Re: approaches to content

[Peter Moxhay <moxhap@portlandschools.org>]

Thanks, Nancy, for the very interesting description of your work. This 
has been
helpful to me as an example of structuring the content. In particular, 
you wrote

>  For
> example, instead of starting with linear functions, we tend to work 
> with
> students on activities that involve varying quantities of amount and 
> rate,
> so that when they do encounter the 'degenerate' case of a linear
> relationship, it is seen as a simpler case rather than a different
> phenomenon. Students don't encounter linear relationships nearly as 
> often as
> they do more complex ones in life, so it is much easier to draw on 
> their
> prior knowledge in supporting their thinking about and mathematizing
> relationships when we start with more complexity.

which sounds to me like moving from the general to the particular, that 
is,
that the kids are acquiring a general concept or method.

Again, as with Peg's work, I'm wondering if the students "model" the 
general
relation (between amount and rate, etc.) with some sort of special
diagrams, or formulas, or tables, or special algebraic notation, and 
whether
this helps them assimilate the general concept. I'd also be interested 
if you have
methods for *assessing* whether students are actually solving problems 
by moving
from the general to the particular, or just solving each particular 
problem as a new
case.

That is, how can you assess that:

> , it is seen as a simpler case rather than a different
> phenomenon.

Peter