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Re: CMP Connected Mathematics does not introduce Lowest Common
Denominator
Posted:
Dec 7, 2007 10:01 AM
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> Hey, if they can somewhere in your curriculum prove
> all or almost all the algorithms/formulas/theorems
> they use, then you got my vote. That would completely
> satisfy me in the understanding stuff department. In
> other words, if they can do that, then you'll hear
> none of this "yes, but can they" stuff from me.
>
> Paul
We've talked about this before in this archive.
Whereas we go through a simple visual proof of
Euler's V + F = E + 2 (very basic to our curriculum),
or N(N+1)/N for the Nth triangular number (we do a
lot with sequences), we don't necessarily care to
prove every theorem we use.
On the contrary, at this pre-college level, the
relevance and importance of a proved result may be
accessible and usable, long before the actual proof
itself becomes so.
So like in a very early class I'll give 'em Fermat's
Little Theorem to play with: base**(prime-1) mod prime
== 1 but the important thing is less to prove it
formally, than to figure out what it *means* in the
first place (e.g. "what's this 'mod stuff' again?),
which is where a Python program becomes useful, and
lots of digits e.g.
>>> def gcd(a,b):
while b:
a, b = b, a % b
return a
>>> def totient(n):
return len([k for k in range(n) if gcd(k,n)==1])
>>> totient(211)
210
>>> pow(base, totient(211), 211)
1
>>> base**totient(211)
1645504557321206042154969182557350504982735865633579863348609024L
Then we get into the important logical point that
if p then q doesn't mean ~p therefore ~q i.e. whereas
IF p is prime, THEN it passes the Fermat test (above),
many non-primes also pass, including for any base
(the Carmichael Numbers). Already we're projecting
off the Internet, showing them literature, developing
fluency around number theory (e.g. building towards RSA),
and yet we're *not* obligated to prove every stepping
stone we use.
This is a literacy course, not a college math course.
You *read* Shakespeare without needing to write the way
he did. You *read* and *use* proved results without
necessarily needing to understand the proofs. Rather,
you should know the *relevance* of the results -- that
gives more motivation to even *care* there's a proof.
And again, I'm not saying we bleep over all proofs. We
don't do a lot of these algebraic manipulations you tend
to call proofs. This isn't math according to Paul Tanner
and we don't cite you in our footnotes (though geeks
reading over my shoulder may encounter my Math Forum
conversations, so your views will tend to be known to
them).
Kirby
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