Thinking and Deciding

KNOWLEDGE, THINKING, AND UNDERSTANDING 25
to which it refers. An example would be a formula for the area that reduced to the
simple formula only by some algebraic manipulation:
A =
(b − h)
(1/h − 1/b)
Another way to put this, perhaps, is that the process that leads to understanding will
fail to learn what cannot be understood. This process is unlikely to accept falsehood,
even when propounded by authority, because falsehood is usually incomprehensible.
4 Of course, many facts are essentially arbitrary, so that “understanding,” in the
sense in which we are using the term, is impossible. A statement such as this — “The
Battle of Hastings was fought in 1066” — must be accepted without understanding.
Katona (1940), a follower of Wertheimer, made several additional observations
concerning the relation between understanding and learning. Katona taught subjects
how to solve different kinds of problems under various conditions, some designed
to promote understanding and others designed not to do so. Certain of his problems
concerned rearranging squares made of matchsticks so that a different number of
squares could be made from the same number of matchsticks with a minimal number
of moves. For example, make the following five squares into four squares by moving
only three sticks:
The “no understanding” groups simply learned the solutions to a few such problems.
The “understanding” groups were given a lesson in the relationship between
number of matches, number of squares, and the number of matches that served as
the border between two squares. The most efficient way to decrease the number of
squares is to eliminate squares that share sides with other squares, so that the resulting
squares touch only at their corners. For example, the square in the lower right
is a good one to remove. You can then build a new square on top of the second
by using the bottom of the second square for the top side of the new one. Groups
that learned with understanding of the common-side principle were able to transfer
better to new problems, and they even “remembered” the solutions to example problems
more accurately after a delay, even though these examples were not part of the
lesson itself. The difference between Katona’s demonstrations and Wertheimer’s is
4 Scheffler (1965) makes related arguments.