I’m very impressed by those of you who are having your kids do 100 problems a day, or whatever. We do math almost every day, Monday through Friday, but not usually for more than 20-30 minutes (at the very most), often closer to 15. I demand focused attention during this time, however. I think the most we ever do is, like, 60, and often it’s more like 30.
We use Singapore Math half the time and Two Plus Two Is Not Five (this is what involves many problems per day) the other half. My wife isn’t very supportive of of the 2+2=/=5 addition drilling, because H. doesn’t like it very much. She thinks that he’ll pick up the addition facts by osmosis, while he will have to study multiplication facts. I am not so sure. Lots of people go to adulthood counting on fingers, and are greatly hampered by not having even basic addition facts memorized. One way or another, a little math student has to have these down cold, and the colder, the better.
2+2=/=5 appeals to me because it teaches addition facts in more or less the same way that I remember addition facts. They are divided into groups of tricks, which are not merely tricks but actually quite valid ways of thinking about what’s going on in a certain kind of addition and subtraction problem. Combining this systematic, rule-based (read: extremely left-brained) approach with plain old drilling tends to get the facts memorized. In the book’s method, it’s actually essential what order you learn the addition and subtraction facts in. You start with the simplest, and you also start with the math facts that involve the simplest-to-use rules/tricks. I don’t actually know if this is important for memory, but it does seem to help H., and it seems to be an important part of the method the book authors designed. If that’s right, then it makes it a lot easier to learn the math facts in a specific sort of order.
The other feature of the book which I didn’t appreciate until the last couple weeks is how important it is to do plain old drilling. That’s why they keep drilling the same problems over and over again, in different orders and groupings and so forth. Basically, as long as you do most of the problems and don’t take too long of a break, the student isn’t given an opportunity to forget facts (and rules) once learned. Learning facts for the first time isn’t easy, and getting them down cold is probably harder, but once they are in long-term memory, at least for H., they seem hard to forget. For some reason, he has his doubles down better than any other problem type (1+1 through 5+5 so far). He can usually do +1 and -1 problems quickly, though sometimes he still has to think about these. +2 even problems (we haven’t learned +2 odd problems) are more challenging but still doable. Now he’s started “number in the middle”: you figure 3+5 by seeing that there is just one number in between 3 and 5, and you double that (he has 4+4 memorized very well). There are other complementary rules which make other facts doable. Sample: http://longevitypublishing.com/sample.pdf
I wish I knew all the different methods and could pick the best one. I’m not at all convinced that this is the best one, but it seems to be effective, even if it requires quite a bit of effort. If H. had any less patience, and he doesn’t have much, we wouldn’t be able to do it.
In other math learning news, by the way, H. is now on the last chapter of Kindergarten Math B, so the combination of A and B will have taken us, let’s see, not quite two years or so. That’s OK with me. He’ll be finishing just before he’d start kindergarten (that would be this fall, but we’re homeschooling). We went through long periods (sometimes months long) in which we didn’t work out of the books (basically, I didn’t want to push him), and other periods where we did only one or two lessons a week. He has certainly progressed well, although to be honest he doesn’t always like math. He actually has liked many chapters (not all) of Singapore Math so far, but he resists the more difficult drilling. (So far, though, this has gotten easier–it feels like we’ve gotten over a hump and are pedaling a little easier now.) He gets motivated by external rewards, doing hands-on math (we have a tub of “connect-a-cubes” which are very useful; today, we played with money and he looks forward to spending some), and now, he’s all excited to finish the book, so he wants to do eight pages tomorrow, and then we’ll be done. We now have Singapore Math 1A textbook and workbook, and I’ve looked through it. As I suspected (and as I wrote here), Primary Mathematics 1A is really mostly a review of Kindergarten Math A & B, so we might end up skipping past a lot of it. 1B definitely introduces some more advanced concepts (basic multiplication and division).
The “goal” of math is easiest to conceive (for me, anyway) as getting through a long series of math textbooks and understanding what they contain very well. The thing is, especially if you’re homeschooling, you can’t depend on one textbook or workbook (series). In addition, you have to identify the problem spots and attack them directly, as well as making use of “teachable moments” of practical math, which help solidify things. Both of these require that you pay close attention to how your child is thinking about math, what tricks he uses and exactly what he does or does not understand. Also, taking breaks of days or maybe weeks strategically can really boost motivation.