I agree completely that deeper not faster is the ideal. But I also think that at younger ages, that's a lot harder to do.
We did the challenging word problems and the intensive math to ramp up the problem solving and slow down the speed. It was okay for a while, but there are only so many ways you can add 2+2 or 55+78 or whatever. Eventually, even that "challenging" stuff just isn't very interesting. Same stuff, different day, you know?
As long as the kids are still only getting calculation and haven't been allowed to dig into concepts (beyond the basics of sets and arrays and that sort of number concept stuff that kids do get in 1st or second grade), I really haven't seen much deeper you can go. (And I've been looking!) A little deeper than just a page of problems, sure, and every little bit helps. I am certainly a firm believer that problem solving is the way to approach pretty much *ALL* math. Word problems and tricky thought experiments are GREAT! But I'm not sure I believe that you *can* effectively go a whole lot deeper than the norm in early math without going pretty far from the normal ways to teach math. Calculation is calculation is calculation...
OTOH, so far geometry is a smashing success for DS6. Seriously, it's been tremendous! He *adores* doing the thought experiments that geometry require--and I'm approaching it from a problem-solving/experimental angle, so he has to figure stuff out for himself--and it seems to be precisely where his head is right now. He just shines when we work on it.
As for next year...Well, we'll see how things go with our goofy physics/calculus experiment. DH and I are kind of excited to give it a try, and if the geometry work from this year is any indication--and that seemed pretty outlandish when we first considered it--we may be on to something for our particular kid, But at least I know that what we're planning WILL be problem solving, with actual, real world problems, and that it WILL be deeper, conceptual math, not merely a faster sheet of the next kind of problems.
I'm babbling...Can you tell that I'm thinking a lot about next year's plans?
