As a mathematician, I just wanted to post some random thoughts after reading this thread.

I decided to major in math because it was the easiest subject for me--nothing had to be memorized! I am not good at rote memorization but I could "see" how to derive formulas from first principles. I did learn my math facts in grade school but it wasn't exactly by rote. It was more that I repeatedly thought about them until the answers were clear. For example, 7+8=15 because 8+8=16 and 7 is one less than 8. This process builds upon itself--i.e. at some point I had to become convinced that 8+8=16 before I could use that to conclude that 7+8=15. After a while, I felt that I could skip the reasoning part and just go straight to the answer.

I remember having "aha" moments like understanding long division as repeated subtraction and how to find the area of a triangle. I was not discovering these things, however. The teacher was presenting that material. Still, there was a moment where I "got" it.

In order to get a concept, a student has to make a habit of understanding each step in the process. If we try to replace understanding with rote learning it introduces a gap in the chain of reasoning. Rote learning can be a useful tool for increasing math fluency. But it should never be a substitute for "getting" a concept.

Lately, I have been teaching third graders fractions. Their teachers had told them that a fraction represents a certain number of parts out of the total number of parts. This is true, but it doesn't seem to be intuitive to third graders. My approach is to look at math like a language to learn. When we cut something into pieces we give those pieces a name depending on how many pieces we made. (You would be surprised at how many kids are not making this connection.) I.e. if we cut something into 4 equal pieces we call each piece a "fourth". In math language we write that as "1/4". "A" means "1" and "/4" represents "fourth". Now if we have 3 such pieces we say we have "three fourths". That is written "3/4". This lays down the foundation for understanding how to add fractions. Now that we really "get" what a fraction means, the only thing that makes sense is to count up how many of each kind of piece we have by adding the numerators. If we need to cut some of the pieces into smaller pieces so that all the pieces are the same size, it makes sense to do that.

Now many of your kids are already beyond this kind of thing. My point (I think I have one smile ) is that verbal reasoning can be used to understand math as a language for representing real problems. This is SO important for kids to understand. The way math is taught in school you would think that the math and verbal domains were completely seperate.

I think that GT kids have the ability to intuit this connection. Mathy kids don't need to have things translated for them this way. Exposing kids like this to math is like immersing them in a foreign language. They will soak it up. Exposing them to calculus at a young age is like letting them read books with big words in them. They may not understand them right away but that's ok.

My son seems to be a mathy kid and he LOVES that Descartes' Cove program. He can't solve the problems on his own but it is still a good teaching tool because it is exposing him to what is possible. My own dad did stuff like that with me, like showing me how to use his slide rule, teaching me about logarithms and exponents and teaching me Newton's method for approximating square roots. We also did set theory and Venn diagrams which I loved. No, I did not completely get the stuff he was teaching me until I was older but I think that early exposure was very valuable and helped to lay down pathways in my mind for future understanding. It also whetted my appetite for more math! I teach a mathlab at my kids' school where my main goal is to expose the kids to stuff beyond arithmetic. This kind of enrichment is beneficial to kids at all levels--without it, math just seems like an arithmetic wasteland to them and they lose interest.