Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER X. WHY IS TEACHING AND LEARNING MATHEMATICS SO HARD?

71. THE MANY FACETS OF MATHEMATICS

One of the aims of teaching mathematics, as we have said, is to arouse an interest in it among the students, firstly so that they should enjoy learning, but also so that those who want to carry on studying to qualify in professions requiring knowledge of mathematics, or even to continue to higher mathematics studies, will not be deterred from doing so. Here we can identify a failure related to the very perception of the profession. Like the elephant described by six blind people basing their description on the sense of touch and coming up with six totally different descriptions, mathematics is a huge elephant with many facets. If we present mathematics from one narrow aspect, we turn away all those not interested in that particular aspect, although they may find other sides of the “elephant” attractive.

First, we should know and describe the various aspects of mathematics. I have already confessed to my weakness in solving the type of problem that comes up in the Mathematics Olympiad, and the type whose solution requires the use of some sort of trick. At the same time I mentioned John von Neumann, one of the greatest mathematicians of the twentieth century, whose method of solving the problem he was facing would be belittled by every trainer of competitors in the Mathematics Olympiad. Mathematics does indeed have the aspect of solving problems by means of tricks, but it also has one of revealing patterns, and an aspect of constructing logical structures, and of course it plays a role in explaining natural phenomena, and in technological developments, and it also has a historical-philosophical aspect. All these should appear in the curriculum. Students should know that if they find difficulty or feel bored with a part of mathematics, they may well find another part very interesting. Someone who does not like classical music can still enjoy jazz.

The main element lacking in the mathematics teaching in schools is the broad perspective that encompasses the subject. Mathematics in schools has, to a great extent, become a presentation of a collection of solutions alongside a collection of questions. Confronted with a question, the student must learn how to find the connection between it and the right formula that will give the answer. That is indeed a natural way of thinking for the human brain, thinking by comparison or comparative thinking, but there is a great difference between a situation in which the brain constructs such a system of comparison for and by itself, and that in which the student must learn by heart a list given by the teacher. This is something that goes beyond the confines of the school, as can be seen from the following.

I recently taught in a higher-education course for mathematics teachers. As the time for the examination drew near, the question arose as to how the students could prepare themselves for the exam. One of them (like the others, a practicing teacher), seeing that my approach to teaching did not seem to fit the mold he was used to, suggested that I should give them an advance copy of the questions, but with different numbers in the questions! He was serious. Apparently that is the practice in secondary schools today. There are many reasons for the development of this practice. One of them might be that the assessment of success in teaching and of the quality of the teachers is carried out by means of standard examinations. As a result, the study focuses on the techniques for solving standard exercises, at the expense of developing a broad appreciation of the subject and introducing other interesting and important aspects of mathematics. We will not expand on this but will just note that this does a great disservice to mathematics and to the students and their future.

Clearly it will not be possible to broaden the range of mathematics teaching and turn it into an interesting field of study as long as the teachers themselves do not recognize the cognitive foundations of dealing with mathematics. For example, one may make a mistake! If a history student gives a wrong date for a certain event, or if a chemistry student misidentifies a certain chemical in a compound, the teachers do not come to the conclusion that the students do not understand history or chemistry. In mathematics, if a student does not answer a question correctly, it is taken to mean that he does not understand. This intolerance is harmful. I have an ongoing dispute with colleagues about questions that should be asked of applicants for entrance to the institution in which I work. Some of them give the applicants mathematical exercises and check the degree to which they manage to solve them. I strongly object to that and complain to my colleagues that they always ask questions to which they themselves know the answers. To succeed in completing exercises, especially in an examination, is a very small and nonessential part of mathematical capabilities.

Finally, the views I have expressed about mathematics teaching are the result of years of interest in the subject, following developments, and activity in the field. The defects I have described are only part, a small part, of the problems of the educational system. I have not referred to the difficult physical conditions, overcrowded classrooms, or the lack of motivation of some of the teachers, and so on. The aspect of the curriculum, however, can and should be improved. To teach mathematics successfully, the teacher must be aware of the difficulties arising from the conflict between healthy intuition and the logical structure of mathematical discussion. Special teaching methods should be devised to impart the technical ability to analyze the logical, nonintuitive aspects of mathematics, and the students should not be expected at the same time to develop an intuitive ability to use such material. Together with achieving broad recognition of the many facets of mathematics and its role in human culture, it will be possible at last to shake off mathematics’ ill-deserved reputation as the hardest subject in school, and it certainly does not need to be the least interesting.