertion and its proof
can become so well structured, so natural, and each part so well
tried, that the whole carries conviction. What one calls a good
proof is not so much like finding a route through a maze, but
like following a walk through a natural seeming landscape, to a
suprising viewpoint.
The output of postgraduate education in mathematics
Postgraduate education represents perhaps an extreme of
individual involvement in mathematics training. I have had the
privilege of taking nineteen people through to a successful
doctorate degree. From all these students I have learned a lot,
since their problems and approaches to mathematics have all
differed considerably. But this number does contrast with the
hundreds of undergraduates whom I have taught over the years.
I still find training postgraduates to work for a doctorate a
risky business. My overall method has been to involve students
in the problems in which I happen to be interested at the time,
and to discuss frankly how one would reasonably assess progress
in this particular field. In this process, some students have
made quite crucial contributions to the overall research
programmes, in ways which I would not have foreseen and quite
possibly would not have worked out for myself.
I think that in all cases, work has been done which otherwise
would not have been done. It was partly forced by the necessity,
particularly from the student's point of view, of some kind of
progress, and so for an analysis of how we should proceed. If
you have a lot of available questions, and there is no real lack
of them in this area, then each one has rather a lesser priority
and can perhaps be replaced by another. Some kind of ranking, of
value judgement, is crucial for the researcher in deciding what
to tackle next. For the student, though, the most important
problem in the world is one that he or she is tackling, and
there is a special urgency about getting somewhere.
The process of training and discussion, of making students
aware of available strategies of work and study, is of course
individually intensive, involving in some cases many hours of
discussion and of reading student's work. In some cases, the
opposite has taken place, and I have been instructed how the
problem should be tackled! What I have provided then is the
context and the problem, the reason for wanting this problem
solved.
All the problems given to students have tended to be my
problems, that is the problems thrown up in studying this area.
This has the possible danger then of leading students into a
byway of mathematical progress. Fortunately, these sets of ideas
have come more and more to link with, to require and to
illuminate known areas. This fact was comforting in judging the
progress and prospects for the overall area.
A severe problem in postgraduate training is that of
background. A certain knowledge is necessary to understand the
problem and its context. An even greater background is necessary
to understand and master the tools which should be relevant for
the study of the problem. Many of these tools are learned "on
the job", and, there is a judgement required as to a "need to
know". There is no easy answer to this. Most problems require
for their solution a degree of skill in certain specific areas,
from say group theory to programming in C, and without these
skills at a professional level no progress which can be judged
worthy will be made.
There are two main procedures which have over the years evolved
as important for postgraduate training.
The benefits of the process of writing mathematics
The main idea is that writing mathematics to a high standard of
exposition is a crucial element of doing mathematics. It took a
long time for me to realise for myself that this was crucial in
my own mathematical work. As explained earlier, the writing of a
book on topology set a course of many years of research work.
So our postgraduate students are set as part of their work the
task of writing up a piece of mathematics, not just by copying
from a text or from various papers, but to give an exposition of
one area from a different viewpoint.
The growth in the use of mathematical wordprocessors has been a
great help in this process. The tutorial process of instruction
can work on a readable typed text, which can be improved, and
the process of making mathematics is seen as the production of a
finished, accurate, clear and readable work. This allows for an
emphasis on the craft of mathematics, and so on its nature as a
process. The art comes in the analysis of the qualities in the
finished product for which one is looking, and the decisions on
how to achieve them. It is very helpful to students to have
these matters discussed.
The analysis of aims
The idea here is to explain to a student what it is that might
be done, and then to discuss the following:
Why have I, the professor, not done it before? Answers might
be:
Just thought of it
Forgot about it
No time
Considered it too hard
Thought it not worth while
More study of the background literature needed
Not clever enough
No time to learn the required skills
How should the answer or answers to the previous questions be
used to influence the immediate tactics?
For example, it might be necessary to do a serious study of a
particular part of the literature, to learn some skills from a
given area, or to evaluate new evidence that the problem might
be more important than previously thought.
What would be the expected results of achievement of the
immediate aims?
Some judgement as to the value of the proposed achievement must
be made, if a student is to spend some time on it.
There needs to be a "fall forward" position: what do we do if
the problem is far easier than had been thought? Where do we go
on?
There needs to be "fall back" position: what do we do if the
problem as stated is far harder than anticipated, or, even
worse, not as sensible as originally thought?
Is the advice of the supervisor sensible?
It is often hard for students to realise that while it may be
the job of the supervisor to have ideas, and to suggest ways
forward, it is for the student to evaluate them.
It is useful to work on the following analysis. If 3% of your
ideas are good, and you have 100 ideas, then you have 3 good
ones: result, happiness. If 3% of your ideas are good, and you
have 10 ideas, then you have problems: result, misery.
For this reason, it is useful to see how the supervisor copes
with failure. Indeed, a part of the success in research has to
be the successful management of failure.
Is the problem a natural one for the student?
Jose Montesinos said that his advice to students was to
continue with those aspects they found easy! It is difficult to
describe exactly what it means to understand mathematics, and
even von Neumann commented: "You don't understand mathematics,
laddy, you just do it!" Each person's mental equipment, and
natural mode of thought, differs from those of others, by
reason of both genetics and of experience. Only trial and
observation can find which problems are the most appropriate.
Skill learning
Of course, if new skills have to be learned to carry out the
work, then the usual methodology of skill learning applies,
namely:
Task analysis
Practice of basic skills
Moving from the very easy in gradual stages as skill level
increases
Putting together basic skills
Observation and analysis of performance
Notions of style and quality
Conceptualisation
Internalisation
The motivation for the skill learning is important, and that is
why I have put task analysis at the beginning. A problem of
mathematical teaching is to give motivation for the skills which
are learnt.
Conclusion
The output of postgraduate education is, in theory, a trained
independent worker, with a proven battery of skills and
knowledge, and with some idea of how to make judgements in
seeking out and inventing problems, reading and evaluating the
literature, making progress with problems, writing up the
results, and evaluating the results achieved.
The problems my students have been asked to tackle have
sometimes turned out remarkably hard, so that the discussion of
what might be partial progress has been important. One student
said that a good aspect of this programme has been the
combination of grand prospect, strong intuitive base, and the
technical problems that need to be overcome to make this
intuition work. Also, I cared strongly about the results, and
was delighted that on many occasions I was shown how to do
things.
The supervisor has a great advantage over the research student
in the knowledge of background, context and notion of value. On
the other hand, each student is an individual, and each is
likely to respond differently to different kinds of mathematics.
I do not think there is any final answer to the methods of
postgraduate education. As with many activities, it is possible
to point to some avoidable mistakes, such as that of assuming
something is "obvious" without writing down all the details, and
also to give some kind of framework to the doctoral process.
Part of the problem seems to be to combine a sense of direction,
with the ability to take note of promising lines if and when
they appear. A considerable part of the difficulty is to acquire
the necessary background and skills to understand, evaluate and
tackle the problems. Since a lifetime can be spent in acquiring
knowledge that might be useful, a pragmatic attitude has to be
taken of learning what seems to be necessary to get on with the
job.
On the other hand, when a skill is necessary, then it has to be
learned. You cannot make an analogy with, or use a method from,
an area of which you know nothing.
There is no advantage in reinventing the wheel, except as a
learning method. The spirit of Polya's "How to solve it" is
relevant, but we need a further trick, against the spirit of his
book, namely "Look it up in the literature". A student would
feel aggrieved if he had been slogging away for six months at a
problem the experts know has been already solved, or would fall
easily to a range of standard techniques. So there has to be a
search for those methods which are in the literature and which
might be relevant. We are not playing a party game, and doing
research is difficult enough without artificial restrictions. A
paper reproving something already known will get short shrift
from a referee, unless a new viewpoint or simplification is
apparent.
The acquiring of necessary skills is no easy task. The teacher
and the student have to allow for time, practice, repetition,
and thought. Persistence is important, since it may take a long
time of apparent no progress before it is apparent that real
improvement takes place.
There is also the question of what level of skill is required
for the problem at hand. I like the comment of a magician who
explained:
I practice till the difficult becomes easy; the easy becomes
habit; and the habit becomes beautiful.
Even if practice does not necessarily make perfect, it is clear
that practice is an essential element of perfection. There is no
way to become a good swimmer, a good musician, without putting
in the hours to get the feel of the activity, on the basis of
sensible coaching and teaching. But how good do we need to be
for the purposes at hand? What do we have time for?
Is there a danger that the baby of practice has been thrown out
with the bath water of rote learning? You tend to get funny
looks at educational conferences when you bring up the topic of
rote learning, but have the psychologists analysed for us the
differences between rote learning and practice? I have met
education students who have learned by rote that rote learning
is a bad thing! Has anyone shown us a royal road to mathematics?
It may be that if we ourselves cannot understand the processes
involved in carrying out an activity, then we are left the
notion of practice until we get the feel of it ourselves. How
many activities can we really understand in a way which helps
with the teaching of them? This is the advantage of
concentrating on the notion of output, and its quality.
The popularisation of mathematics
I have been involved in mathematical popularisation. I have
given three addresses to the British Association for the
Advancement of Science (1983, 1987, 1992), in the last case as
President of the Mathematics section. I have given a London
Mathematical Society Popular Lecture (1984), a Mermaid Molecule
Discussion (1985), and a Royal Institution Friday Evening
Discourse (1992). This last was particularly exciting and
taxing, since it involved not only a lecture but also a library
exhibition. I have also been involved for ten years in the Royal
Institution Mathematics Masterclasses for Young People in
Gwynedd, where we take about fifty five school children aged
thirteen for five Saturdays fortnightly. A team at Bangor spent
four years preparing an exhibition Mathematics and Knots, for
the PopMaths Roadshow, which circulated the UK in 1989 and 1990,
starting at the ICMI89 meeting at Leeds on The Popularisation of
Mathematics.
Problems of the popularisation of mathematics
The aim overall in these lectures and exhibitions was to convey
something of the methodology and nature of mathematics to a
broad audience. Part of the problem is thus to make abstract
ideas concrete.
Among all the subject areas, mathematics has a special
difficulty in popularisation since the general public have
little ideas on what are even some of the most basic objects in
mathematics, such as that of a group. Part of the theme of this
article is that this lack of knowledge is in part traceable to a
lack of clarity in the mathematical community, by both
researchers and teachers, as to the nature of mathematics
itself, and so to a lack of clarity about conveying this to the
general public.
The nature of mathematics
My popular lectures aimed to convey some related aspects of
mathematics. These were exemplified with two demonstrations in
the Washington talk, which it is worth explaining here, and
which were necessary to give on that occasion, since I had
brought the things 2000 miles!
Mathematicisation
One of these was the aim in mathematics of mathematicisation,
that is of expressing an intuitive idea in a format which is
sufficiently precise to enable deduction, calculation and proof.
There is usually a prior stage to this, that of
conceptualisation, which also requires its own analysis.
Rules and laws
Mathematicisation usually involves the notion of a rule or law.
One example of this which I used in the lectures was the theory
of groups, and the calculation with relations on symmetry
operations.
The mathematicisation of symmetry can be demonstrated through
the particular example of the symmetries of a square, which can
easily be done for a large audience with a large cut out square
with labelled corners. It is easy to show the rules where x
denotes rotation of the square through 90ø, clockwise, say, y
denotes reflection in a bisector of two edges, 1 means the
operation which leaves the square alone, and xyxy means do
first x, then y, then x, then y. The point here is that the
representation of an action by symbols, of consecutive actions
by concatenation of symbols, and the use of the symbolic method
generally, is one of the greatest difficulties the general
public find in getting a glimpse of what mathematics does. So it
is necessary to show this feature in a concrete situation, and
then to go further and show the value of the symbolic
representation through the use of rules on combinations of these
symbols for explicit calculation, and to show how this
calculation models real operations.
Another nice feature of this example is the importance of
pedantry, in this case the importance of taking note of the
operation 1 of leaving the square alone. Without this, the rules
for the symmetry could not be properly expressed, just as our
counting system would not work without the use of the number
zero.
There is a nice trap here about this representation, which
shows the importance of precision. When you rotate the square,
the labels on the corners change their position. But one wants
to iterate the operation x. So x has to be an operation which
works on positions, that is it moves all the elements in their
various positions one place around clockwise.
One has also to be careful, because the notion of clockwise
differs between the audience and the demonstrator. This gives an
opportunity for another remark on symmetry.
Idealisation
This is a standard procedure. The real square is not completely
symmetrical. For the mathematicisation, we think instead of an
ideal square.
Abstraction
This symmetry example also shows the importance of abstraction,
namely representing the real operation of rotation on an ideal
square by a symbol, and representing the combinations of
operations by combinations of symbols.
This abstraction is usually a stumbling block for the general
public. However it is crucial to the progress of the subject
since it allows the notion of analogy, an aspect of mathematics
not commonly stressed.
For example, when we write x+y = y+x, and xy = yx, we are
illustrating the commutative law and so illustrating an analogy
between addition and multiplication. This process of analysing
and using laws at various levels in the subject is very
important. For example, algebraic structures are an important
part of mathematics, but there is also a mathematics of
algebraic structures, in which such systems are looked at as a
whole. Thus analogies work between levels of abstractions as
well as at one given level. The symbolic method is a crucial
part of the process of abstraction, since symbols can represent
a variety of things.
Deduction and calculation
A further part of mathematics is the deduction from and
calculation with rules. In the case of the symmetries of a
square, we want to deduce from the rules given above that, for
example, xy = yx^3 , yx = ^3y. This shows that whenever x is
taken past y it changes to x^3. The carrying out of these kinds
of deductions and calculations is an important part of
mathematics.
The learning of algorithms
In some circles, the ability to learn and carry out algorithms
is rather decried. To the contrary, I believe it is a basic part
of mathematical skills, just as hitting a ball consistently is a
basic part of tennis. Mathematics involves of course far more,
namely the ability to solve problems and develop theories by the
planned use of basic algorithms and methods, and also by the
effort to see their scope and limitations.
Surprise
Some of the best mathematics has about it the element of
surprise, the revealing of a fact one would not have thought
possible. Such a surprise makes one want to explain why this
happens. Here is an example.
Dirac String Trick
The Dirac string trick illustrates a surprising feature of
space, and also the notion of a rule or law. The apparatus for
this is as follows.
Take two squares of card or board, say 18" square (0.5m), and
on one you draw an arrow, to indicate direction, or place a
picture. The corners of the top square are then connected to
those of the bottom square by string, or, better still,
different coloured ribbon. It is a good idea
Figure 1
to clip the ribbon to the board by bulldog clips, so that the
apparatus can be untangled easily.
Hold the bottom square on the floor by your foot, and rotate
the top square through 360ø, keeping it horizontal. The ribbons
become tangled.
Now rotate the top square in the same direction through another
360ø. It appears that the ribbons become more tangled. However,
it is possible to untangle them completely, moving the top
square up and down to allow room for the ribbons to be
manipulated, but without altering the direction of the top or of
the bottom square.
This illustrates the law x^2 = 1, where x is now the rotation
of the square through 360ø.
There is a more subtle point which accords well with my
research interests. The proof that the rule holds is obtained by
untangling the ribbons. However, the finding of such an
untangling, and the classification of these untanglings, is a
"higher dimensional problem", of a much greater difficulty.
It is amusing to try variations on the above, such as rotating
the square about a different axis than a vertical one.
Dirac's interest in this trick was the argument that it was a
model of the spin of an electron. This analogy is quite a good
one, once the exact mathematics of the situations has been
spelled out.
Knot theory
Here is another "trick" which also illustrates some
mathematical methodology.
Make out of copper tubing a pentoil knot as shown in thick
lines in Figure 2. (This needs professional help, such as an
engineering workshop, to make it look good, but you can make it
yourself with thick wire.) Now tie string on according to the
following rule:
xyxyxy^{-1}x^{-1}y^{-1}x^{-1}y^{-1}
as shown in the picture, and tie the ends of the string
together. It is then possible to take the loop of string off the
knot without cutting or untying it. That is, the above
complicated formula also in this situation represents, or
equals, 1. It is even possible to give the proof of the formula
for bright youngsters familiar with some algebra, such as
cancellation.
One of the reasons for the choice of knot theory for public
lectures is that it illustrates a number of mathematical
principles, and yet the basic problems can be put over in a
direct fashion by using a piece of rope.
Conclusion
There is no space or time here to discuss the wide topics of
output at undergraduate and at school level. Each of these
represents a very big problem. It is an important fact about the
undergraduate output that our teachers of mathematics, at least
in the UK, largely come from this group, so that attitudes
fostered at Universities have a large influence. There is even a
question as to whether in fact any attitudes are deliberately
fostered.
Here is the elephant analogy: We can teach the structure of
the elephant's trunk, tusks, skin, feet, stomach, and so on. But
can the elephant really be understood without a global approach,
an ecological and evolutionary approach?
There is a worry that students of mathematics are starved of
any attempt at a global viewpoint, and a sense of value and of
context, an understanding of the place of mathematics as a human
endeavour. I am sure that those who came to ICMI94 are aware of
these concerns, and that the work of ICMI will help to develop
not only teaching in mathematics but also a general awareness of
these issues.
Figure 2