[Edu-sig] a non-rhetorical question

[Jay Bloodworth]

On Tue, 2007-07-10 at 01:22 +0300, kirby urner wrote:

> 
> Yes indeed.  And I think many of us are making the point that
> students develop differently, such that they might use their
> strengths to address their weaknesses (with guidance from a
> teacher/mentor should they be lucky enough to have one).
> 

I agree with what you're saying in principle.  I'm just not sure that
talents in one area are always infinitely marshalable to the service of
other tasks.  For example, I'm good at math but suck at basketball.  But
I doubt that even the genetically engineered lovechild of Jaime
Escalante and Phil Jackson could turn my mathematics talent into success
on the basketball court to any significant extent.

So maybe basketball and math is too big a gap to bridge.  But surely
algebra and geometry is doable, right?  Certainly, there are numberless
fascinating connections that might be productively studied in either
course.  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.

There are certainly many beautiful connections among the branches of
math and computer science, and I agree that mathematics curricula should
include some of them.  I'm just skeptical that these connections are the
golden door to learning formal math.  Information and computational
theory teaches us that data and algorithms have a certain irreducible
complexity.  If your Turing machine doesn't have enough states or enough
tape, there are things it just can't do.  I don't think in undamaged
human brains difficulties in learning algebra can be traced to such
gross level deficiencies in computational power, but I do think computer
science may have something to say here.  Not every Turing machine is a
universal Turing machine.  If your algebra machine is working, I believe
it's doubtful your geometry machine will work in the pinch.

Kirby, I have skimmingly followed your work the past couple of years,
and I don't mean for any of this to be a criticism of what you're doing
in particular, just the general notion that models and applications are
the answer to everything in education.

Jay