I'm changing pace slightly, after making 2 points: (1) Dot notation *does* exist in Maths. Its just called "subscript" notation instead. But its the same thing. I try to make a habit of using subscripts (and sub-subscripts) as much as possible because it shows the same logical relationship.
(2) The fractions you show don't have beauty. In continued fractions, its *recursion *that has the mathematical beauty. (3) *A practical question* - can any *high/middle-school *teachers give me clear pros/cons of using programming as a tool to teach algebra? I'm rewriting the Year 8 maths program for next year (13-14 year olds) and am considering trialling using Python. The students are the school are very tech-savvy and I wouldn't aim to teach them anything more than formulas really... formulas, basic IO and some ifs. Maybe (maybe) could do a basic for i in range(20) loop, but nothing more than that. Note that I am the computing teacher in the school; the majority of my teaching is the computing studies subjects for older students (15-18 year olds). On Tue, Oct 7, 2008 at 3:42 PM, kirby urner <[EMAIL PROTECTED]> wrote: > 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 >
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