Rodney

I've been working with some elementary school kids on math; they're not calculator users, and their math skills are still lower than I expected.

I took a not-very-good math education course recently, but one good point that it made was that what kids really need to develop is a "number sense:" the almost instinctive knowledge of how to break numbers down into other numbers, so that you can "slice and dice" the numbers in any problem in your head to make the problem solvable. For example, if asked to multiple 4 times 95, you might:

*Note that 95 is 5 less than 100; so, multiple 4 x100 (a number fact that everyone is taught), then multiple 5 x 4 (the different between 4 95s and 4 100s, and subtract).

*Know from your multiplication tables that 9 x4 is 36, so that 9 x 40 must be 360; then multiple the extra 5 x 4, then add both products.

How do you get a number sense like this? Some kids develop it themselves (I did), but you also get it from working on problems without the teacher providing a set technique; what a teacher might do is divide a class into pairs and small groups and say, "solve this" and let them all go at it. This is not the only component in a good math education, but it's a necessary supplement to teaching them the standard algorithms for addition, subtraction, multiplication, and division. If they just learn them by rote, without understanding why the numbers move as they do, they're still math illiterates. Some kids can add three digit numbers just fine and _still_ not be able to tell you why they're "carrying the one." That's just what they've been taught to do.

Anyway, the course went on to say that in other countries, such as Japan and England, math is taught at least partly with the problem-solving approach, and it leads to better mental math skills. So I went over to talk to my then-neighbor down the block, a Brit who works as a graphics artist. He never took math past age 12 or 13. But he did have the kind of problem-based math training that I was asking about -- said it was fairly common in England -- and said that he and most of his Brit friends could do problems in their head that his American buddies had to drag out the calculator for.

Of course, the fact that they can't even do it with paper and pencil anymore is another shame. But the math course did say that if _all_ you learn about math is the algorithms, and not the number sense, that forgetting the details of the algorithms means that you can't do the math anymore, even with paper and pencil. You don't have enough background knowledge to fill in the gaps in your memory and remember how the algorithm _must_ work. I've probably retaught myself long division on paper two or three times, because although I might have forgotten the details, I remember how it's supposed to work and thus can infer those lost details.