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August 16, 2004
American children come off badly in international comparisons of mathematics performance, and they do worse the longer they're in school.
One such comparison, the Third International Mathematics and Science Study, tested more than 500,000 children in 41 countries, starting in 1995. As part of the study, researchers videotaped more than 200 eighth-grade math lessons.
These lessons have been studied intensively in an effort to figure out why Japanese students do so well in math while American students do so badly. Alan Siegel, a professor of computer science at New York University, has reviewed the videos and calls the teaching "masterful."
He also believes that many of the TIMSS studies contain "serious errors and misunderstandings." If you have doubts, he says on his Web site, "go review the tapes and check out the references. After all, that's what I did." (www.cs.nyu.edu/faculty/siegel/) His paper also appears in a recent volume of essays on testing published by the Hoover Institution, Testing Student Learning, Evaluating Teacher Effectiveness.
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. This is, of course, highly relevant to urban Japanese youngsters, who are likely to be called upon frequently to accomplish this task.
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."
But that's not the way the lesson has been described in the literature. A 2000 commission report from the U.S. Department of Education, Before It's Too Late, gushes that in Japan, "teachers begin by presenting students with a mathematics problem employing principles they have not yet learned. They then work alone or in small groups to devise a solution. After a few minutes, students are called on to present their answers; the whole class works through the problems and solutions, uncovering the related mathematical concepts and reasoning."
How could Japanese children solve problems based on "principles they have not yet learned"? Why, in the same way that Meno's slave solved a mathematical problem on the exact same day that Socrates happened to be asking him questions.
As to how this confusion might arise, Siegel notes that a report by J.W. Stigler and others for the National Center for Education Statistics, The TIMSS Videotape Classroom Study, uses this very lesson as an example of how their data analysts were trained to identify solutions discovered by students.
"Altogether, this lesson is counted as having 10 student-generated alternative solution methods, even though it contains no student-discovered methods whatsoever," Siegel says.
Furthermore, the mathematicians who wrote about the study subsequently didn't see the original tapes; they relied on the misleading coding done by the data analysts.Why does it matter? Because so-called "discovery learning" is the promised land of mathematics reform, and if only we follow the prophecies of the National Council of Teachers of Mathematics across the River Jordan, all our failings and failures as a nation will vanish away. And we know the prophecies are true, because the Japanese have gone before us.
Only they haven't. This is teaching in the traditional mode, beautifully designed and superbly executed, but nothing like the parody of instruction that goes by the term "discovery learning" in math-reform circles in the United States.
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."
Illusion prevails in far too many American classrooms.
Linda Seebach is an editorial writer for the News. She can be reached by telephone at (303) 892-2519 or by e-mail at seebach@RockyMountainNews.com.http://www.rockymountainnews.com/drmn/news_columnists/article/0,1299,DRMN_86_3093879,00.html
This article was sent to us courtesy of NYC HOLD (Honest Open Logical Debate) on Mathematics Education Reform www.nychold.comTelling Lessons from the TIMSS Videotape: remarkable teaching practices as recorded from eighth-grade mathematics classes in Japan, Germany and the US, by Alan Siegel available at http://www.cs.nyu.edu/faculty/siegel/ST11.pdf.
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