THE "MODERN MATH" EPOCH IN THE UNITED STATES.
W. W. Sawyer
My presence in U.S.A. at this time came about in the following way. In 1954 Professor Tucker, head of mathematics at Princeton, came to see me in New Zealand. he said that my Penguin Prelude to mathematics gave an excellent account of the spirit of modern mathematics amd a group of mathematicians, who were concerned to make modern mathematics more widely known in the States, invited me to come and work there. As a result of this, I went and was appointed Associate Professor of Mathematics at the University of Illinois, where a group of teachers under the leadership of Max Bebermann were preparing a set of high school textbooks embodying the new outlook. They were behind schedule in this and hoped that, as a writer, I would help them to catch up quickly.
The existing situation in mathematics education at that time was extremely unsatisfactory. I cannot produce documemtary evidence in support of the following account, as I would have to do if I were writing a thesis on this period, but what follows represents the general view of the situation that peiole had at that time. Everything that I heard or observed was in full agreement with what they said.
It seems that a group of professors of education held the view that there was a general theory of communication; if you understood this, it was unnecessary to know any more about the subject you were teaching. There were a few professors of education, known by a name something like "Content Men", who believed that you did need to know the subject you were teaching, but they had little influence. Their weird beliefs did not prevent the educationists from having an excellent understanding of the way to manipulate the political system; in many States the educationists secured legislation which insisted that teacher training must be done in accordance with their views. The effect of this was to provide jobs for educationists and to ensure that most teachers of mathematics knew nothing of their subject.
The writers of school textbooks on mathematics produced special copies for teachers, in which the correct answers were written in red ink under the questions. These books were the teacher's bible. If a student asked some question such as "Why is minus one times minus one equal to plus one?" the teacher would reply, "If you don't keep the rules, you will not get the right answer." No one seems to have considered the question where these rules came from or the means by which they had become known to mankind. Teaching by rote inevitably became universal.
There were thus many things in mathematics teaching that called for change, but the mathematicians did not seem concerned with these. Max Bebermann, who, as mentioned above, was the leader of the group producing new material, seems to have been swamped by a flood of mathematicians telling him what points to stress. These all seemed to arise from their experience of teaching in graduate school; terminology and symbolism needed or subtle distinctions that were important at that level. Max Bebermann was a good teacher but was in no position to question what the mathematicians told him. Polya's description of the situation was, "He is a good man who has fallen among logicians."
In this way it came about that in the book the Illinois group were writing, at the stage in algebra where one would expect some sort of explanaion of how the letter "x" came to be related to arithmetic, there was a long disquisition on the difference between use and mention. Students were told to distinguish between the statement there is milk on the blackboard and the statement there is "milk" on the blackboard, the latter referring to the word "milk" written there, and the former to the actual substance spilt on the board. I cannot see any rational explanation why this should be considered at this stage. I have wondered whether it comes from Goedel's theorem, where numbers are used to encode mathematics and one has to distinguish between the actual number and the message it is carrying. Whatever the explanation, it clearly reflects something arising in graduate mathemaics and entirely inappropriate in high shool teaching.
This obsession with advanced topics was widespread. My first summer in America I was asked if I would like to teach modern algebra to a summer school of secondary school teachers. I said I should very much like to do so. I assumed of course that those attending a course on modern higher algebra would have a good grip on ancient lower algebra. This was far from being the case. One man in the course was the head of mathematics in a high school. He could not solve a quadratic equation, and he could not solve a linear equation if it was not put in the form he he was used to. There was one woman in the class who was competent in elementary algebra. She had been at school in Canada, which educationist legislation had not reached. The impression this experience made on me was that the Americans had simply gone mad.
Numbers and Numerals.
Several of the mathematical reformers were dissatisfied with the way teachers thought about numbers; they said that teachers thought numbers were the symbols used to represent them. It was therefore stressed that the distinction should between numbers and numerals should be stressed. I remember a passage which Max Bebermann produced to point this point across. He invented a mythical schoolboy who answered a serious of questions wrongly. For instance, when asked to write a number larger than 2 he wrote an enlarged numeral 2. when asked to write half of 8, he wrote 3, this being the right-hand half of the numeral 8. The story does not sound vary plausible, but in any case with so much needing to be explained, this particular distinction did not seem to have a high priority.
Escaping from Rote Learning.
What did need to be done was to take the material that was actually being taught in the classroom and show one could see and understand its meaning. In fact the teachers wanted this and one could get an extremely positive reaction by providing it. Once, when speaking to a large meeting of elementary school teachers in Connecticut, I gave a very simple way of seeing the properties of even and odd numbers by considering the packing of eggs. An even number of eggs fit nicely into a rectangular case
If it were required to adapt this type of container to take an odd numbers of eggs (which of course is never done)
the case would have to grow a projection to accomodate the extra egg. The properties of even and odd numbers now become intuitive. If rectangles for two even numbers are juxtaposed we the result is a rectangle. If we put a rectangle with the odd shape, the combined shape is odd. if two odd shapes are combined a rectangle can be obtained by rotating one of them.
Surely there is nothing very profound here but the effect on the teachers was extraordinary. The atmosphere became almost that of a religious revival. They had never before seen a mathematical result. They asked, "Why did nobody ever tell us this before?" There was a very real desire for enlightenment which could have led to far-reaching improvements. Unfortunately there were powerful currents streaming in quite a different direction.
So far the events were purely local. It became clear to me from my first view of their writing
that their approach was the exact opposite of how I believed mathematics should be taught. They were disappointed
that I could not sit down and rapidly write the volumes they were behind schedule. I was in the uncomfortable position
of having to oppose those who had invited me to America.
Copyright © W. W. Sawyer & Mark Alder 2000
Version: 22nd March 2001