Mathematics and the Real World (66 page)

One of the greatest stumbling blocks in mathematics instruction is the misunderstanding of the role of definitions and axioms. In presenting mathematics, generally the definitions come first, and then the axioms, and next come theorems and proofs. But it must be stated again: that is just a way of presenting the subject. The role of definitions and axioms is to clarify and to draw a precise outline of the limits of the discussion. For the definitions to clarify something, that something must be there, must be in the brain, even if it requires clarification. Mathematicians and teachers of mathematics have become accustomed to definitions coming before the subject itself, and they are prepared to accept it on the understanding that the subject will be presented thereafter. For a “trained” audience, it is actually easier to follow mathematical material when the presentation starts with definitions and axioms. Students of mathematics (generally the less-gifted students) sometimes even ask the lecturers for them. Once I was asked by a colleague in a non-mathematical field to explain what groups and fields are. I did the best I could, with little success. The same colleague sought help elsewhere. Some time later he asked me with some scorn in his voice, “Did you even know that a field is a triplet?” Many mathematics books do indeed begin the chapter on fields with the declaration that a field is defined as a set with two operations performed on it, and that is the triplet my colleague referred to. That same colleague, apparently in light of the explanation he had been given, ascribed the greatest importance to that part that indicates a field is a triplet to the extent that the fact that I had not stressed that aspect led him to doubt my skills. That is professional distortion. In a mathematics book, in which it is difficult to present intuition, there may be no choice but to begin with dry formalism. It must be borne in mind, however, that in the natural structure of mathematics, the definitions and axioms came after the intuitive understanding of the material had been absorbed. Try beginning an explanation to a non-mathematical audience, even engineers and scientists, with a list of definitions and axioms,
and see their glazed eyes. The structure of definition, theorem, and proof exists only in the study of mathematics and is apparently necessary when reporting mathematical results to a professional audience. This structure used in schools, however, often causes harm and contributes to students’ alienation from mathematics. The right way is to start with an intuitive discussion, and when the subject has been presented, at that point it can be shown that it is important to be clear and precise, and only then should the axioms be given. I am aware that presenting the purpose and content of a mathematical subject first on an intuitive level is more difficult than a logical presentation of it, but that is a challenge that mathematics teachers ignore on the grounds that mathematics is taught differently; that is a failure of mathematics teaching.

What position, then, should the foundations of mathematics, that is, the axiomatic and logical structure on which mathematics can be based, occupy in teaching? When it is necessary to present a complete picture of a certain topic in mathematics to experienced professional mathematicians, one can certainly start with definitions and axioms and only afterward deal with applications. If the subject is the general structure of mathematics known today, it may be worthwhile to start with set theory, follow that with natural numbers, and then rational and irrational numbers, as described in section 59. It should be borne in mind, however, that this theory was created simply to establish mathematics on solid fundamentals, and not so that we should better understand how to use mathematics, and certainly not in order for us to teach mathematics better.

For example, for every practical requirement and possible use, it is preferable to introduce the numbers directly, that is, 1, 2, 3, and so on, and to assume that they are something given and clear and do not need to be defined. Similarly, it is easy to explain what a real number is in geometrical terms of distance. The basis of arithmetic and basic fundamentals of geometry are already present in the human brain and should be used. Although comparing sizes, for example determining intuitively which is the larger set, was embedded in the human brain over the course of evolution, this intuitive comparison, which apparently predated counting, did not
develop via one-to-one matching. The human brain works in such a way that it compares new concepts to familiar ones. If natural numbers are taught via sets, or if real numbers are taught by means of Dedekind cuts (see section 58), the mind must “erase” what it already knows about numbers and then absorb the new concept. That is no easy task and is generally not required. To teach a subject while ignoring what the student already knows is a didactic error. This principle is clearly relevant to the study of all subjects, especially learning a language, and is not confined to mathematics. Hence, if the aim is to teach simple arithmetic or calculus, it is worthwhile to use the knowledge the student already has. It would be a pedagogic error to try to teach grade 1 students to count by means of comparing sets; indeed, they already know what numbers are. It would be a mistake to start teaching calculus by presenting real numbers as Dedekind cuts; indeed, the students already know the numbers as means of measuring distances.

That said, there is a better reason for not using sets to describe the natural numbers as sets constructed on a basis of the empty set, or not teaching rational numbers as equivalence classes of pairs of natural numbers, or irrational numbers as Dedekind cuts. Those concepts came into being not to improve the understanding of numbers but to prevent the logical structure of numbers, and thereafter calculus, from being based on geometry. A structure not based on geometry is much more complex and less comprehensible than the classic geometry-based structures. For all the uses that will follow from the meeting of students with these concepts, a geometric structure is sufficient, is more efficient, and is certainly correct from the aspect of logic. When the purpose is to provide the student with mathematical tools to enable him to function in the technological world and also to advance in the world of mathematics, it is advisable that the concepts be based on geometry.

Why, then, did Israel's Ministry of Education decide to base the teaching of numbers on sets? How were teachers and parents persuaded that without sets their children would lack a vital stage in the understanding of mathematics? Why, in the first lecture I heard in university, did the lecturer define the rational numbers as equivalence classes of pairs of whole numbers, and irrational numbers as Dedekind cuts, without explaining the
historical reasons for those structures? Is this lack of perspective of the profession the reason that in various institutions, and even in a book I saw recently intended for engineers, the irrational numbers are presented as Dedekind cuts? Those who teach in this way are apparently convinced that without these “foundations” their teaching would be deficient. They are wrong. Basing numbers on geometry is no less “complete” than basing them on sets.

On a similar subject, why does a college lecturer teach his students that the sequence
a
₁
,
a
₂
,
a
₃
, and so on…of the real numbers is actually a function from the natural numbers to the real numbers? From the aspect of the foundation of mathematics, he is right. If you do not know or cannot understand what a sequence is (and for some reason you do know what a function from the natural to the real numbers is), that is indeed the way to define a sequence. I am not convinced, however, that the lecturer ascertained for himself whether it is at all necessary, and useful, to define a sequence instead of relying on the fact that we all understand intuitively what a sequence is.

Considering set theory as a step that cannot be skipped in trying to understand concepts constitutes a barrier to the teaching of mathematics. We showed this in the previous section when we discussed the teaching of numbers in elementary school. The failure goes deeper than that, however. Not long ago I participated in the entrance examinations for graduate students of mathematics teaching. A graduate of one of the teacher-training colleges said that she gained a great deal from set theory, which she learned in college. We asked her to explain, and she said that now she had a better understanding of the meaning that a child belongs to a particular class. In other words, in her opinion, or in the opinion of her teachers in college, the sentence “Benjamin is a student in Class 2A” is not clear enough. The sentence “Benjamin is a member of the set of children in Class 2A” would apparently clarify the situation. This is a distortion and misrepresentation of the role of mathematics and is harmful indoctrination. It is important to understand the logical structure of mathematics as part of human culture, but it is not right to start teaching by means of that structure. The need to base mathematics on sets arose as a result of the previous foundations
being questioned and followed the lack of precision of the intuition used previously. But the study of the logical structure without understanding the need for it does not help and actually causes harm. The purpose of axioms and definitions is to check intuition and to prevent errors and illusions, even at the cost of complexity that is difficult to understand. They check intuition and do not render it superfluous.

That role is not always clear to teachers and to their teachers. The story of the parallel-lines postulate (section 27), for example, is fascinating as part of understanding human culture, and it also illustrates what mathematics is and the role in it of axioms. A committee appointed recently in Israel's Ministry of Education to review the mathematics curriculum reached the conclusion that the parallel-lines axiom was difficult for Israeli students to understand, as it dealt with infinite lines. The committee proposed replacing the parallel-lines axiom with another one, the rectangle axiom, the details of which do not concern us here. They went even further and put forward a detailed program of relevant geometric studies, in which the rectangle axiom takes the place of the parallel-lines axiom. Again, we are dealing with a failure. First, the difficulty that the members of the committee imagined that students encountered is artificial. “Does infinity exist?” and “If it does, what is it?” are questions for philosophers and mathematicians. Not all philosophical contemplation bothers people who are not philosophers. The average student, or even outstanding ones, will probably not identify a line that extends to infinity as a problem, unless it is explained to them that actually there is a problem of definition. A typical student will have no difficulty understanding the concept of a line that continues to infinity. If there is a problem, it is a philosophical, not intuitive, one. Moreover, the axioms are meant to express a property that is itself a natural, intuitive one, and in my view, reference to parallel lines is more natural than reference to a rectangle. Even if the members of the committee are right that the rectangle axiom is more natural than the parallel-lines one, it should not be adopted for teaching purposes. The story of the parallel-lines postulate is today part of human culture. It is related in countless (I almost wrote
an infinite number of
…) books, and it occupies a prominent position on the Internet. Teaching geometric axioms in schools by substituting another
axiom in place of the parallel-lines axiom would result in the material to be learned being set apart from general culture.

70. WHAT IS HARD IN TEACHING MATHEMATICS?

We wrote in sections 4 and 5 that some subjects in mathematics are compatible with human intuition as it developed over millions of years of evolution, but others offered no advantage in the evolutionary struggle and are contrary to natural intuition. This distinction should be reflected in the methods of teaching and study. Unfortunately, that is not the case. Turning a blind eye to the problem causes conflicts and difficulties.

The following is an imaginary but definitely realistic exchange between a teacher trying to explain that the square root of two is irrational and a student (we discussed the proof in section 7).

Teacher:
We will prove that √2 is irrational. We first assume that that it is rational.

Student:
But how can we assume that it is rational if we want to prove that it is irrational?

Teacher:
Wait a minute and you'll see. Assume that it is rational, and we will write it as √2 =
, where a and
b
are positive integers. We can assume that one of them is odd, because if they were both even, we could divide them by 2 and would continue to do so until at least one, either the numerator or the denominator, is odd.

Student:
I understand all that, so far.