[Edu-sig] a non-rhetorical question

[Laura Creighton]

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