The more I learn higher mathematics, the more I wonder how little benefit elementary mathematics education has brought to children. In light of the advance in computers, they are borderline useless and not indicative of the innate talent of the students.
Children spend a lot of time in school learning arithmetic, chiefly the four operations on real numbers, solution of linear systems, and manipulation involving polynomials. As the level progresses, some of them struggle, and others appear more capable. Consequently, they either choose or abandon majors or occupations related to mathematics.
But in a broader perspective, these calculations are some simple algorithms in disguise. Operations on a field, changing basis for a matrix, Euclid algorithm on a polynomial ring, and so on—we know that they are straightforward, if tedious, to program from scratch, since they are merely a bunch of rigid rules. Geometric auxiliary lines are a nightmare to guess, but they can be determined on a ring with Gröbner basis. Trigonometry identities are said to be a torture, but they are a nothing more than polynomial identities of exponential functions. These matters are easy to program, but hard to calculate in head.
Presently, children who are faster and more accurate in exams, win. What is the big deal about that? What difference does it make, as long as most of them can do it correctly in sufficient time? Today such basic calculations are what computers should do, and apart from them, there is a lot more to explore in mathematics.
Indeed, the students are implementing a computer with their brain. We practically require the children to invoke their vague intuition by themselves. Unfortunately, children surely don’t know they are running a program with their brain when they take an exam. Some children are better, and some are slower to grasp the insight. For the latter, if they quit, they will never be revealed of the mystery, and they will remain to think mathematics as an enigma. While it isn’t ideal to obscure children from the concept of computation, undoubtedly many students in middle school aren’t ready, either, to learn programming and computation theory before they are exposed to concrete objects for several years.
Elementary mathematics education is in dire need of reformation, and what can we do about the situation? There, I hope, could be a way to somehow educate children more about computation, so that they are self-aware when they do it, and it might turn out that the way we classify students’ inclination has been the very opposite of unleashing their gifts.
❧ September 19, 2020; revised August 3, 2021