“Discovery learning” advocates claim Japan’s high-scoring math students come up with original answers without being guided by the teacher. That’s not so, says Alan Siegel, a computer science professor at NYU who’s studied the videotapes of Japanese math lessons. Columnist Linda Seebach explains:
The eighth-grade geometry lesson Siegel discusses is based on the theorem that two triangles with the same base and the same altitude have the same area, and it is framed in nominally “real world” terms as a problem in figuring out how to straighten the boundary fence between two farmers’ fields so that neither farmer loses any land.
. . . The teacher first primes the class by reminding them of the theorem, which they had studied the previous day. Then he playfully suggests with a pointer some ways to draw a new boundary, most of them amusingly wrong but a couple that are in fact the lines students will have to draw to solve the problem (though they aren’t identified as such).
Then he gives the students a brief time, three minutes, to wrestle with the problem by themselves, and another few minutes for those who have figured out a solution based on his broad hints to present it. Then he explains the solution, and then he extends the explanation to a slightly more complex problem, and finally assigns yet another extension for homework.
As Siegel describes it, “The teacher-led study of all possible solutions masked direct instruction and repetitive practice in an interesting and enlightening problem space.
“Evidently, no student ever developed a new mathematical method or principle that differed from the technique introduced at the beginning of the lesson. In all, the teacher showed 10 times how to apply the method.”
A U.S. Department of Education report claims Japanese students devise their own solutions to “mathematics problem employing principles they have not yet learned.” Siegel says analysts who watched the videos were poorly trained. They came up with “10 student-generated alternative solution methods, even though it contains no student-discovered methods whatsoever.”
Discovery learning is fashionable in math reform circles, writes Seebach. The Japanese are supposed to be the models. But the Japanese teach traditionally — with “beautifully designed and superbly executed” lessons.
The videotape shows, Siegel says, that “a master teacher can present every step of a solution without divulging the answer, and can, by so doing, help students learn to think deeply. In such circumstances, the notion that students might have discovered the ideas on their own becomes an enticing mix of illusion intertwined with threads of truth.”
We’re short of master teachers, especially in math.
Overall, students had become weaker in nine of the 11 areas that the survey asked about. The most striking declines in students’ scholastic aptitude were in the ability to calculate, which did not feature in the society’s survey in the mid-1990s, and in logical expression. The most alarming deterioration was among students at teacher-training universities.
That bodes ill.