Response to Maria Ong Wenbourne <firstname.lastname@example.org>:
> 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 step?
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