Monday, December 25, 2006

A Message to Edspresso (John Dewey)

The following is a comment made on edspresso's blog and expanded on here:

Happy Holidays Mr. Dewey!
I invite you and your faithful to read my expanded comments in my 'newish' blog MathNotations.
At any rate, I've just become aware of your excellent writings and I commend you for your thoughtful insights. Ok, enough of the brownnosing... Here's the skinny. I believe you're on the right track with your thoughts about constructivism. My comments regarding this appeared in Joanne Jacob's blog so I won't go into detail here. As children move up the ladder, they should need less of the hands-on manipulatives and tactile experiences like the one described in my post. This is why I suggested that cutting off the corners from the vertices of a polygon is a worthwhile activity for FOURTH graders, but I would not spend that kind of time for middle schoolers. Once they've experienced the hands-on approach, they can quickly revisit this for a triangle, then move onto a more abstract pattern-based approach in the middle grades, e.g., dividing a polygon of n sides into n-2 triangles (they can formulate this for themselves within 5 minutes), thereby developing the standard formula. By the time they reach a traditional geometry course in hs, they've had the spatial experience from 4th grade, the pattern-based approach in middle school and therefore they can quickly review this and focus on APPLYING the formula to regular polygons and more sophisticated algebraic exercises. But this discussion has important implications:
0. None of my remarks makes any sense unless 4th, 8th and 10th graders in downtown Chicago are exposed to the same learnings as those in the affluent suburbs of Chicago off Lake Michigan, if you get my point. THERE MUST BE ONE NATIONAL MATH CURRICULUM and it cannot represent one side or the other in the Math Wars. Your approach is a good one, Mr. Dewey, because it is a blend, but you might need a bit more field experience before deducing general principles. I'm not being patronizing or condescending here, so pls don't take it that way. Radical solutions from either camp can be dismissed but how we combine the best of traditional and reformed approaches is not so obvious.
1. More hands-on in lower grades (assign any label you want!) with teachers who are committed to this and properly trained
2. Gradual development of abstract formulations of patterns with algebraic representations starting much earlier in Middle School than is the norm in the USA and ONE of the reasons why we lag behind other nations.
3. More challenging applied problems for the hs students instead of merely rehashing formulas and doing the standard problems
FINALLY, SOME APHORISMS (mock me if you will!):
4. Despite 'cutting the corners' for polygons, THERE ARE NO SHORTCUTS FOR DEVELOPING A PROFOUND UNDERSTANDING OF FUNDAMENTAL MATHEMATICS! Here's what I tell my students and they don't think the reference to their grades is amusing:
The only shortcut in math is from 'A' to 'F'! [ok, you can groan loudly now!]
Dave M

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