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Topic: Re: New Post at RME: "Are 'Both Sides' in the Math Wars Dogmatic

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Re: New Post at RME: "Are 'Both Sides' in the Math Wars Dogmatic

Posted: Feb 25, 2009 10:01 AM
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> Why do you belittle chalk and talk. More people have
> learned more math that way than by button pushing on
> calculators. At least in college that is. Of course
> High School math is so dominated by TI that the
> students can no longer add 2/3 plus 3/4 plus 5/6
> without a calculator. A real pitty.

To whom this question was directed I'm not sure, and I do
use a white board sometimes (like chalk).

However, understanding fractions in depth is going to take
algorithms and those you should study and *test* in a more
generic fashion, making use of many skills.

With a calculator, you might just punch these in, but in
an interactive computer language you could (and should)

For example, Euclid's Method for the GCD (who teaches it?
one of the oldest) will let you "reduce to lowest terms"
so maybe start with that? First understand why it works,
using cartoons?

Then it's easy to explain about common denominators, the
need for. "Denomination" has lots of good connotations
to begin with, links to nouns (nominate, "to name"), so
when people say "like comparing apples and oranges"
they're complaining about "different denominations"
(must convert first). Links to money, currencies.
Like you could add pesos and yen by converting both to
dollars first (a side conversation, looking ahead...).

Multiplicative identity: 1/1, 2/2, 3/3, 4/4 -- all the
same number (1). So when adding two fractions, forget
about "lowest terms", just use the other guy's
denominator if it's not yours, using multiplicative

(2/3)+(3/4) = (2/3)*(4/4) + (3/4)*(3/3) = (8/12)+(9/12).

So that's all chalk 'n talk so far. But because we've
also been teaching Python or one of those FOSS languages
all this time, the exercises move towards writing out
(and testing) the algorithms in Machine Logic. They're
thinking in terms of "objects" (like apples, like oranges)
and there's an "abstract algebra" spin because we're
going to actually code for the + and * operators, which
under the hood become "methods of the Fraction type."

Doesn't Python have a built-in Fraction type? Yes it
does. But here we're learning, so want to write one.
Here's some code for ya, the kind of stuff my students
might eat for breakfast:

That looks really hard and cryptic (kids love to impress
their parents with this stuff), but we started months
ago coding a Dog and Monkey class. They understand about
classes and objects. Fractions are but one more
application of a well worn set of concepts by now. Plus
this is all very job-relevant. You can use Python to
write air-traffic control systems, eCommerce sites, plus
it's what computer science courses are tending to start
with -- you'll be way ahead, maybe have early college
credit, thanks to our approach, with no worries about
'relevance' even though your black box calculator could
have done it for you (while teaching nothing about
Machine Logic).

So yes, there's "chalk 'n talk" in this picture, and/or
cartoons and Youtubes (passive viewing, listening to this
and that). But we've also got hands on, lots of
scaffolding. In coding algorithms we're getting into the
guts of "the fraction type", linking in the GCD etc.
This is a smart way to go. We're leaving our competition
in the dust, is how we market it ("poor you if they
expect you to just use calculators, how lame!").

If all you have is paper and pencil, like some schools in
Baghdad (home of Mr. Algorithm aka Algebra City), we
still might use Python, anticipating someday we'll be
able to afford the real deal. Commodity hardware is
inexpensive. If you already have the server, one more
dumb terminal costs less than many a high end TI.


Python for Teachers (what self teachers might use):

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