A few months ago, in San Diego Susan Addington acquainted me with Robert and Ellen Kaplan who teach a mathematical circle in Boston. I thank Susan very much. So, when I planned a trip to Boston, I sent a message to Kaplan saying that I would like to attend his circle. I have just returned from there. Here are my impressions. Usually children are separated into three groups: beginners, middle and advanced, but this time was beyond the program (arranged especially for me). Only some students came, so all of them (about 20-30) were in one room plus Kaplan's wife Ellen and a couple of parents. First Robert discussed the possibility to cut a rectangle into squares. It was immediately noticed that it is easy to do if the squares are allowed to be equal or there is just one square. Then Robert concentrated attention on the case when there is more than one square and all squares are different. Students said something from their places, Robert standing before the blackboard and putting everything together. The most active were students around 14 years old, although some others were older and some were younger. After some efforts the active students said that they thought it was impossible. The argumentation involved the smallest square and it seemed proved that the smallest square could not be adjacent to the border. Then Robert proposed to consider the case when the smallest square is surrounded by bigger ones and distributed a scheme in which lengths of sides still needed to be evaluated. He wrote several equations on the board and showed that they had a solution, thus showing that a rectangle could be cut into unequal squares. Then he discussed some similar problems, for example he easily showed that a parallelepiped cannot be cut into unequal cubes. All this time I was sitting in a corner watching students from behind. I liked the circle very much (I knew in advance that I would like it, that is why I wanted to visit it), but I did not like that students made no notes and that Robert wrote and solved the equations by himself. Perhaps, it is different at regular classes. Then there was a break with cookies and drinks brought by Ellen. I used this time to talk with a boy who was sitting just before me. He seemed the youngest and did not say a word thoroughout the class and I was afraid that he was bored. But he was not. He told me (answering my questions) that he was 11 years old, that he liked all this and that he would attend this circle in the next year and that his father was a chemical engineer. Another student who attracted my attention also was among the youngest ones: it was a shy black girl who also said nothing during class. She told me that she was 11, that she liked it and would attend it the next year. But after that she avoided talking to me (fearing that I might question the appropriateness of her presence there, as I think), and sat in the midst of other students besides a bigger girl. After the break Robert proposed me to do something. I chose two themes: First, I `proved' that 64=65 by cutting a 8x8 square into 4 parts and making a 13x5 reactangle of these parts. I had used this in the most advanced classes I taught in a university in San Antonio, and every time it was hard (and useful) for my students to figure this out. This time somebody immediately said what was the matter, so the full explanation took only 15 minutes. Then I proved that all triangles are isosceles. This time nobody knew in advance what was the matter, so I enjoyed bamboozling the students about 10 minutes successfully answering their questions, but at last somebody noticed that I failed to consider one case, and then I explained everything. Overall my impression of Kaplan's circle is excellent. (I knew in advance that it would be). Two of students were Russians. I spoke to one of their fathers, Misha Malyutov, whom I had known in Moscow. Maliutov told me that he taught as Northeastern University (which provided place for the circle) and that he taught his class for graduate students in Russian because all of them knew it.
After that we had lunch and discussed math. education. Kaplan said that all children should be taught math. in the style in which he taught his circle, I replied that in the present civilization it is impossible and that I would be very happy if everybody would be taught to solve word problems. I also observed that the `reform' people, e.g. authors of NCTM `standards' have a very poor idea of human motivation: they seem to think that students can be interested only in such problems which provide some immediate material gain, mostly problems about bank accounts and pizzas. Ellen answered that, according to her observations, students take offence when they are taught in this way. I think that she is right and that it is important.
Andre Toom Department of Mathematics firstname.lastname@example.org University of the Incarnate Word Tel. 210-646-0500 (h) 4301 Broadway 210-829-3170 (o) San Antonio, Texas 78209-6318 Fax 210-829-3153