By the Numbers
Have you run across "repeat-subtraction" yet? It's a generational question, because repeat-subtraction is actually a way of doing division - a way no one much older than this generation of fifth or sixth graders (math teachers excepted) is likely to have experienced.
Unlike traditional long division, this approach considers the dividend (the number to be divided) as a whole and subtracts from it a large multiple of the divisor (say, 10 times the divisor). The process continues, with progressively smaller multiples, until the subtraction ends in zero, or in a remainder smaller than the divisor. The quotient (answer) is obtained by adding up the numbers used to multiply the divisor before each step of subtraction.
To parents raised on long division and its assiduously learned estimating of how many times one number goes into another, this is Greek, not math. But, in fact, it is math - or at least a math pedagogy that emphasizes different strategies for arriving at an answer. In the case of repeat subtraction, the strategy is to get students to see the relationship between subtraction and division - just as teachers have long drawn the link between addition and multiplication.
But for adults trying to help with homework, appreciating these new strategies, and leaving the old ways behind, can be a tough assignment. Some communities have engaged in virtual "math wars," especially since a list of standards published by the National Council of Teachers of Mathematics has come into wider use.
The standards recommend, for example, that students work in teams to solve problems, make use of calculators and computers, and learn how to explain their answers. (The paragraph above, explaining repeat subtraction, indicates how hard that can be.)
Underlying the council's standards, and other updated approaches to math, is a conviction that past methods based on repeated memorization of facts and processes don't meet today's demand for people who can reason mathematically. We're willing to go some distance down that road, but we also suspect that reasoning in mathematical terms is a little like learning to reason in a foreign language. The structural building blocks, the mathematical facts and processes, have to be firmly in hand. (And, in fact, the council strongly endorses knowing the basics, like the multiplication table.) Ultimately, some people - because of inclination and career orientation - will attain much greater fluency than others. Yet all need a grounding.
Reasonable parents, and educators, appreciate this. After all, many of us can attribute what math know-how we have to that patient fifth-grade teacher, that demanding high-school instructor, or perhaps that game mom or dad who got us past computational logjams and communicated the exhilaration of getting the numbers right.
Today's kids may be learning different ways of using numbers to solve problems. But the crucial word, certainly, is learning. There are usually multiple approaches to solving a problem, and if today's young numbers-smiths grasp more of these than previous fifth graders, that can't hurt. But let's keep a close eye on how they do when it's time to step up to algebra, geometry, calculus - or simply giving correct change on the job. That's the final exam.