... I recently read a paper which found that for children who do understand abstract concepts, that physically representing them with rods etc can actually be limiting. I wish wish wish I had written it down - I have no idea where I read it now and I can't find it again.
Not the exactly same thing but quite related ... when DS5.2 went into public preschool for the first time shortly after turning 4 he went in being able to do basic addition to 20 (though not 100% correct yet) and basic subtraction within 10 and no more than two or three weeks into school he started coming home telling me he doesn't know any of it because he hasn't learned it yet. They were using counters to learn basic counting and he would tell me he can't count in his head ... he has to use counters or other objects to count. Long story short, he seriously regressed in math while in preschool. His love for math came back once preschool was over. Now he's in K where they are learning counting objects to 10 and finding one and two more or and one and two fewer objects. I made it a BIG thing for him to know that what he's doing at school is one thing and what he's doing at home because he's just curious is another thing and it's perfectly ok to know more than what the school is asking of him. So two weeks ago he was playing with double digit numbers, last week he started to ask me about hundreds and yesterday he was all about thousands! lol He can't make calculations with those big numbers but he absolutely gets the point. So I 100% believe that the ways that might work for most kids can be very limiting for kids like ours who just "get" things.
