[Edu-sig] Algebra 2

[kirby urner]

On Mon, Oct 6, 2008 at 10:05 PM, DiPierro, Massimo
<MDiPierro at cs.depaul.edu> wrote:
>
> I agree with this
>
> 1.  The importance of 'computational thinking' as a math standard
> 2.  Python as a vehicle for this
>
> But it is important to make a distinction:
>
> a) a math formula represents a relation between objects and the objects math speaks about (with very few exceptions) do not have a finite representation, only an approximate representation (think of rational numbers, Hilbert spaces, etc.)
> b) an algorithm represents a process on how to manipulate those objects and/or their approximate representation.
>

There's a whole philosophy of mathematics, and of language more
generally, implicit in your (a) and (b), inheriting from both realism
(as in "the reality of Platonic objects") and nominalism (as in "nouns
point to things" -- with "pointing" considered entirely
non-problematic).

The linguistic turn (named by Rorty), launched by Nietzsche and
culminating in Wittgenstein's later works, is about undoing some of
these gestalts, returning us to a more operational view of how
language works in the world (or doesn't).

This is getting way off topic I'm sure some are thinking, and I agree,
so just lets admit we don't all come to mathematics from the same
perspective, and that this is as it should be.

> While math and math teaching could benefit from focusing more on process and computations (and there python can play an important role) rather than relations, it is important not to trivialize things. For example:
>
> In math a fraction is an equivalence class containing an infinite number of couples (x,y) equivalent under (x,y)~(x',y') iff x*y' = y*x'.
> Any element of the class can be described using, for example, a python tuple or other python object. The faction itself cannot.

The way I'd put it is the class Rat (rational number class) spells out
what fractions might do, in terms of __add__, __mul__ and so on, but
then there's no limit on the number of fraction objects you might want
to build from this blueprint, i.e. the type of object is distinct from
the instances, in a pleasing, teachable, lexical way.  At least as
relevant as Bertrand Russell's stuff if you ask me, this object
oriented paradigm.

And yes, no limit on the number of tuples that map to that tuple in
lowest terms, which is where gcd comes in, gotta teach that.  Pre
college algebra with no introduction to Euclid's Algorithm for the GCD
is laughably idiotic and I openly sneer at the idea when I think no
one is looking.

>
> It is important to not to loose sight of the distinctions. Math is gives us the ability to handle and tame the concept of infinite, something that computers have never been good at.
>
> Massimo

I like Knuth's take, lectures at MIT (audio on the web, maybe video
too as I recall), which is very into finitude.

Accepting finitude takes courage too.  I'm glad our computers are
harnessing it, leaving humans to their fantasies of greater greatness.

Kirby