On Mon, Oct 6, 2008 at 10:05 PM, DiPierro, Massimo <[EMAIL PROTECTED]> 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 _______________________________________________ Edu-sig mailing list [email protected] http://mail.python.org/mailman/listinfo/edu-sig
