Designing Learning Environments for Developing Understanding of Geometry and Space

By Richard Lehrer; Daniel Chazan | Go to book overview

19
Teachers and Students Investigating and Communicating About Geometry: The Math Forum

K. Ann Renninger, Stephen A. Weimar, and Eugene A. Klotz Swarthmore College

The mere memorizing of a demonstration in geometry has about the
same educational value as the memorizing of a page from a city direc-
tory. And yet it must be admitted that a very large number of our
pupils do study mathematics in just this way. There can be no doubt
that the fault lies with the teaching. ( Young, 1925, p. 5)

Although it may seem obvious that teachers (and their students) need to appreciate that learning must be much more than memorization, this realization is almost certainly not enough to enable teachers to move beyond a "review the homework, go over new material, do your homework" approach. Those who come to know the elegance, intuition, and sense of discovery that can characterize the doing of geometry may apprehend it regardless of how they are taught, whereas those who mostly memorize will be convinced that geometry (and, more generally, mathematics) is equivalent to getting right answers.

There are several reasons why this state of affairs has continued in geometry. First, there are many individuals for whom construction of correct proofs is an end in itself because they have been assigned proof after proof, have no questions themselves, and are not really engaged in the questions the proofs represent. These individuals stand in contrast to those who enjoy and continue to work with geometry and for whom proofs are an effective means of communicating what they have figured out.

Second, one function of schooling in the industrial model has been about finding efficient ways to transmit knowledge so that the time it takes to learn things can be dramatically shortened and more can be

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