On Learning from Arnold's talk on the Teaching of Mathematics

Kapil Hari Paranjape

On reading the transcript 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.

Don't emulate Arnold

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 who shout.

Keep your eyes open

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.

Can math be ugly?

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.

Exemplary Teaching

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 for details''.

The failings of non-mathematicians

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.

No royal road

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 poor mathematicians.

If you want credit go to a bank

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!

Collaborate and join the gang

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.

Geometry is about abstraction

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.

Footnotes

... student,¹

and I hope that I can still be characterised as a student, albeit an older one!