Don't forget that for DS, it was his early verbal skills that first brought us to testing...

Well, I see what you're saying, CFK. Some PG kids are very good at this form of self-teaching.

But I think it's a mistake to assume that PG = needs no instruction. I don't accept this as true for all PG kids, nor do I think it is a necessary part of the definition for PG.

In fact, I think this is a pretty dangerous line to take, since it sounds very much like the line the schools often take: "If he's so smart, then why doesn't he know it already? If he's so smart, he doesn't need to be taught."

I think the notation thing is huge -- it took me a long time to realize that even though KG could add and subtract double digits in his head, he didn't know what the signs were until last year! Duh.

With reading, it's fairly cut and dry--they can read it with comprehension and answer questions, or they can't. Of course, there's the issue of inference too, but I'll skate over that. :-)

With math, I think there's more gray area since math is essentially its own "language." If no one teaches you the notation, you're seriously limited. I'd like to see the kid that figures out calculus on his own, in his head, with no formal math instruction at all! There may be one out there, but wow--that's the PGest of PG kids!

No, that's not what I'm saying at all, I was just refuting your comment that PG kids can only achieve up to the level of which they are instructed or exposed. I think that a lot of PG kids can self teach elementary school concepts or else learn them with minimum instruction. Everyone (well most everyone, I don't like to make generalities), no matter their intelligence, eventually will reach the point where they have to be taught if they are ever going to reach the top of their field.

I think we're all in agreement there (or are we? Maybe not ...) -- any of us with a child who taught themselves to read (and I know there are lots of us!) can hardly deny that!

Yes, I got that. Some can teach themselves.

But some cannot. I'm saying that the some who cannot are important because they mean that the *definition* of PG is NOT "A child who can learn things they've never been taught."

You're stressing the "can," and I'm stressing the "cannot" because I think it's potentially dangerous to stress the "can" for the kids who "cannot."

But of course some can. And it's cool that they can.

[quote=Mia
With reading, it's fairly cut and dry--they can read it with comprehension and answer questions, or they can't. Of course, there's the issue of inference too, but I'll skate over that. :-)

With math, I think there's more gray area since math is essentially its own "language." If no one teaches you the notation, you're seriously limited.

[/quote]

I tihnk that's true with written math, but I think the math mental component is the same as in reading. For those really early self taught readers someone probably showed them the letters of the alphabet, then they were able to make inferences on their own. With math, there are probably a lot of young unschooled kids that could answer a math problem if given verbally, but would still need to learn the "alphabet" of math in order to write it down.

So yes, you would eventually be seriously limited without learning the notation, but you could still be extremely "math-able"

Yes, we're fighting on the same side here, just from a different angle!

I'm coming from my angle because my children have had more difficulty proving that they "can" learn something in school without being taught than the other way around. DS11 spent an entire wasted year in pre-algebra because the school did not understand how he could have grasped the concepts without going through the instruction. Not even a placement test could convince them that he had mastered a class that he had never taken. My youngest would tell me what the 6% sales tax was on a purchase when he was 4 years old (and therefore, by inference, could easily make change from a dollar!) but still the school did not beleive that he could have learned anything like that before ever entering school ( and therefore spent weeks teaching him the values of quarters, nickels and dimes). I have never had a problem with a school thinking my child was so smart that they didn't need to spend days and days, weeks and weeks, teaching the same concept, LOL!

So that's my angle, but I understand yours, too. There's all kinds of flavors in the upper end of the bell curve.

We're usually always in agreement here. We just like to hash things out, lol!

My DS7 in particular just has real confidence issues. He actually knows ahead of time a lot more than I expect him too, but panics immediately if he sees something in a format he hasn't seen before and shuts down. Of course, 5 minutes later he might be breezing through the same thing like he's known it forever.

I do agree about the mental math component. I think I'm not actually teaching DS anything about math. More about notation and showing his work.