We finally got a chance to read Liping Ma's Knowing and Teaching Elementary Mathematics, and Wow, what a book! This book (Ma's PhD thesis) used interviews with U.S. and Chinese mathematics teachers to compare approaches and understanding about basic elementary mathematics: subtraction with regrouping, multidigit number multiplication, division by fractions, and explorations of new knowledge. Her findings have direct and practical implications for teachers, parents, and tutors.
- U.S. teachers tend to focus on procedures rather than understanding the conceptual foundations of basic math operations(in one testing situation, 83% U.S., 14% Chinese used procedural knowledge only)
- Impaired conceptual knowledge impaired U.S. teachers' abilities to understand student mistakes and apply math reasoning to new problems or scenarios.
- U.S. teachers' use of manipulatives could be misapplied, for instance, removing the need for a child to understand regrouping, rather than illustrating principles regrouping or borrowing.
-U.S. teachers showed less flexibility and alternative problem solving than their Chinese counterparts.
-U.S. teachers seemed more likely to introduce entertaining or visually appealing aspects of math calculations, but sometimes at the expense of accurate conceptual teaching.
Example: "One thing I would do is....put either an apple, orange or whatever, in the spaces...I mean it could be some weird thing, even pictures of elephants. I do not care what it was. But the children memorized this and they said, oh I remember that [my teacher] said do not put anything there because that is where the orange was or that was the apple...Just put something different there so that it will hit their eye."
- Some Chinese teachers regularly showed a math problem with errors on the board, and challenging students to 'find the problem' and 'summarize the rule'. A discussion would then follow about the correct underlying concept.
- Only 43% U.S. math teachers arrived at the correct answer to: 1 3/4 divided by 1/2 compared to 100% of the Chinese teachers. Problems included over-reliance on a mneumonic (without conceptual understanding), a lack of alternative ways to solve the problems, and limited analogical understanding. For the latter, for instance, U.S. teachers tended to rely on pizza-type examples for fractions, but became stumped with dividing by the 1/2.
- Finally, U.S. teachers also fared poorly in a scenario of a student suggesting a novel theory that increasing the perimeter of a closed figure would necessarily increase the area. 9% accepted the theory without a doubt, 78% would not have conducted any mathematical investigation ("I'd try to look it up in a book"), and only 13% would have investigated the claim. Only 1 teacher out of the 23 was able to solve the answer. In comparison, of the Chinese teachers, 8% accepted the claim outright, 70% arrived at the correct answer, and the remaining at least tried with mathematical reasoning, but solved the problem incorrectly.
Clearly we have a long way to go with teaching conceptual foundations in math. Bravo, for the math teachers at NCTM for taking a hard look at the book, and considering reforms.
NCTM: Comparing U.S. and Chinese Elementary Math