Challenge the ego

Mumbai: Considerable thought has gone into the question of how to make teaching interesting to the taught. There is no unique answer to this question, nor is it the case that the sum total of all suggestions is fully exhaustive. It may also happen that a particular suggestion made by a person new to the field of discussion would receive this reply from the experts: “Yes, it has been tried before but it has limitations.”

At the risk of inviting such a response, here is a suggestion for what it is worth. It is more suitable for the relatively bright pupils. The keyword behind this suggestion is “challenge”. Ask a student to solve some routine problem and he or she may do the job as a routine matter, without any enthusiasm. However, if we raise the level of difficulty and also let it out that finding a solution is something of a challenge to the ability of the solver, then it may well receive a more serious response. For, every human being possesses an ego and the notion of a challenge is an attack on that ego. The teacher saying “Do this problem” will not get as enthusiastic a response as the teacher rewording the statement to “I bet you will not be able to do this one.”

Of course, as stated before, this may work better for the students who are good achievers and who have acquired high enough ego to be disturbed by a challenge. Given this condition, challenges have stimulated efforts all the way to the highest strata of intellectual achievements. Take, for example, the famous problem of trisection of an angle.

This problem is very simple to state. If we draw any angle on a plane sheet of paper, how do we trisect it using a ruler and a compass? Thus, if we have two lines OA and OB meeting at a point O, they form an angle that we denote by AOB. The question is, can we construct an angle equal to one-third of this angle?

For this construction, we are allowed to use a ruler for drawing straight lines and a compass for drawing arcs of circles. If the question were concerned with the bisection of an angle, that is division of the angle into two equal parts, then this is possible to achieve by using a ruler and a compass. In school geometry, this is one of the first few simple constructions that the children are taught. Having bisected the angle, if we further bisect each half then we get four angles, each equaling one-fourth of the original angle.

The simplicity of these constructions often gives the impression that dividing the original angle into three equal parts must also be a simple construction. Far from it! In fact, mathematicians have now proved by rigorous argument that any general angle cannot be trisected by using a ruler and a compass. (In special cases, such as angle of 90° this is possible, but not so for an arbitrary angle. This proof was given by French mathematician, Pierre Wantzel, in 1837.)

Nevertheless, there are people who believe otherwise and some of them spend a lot of time and energy to find elaborate constructions using ruler and compass only for trisecting an angle! Needless to say, these constructions turn out to have errors, often very subtle ones. Even today, professional mathematicians may receive such proofs in their post with the request from the senders to check and authenticate them. Often the sender fails to see the error in the construction despite being shown where it lies!

The distinguished mathematician of last century, David Hilbert, had announced a list of some 20-odd problems of difficult nature as challenges for the community to solve. They are not yet all solved and whenever a problem from Hilbert’s list is solved, it becomes an event to celebrate. Of course, a claim to have solved such a challenge gets examined carefully by other mathematicians and, in some cases, the solutions are declared incorrect.

This is an example (perhaps an extreme one) showing how tough challenges can be. But descending to the school level, one can find problems tougher than the average, to challenge the ego of school children. I myself recall a similar experience I had in high school. My uncle (Moru mama) was staying with us while studying for his masters in mathematics at the Banaras Hindu University. He began setting problems to challenge me. We had a couple of blackboards on a wall of our house and one of those began to carry “a challenge problem for JVN”. It started as a game between us which I won if I did solve the problem, and lost if I gave up and he revealed to me the solution.

It was great fun while it lasted. But the process developed into a teaching medium which showed me glimpses of the wide vista of maths, far wider than suggested by school texts. I think challenges are under-utilised in our school teaching. In fact, not only in the quantitative subjects, like mathematics and the sciences, but also in other subjects, questions could be framed so as to make the student think and come up with original ideas. In fact, the approach could be two-sided, with the students also raising issues for the teacher to settle issues that may make the teacher also think afresh.

Elsewhere I have advocated the custom of having one period per week in school classes wherein there is a free-for-all discussion between the teacher and the pupil. The pupils would be encouraged to raise any question and if the teacher does not know the answer, Web-search is done by the class to discover the answer.

For, in the last analysis, while it is pleasing to be told the answer to a question that has been bothering you, it is even more pleasing to discover it yourself!