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Kaplan's circle
Posted:
May 26, 1997 8:13 AM
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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 toom@universe.iwctx.edu
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
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