If you're wondering why your kids aren't picking up how to divide fractions, it could be that their teachers don't quite get it, either.
That's the message from "Knowing and Teaching Elementary Mathematics" (Lawrence Erlbaum), a book that is becoming a stealth hit for math junkies on both sides of the "math wars," and a must read for anyone interested in solving the problems of public schools.
The key insight in the book is that what counts isn't whether the kids sit in rows or face each other at tables. Or whether the teacher adopts to New Pedagogy X or Y. What counts is whether the teacher really knows math. It's that understanding that makes a classroom genuinely helpful to kids.
Author Liping Ma was an eighth-grader in Shanghai when she was ordered to a poor village in south China to work in the fields during the Cultural Revolution. Instead of providing "re-education" for the kid from the city, illiterate villagers asked her to teach their children. Baffled by the huge needs she saw in her classes, she sought out other teachers for help, then taught herself English so that she could read education classics, including "anything by John Dewey." Seven years later, she was promoted to principal, then county superintendent of schools.
After the Cultural Revolution, she was too old to go to college, but succeeded in passing exams to get into grad school at East China Normal University. She arrived in East Lansing, Mich., in 1988, with only $30 and an acceptance letter to Michigan State University to study education. It said that the school would waive the in-state tuition fees. What she hadn't understood was that she would still have to come up with out-of-state tuition.
"I worked at the school cafeteria and doing housekeeping at a hotel," she recalls. Finally, a faculty member helped her find a research assistantship at the school of education - a project that led her to the subject of this book. The project, under the direction of Michigan State Professor Deborah Ball, involved coding data from a national survey of elementary school teachers. The goal was to determine how well the teachers understood elementary mathematics. Not very well, it turned out.
"The first thing I found was that among the 136 teachers, I coded only one who could correctly describe the meaning of a question involving division by fractions," she says.
Most respondents thought that dividing 1 3/4 by 1/2 was the same thing as dividing by 2. Moreover, most were confident that they understood elementary math very well.
Michigan State gave her a $1,000 grant to return to China and pose the same questions to Chinese elementary schools. A big difference, she noticed, was cultural: "Most American teachers said, while they did not know advanced math, elementary math was simple; they already knew it, and the only need was to learn how to teach. But Chinese teachers thought that they still needed to learn about the subject - not only about how to teach. They saw teaching as a way to learn more about math."
In fact, US elementary school teachers appeared better educated than their Chinese counterparts. American teachers were all college graduates, while Chinese teachers had the equivalent of a high school education. Yet they consistently failed to demonstrate the grasp of basic math concepts that were standard in a Chinese classroom.
That difference shows up in the simplest problems. "We can't subtract a bigger number from a smaller one," said one US teacher in explaining how to solve 62-49=13.
That's a false mathematical statement, which could be confusing to children down the line, says Ms. Ma. "In fact, young students will learn how to subtract a bigger number from a smaller number in the future. Although this advanced skill is not taught in second grade, a student's future learning should not be confused by emphasizing a misconception," she writes.
Another misleading - and common - technique for helping US kids solve this problem: the 2 "borrows" 10 from the 6. "It suggests that the two digits of the minuend are two independent numbers rather than two parts of one number," she says.
In China, children learn that subtraction involves "decomposing a unit of higher value," much as on an abacus, one takes a bead on a left wire and changes it into 10 beads on the right. It's a subtle but important point, and helps kids relate subtraction (decomposing a unit) to addition (composing a higher value unit).
Indeed, Ma says, language is a key defining difference between American and Chinese teachers. "American teachers ... speak like a lay person," she says. "Teachers with an understanding use math terms that would make the argument more clear...."
US teachers aimed to teach kids correct procedural knowledge, while the Chinese taught problem-solving strategies. For more-complex problems, such as dividing by fractions, most US teachers didn't even get the calculations right.
In America, there's a "vicious circle formed by low-quality mathematics education and low-quality teacher knowledge of school mathematics," she wrote. "In US elementary schools ... regarded as very good ... I saw such obvious math mistakes on the board...." she says.
Ma's work has gotten attention on both sides of the math wars. No matter which side "wins," she says, they ignore the need to improve teacher knowledge of math. "Math is ... not a mystery that cannot be solved.... I believe that anyone can learn math. The problem is how we teach them. We have to build math concepts and skills step by step."
(c) Copyright 2000. The Christian Science Publishing Society