Mathematics and the Real World (64 page)

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.

67. MATHEMATICAL THINKING: THERE IS NO SUCH THING

The market is swamped with books, courses, advanced-study courses, and similar tempting offers that promise to teach and improve our and our children's mathematical thinking. All you need to do is to agree to pay the stated sum, and your thinking will improve and will become proper mathematical thinking. Are there any parents who would refuse to make the effort and pay the required amounts and prevent their children from benefiting from the opportunity to think mathematically? Without mathematical thinking it would be hard to function in the modern world. (Just to remove any possible doubt, the above is written in irony.) This went so far that one newspaper published an article about courses in mathematical thinking for two- and three-year-old children. I searched the Internet for the term
mathematical thinking
together with the word
classes
, and found that extracurricular classes for first graders and higher grades headed the list, with classes also for three-year-olds up to kindergarten age that included
“mathematics as a mother tongue.” Parents in one of the schools in the central part of Israel complained to the teachers and the headmistress that the school did not offer extracurricular courses in mathematical thinking. Our children are being left behind, they argued. The institute where I work and other institutes of higher learning offer courses for youngsters on mathematical thinking. The truth is,
there is no such thing as mathematical thinking
. If that is the case, you may ask, what is taught in all those classes and courses? The answer is, they simply teach certain mathematical subjects, generally the more logical parts of mathematics, as well as tricks for solving problems, most of which are called problems of logic.

At this point a number of questions come to mind, which we shall consider one by one. What does “there is no such thing as mathematical thinking” mean? What is wrong with calling the activity that takes place in these classes and courses mathematical thinking? Does participation in such classes cause any harm? Are they helpful in any way?

In various sections earlier in this book we discussed ways of thinking, in particular in section 62. We stated that thinking is mainly an activity of the brain intended to analyze situations and to make decisions. A rough-and-ready distinction divides that activity into two types of thinking. One is comparative thinking, in which people compare the situation they are facing to other similar situations with known solutions. The second type of thinking requires a new, more creative response to an unfamiliar situation. There too the brain uses the range of reactions that it knows, but it has to adapt and update them as necessary, and sometimes it will have to come up with a new and creative response. These two elements of thinking are
common to all disciplines
and are not confined to mathematics. These types of thinking cannot be taught. The more you know and the more experience you have in the subject you are thinking about, the more efficient your comparative thinking will be and the better your reactions. The more you know, the more efficient your creativity will be too. It is thus worthwhile to learn a variety of subjects in order to enrich your pool of knowledge and experience, and to help enhance your thinking. The same applies to mathematics. The more you learn and practice, the more you will succeed. This rule is valid for all subjects; you do not learn thinking, but content.

Friends who teach mathematics ask me why I am so pedantic. They say that the public calls these subjects, that is, exercises in logical problems, and so on, “mathematical thinking”; they say it is just a matter of what it is called, not of essence. Those who make this claim are the victims of the mathematics profession, and they are ignoring the effect of language on learning. Let me explain: it is the accepted norm in mathematics to give names, that is, to make up names, for new mathematical entities or specific operations. We have come across the terms group and field earlier in the book. Trees, matrices, manifolds, automata, and machines also appear in mathematics. The motive for giving a new and inclusive name to a characteristic, or to a certain collection, is clear. As soon as we have given a mathematical name to something, and we have absorbed what underlies that name, it can be referred to without having to explain from scratch what we are talking about. Giving a name is not arbitrary (a fact that is not clarified sufficiently in mathematics lessons, even in higher education). No authority can forbid you from giving any name you choose to a mathematical concept. Instead of a group, you could call it an elephant, or you could give the name Moses to what is generally called a mathematical tree, or you could concoct a name consisting of meaningless syllables. Nevertheless, that is not the practice in mathematics. A mathematical tree to some extent brings to mind a tree, and the word
group
was chosen because it refers to a collection of entities and the relations between them, and likewise with other names. Mental processes are associative. If we were to call a mathematical tree a steamboat, or replace the term
function
with the word
elephant
, it would disturb the users of the terms because the names would probably conjure up in their minds images of rivers or zoos. It is true that even if there is a good reason for choosing a particular name, it could still cause confusion. A mathematical tree is not a botanical tree. The term would not confuse a mathematics graduate but could well confuse someone not involved in mathematics. (The story is told that after the publication of an important paper by the mathematician Michael Rabin on automata on trees, he was invited to give a lecture on the subject by a faculty of agriculture!) Hence the importance of choosing a nonarbitrary name that will not mislead, that is, a name that would allow the relevant intuition regarding the concept to develop.

Those with higher degrees in mathematics are sufficiently well trained so that when certain subjects in mathematics are called “mathematical thinking” they will repress the intuitive meaning of the term “thinking.” This apparently also applies to those who offer courses in mathematical thinking. To the general public, however, the concept of thinking means more than the learning of content. Using the term
mathematical thinking
for certain aspects of mathematics or logic is misleading, or at least an error, especially if there is a hint that the course offers more than just additional (and very partial) knowledge in mathematics. This is also a matter of raising unrealistic hopes. The result of the disappointment due to their non-realization can cause real harm: if in the future children or their parents discover that they have difficulty in a certain topic in mathematics, they are likely to conclude that they are suffering from some disability. After all, they are graduates of a course that is not a vague general course in mathematics, but a course in mathematical thinking!

If we ignore the deception, we could ask whether there is something wrong in itself in learning some parts of mathematics, puzzles in logic, and various mathematical tricks. Of course there is nothing wrong in learning, per se; it depends on the parents’ economic situation, the alternative cost of a babysitter, and the amount of pleasure the child (not the parents) gains from participating. There is something wrong, however, in presenting this mathematics as if it should guide daily conduct outside the classroom. Presenting “mathematical thinking” as a necessity, or as beneficial to life beyond mathematics and its applications, is likely to cause harm. Some time ago I met a teacher in one of the well-known secondary schools with a good reputation for its teaching of mathematics and known for preparing its outstanding students for the Mathematics Olympiad. That teacher was really angry with me when he heard the views that I am expressing here on mathematical thinking. He claimed that he was more than a mathematics teacher, rather, he prepared his students to think correctly in all areas of life. He argued that they must examine each step they take by means of mathematical thinking. Poor students, I thought to myself. Just like the centipede that was strolling along quite comfortably until it was asked: How do you manage to coordinate your steps so that your right leg number
23 and left leg number 12 are forward while left leg number 17 and right leg 19 are pointing backward? The creature stopped in its tracks and could not move any of its legs. In our daily lives there is no time for a logical analysis of every step we take. We have to use our intuition, and it is certainly acceptable, and sometimes even preferable, for mistakes to occur. Mathematics and logic do not help, and are likely even to hinder, if they are used as a tool to manage our lives.

Is there then any benefit in participating in classes for mathematical thinking? Again, that depends on the alternatives. If the choice is between that and watching a so-called reality show on television, the mathematics class would seem to be preferable. If the alternative is a theater group or literary circle, or an athletics or football team, the choice will depend on the child's preferences or what is better for his physical or mental wellbeing.

Such enrichment courses may actually cause indirect harm, as they are likely to give the impression that exercises of that type reflect the whole of mathematics, and they pressurize the outstanding students to advance as quickly as possible. This striving for excellence sometimes results in the student learning advanced material too soon, before he or she is ready for it. A child who excels in such groups is sometimes sent to university to participate in special courses for outstanding students. Some of the students are mature enough for such courses, but for others (the majority, in fact), the early excellence is in too narrow a field, and the push to advanced studies at that early stage is harmful. These students do not absorb the required material properly and form a misguided picture of mathematics not because they are less talented, but because they started before they possessed the necessary degree of maturity. Recently I participated in a number of interviews with candidates for graduate studies in the academic institute where I work; these candidates had obtained their bachelor degrees at a very early age in one of the special programs for outstanding students. For some of them, the programs had destroyed their chances for a real scientific career; more's the pity.

Here is another example illustrating the importance of accurate terminology. It was Galileo who came up with the saying that mathematics is the language of nature. That is a lovely analogy, but nothing more. “Language”
in that sense is not a means of communication as we understand the word in the context of interpersonal expression. A committee appointed by the Ministry of Education in Israel headed by the scientist and educator Haim Harari used the metaphor of mathematics as a language when it recommended that mathematics teaching in schools needed greater emphasis and reinforcement. From that point it was apparently only a short step to the message from the principal of a well-known school to new students and parents that three foreign languages were taught in his school, English, French, and mathematics! Calling mathematics a foreign language is simply inappropriate. To me, that is similar to the promise given by the organizers of those courses I have spoken of, that they undertake to impart mathematics to children of three to six years of age to a mother-tongue level. That would be fine if all they meant was that the children would be a bit more familiar with mathematics, although I do not see the advantage of special lessons specifically in that subject. If, however, underlying the term
mother tongue
is a promise that the children would gain intuitive insight into mathematics in the way that we absorb a mother tongue and use it, then I am worried.

68. A TEACHER-PARENT MEETING

Here is a description of a parent-teacher meeting in which I took part, this time as a parent of a child in first grade. The description is quite accurate. The meeting took place many years ago, and the specific curriculum that we discussed in the meeting has changed since then, but the problems that arose in the meeting still apply. I should add that I did not play an active part in the discussion, nor did I reveal that I was a mathematician.