>>>>>>>>>>>>>. . . . .(snip) . . .(snip) In summary, to my opinion, there are pure math and applied math, and each of them have algorithms and concepts. In general, each of them can help the development of others, and they are equally important. For students who have unbalanced growth of these abilities, teacher should use their strength to help them develop other parts. But sometimes, it is very individualized especially for some very talented kids. One should not conclude that the kids is only an algorithm machine just because their practical number sense develop slower. Similarly for kids with strength in practical sense but slower in abstract number sense, we can help, but sometimes we just have to be patient, with constant encouragement and appreciation toward their talent.
Very well said, in my opinion. As Piaget has shown us many kids are just developing their abstract thinking abilities at the same time we are trying to teach some very abstract concepts. No wonder many would prefer we just give them the algorithm right off the top and let them do their homework!
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