[Edu-sig] Algebra 2
DiPierro, Massimo
MDiPierro at cs.depaul.edu
Tue Oct 7 07:05:52 CEST 2008
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.
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.
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
________________________________________
From: edu-sig-bounces+mdipierro=cs.depaul.edu at python.org [edu-sig-bounces+mdipierro=cs.depaul.edu at python.org] On Behalf Of michel paul [mpaul213 at gmail.com]
Sent: Monday, October 06, 2008 10:09 PM
To: kirby urner
Cc: edu-sig at python.org
Subject: Re: [Edu-sig] Algebra 2
My spin in Pythonic Math has been to suggest "dot notation" become accepted as math notation
I absolutely agree with this.
In about 5 weeks I'll be giving a California Math Council presentation that I titled Fractions are Objects, not Unfinished Division Problems.
I submitted the proposal with the attitude of 'who cares? Let's just see what happens.'
Surprise! They accepted.
OK, so now I have to have something to say. : )
I think the theme of 'dot notation' as a kind of standard math notation would be valuable.
Generally, I want to present
1. The importance of 'computational thinking' as a math standard
2. Python as a vehicle for this
Thanks very much for any helpful suggestions along these lines.
- Michel
2008/10/6 kirby urner <kirby.urner at gmail.com<mailto:kirby.urner at gmail.com>>
2008/10/4 michel paul <mpaul213 at gmail.com<mailto:mpaul213 at gmail.com>>
For math classes I think it's more pertinent to focus on functional interactions and not on IO issues, and that was what I was trying to get at.
I'm enjoying this thread.
My spin in Pythonic Math has been to suggest "dot notation" become accepted as math notation, and with it the concept of namespaces, which I tell my students is one of the most important mathematical concepts they'll ever learn. We look at how several languages deal with the problem (of name collisions, of disambiguation), including Java's "reverse URL" strategy e.g. net.4dsolutions.mathobjects.vector or whatever.[1]
I tend to look at .py modules as "fish tanks" i.e. ecosystems, with both internal and external (import) dependencies, with the user of said fish tank being somewhat the biologist, in testing to find out what's in there, what the behaviors are.
Starting with the math module is of course apropos, discussing the functions, not shying away from trig even pre high school, no reason to withhold about cosine just because they're "young" (this is actually a prime time to gain exposure to these useful and time-tested ideas).
Because of my "fish tank" idea, and using the math module as a model, I don't encourage "self prompting" i.e. using raw_input for much of anything. We need to "feed the fish" directly, i.e. pass arguments directly to functions, with f ( ) looking like a creature with a mouth, ready to eat something. fish( ).
Regarding GOTO, sometime last month I think it was, I told the story of assembler (JMP) and spaghetti code, Djikstra to the rescue, further developments. It's through story telling that we get more of the nuance. I'm a big believer in using this "time dimension" even if not doing anything computer (hard to imagine) i.e. the lives of mathematicians, their historical context, why they did what they did -- critical content, not side-bar dispensible, not optional reading.[2]
Metaphor: education systems are like those old Heinlein moving sidewalks (science fiction), where you can't jump on the fast-moving one at the center from zero, have to slide from walk to walk, each one a little faster, and likewise when a approaching a destination, start to slow down.
By including more content from geek world, getting more of a footprint for the circus I work in, I'm giving a sense of one of those fast moving sidewalks at the core of our infrastructure (coded reflexes, superhumanly fast business processes). Math pre-college should be a door into all sorts of careers (starring roles) that include numerate activities. It's not about Ivory Tower PhD mathematicians having exclusive access to future recruits, shoving the rest of us aside because our skills are "impure" (not pure math).
What passes for "pure math" would be something to study in college, after getting a broad sampling ahead of time, good overview, the job of a pre-specializing curriculum. In the meantime, if your school doesn't give a clear window into computer science in over four years of numeracy training, then hey, its probably a *very* slow moving sidewalk (more 1900s pedantic and plodding than fast paced like TV).[3]
Kirby
[1] Like when I do the IEEE lecture on Nov 4 at the Armory (theater), I'll be talking about coxeter.4d versus einstein.4d versus bucky.4d -- three namespaces, named for thinkers, in which the concept of "four dimensional" makes sense -- but in quite different language games. (a)
[2] I like telling the story of those Italian Renaissance era polynomial solvers, a proprietary model in which mathematicians were like race horses, gained owner-patrons who would stable them, let them work out, then they'd have like "cock fights" in the village square, to see how could solve whatever third of fourth degree polynomial fastest. Without this kind of focus, polynomials wouldn't have the momentum they still have to this day, as a key math topic pre-college (and another kind of "math object" from a Pythonic math point of view).(b)
[3] Marshall McLuhan wasn't just blowing smoke. People who grow up on a lot of TV are geared differently and in the early 21st century a lot of what "school" is about is asserting the value system of a pre-TV era (pre computer, pre calculator...). To "side with the kids" would be entirely subversive of traditional classroom thinking, would involve a lot more learning how to make televisions (multi-track) not just passively viewing it. In my model numeracy classes, making "math shorts" (like on Sesame Street) and uploading 'em to YouTube, for peers to admire (peers thousands of miles away perhaps -- no problemo) is a big part of the action.
(a) FYI here's the bio of Kirby that went out to subscribers:
An IEEE Oregon Section event
"R. Buckminster Fuller: The History (and Mystery) of the Universe"
with exclusive presentation by local Buckminster scholar and consultant to the playwright, Kirby Urner Tuesday, November 4, 2008 on the Mezzanine at Portland Center Stage Gerding Theater at the Armory
128 NW Eleventh Avenue, Portland, OR 97209
Hors d'oeuvres Reception: 5:30 p.m.
Presentation and Discussion: 6:00 p.m.
Theater Performance: 7:30 p.m.
$49 per person. Tickets are limited.
Please register by October 14, 2008. For more information and to register go to <link here>.
We regret that we cannot offer refunds for cancellations received after October 14.
-----
R. Buckminster Fuller: The History (and Mystery) of the Universe
Written and directed by D.W. Jacobs from the life, work and writings of R. Buckminster Fuller
Play:
"Everything you've learned in school as 'obvious' becomes less and less obvious as you begin to study the universe." - Buckminster Fuller
Does humanity have the chance to endure successfully on planet Earth, and if so, how? This is the question framed by Buckminster Fuller, the engineer, designer, poet, and philosopher who, among other things, was Mensa's second president and invented the geodesic dome. Join us for an unforgettable journey inside one of the most remarkable minds of the 20th century in a one-man show that blends videos, lectures, poetry and a healthy dash of humanist humor. A hero of the sustainability movement, Bucky framed many of the great ideas of his time and ours. This is your chance to get to know the man behind the world-saving mission.
Presentation:
How has the literature developed since the publication of 'Grunch of Giants' in 1983 and what are likely outcomes and future directions projects Fuller started over a lifetime of heavy lifting?
Speaker:
Kirby Urner started exploring Fuller's work in earnest following his earning a BA in philosophy from Princeton University, while serving as a high school math teacher in Jersey City. He's served as a contributing editor for McGraw-Hill, Rockefeller Center, political activist for Project VOTE! in Washington DC, and computer programmer for myriad governmental and nonprofit organizations in Greater Portland. Working in cahoots with Kiyoshi Kuromiya, Fuller's lieutenant on a couple of key books, he snagged the domain name bfi.org<http://bfi.org> and served as the Buckminster Fuller Institute's first web wrangler. His 'Synergetics on the Web' is one of the main stops for Bucky scholars to this day (www.grunch.net/synergetics). Kirby is an IEEE member.
(b) yes, tell them early that we have no "closed form algebraic solution" to fifth degree polynomials, but that doesn't keep Python from being useful in implementing some of the progessive approximations for root-finding, such as you get under the hood with Mathematica et al. I've got a prototypical Polynomial class out there somewhere that self solves pretty well, maybe others here do too.
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