Adrian
Posts:
197
From:
Houston, TX
Registered:
1/1/07
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Re: If you really want math reform....
Posted:
Jan 7, 2007 2:09 PM
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> At 08:31 PM 1/6/2007, Kirby Urner wrote:
>
> >And we go over why N(N+1)/2 generates the
> triangulars
> >in the first place, using a story about the young
> Gauss
> >being forced to add 1...100 as busy work.
> >
> >This *might* be construed as a proof,
>
> Presented reasonably, this *is* proof. God gave us
> the natural
> numbers, not Peano's Axioms.
>
I think he's talking about listing out the numbers:
1 + 2 + ... + n
n + n-1 + ... + 1
_____________________
n+1 + n+1 + ... + n+1
and observing that n+1 appears n times in the sum and so on. I don't know that this is that bad of an argument. You can dress it up by writing it down with summation symbols and rearranging everything to accomplish what we see here visually. And so, that might actually work out in this particular case.
But, that's still not the way a mathematician would handle a real problem that is similar to this one -- by some sort of similar trickery. They would use mathematical induction. Why not just teach that or just start using that argument? I don't think it is that dense -- you don't have to hold them to the same standards you would a graduate student. You could just do the inductive step...
"Suppose the sum of n numbers is, indeed, n(n+1)/2. Then, what would happen if we added the next number to that sum? We would have that the sum of n+1 numbers is
n(n+1)/2 + (n+1)
= [n(n+1) + 2(n+1)]/2
= [(n+2)(n+1)]/2
= (n+1)[(n+1) + 1]/2
So, if the sum of the first, say, 3 numbers is 3*4/2=6 then we will automatically get the sum of the first 4 numbers is (3+1)[(3+1) + 1]/2 = 4*5/2 = 10. And so on."
In this case 8th grade students might get lost on the manipulation in the middle, so you have to wait until they are really comfortable with some basic algebraic manipulations. The point here is that it doesn't have to be a seemless derivation from Peano's Axioms to every single mathematical fact the student knows (like we tend to attempt with graduate students, for instance). The idea is to try to build the students' abilty to prove their own theorems which means you could "start in the middle" just as well as start with Peano's Axioms. But, you try to stay close to what mathematicians actually *do* -- to the profession it all comes from. (And, that is another reason why there's no particular reason to do Peano's Axioms -- because that isn't what most mathematicians do -- that's just what logicians do, really.)
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