Mathematics in ?depth?

'Depth.' Say the word and mathematics researchers nod their heads.

Other words, particularly 'constructivist' and 'drill,' are apt to spark controversy, but most educators agree on the need for 'depth' in American math teaching.

By depth, they mean considering math problems from many different angles, inventing new ways of solving problems and discussing strategies for tackling ambiguous problems.

'I show kids there are a number of strategies for working the problem,' says John Phillips, a middle school mathematics teacher in Virginia?s Arlington Public Schools. 'That way they can try different ways and pick the one that works best for them.'

The idea of depth garnered attention when research revealed that American math curricula cover more concepts than curricula in other countries. As a result, the teaching coverage tends to be shallower, according to a study by psychologist James Stigler, PhD, and several colleagues.

As part of the Third International Mathematics and Science Study, Stigler and his research team videotaped American, German and Japanese math teachers instructing eighth-graders, then analyzed the results.

While American teachers raced through math skills, Japanese teachers fostered in-depth understanding, the researchers concluded when they released the results in 1995. Japanese classes sometimes spend a whole class period trying alternate ways to tackle one problem, the researchers found. The study spawned a popular criticism that American teaching is 'a mile wide and an inch deep.'

Cognitive research suggests students need more grounding in mathematical analysis. For example, a study of elementary school students by psychologist Robert Siegler, PhD, indicates that children learn by working through the reasoning in problems?telling the teacher what makes them right or wrong, for example. The study appeared in Cognitive Psychology (Vol. 28, No. 3, p. 225?273).

'If kids can explain why the wrong answers are wrong and the right answers are right, they?ve really learned something,' says Siegler, a math researcher at Carnegie Mellon University.

Siegler says the American educational system falls short of teaching for understanding because it undervalues, underprepares and overworks teachers. 'It?s ironic that North America has the highest quality math research but very poor quality math instruction,' says Siegler. 'We need to take the research more seriously.'

?Bridget Murray