Date: Feb 24, 1995 2:47 AM
Author: W Gary Martin
Subject: Re: Learning and Mathematics: Lampert- Knowing, Teaching

Response to Maria Ong Wenbourne <>:

> I believe that Lampert's article is an outstanding contribution
>to mathematics education because it blows up the long-standing
>"procedural knowledge (computational/symbolic) vs. conceptual knowledge
>(understanding)" debate....

YES YES YES YES YES YES. Actually, take a look at Bruner's old stuff
sometime. He advocates the same general idea of symbols as a recording of
students' operations at lower levels of abstraction. Let's see if I deserve
to keep my degree: concrete -> iconic/pictorial -> symbolic. Did I miss a

This also ties into the discussion on long division; the symbolic recording
should be closely linked to the more concrete actions and understandings.
Thus, the standard algorithm may be too abridged to be a good recording of
how the child is thinking/operating.

>My question is two-fold in considering how we may extrapolate
>Lampert's innovative ideas on teaching multi-digit multiplication to
>teaching concepts in geometry, algebra, trigonometry, and calculus:
> 1) What kinds of "intuitive understandings" do students have
>about these more complex mathematics domains, and
> 2)Is it possible to foster students' understanding in, say,
>geometry or calc by encouraging the finding of alternative solutions? The
>way I remember high school math, there was only one right answer and
>usually only one way to arrive at that answer.

This is my life work, looking at high school geometry from this general
perspective. And, YES YES YES! It does work! But the answer to your first
question has not been trivial. We had a hard time finding strong entry
points in standard synthetic geometry and finally ended up basing the
course on transformations because they are intuitive, can effectively
generate important geometry (aside from group theory), and provide useful
"thinking tools" that allow them to approach problems from many different
directions. It has taken six years of research to put this together into a
curriculum that WORKS!


W. Gary Martin 1776 University Ave.
University of Hawaii Honolulu, HI 96822
Curriculum R & D Group (808)956-9956; FAX (808)956-4984