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?

Is it helpful to join a class in mathematical thinking? • How can you learn mathematics to a mother-tongue level? • How did prehistoric man discover mathematics? • What is the essence of a triangle? • Is there a link between a mathematical tree and a botanical one? • How does a centipede walk? • Do students have difficulty in understanding the parallel-lines axiom? • What are the chances that in a family with three children the third will be a boy?

66. WHY LEARN MATHEMATICS?

There is no doubt that mathematics is considered one of the hardest subjects in the education system, from elementary school to university, and achievements in the study of mathematics generally fall short of expectations. However, before we turn to the question of how to improve mathematics teaching, we should clarify what we expect from it. Once we agree on its aims, we may be able to examine the extent to which teaching practice achieves those objectives and improve the system to enhance its performance. The targets that we will formulate relate to students in elementary and secondary schools. We will also relate to higher education, from the aspect of teacher training.

The first objective is to provide the student with the basic mathematical tools needed to function in the modern world. Functioning clearly includes the ability to operate in a world driven by commerce, money, loans, investments, and the like. It is also advisable to be able to understand a world in which we are assailed by a mass of statistics, some of them important, and some unreliable. It is also important to possess the ability to estimate and calculate areas and volumes, that is, the ability to use basic geometric tools relevant to our environment. Lastly, in order to be able to function properly in the modern world it is worthwhile gaining at least an elementary mathematical understanding of the developing technological world.

The second objective is to become familiar with a system of logic that rigorously checks claims. This applies despite the fact that we have argued several times in this book that our natural tendency, resulting from evolution, is to believe what we are told. Yet when there is doubt whether the claims put to you have a firm basis, it is worthwhile being able to check them. No system can match mathematics in being able to clarify the difference between an assumption and a conclusion, between a conclusion drawn by deduction and a hypothesis reached via induction, and so on. Mathematics develops the ability to perform an ordered analysis using the tools of logic, to examine assumptions with minimal use of metaphors, and to query the quality of the information by using very precise language. It is true that people's day-to-day activity is based on intuition, and that is something that is impossible and unnecessary to change. Yet it is important that a graduate of the education system should have the ability to follow a logical analysis of claims, and even to perform such a logical analysis in cases where it is important.

The third objective of mathematics teaching is to recognize mathematics as part of human culture. Alongside history, literature, music, the plastic arts, social philosophy, and so on, mathematics and the sciences played a major role in human development. An educated person should know the part fulfilled and still being fulfilled by mathematics in the understanding and development of the sciences, the arts, technology, and society.

The last objective we shall mention here is to open a window for anyone interested in the subject and for future researchers in sciences and engineering in general, and in mathematics in particular. We are not claiming that secondary education should train scientists and engineers. The least that can be expected of the school is that it should arouse enough interest in its students and should show them how to learn and the possibilities of studying sciences and mathematics so that when they have to choose the path they wish to follow, they have the information necessary to make the right decision, in accordance with their aims and capabilities. The system should also impart to those who will continue to follow the scientific path the basic abilities to realize their potential.

Unfortunately the education system at present falls short in all of these objectives. The following sections will try to indicate some of the defects that lead to this sorry state of affairs. We will not presume to propose a detailed system of what and how to teach. That is too daunting a task that requires means, manpower, further development of didactic systems, and other similar serious issues and limitations. The range of reasons for the problem is beyond the scope of this book, but understanding some of the defects, particularly those related to the question of what is mathematics, may help to correct them.