Mathematics and the Real World (65 page)

The parents had been invited by the teachers of that grade to talk about the new mathematics textbooks the children would be using that year. The teacher who opened the meeting explained that the material in the books was new, and even the teachers were not very familiar with it. Hence, a senior teacher from the Ministry of Education, named Batya, had been invited to present and to explain to the parents what their children would
experience in the mathematics lessons during that year. After a few polite introductory words, greetings, and the like, the guest speaker started.

“It has recently been discovered how prehistoric man developed mathematics.” At this one of the parents was heard to murmur, “I'd like to know just how it was discovered. Did they find a book left by prehistoric man?” It was said at a stage-whisper level loud enough for others to hear, but I do not know if Batya heard. In any event, she did not react.

She continued: “Once there was a cruel tribal chief who every morning would send one of the children of the tribe to herd the sheep. At the end of the day, when the young shepherd brought the flock back, the chief would beat him severely, claiming that he had lost some sheep. One of the children, called Ogbu” (here Batya clarified, apparently to add credibility to her story, that perhaps that wasn't really the child's name) “was very clever and found a way to avoid being beaten. Every morning, when he took charge of the sheep, he gave the chief a stone for each sheep. When he brought them back, he would take one stone back for each sheep. When the chief realized he had no stones left, he understood that all the sheep were back, and he didn't beat Ogbu.”

My reaction to the story was that it was not very convincing. Every prehistoric tribal chief that I knew would have beaten Ogbu regardless of whether he was left with any stones or not. Moreover, I did not know a single tribal chief, cruel or kind, from prehistoric times until today, who would accept the idea put forward by a clever child (or a clever adult for that matter) for an innovative process of stones representing sheep to prove that no sheep had been lost. Ogbu would have been beaten just for his cheek in making the suggestion. I kept my thoughts to myself.

Batya went on: “Thus, Ogbu's clever idea led to the development over the years of the concept of a one-to-one correspondence (between stones and sheep in prehistoric times), and the correspondence between a set and a subset of another set, thereby laying the foundations for the natural numbers as we know them, and hence to addition and subtraction, and so on.” I thought to myself, what a pity that the chief had enough stones to give back to Ogbu. If there had been a few short, that would clearly have led to the discovery of negative numbers thousands of years earlier than their actual discovery.

We will skip the questions parents asked. The approach and the material were new to them. Toward the end of her talk, Batya claimed that once it became clear how prehistoric man had developed the concept of numbers based on sets and matching them, we can use that today and teach the basic concepts better so that children will understand and imbibe the right concepts and will not just do arithmetic exercises mechanically. She went further and said, “The purpose of learning is to understand. It is not so important that the calculations come out exact. Understanding is the main thing.”

One of the parents nevertheless asked how does one check understanding if the calculations are not exact. Batya did not have an answer. Another parent suggested, probably sarcastically, that if a child hesitates over the question how much is three plus four, he may well understand, while the child who immediately answers correctly is presumably answering automatically and does not really understand how to add. Another frustrated father banged the table and declared loudly that he hadn't understood much of what Batya had said (he said this in somewhat less diplomatic language), but what he wanted was that when he sent his daughter to buy a newspaper, she should come back with the right change, regardless of what she “understood.” It took some time before a calm atmosphere was restored, and the meeting broke up with the general feeling “We will see what happens.”

The children used the new textbooks throughout the year, and they and their parents were indeed deeply confused by them. The fun that the books offered the children when they were asked to match collections of items, generally sets of animals or flowers, was a marvelous opportunity to fill the pages with beautiful colors, each child according to his artistic ability. However, when the book tried to move on to develop natural numbers, several mental obstacles were encountered.

For example: one day my son came home with some friends and told me that the teacher had handed out blocks in the classroom, one block per child. Then the blocks were collected, and put into a bag, and the class went to visit the class next door. They gave the blocks out to the children in that class, and found that they did not have enough. The conclusion was,
they explained, that there were more children in the second class than in their class. Obviously the purpose was to illustrate a one-to-one correspondence before the children learned the numbers themselves. In all innocence (or not) I asked them how they knew that there were more children in the second class. Perhaps some blocks had fallen out of the bag. “Oh, no,” they answered together, “they have thirty-two children in their class, and we have only thirty-one!”

Another problem came to light toward the Hanukkah holiday in December, by which time the children had learned, needless to say on a fundamental level, the numbers from one to four. But on Hanukkah, the Festival of Lights, we are supposed to light eight candles. The creative solution proposed was that eight was a guest, a visiting number, who left immediately after the festival. Of course nearly all the children were familiar with and could use much larger numbers. But the teachers made them repress that knowledge because of the Ministry of Education indoctrination, according to which there was something defective in the children's intuitive understanding. It is clear there is a defect somewhere, but not with the children.

At a later stage, the teacher had to present the whole system of numbers, including the law of commutation, that is, the rule
a
+
b
=
b
+
a
holds for all two numbers. But this property was already known, so why was it necessary to formulate a law? It took about two months to persuade the children that the commutative property was not self-evident and that there were other possibilities. For example, the order according to which first we put our socks on and then our shoes is important, and so that is not a commutative operation. When the whole class was convinced that it is not self-evident that
a
+
b
=
b
+
a
, then the punch line was delivered: nevertheless, the equality does hold. Why? The answer to that question was not given in the class. In the background material provided for the teacher, it states that it is an axiom. It also says there that the concept of equality is not obvious, and that it is also an axiom. We repeat: teachers are simply told that
a
=
a
is an axiom, just like that; that is the “support” the teachers get from the Ministry of Education to help them understand the new teaching program.

These and similar instances made me realize how the following incident
could occur. It happened before the school year had begun. My son was due to start first grade and had to meet the staff for them to assess him to be sure that he was at the right level. At home he would amuse himself (without encouragement from his parents) by counting forward and backward, 1, 2, 3,…and 9, 8, 7, and so on. The principal was the head of the assessment team, and one of her first questions was what number comes after six. My son answered straight away that it depends in which direction you are counting. It was clear from the expression on her face that she had doubts as to whether the child was ready for school, until I stepped in and explained the background and what my son meant. The principal (who knew my profession) said, “You mathematicians have to clarify for us the essence of the numbers.” When I answered that numbers have no essence other than counting, she reacted as if I were belittling or mocking her. That was not the place or time to explain to her that the “mockery” was coming from another source.

That incident occurred a long time ago, but the strange system of confusing teachers by making them look for something in mathematics that is not there still continues. In a conference devoted to mathematics teaching, the lecturers complained recently that the students, very young children, did not understand the essence of a triangle. I confess, for me a triangle is a triangle, and I am not aware of any other hidden essence. When teachers and their instructors in teacher-training colleges try to achieve a situation where students will understand the essence of a triangle, they mean some sort of essence, a collection of properties, that they have decided upon arbitrarily. In fact, they want the children to take on the opinion of those who have made that arbitrary decision. Perhaps teachers should be required to impart to their young students the essence of a house or a bus? Why does the idea of “essence” arise, in the education of youngsters, only in mathematics?

I am pleased to say that some of the examples quoted here have been generally recognized and acknowledged as defects and are no longer part of the current curriculum. One-to-one correspondence, for example, has been taken out of most first-grade textbooks. Unfortunately the shortcomings
that resulted in absurdities remain and have spread beyond the confines of elementary schools. One of the main failures is the result of the freedom that the curriculum developers take in defining mathematical content and approaches that they consider offer better content or are more instructive, regardless of material that the children or parents already know. One of the reasons is apparently that it is acceptable in mathematical research to present and use new definitions, and even new systems, as necessary. The difference is that in advanced research this is done carefully and in moderation, and in a manner appropriate to the researcher's target audience, while in school, teaching the target audience is not taken into consideration. The outcome is that parents cannot understand the textbooks of their six- and eight-year-old children. A book was published recently intended for mathematics teachers, with the purpose of placing the background to the pedagogic material on a sound basis. Professors of mathematics to whom I showed the first exercises in the book did not even understand the questions! To be able to answer them, one apparently had first to learn the terms that the authors themselves had invented. That is absurd. The damage caused by such alienation far outweighs any benefit of the new system, which in many areas is rather suspect.

At about the same time that I became aware of the above problems in teaching, a solution also came into my mind: maybe it is not worthwhile to teach mathematics formally in the low grades. I suggested to the supervisor of mathematics teaching in the Ministry of Education that he should try a system in which no formal lessons are given to the first three grades. The teachers could teach and give exercises in subjects like counting, addition, subtraction, multiplication, and the like, by means of contact with the real world, games, stories, and so on. This would rely on what the child had already encountered and recognized, and apparently absorbed, without compulsory lessons. The supervisor loved the idea and suggested it to a well-known school in Jerusalem. And then the idea was rejected by the parents. They were concerned that the delay in learning the basics would have an adverse effect on their children's future understanding of mathematics.

Several years later I learned that I did not pioneer that idea. In 1929 the superintendent of schools in Manchester, New Hampshire, Louis P. Bénézet, recorded a broad experiment covering mathematics teaching in the six grades of elementary school. The studies/lessons were conducted without any formal material, books, and the like. Instead, the teacher had to make the children aware of the connection to what they needed to learn, such as counting, numbers, estimating sizes, calculations, geometry, and so on, by means of games and trial and error. Occasionally the need would arise to present a mathematical symbol and the formal method of derivation, but then that was done only in the context of what had come up in practice, with a mixture of intuition and formalism. After seven years of the experimental method, comprehensive tests were carried out, and the achievements of the students who had participated were compared with those of schools that had continued with the old method. The unequivocal result was that students who had learned with the Bénézet method performed better. I do not know what happened following the experiment. In general, one can always question the validity of tests comparing the results of an innovative teaching method that attracts teachers’ and supervisors’ attention and boosts the efforts they invest with those of the commonplace system, which the teachers are already tired of using. The basic insight obtained, however, that understanding mathematics is not achieved via formal logic, is still valid today.

69. A LOGICAL STRUCTURE VIS-À-VIS A STRUCTURE FOR TEACHING

Many parts of mathematics consist of layer upon layer, that is, they are derived from previous developments, which leads to the belief that in mathematics, unlike in many other disciplines, you cannot advance to the next level without mastering the one you are on currently. That belief is true to some extent, but before drawing conclusions from it relevant to teaching, one must understand the stages on which mathematics is constructed. First, one must realize that there is a difference between the levels
of mathematics as a complete structure of logic and the levels through which one understands mathematics, according to which it must be taught. The lack of realization of this difference is the cause of several of the difficulties in mathematics teaching.