Date: Dec 30, 2012 12:05 PM
Author: kirby urner
Subject: Re: A Point on Understanding

On Sun, Dec 30, 2012 at 12:16 AM, Paul Tanner <upprho@gmail.com> wrote:

>> So my I conclude you're against this strategy of arguing pro and con a
>> mathematical proposition?

>
> No, and in my last post
>
> http://mathforum.org/kb/message.jspa?messageID=7944754
>
> I communicated so:
>


<< snipping repeated prose >>

I don't see where I've denied any theorems in setting up a tension
between the deficit at each point and an argument that gets rid of the
deficit at some limit, using an epsilon / delta approach. Rather,
I've introduced several new theorems and suggested we prove them.

> One of the responsibilities of [a] teacher [is] to teach his or her
> students how to think such that they can avoid becoming cranks.
>


Lets not forget that mathematics is more than a scaffolding of proved
theorems. It consists of conjectures (unproved) upon which other
scaffolding may be contingent, and it depends on definitions which
alter what's proved and/or provable.

For example, per standard definitions, a cube with edges SQRT(2) must
have a volume of SQRT(8) because our definition of 3rd powering
incorporates the geometric hexahedron as a model of 3rd powering.

In contrast, a Martian civilization might consider the regular
tetrahedron their 3rd powering model. There'd be a conversion
constant and all volume ratios would remain the same (e.g. cube to
tetrahedron inscribed as face diagonals = 3:1).

The argument *against* teaching both Earthling and Martian mathematics
side by side would be we don't want to confuse students. We want them
to reflexively think and say "squared" and "cubed" in connection with
2nd and 3rd powering, not "triangled" and "tetrahedroned". PhDs might
explore alternatives, but such forays into non-orthogonal (aka
unorthodox) thinking would only scramble the brains of the young.

The argument *for* teaching the two paradigms in close conjunction is
this approach retains more flexibility in thought and allows the role
of definition and cultural choice to remain in the foreground. I'd
consider such multi-paradigm training more useful for students
planning a career in diplomacy / international relations, where rigid
adherence to the "one right way" is too close to mindless
fundamentalism for comfort. Better to scramble their brains a little
than to encourage strait-jacketed habits of mind.

So here's another case where there's legitimate debate and yet no one
is denying any proved theorems. There's no need to fling any
accusations that one side or the other are just crackpots.

In my view, any "one size fits all" approach to math pedagogy is ill
advised. This is different from saying we should have a fast track
for the talented and gifted and various slower tracks for those less
good at math. I'm saying not all who are stellar at math need the
same training, even in early childhood. Those trying to impose the
"one right way" (what topics to include or exclude, and in what
sequence) are the closest to crackpots in my vista.

> This "arguing pro and con" is fine as long as theorems and proofs are
> not denied - if it's only a pedagogical technique for introducing
> theorems and proofs, then OK. But if the agenda is to deny
> mathematical theorems and proofs, then it's not fine - it's not fine
> if this "pedagogical technique" is a smokescreen for denying these
> facts.
>


There's more to mathematics than theorems and proofs.

Kirby