Thinking and Deciding

24 WHAT IS THINKING?
In contrast to these students, Wertheimer reported other cases of real understanding,
some in children much younger than those in the class just described. Most of
the time, these children solved the problem for themselves rather than having the
formula explained to them. They figured out for themselves that the parallelogram
could be converted into a rectangle of the same area without changing b or h. One
child bent the parallelogram into a belt and then made it into a rectangle by cutting
straight across the middle. Wertheimer does not insist that understanding arises only
from personal problem solving, but he implies that there is a connection. When one
solves a problem oneself, one usually understands why the solution is the solution.
Learning with understanding is not the same as discovery by induction. Wertheimer
(pp. 27–28) made this philosophical point concretely by giving the class the
values of a (the side other than the base), b, and h from each of the following problems
(with parallelograms drawn) and asking the students to compute the area of
each parallelogram:
Wertheimer describes what happened:
a b h Area to be computed
2.5 5 1.5 7.5
2.0 10 1.2 12.0
20.0 1 1/3 16.0 21 1/3
15.0 1 7/8 9.0 16 7/8
The pupils worked at the problems, experiencing a certain amount of
difficulty with the multiplication.
Suddenly a boy raised his hand. Looking somewhat superciliously at
the others who had not yet finished, he burst out: “It’s foolish to bother
with multiplication and measuring the altitude. I’ve got a better method
for finding the area — it’s so simple. The area is a + b.”
“Have you any idea why the area is equal to a + b?” I asked.
“I can prove it,” he answered. “I counted it out in all the examples.
Why bother with b · h. The area equals a + b.”
I then gave him a fifth problem: a =2.5, b =5, h =2. The boy
began to figure, became somewhat flustered, then said pleasantly: “Here
adding the two does not give the area. I am sorry; it would have been so
nice.”
“Would it?” I asked.
I may add that the real purpose of this “mean” experiment was not
simply to mislead. Visiting the class earlier, I had noticed that there was
a real danger of their dealing superficially with the method of induction.
My purpose was to give these pupils — and their teacher — a striking
experience of the hazards of this attitude.
Wertheimer also pointed out that it is difficult or impossible to understand a principle
that is “ugly” — that is, not revealing of certain important relationships in the matter