Posts
GK proves negative exponents.
It is Catherine's theory that how our schools treat our gifted can be a pretty good measure of academic rigor in general.
I would add that we can learn a good deal about mathematical thinking, and all its various forms, from observing the gifted kids. The problem is keeping up with them. That's why my blog is called "Clueless Mom of Gifted Kid."
update from Catherine:
I asked Barry whether this is a proof - it is!
Yes, that would constitute a proof, and even though it proves it for 3 ^(-2) one can see that it extends to all numbers. A more general proof would be that since a^m/a^n = a^(m-n). If n > m, then m-n is negative. Since it's the same as dividing a^m by a^n, one can see that there are m a's in the numerator and n a's in the denominator. Through cancellation one is left with 1/a^(m-n).
There are more rigorous and formal proofs but the above is suitable for an algebra 1 course.
11 comments:
THANK YOU THANK YOU THANK YOU!!!
I absolutely stand by this argument.
When a school "blows off" the academic needs of an entire class of kids, as a class, it has established a precedent of blowing off academic needs.
Blowing off academic needs is "on the menu," as psychiatrists say about families that include members who've committed suicide.
The district will do it again.
I've seen it.
Beyond this, as far as I can tell the bell curve RULES inside educators' (and parents') minds: individual and group differences can't be erased.
I think what may happen (not sure what the mechanism is) is that when you deliberately suppress the learning of gifted children, which has been done in my district and in many others, you push everyone else down, too.
In other words, parents of gifted kids usually assume that the school is ignoring their kids and "teaching to the middle" - or, in the era of NCLB, ignoring everyone in an effort to bring the kids on the bottom of the ladder up.
But I'm pretty sure that's not what happens - at least it's not what I see happening in my own district.
What I've seen happening here and, I think, elsewhere, is that when the gifted kids are "pushed down" everyone else is pushed down, too, perhaps by the same amount.
Pushing down the middle kids doesn't put them in the middle with everyone else.
Instead, pushing down the middle kids creates a new, lower middle.
The gifted kids are still on top, and the bright & average kids are lower than they were "before" - or than they should be.
It's blindingly clear to me that across the board kids in my own district are working below capacity - and that this is the effect of the school, not the families, who are in a constant struggle to "push" their kids back up.
Hence: pushy parents.
I told my mother-in-law yesterday that Christopher had been "accepted" into Earth Science.
She said, "What? Accepted? He had to apply?"
(There's no application process; kids get selected or not-selected.)
She couldn't even imagine such a situation.
I'm not sure this is what happens in other districts; maybe other districts really do manage to level everyone's performance.
But I wouldn't be surprised to see that the bell curve continues to exist everywhere, no matter where you set the top performers.
Is there a math-head here that could answer the question that's kind of embedded in my blog? (Click on the link in my entry about gifted kids to read the actual entry in my own blog.)
I'm looking for whether what my kid did is a "proof" or is it something less formal than that. I just don't know terminology well enough. And would like to learn.
I just asked Barry!
It's a proof!
Yes, that would constitute a proof, and even though it proves it for 3 ^(-2) one can see that it extends to all numbers. A more general proof would be that since a^m/a^n = a^(m-n). If n > m, then m-n is negative. Since it's the same as dividing a^m by a^n, one can see that there are m a's in the numerator and n a's in the denominator. Through cancellation one is left with 1/a^(m-n).
There are more rigorous and formal proofs but the above is suitable for an algebra 1 course.
Tell Barry thanks. That was a clear explanation, too. Now I know it's a proof, at the algebra 1 level.
And I notice I made a typo. THe last line of 1st paragraph should read "Through cancellation one is left with 1/a^(n-m)"
Sorry.
"And I notice I made a typo."
Barry, my brain rearranged it behind my back. So it never looked wrong to me.
Isn't amazing how our minds do that? Or maybe I'm just too clueless to have noticed it. LOL
Either way, I got the idea that at my child's level, we could think of what he did as a sort of informal proof.
Post a Comment