On Learning from Arnold's talk on the Teaching of Mathematics
Kapil Hari Paranjape
On reading the
of the talk given by V. I. Arnold at the Palais de Decouverte, Paris
in 1997, one's first reaction could be that of glee (if one
agrees with him) or annoyance (if one does not). If one belongs to the
second camp (like I do) then one might react with an equally polemical,
barb-laced article--indeed that was how the first draft of this piece
looked. On further reflection I decided that this would be reaction to
the style rather than the substance of Arnold's talk; for there
are some substantial points that Arnold is making. Hence the title
of this article.
The first thing we can learn by reading Arnold's article is that one
should choose one's words carefully if one's intention is to reform.
Angry name-calling (ad hominem) arguments using words like
``ugly'' and prefixes like ``pseudo-'' can only detract from the message
one is trying to get across. Of course, one can speak and write
to express one's anger and attack those whose methods we oppose, but we
should remember the maxim that the world often does not listen to those
Most disciplines that are the subject of scholastic activity (be it teaching or
research) have a lot of links to other disciplines and even more so to the
world outside of scholastic activity. A scholar may occasionally choose to keep
distant and monk-like in order to avoid distractions--both physical and mental,
but must eventually and continually return to these ``roots''. To my
mind it is imperative that any person who is not deep in the throes of a
difficult calculation must keep her eyes and ears open for
interesting information from these ``side-lights''. The joy of research
comes as much from tracking the highway to one's goal as from following
the scenic village by-roads of other people's creations.
Very often we come across people who declare that some classes of
mathematics is ``ugly'' or ``boring''. When I was a young
student,1 such a class was non-existent--there was
only mathematics that I could understand and that which I could not.
That which I could not understand was of course more challenging and
required greater effort. Of course, some mathematics becomes boring
because we understand it well and do not see any challenges in pursuing
it. However, a more common context in which people call some subject
ugly is when they do not understand the point of the exercise. In
this case it probably needs more study and a clearer perspective to see
the worth of what one's colleagues are doing. Ugliness, like beauty is
often in the eye of the beholder.
One of Arnold's more substantive points, this one is made with some clarity in his
talk. Today mathematics is often taught in the ``Definition, Theorem, Lemma,
Proof ...and repeat''-style. Many people (dis-)credit Bourbaki with this
approach. Good teachers know the importance of pertinent examples as a way of
underlining the key points of theory. Detailed calculations and proofs are best
checked on one's own so there is not much point presenting them in class. If
students get stuck while working them out then the teacher can help them or
point them to a complete reference such as Bourbaki! In fact, it is my belief
that it is for this purpose that the Bourbaki texts were written. To paraphrase
Weil, ``after Bourbaki no one can argue that a proof is essentially there but
Another flaw often seen in mathematics teaching is that lectures are aimed at
only the best and brightest students--those who will take up research in the
subject. If we enjoy our mathematics then we should be able to convey this
enjoyment to everyone. While it is true that students who do not do the
necessary paper and pencil work will eventually drop out or fail, we must
motivate them to put in the necessary effort. In the modern scenario, where a
good knowledge of mathematics (and the sciences) is fundamental in order to
understand the world around us, the teachers who fail their students (in both
senses!), are doing us all a dis-service.
As regards ``doing one sums'' as a method for learning mathematics; there is
indeed ``no royal road to mathematics''. Students who wish to pursue a career
in mathematics or a related science must do a large number of seemingly mundane
calculations, just as a prospective artist spends a lot of time at the art
museums ``copying'' the lines drawn by those who have ``gone before''. The
precise workload may vary from student to student, but a good thumb-rule is
that one shouldn't stop until one has reached beyond difficulty and into
boredom. Unfortunately, students who are inundated with theory in class end up
not working out either the examples or the details of the proofs. This makes
Arnold's experience (or more precisely his information about other people's
experience) shows us that research in mathematics (or probably any other area)
is not the place to be if you are looking for credit. On the one
hand we should try our best to acknowledge the great public realm from
which we derive so many of our insights. On the other hand, we should
not be particularly perturbed if someone learns of our work through a
different path and mis-directs credit. Our real (mental) gain comes from
study itself and the understanding acquired in consequence. To obtain
real (physical) credit, one must go to the financial authorities!
To a person of World War II vintage the words ``collaborator'' and ``gangs''
probably bring up unpleasant memories. However, in research today there is no
easier way to keep in touch with numerous areas than to be part of a group.
There is an additional benefit. In meetings where most people do not know each
other well, many are loth to criticise or point out any but the most obvious
errors. On the other hand, within a tight-knit group even acrimonious criticism
does not jeopardise the activity. A hierarchical group that consists of a
professor, his junior faculty, post-doctoral fellows and students, is unlikely
to have free flowing discussions in the same manner as (by all accounts) the
Bourbaki group did.
Arnold's description of how ``modelling'' works shows that geometry is all
about creating abstract mental images. On the other hand algebra only exists in
the physical world--by means of language, paper, pencil, ruler, compass and
more recently the computer. While algebra (like language) is symbolic and
appears to be abstract, it provides the only means of transmitting the
``specifications'' for the ``glasses'' that allow others to view the same
patterns that we have seen. Occasionally, when patterns are already half-formed
for others through shared experiences, one can dispense with much of the
algebra; in such case people call a proof ``intuitive''. But the challenge in
learning or teaching something that is already half-understood is surely lesser
than that involved in making ``the tube-light light up''!
I will end with a point that I often like to make. About a thousand years ago,
the proof of Pythagoras' theorem was thought to be difficult and abstract and
only for the ``scholastic'' geometers. Today the same theorem is considered an
intrinsic component of school education. Perhaps one day algebra, sheaf theory
and cohomology will be so well understood that we will be able to explain the
proof of Wiles' theorem to school children as well.
- and I hope that I can still be characterised as a
student, albeit an older one!
Kapil Hari Paranjape