WHAT IS HARD IN TEACHING MATHEMATICS - WHY IS TEACHING AND LEARNING MATHEMATICS SO HARD - The Remarkable Role of Evolution in the Making of Mathematics - Mathematics and the Real World

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?

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 = images, 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.

Teacher: Now we square both sides of the equation, and we get 2 = images, or a2 = 2b2. So a is even, and can be written as 2c.

Student: Fine.

Teacher: Substitute 2c in the previous equation, and we get 4c2 = 2b2. Dividing both sides by 2 leads to the conclusion that b is also even. But we started by justifying the assumption that either a or b must be odd. We have arrived at a contradiction.

Student: So what?

Teacher: The contradiction derived from the assumption that the square root of 2 is rational.

Student: I said at the beginning that we couldn't assume that.

The above was an imaginary conversation, but similar ones take place very often in the study of mathematics. The difficulty that it reflects derives from the fact that the teacher has become accustomed to the argument, may have taught the proof many times, and has simply forgotten the basic difficulty based on proof by contradiction. We should be aware of and remember the obstacles the proof had to overcome among the Pythagoreans, who concealed it for many years for reasons of their own, and it may well be that they did so because the proof is not easy to understand. We should also remember that even in the twentieth century there were those who argued against the method of proof by contradiction. The objections to the method would not have arisen if such proofs were obvious, that is, if they were compatible with intuition. I am not suggesting that young students be exposed to the tradition that views such proofs as problematic, but the recognition that such proofs can be memorized but are difficult to absorb should be reflected in the way they are taught. This difficulty cannot be completely overcome. The best way to deal with it is to separate the explanation of the principle underlying these claims from the proof itself, to be patient, and not to expect the student to arrive at such claims by himself.

The student will encounter similar difficulties in understanding claims containing logical quantifiers such as “all,” “there exists” or “there does not exist.” For example, if the teacher proves that a certain property, say of triangles, holds for all triangles, he cannot expect that henceforth the student will use such knowledge. The lack of accessibility to a fact that has already been proven is not due to the student possibly forgetting that property but is the result of the fact that the claim “all” does not sit naturally in the human brain. The student will face the same difficulty when the teacher uses the third of the classic rules of thought, the rule of the excluded middle (“either P [the proposition] is true or its negation is true”). The use of this rule is not intuitive. The only way to succeed in using it is to isolate it, and to draw attention to it whenever using it, and lessons must be planned accordingly.

One of the central difficulties deriving from the way the brain works is its failure to relate to stipulations or conditions. The mind does not think conditionally and generally does not discover that it lacks data required for an analysis of the situation. Evolution trained us to complete an incomplete picture somehow or other. A delay intended to enable the brain to discover the missing condition could have had dire results for the human race in the course of evolution, as a result of which, today we usually skip that stage. In mathematics the result is likely to lead to errors. I will illustrate with an example from probability studies.

We described the logical approach to probability and difficulties arising from it in sections 40 to 43. We will now see how experienced teachers are exposed to errors. It must be stressed, the example is not given to illustrate the error or to embarrass or belittle the writer of the article we shall analyze. We give the example in detail because we think that, in that way, we can identify the source of the error, which is the difficulty in identifying provisions and the completion of the picture by the brain. We will extend the discussion and go into detail in order to emphasize the refinement required in analyzing such problems.

The title of the article that appeared in a teacher's magazine some years ago was a promising one: “Conditional Probability as a Source of Paradoxes and Surprising Results.” The introduction was in the same vein: “Mathematical thinking is an important means for discovering the world…. Phenomena that seem mysterious and paradoxical are explained rationally…,” and so on. The article itself gave examples of apparent paradoxes, which it then explains rationally, apparently. Here is one of the examples.

Boys and girls. We are given that the probability of the birth of a son is images. A certain family has three children. The example has two stages. The first is that outside their apartment we see two girls of that family. What is the probability that the other child is a boy? The second stage is that in addition to the two girls, we can see the outline of a baby, in the apartment, a younger brother or sister of the two girls we met. What then is the probability that the third child is a boy?

The approach of the article to the answer to the question is flawed. We will show the answer given in the article and will then point out the error. We reemphasize: it is not our purpose to belittle the author of the article. Errors are standard fare in mathematics. Our objective is to indicate the source of the error. First we will show the solution given in the article.

The writer solves the exercise only after stating that many students are convinced that the answer to the first part of the question is images. Their explanation, he claims, is that because of symmetry: the chance that the third child is a boy is equal to the chance that it is a girl. Then the author gives the “right” answer. He assigns the number 1 to a male child, and 0 to a female child. With the help of these symbols we can record eight three-digit permutations that describe all the possible situations for the three-child family:

Ω = {000, 100, 010, 001, 110, 101, 011, 111},

where for example 011 indicates that the first child was a girl, followed by two boys. The article continues along the formal lines we described in section 41. We will use the symbol A for the event in which “the family has exactly one son,” and the symbol B for the situation in which “the family has at least two girls.” We must calculate the conditional probability P(BA), which is images. This is a fairly simple calculation. The article gives two methods, which give the same result. We will give the shorter one here: as it is known that there are two girls in the family, the relevant event, so claims the author, consists of four out of the eight possibilities listed above, and they are {000, 100, 010, 001}. In three of these there is a son in addition to the two daughters, so that the probability is images.

The article then addresses the second part of the question. It would seem, he warns, that in light of the above result, we would expect the probability to be images as well. That is not correct (according to the article). Note: we know that the question relates to the child born last, so that the range of possibilities is just {001, 000}, and the probability is now images, and not images. As stated above, the writer considers this a surprising result, or a paradox, which mathematics clarifies.

The author's approach to the solution is incorrect, and the paradox, or apparent surprise, does not exist. The writer did not notice that we have insufficient data to reach an unequivocal answer to the question. First, the reason that we cannot accept the proposed solution is that the way the question is formulated, the solutions to the two parts contradict each other. If the answer is images when we know that the child in the house is the youngest, then that would be the answer also if the child in the house is the middle child of the three, and that is also the answer if we know that the oldest of the three is in the house. These are three separate cases, and together they exhaust all the possibilities. And if the probability of each of these cases is images, the probability would also be images if we do not know which of the three children is in the house, that is, the youngest, the middle one, or the oldest. Therefore, from the fact that we get images as the correct answer to the second part of the question, we derive that images is also the correct answer to the first part.

Where, then, is the author's mistake? I could make life easy for myself and claim that the writer was misled by the term conditional probability in the title, which is sometimes interpreted in the literature to mean the probability of A when it is known that B holds (see section 40). This is an incorrect interpretation of the concept of conditional probability, as it ignores the question of how it becomes known that B holds. Those who read section 40 will know that to solve problems like these we must use Bayes's thought process. If the author of the article had tried to apply Bayes's scheme, he would have seen that without more information it is impossible to arrive at an unequivocal solution. (We gave an example, the tale of the six competitors in a beauty contest, in section 42. A similar example was also “solved” incorrectly in the article.) As we have stated, if data are lacking, the brain of the person solving the problem supplies what is missing itself, generally unknowingly. The problem is that completing the picture in different ways yields different results.

We will put forward three different versions of how the picture can be completed, that is, how the missing information can be supplied, each of which yields a different answer. In the first, assume that in a family with three children, two, chosen at random, go outside to play. Then the solution to both parts of the question is images (we skip the calculations). In the second version, assume that among the families in that neighborhood children always go out to play with another child of the same sex, meaning boys play with boys, and girls with girls, and if there are three boys or three girls in a family, the two older ones go out to play together. In that case, in the first part of the question the probability that the child who stayed in the house is a boy is images, and the probability in the second part of the question is images, as was claimed in the quoted article (again we skip the calculation, but note that here the calculation quoted in the article is correct). The third possibility is that two children of the same sex always go out to play together, and if there are three of the same sex, then the two who go outside to play are chosen randomly. In that case, in both parts of the question the probability that the child in the house is a boy is images. Thus we see that the information provided in the question can be completed in different ways, which all give different answers to the questions. What is the correct way to fill in the missing data? The formulation of the question does not provide an answer to that. The use that the writer made of the mathematical formula incorporates an unstated assumption about the question itself (for example, one of the assumptions we listed above), an assumption that does not appear in the formulation of the question. That is why I chose my words advisedly and said that the author's approach to the question was wrong, and not that the numerical answer in the article is wrong. (Although, as the writer related only to equations, it is hard to believe that he considered the various possibilities. Herein lies an important lesson for anyone using mathematics: do not use formulae before checking that they are relevant to the situation at hand.)

As we have said, the reason for the error is rooted in the fact that thinking under provisions is not a natural process for the human brain. So much so that the author of the article did not recognize it and thus also did not identify the source of the contradictory results he himself obtained. He preferred to think of the discomfort caused by intuition as a paradox, a paradox that formalism can explain. As we shall see, in this he is not alone.

Possibly the most famous instance of lack of clarity and unknowing completion of information is known from Let's Make a Deal, a television game show in which the competitors are given a mental challenge. The following is an exact formulation of the question:

You are a participant in the game. You are shown three closed doors. You are told that behind one door there is a big prize, while behind each of the other two is a goat. You are invited to select a door. Before it is opened to reveal the big prize or a goat, the host, who knows what is behind each door, opens one of the other doors, and reveals a goat. He now offers you a chance to change your mind and select the other door. Is it worth your while to change your selection, that is, to choose the door that the host did not open?

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.