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Topic: [math-learn] Introduction to functions
Replies: 30   Last Post: Dec 7, 2005 9:13 AM

 Messages: [ Previous | Next ]
 kirby urner Posts: 3,690 Registered: 11/29/05
Re: [math-learn] Re: Wanted: Good Graphing Calculator Questions for Algebra I and II
Posted: Dec 2, 2005 11:13 AM

I spent several formative years growing up in Italy. Not only was I
impressed by the importance of Latin -- which I didn't study formally,
except through other Romance Languages -- but I discovered the
importance of Roman Civilization, building a lot on the Greek.

There's nothing like history to help one understand why an "ideal
circle" should be one we'll never come across, as no real circle is
really ideal. Or why we belabor the "number versus numeral"
distinction (somewhat bogus). Or why we insist points have no
dimension, but if you line up an infinite number of them, suddenly
you've got a line.

Ah philosophy (they tell you it's math, but at bottom, it turns into
something a lot more squishy).

Regarding Algebra, in my neck of the woods we're doing more to
converge exposure to variables with simple computer programming.
Defining little functions, such as to give the nth triangular number,
is a great way to learn why we say "nth" i.e. we don't want to commit
ourselves to just one number, we want to use the very same definition
to return any triangular number, including the 2091092th.

That's *not* the whole of what Algebra is about of course (we mention
it got started in Baghdad by the way -- that belongs in the
standards), but it's a good way to start phasing in a lot of named
variables in place of numbers, getting kids used to *templates* and/or
*recipes*. Plus we don't just work with number types anymore.
Symbolic processing is *so much more* than mere numerical methods.

Re those Triangular numbers:

IDLE 1.1
>>> def tri(n):
return sum([i for i in range(1, n+1)])

>>> tri(2091092)
2186333921778L

>>> ((2091092+1)*2091092)/2
2186333921778L

So why are those two answers the same? (And what's that silly "L"
doing there?). This takes us immediately to the child Gauss, and his
game to add the first n consecutive integers. Kids like this story
because the premise is Gauss was misbehaving (we say the whole class
was) and the assignment to add 1 + 2 + 3 + ... + 100 was for
punishment. The punch line is Gauss discovered this short cut and
*out smarted* the teacher!

Why do I share this story? Because I think it touches on an important
aspect of education. A younger generation likes to feel it outstrips,
outsmarts, out-thinks the generations before it. At least that's a
hallmark of western civ, of the USA. We're not really into ancestor
worship, even though we respect specific ancestors (like some of those
movie actors in those old b/w classics, and like Gauss). We like to
feel we're "making progress" which means we leave older adults behind,
eating our dust (not trying to be mean -- those adults had their
chance), as we race towards new horizons.

What's stultifying are those adult systems which act like their job is
to "bring kids up to speed" in the sense of making carbon copies of
the teachers. After years of work, you'll be no further along than
your fuddy dud mentors. That'd be a horrible fate, by many internal
criteria. To be just like Wayne or Dom or Kirby or Pam in terms of
one's understanding of mathematics -- what a failure that'd be! Math
is more like music. Each generation pioneers its own.

So if you really want to be an effective teacher, you'll not get in
the way of students surpassing you, and understanding in ways you
simply do not. Let them also teach *you*. Make it a two way street,
from the get go.

In any case a superior education system allows and encourages this.
We're not even *trying* to reassure parents that their kids are just
recapitulating the math *they* had in school. No way Jose. These
kids are onto something new. It's not your childhood all over again.
It's theirs and it's different, sometimes extraordinarily so. Get
used to it.

Kirby

DonorsChoose.org helps at-risk students succeed. Fund a student project today!
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Date Subject Author
11/30/05 Stephanie Kolitsch
11/30/05 John Clement
11/30/05 kirby urner
11/30/05 Stephanie Kolitsch
12/1/05 kirby urner
12/1/05 kirby urner
11/30/05 ArPeEs@aol.com
11/30/05 Richard Sisley
11/30/05 Wayne Bishop
12/1/05 John Clement
12/1/05 Wayne Bishop
12/1/05 John Clement
12/1/05 Wayne Bishop
12/1/05 John Clement
12/1/05 Ihor Charischak
12/2/05 Wayne Bishop
12/2/05 Ihor Charischak
12/2/05 Michael Paul Goldenberg
12/2/05 pamkgm7
12/2/05 pamkgm7
12/2/05 Ihor Charischak
12/2/05 Michael Paul Goldenberg
12/2/05 pamkgm7
12/2/05 Michael Paul Goldenberg
12/2/05 kirby urner
12/1/05 kirby urner
12/7/05 Antonija Horvatek
12/7/05 Wayne Bishop
12/7/05 Antonija Horvatek
12/7/05 Antonija Horvatek
12/7/05 Antonija Horvatek