A Solid Foundation, Revisited

As an example of what is implied when people get nostalgic over the way math "used" to be taught, before all this crazy New Math and so forth, when kids learned the "basics" and didn't futz around with silly theory that nobody understands, etc., and got a "solid foundation" in facts, and walked up hill to school both ways in the snow with no shoes -- both of my parents learned to multiply using the multiplication table without being told that multiplication represented a short cut for repeated addition. I have no idea how common this was, but I thought it interesting that it was true for both of them, who attended school in different states.

You know, I agree that it's a useful skill to quickly and accurately come up with the fact that 8 x 9 = 72. Less useful than it was in the pre-calculator/pre-Excel/pre-etc. era, but still useful. Of course, it seems to me awfully useful to know why 8 x 9 = 72 also. I mean, isn't memorizing the multiplication table a lot easier when you realize that 8 x 9 = 2 x 4 x 9 and that 8 x 9 = 8 x 8 plus another 8? When I was learning my table, there were times I couldn't quite remember this particular number (I knew it was something in the range of 71 - 74) but I was able to figure it out again on the fly because I did know that 4 x 9 = 36 and I knew how to double that number. I also knew that 8 x 8 = 64 and could add 8 to that if I forgot 4 x 9 during the same brain freeze. I had a lot of different ways to approach the problem.

I don't see how it's to any kid's advantage to learn that 8 x 9 = 72 in the same way that they learn that "in 1492, Columbus sailed the ocean blue." It gives the impression that all these numbers are what they are through some mysterious process or accident of history while in reality, there's a lot of sense to it. And in my opinion, it is (I hesitate to use this word, but I think it fits) empowering to realize that you are dealing with a sensible system in which things can be figured out using rational thinking. Once you learn why 8 x 9 = 72, you can figure out what is 16 x 9 pretty easily, and 16 x 27, and well, a whole lot of numbers, without ever having to be told by someone else or memorizing a bunch of things.

I don't mean to imply that every person who talks about "solid foundation" really wants to return to an era when children did not know what multiplication means. But I wonder how many people (and here I mean normal people in the general population who are reactant against all stripes of innovative "New Math"-esque curriculum, not math educators who are arguing about these issues at a much higher level and with a lot more knowledge than the rest of us have and who would rightly view my comments as knocking down a strawman in terms of where the informed debate is occuring) who yearn for that supposedly wonderful pre-New Math time realize/remember quite how fact (and not theory) focused the instruction really was. Perhaps some of them would not even recognize the idea of multiplication as repeated addition to even be "theory" now that it's so commonly taught.

I was lucky to learn this stuff during the post-New Math era and to have a dad who thought it was fun to teach me algebra when I was quite young. (He also bought me and worked with me on geology sets, chemistry sets, and electricity sets. The fact that I did not grow up to be a scientist or engineer is not through any lack of exposure, sexist or otherwise. I joke to my younger sister the applied math and computer science major that where math, science, and computers are concerned, I peaked early.)