Sep 19, 2020
Modern children spend a lot of time learning arithmetic in school, chiefly the four operations on real numbers, solution of linear systems, operations on polynomials, and so on. Beginning perhaps at the high school level, quite many of them struggle, and others find themselves relatively good at it. Consequently, they either choose, or abandon, mathematics related university departments and occupations, depending on their performance.
The thing that troubles me is that, I suspect the traditional mathematics curriculum does not explore well, nor predict well, the innate talent of the students’. Indeed, viewed in a broader perspective, these operations are some straightforward algorithms in disguise. The students are really implementing a lambda-calculus or a Turing machine with their brain. If one is capable of basic programming, one would find the operations on a field, changing basis for a matrix, Euclid algorithm on a polynomial ring … they may be tedious to implement, but nothing more than a rigid following of rules. There is a lot more to explore in mathematics than these. Unfortunately, children surely do not know they are themselves Turing machines when they take an exam, and we cannot make them understand before they are exposed to concrete objects — which seems hard to debate too.
Still, the problem is, we practically require the children to invoke their vague intuition on computation. Those who return more correct answers in exams, win. What’s the big deal about that? The reason that some children are doing well is simply that they are quicker to realize, implicitly, the way to run a program with their brain; and the reason that some are not doing well, that they are slower. For the latter, they will never be revealed of the mystery, and will probably remain to think mathematics as an enigma.
What we can do about the situation, I don’t know for sure. There might, I hope, be a way that children may be introduced, somewhat, of the concept of computation, so that they are self-aware when they do it. If I am right, it might turn out in the future that the present way we classify students’ inclination is borderline silly, and far from unleashing their gifts.
Sketchbook 