In a message of Tue, 10 Jul 2007 09:20:15 EDT, Jay Bloodworth writes: <snip>

I'm just not sure that geometric models are likely to help with the difficulties struggling students often have. Two examples:

* It's like pulling teeth to get students to respect order of operations and remember consistently that -3^2 = -9. Where's the geometric model for that?

* (a + b)^2 = a^2 + 2ab + b^2, not a^2 + b^2. Here there is the standard geometric area model for multiplication. Sometimes I present multiplication with the model, sometimes not. It doesn't seem to change the error rate.

<snip>

Jay

Something I have had success with is dividing what I am teaching into a) universal truths about the way the universe works vs b) notational conventions which are true because we all agreed on it. using '+' for add instead of '!' is notation. using ( ) to group things and not {} is notation. but given that we have agreement, the reason that (a + b)^2 = a^2 + 2ab + b^2 is because that is _really the way the world works_. Having a computer helps. You write code and try it yourself for a lot of values of a and b, and can really prove that (a + b)^2 != a^2 + b^2 to your own personal satisfaction. Which is why I want geometry first, to give students a whole lot of exposure to geometric proofs of geometric truths. Laura