But the weaker students would surely benefit from more repetition in some basics of any of the math concepts (long division, fractions, etc).
Well...I think it's like Texas Summer wrote some time ago: there are 2 sets of skills you need for math--the "tools," which are the rote knowledge of math facts, and the "concepts" which are more like the way things are done and why you're doing them. Two separate parts, and often two different ways of learning the parts.
Repetition is usually required for memorization of math facts. In our case, DS6 has mastered virtually every part of the *concept* of even multiple-digit multiplication--that it's about multiple sets of like groupings, carrying, etc.--but he currently lacks the *tools* because he's not yet ready to memorize his times tables. He needs both to progress (or at least a good work-around until he does the memorization). Repetition would/will help him learn those tools when he's ready.
OTOH, repetition was NOT necessary to help him with the concepts. He picked those up right away. Some kids can probably come to understand the concepts through repetition of the tools. But I don't think that this is most kids. I do think the way math concepts are usually taught is to have the class do multiplication problems ad nauseum and hope the child gets why things are done that way. But I do NOT think it is very effective.
For example, when my father was a very young child, he could do multi-digit math problems of all sorts in his head, giving the answer the second you finished saying the problem. Wow, right? But he did it backwards, starting with the 1000s (or millions, or whatever) place instead of the 1s place. When he got to school, the teachers saw that he was using the tools "wrong" (never mind that his conceptual grasp was ideal and that he always got the right answer fast!), so they drummed it out of his head. He can't do it anymore. They focused on the tools, not on the concepts.
For my part, I don't recall anyone explaining to me why we did math, say long division, the way we did. We were just expected to "follow the leader." The better you were at imitating the teacher exactly, the better your grade. When I got to calculus, I was a goner because we were expected to think, not to imitate, and I had no idea how to do that in a math class. It had never been asked of me before.
Of course, these are non-NCLB examples--two old ones and one from home school. Maybe things in school are different now? I just still don't understand what, exactly, is different. Please forgive my denseness, but I really would like to know!
K-
