International comparisons show that American schools do a mediocre job of preparing students in mathematics, which in practice means preparing pupils in basic arithemtic. Here in Colorado, "math" preparation will probably show an improvement as soon as the CSAP tests can be made easy enough. The examples of CSAP questions that I have already seen have been very easy. Nearly all "math" questions concern numerical calculations with whole numbers, and can be solved by the application of rules and formulas.
In the article that prompted these comments, it is reported that "math" educators (I cannot help putting "math" in quotes in this connection) met in Washington and identified a few topics upon which they could agree. These recommendations were: (1) heavy reliance on calculators in the early elementary grades is a bad idea; (2) elementary school children must have automatic recall of number facts, meaning that, yes, they have to memorize multiplication tables; and (3) children must master basic algorithms.
This is a rather stupid list, but typical of American educators, and I rather disagree with each assertion. The first one should be: "mental calculation should be emphasized in all grades." As I point out elsewhere, the old process of algorism, computing with pen and paper, is so slow and error-prone that it should be considered a curiosity. For serious calculation, electronic calculators are superior, and their use is essential knowledge. Algorism is popular only with those who do no calculations, like elementary school teachers. With bankers, engineers and scientists it is seldom used. Mental calculation is a very different thing, and can be quite valuable.
Recommendation (2) shows what educators mean by "number facts"--that is, rote learning. This memorization, of course, is only in support of algorism, for which it is essential. I actually learned my multiplication tables up to 10x10, which is not far enough, but am clumsy with addition and subtraction tables. The reason for this is that in the 1st grade I could calculate faster by other methods, since the standard was low. This has not proved a very important disadvantage. As far as I recall, no efforts were made to foster mental calculation at all in my schools.
Recommendation (3) is another appeal to rote learning. That is, give recipes to solve certain common problems and drill the pupils in them. This, of course, makes "teaching" easy and engenders confidence in pliant students, since it can be mastered by nearly everyone. However, it discourages understanding. If (3) were "children must master the elements of proof (or deduction)," things would be much better. However, this is probably beyond most students, so reliance on "algorithms" is much safer.
Nearly every pupil can be prepared to do easy mental calculations, to work with fractions and percentages, to operate a pocket calculator, and to appreciate certain simple geometric relations (areas, Pythagorean theorem, and such). The failure of American schools to provide this modicum is shameful. It is also shameful to fail to introduce capable students to real mathematics. How there can be any "battles" over such simple requirements is incomprehensible.
The Denver Post, 24 December 2004, p. 6A. "Ongoing math battles offer few answers," by Valerie Strauss, Washington Post.
Composed by J. B. Calvert
Created 25 December 2004