Mathematics and the Real World (69 page)

This problem has given rise to a huge volume of words, discussions, and arguments that turned into insults and abuse. The Internet is swamped with material on this subject. The problem also appears in textbooks, usually showing a solution, with the problem formulated similarly to our formulation, in most cases without indicating that the formulation lacks something. What is missing is the answer to the question whether the host was
obliged
to open one of the doors you did not select. If he is obliged to do so, it is not hard to show that you will not lose and may even gain by switching your choice of door (to know how much your chance of winning increases, you need to know how the host chooses which door to open). If he is not obliged to open a door, the right answer depends on the host's intentions, and these do not appear in the question. If, for example, he opens a door with a goat behind it only if you have chosen the door with the big prize, and he does that only to fool you, then it is not worthwhile for you to change your choice. (People approached Monty Hall, the host on the show, and asked if he was obliged to open a door, and his answer was that he doesn't remember.) Here the lack of clarity in ordinary language is revealed. Many people are convinced that the wording of the question shows that the host must open a door. Others disagree. Most books ignore this point, and the authors simply assume that the interpretation in their minds is the same as their readers’ interpretation.

Teachers and their lecturers in teacher-training colleges ignore the fact that when we teach mathematics we use natural language, which is not subject to the laws of mathematical logic. One of the lecturers at a conference convened to determine what mathematics teachers must know was an expert, a professor of mathematics education. He complained about the lack of precision in the use of the language of mathematics among students and their teachers and claimed that a teacher must teach how to correct these errors. The many examples he brought included a quote of a definition from a textbook, which stated that an even number is a number whose units digit is one of the digits 0, 2, 4, 6, and 8. It was clear from the way
he made this point that he thought everyone in the audience would spot the mistake, whereas I could not see any lack of precision in that definition. The professor went on to insist that the definition lacked precision because it would lead to classifying the number 26.5, say, as an even number. In presenting the issue, and in the discussion itself, he did not explain that when he referred to a number, he meant a real number or a decimal number. In that case, the definition clearly is not right. Yet when I heard the word
number
, what came to mind was whole numbers, so that I did not see an error in the definition. I do not know the background or framework of the book the professor was quoting from and criticizing for imprecision, or its target readership, but most of the examples he gave suffered from the same type of obscurity or lack of clarity. Apparently, he had not imbibed the fact that in describing mathematics we use natural language, and there is
no way
of escaping the lack of accuracy of spoken language.

Herein lies the difficulty in teaching logic-based subjects. Logical analysis does not allow lack of clarity, but living language is based to a large extent on intuition and structural imprecision deriving from the desire for efficiency. In our daily lives we deal with this one way or another, mainly by ignoring the meticulousness of logic. A father warning his son “If you don't eat the banana, I'll punish you,” in some way promises that if his son does eat the banana, he will not be punished, although from the strictly logical aspect he has not made any such commitment. In mathematics lessons, logical precision cannot be ignored. How do we reconcile these two facts? We go back to probability. Some of my colleagues claim that the logic underlying probability theory is so far removed from intuition that it would be preferable to remove it totally from the secondary-school curriculum. I am more optimistic. We can teach, and it is important that we do teach, the logical foundations of mathematics, including the logic of probability theory. But it is vital to understand that it is a difficult subject, and it cannot be imparted intuitively. The teachers too, and their teachers in college, will be exposed to the risk of making mistakes if they do not first clarify the roots of the logical structure of every new exercise or problem in probability they encounter. Mathematics is a combination of an intuitive approach and logical considerations. Awareness of the logical aspects, and
the fact that they must be treated differently than the subjects that are consistent with the wealth of material in our brains, is the first step in correct the teaching of mathematics. The lesson plan must be tailored in accordance with this inherent conflict.

Having said that, intuition should be used to teach and advance the appropriate parts of mathematics. For example, in section 4 we mentioned the sequence 4, 14, 23, 34, 42, 50, 59,…the natural extension of which is 72, as the numbers are the street numbers at which the subway in Manhattan has a station. The mathematician Morris Kline, whose critical book on mathematics teaching is very instructive, quoted this as an example of teaching without any logical basis, like the whole subject of finding the extension of a sequence. Certainly, the question is not appropriate or relevant to nonresidents of New York City, but searching for patterns is deeply embedded in human intuition, and this property served mankind in a way whose importance cannot be overstated. This property is a cornerstone of mathematics research itself and, as such, is respectable mathematics that should be encouraged and practiced, even if the extension of the sequence is not derived by pure logic. Similarly, we can and should take advantage of intuition relating to numbers that students can develop easily. A sense of numbers is innate in human nature, and it should be exploited, but we must be aware that there is no chance that schoolchildren, or indeed anyone, will develop a feeling or an intuition for logical operations, mathematical symbols, or other abstract systems without their being rooted in and backed by arithmetic or geometry. Teachers must also be alert to this, and lesson plans drawn up accordingly.

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.