Math-teaching anecdote from chez Derb. Nellie is doing factorization of quadratic forms–factorizing, for example, 2x^2+5x-3 into (2x-1)(x+3). She had done all the homework exercises, but had a test next day, so came up to my study to beg me to write out some more exercises she might try.
I was back in her room approx. 90 seconds later with my 1926 edition of William Hart’s College Algebra open at pages 18-19, which contain about 70 exercises of this sort. I had marked up a dozen as suitable, and Nellie set to.
These old-style textbooks are indispensible in situations like this, and every house with kids should have a few. (You can buy them for next to nothing on Abebooks–don’t buy anything that costs more than the postage!) They make no pretense that math is “fun” or “relevant” to anything. They just give you pages and pages and pages of drills.
Here is Steven Pinker in his book How the Mind Works, pp. 341-2:
On evolutionary grounds it would be surprising if children were mentally equipped for school mathematics. These tools were invented recently in history and only in a few cultures, too late and too local to stamp the human genome. … The second way to get to mathematical competence is similar to the way to get to Carnegie Hall: practice. … Calculus teachers lament that students find the subject difficult not because derivatives and integrals are anstruse concepts–they’re just rate and accumulation–but because you can’t do calculus unless algebraic operations are second nature, and most students enter the course without having learned algebra properly… Mathematics is ruthlessly cumulative, all the way back to counting to ten. … Evolutionary psychology has implications for pedagogy which are particularly clear in the teaching of mathematics. American children are among the worst performers in the industrialized world on tests of mathematical achievement. They are not born dunces; the problem is that the educational establishment is ignorant of evolution.. … Setting our mental modules to work on material they were not designed for is hard. … [W]ithout the practice that compiles a halting sequence of steps into a mental reflex, a learner will always be building mathematical structures out of the tiniest nuts and bolts, like the watchmaker who never made subassemblies and had to start from scratch every time he put down a watch to answer the phone. Mastery of mathematics is deeply satisfying, but it is a reward for hard work that is not itself always pleassurable.
Although, in defense of U.S. math teachers, I’d note that we don’t do that badly on international comparisons. In this year’s International Math Olympiad, the U.S.A. team came in 5th among the 90 participating countries. I had a note on this in my September Diary, and a link to a picture of our team and their coaches.