A Cartoon Version of Math

Town and Village
January 25, 2001


To the Editor:

We are the ``math experts'' who, as mentioned in your education article of Jan 18, visited some mathematics classes in the Upper Lab School in District 2. Unfortunately, what we saw was quite different from the statements attributed to us in your article.

The classrooms were indeed very sweet environments with dedicated teachers and engaged students eager to learn. However, the teaching style we saw used an approach that gives the children only half of the learning opportunities that they deserve and need.

In one instance, we witnessed a lively classroom discussion where a perfectly correct answer went unrecognized and was soon lost amidst total confusion. It was clear that class foundered because the children lacked a few fundamental skills and understandings that were necessary to recognize a correct argument. Although they were engaged, their nearly 45 minutes of discussion yielded no advance in understanding. Most of the ``lesson'' was a lively debate that failed to answer one basic question.

It was sad to see so much time lost and so little progress, especially when just a few minutes of explanation --- or quite possibly, pointed questioning by the teacher --- could have set the children straight and could have even facilitated their figuring out much of what they needed to know. Based on what we saw, we are convinced they could have made the transition to a complete understanding had they first focused on the fundamentals underlying the problem in a teacher-controlled discussion.

The technical term for the teaching style we saw is cooperative learning, which sometimes includes a resolute approach of non-intervention on the teacher's part. As we saw, this ``guide on the side'' philosophy can come at a high price.

It is worthwhile to compare this teaching style with that used in Japan. The Japanese style --- which is frequently misunderstood by academics who are not mathematicians ---follows a mixed strategy that has more discovery-based learning than we saw in the District 2 classes plus more more direct instruction than we have in the traditional American lecture-based teaching.

How do they do it?

The answer is a long story, but the key point is that Japan uses an outstanding substance-based curriculum. And the teachers are free to use whatever it takes to open the eyes of the students. Before introducing a challenge problem, they review the key ideas needed to solve it. But the problem often requires such a deep understanding of the mathematics that many of the students will fail to solve it in the three to six minutes allotted to the question. Then the class goes over the answers with teacher-assisted presentations by students and with the teacher subsequently presenting a complete solution in an organized and insightful manner. Because of the careful time management, the Japanese classes can then cover another problem of the same type to give all students a chance to walk in their teacher's footsteps. But the new problem is no plug-in question. It is designed to be just as challenging and educational as the first. Moreover, the problems cover substantial mathematics that reappear in more advanced settings. The mathematics is real, it is tough, and even deep. Yet the students are engaged, ask questions, and learn to think. The teachers are ready to give away ideas when the circumstances warrant, to redirect missteps with leading question whenever appropriate, and even to express doubts about incorrect answers.

Now, the traditional American mathematics texts are wretched, and we would not advocate a return to them or to the ill-conceived Sequential I, II and III that was once the latest math reform in New York State. Frankly, it is difficult to understand how such a weak and disorganized program could have ever been adopted. But the current reforms are no better; they are just different. Not all problem solving is the same. Compared to the substantive problem solving taught in Japan, China and the better programs in Hungary and Russia, the latest American approach is one dimensional and rigid. It is closer to a cartoon version of mathematics than a world-class program.

We believe that there are engaging math programs vastly superior to the sample classes we saw in District 2, and are saddened to see that they are not being used. We are also at a loss to understand why the principles of a democratic society would not suggest that there be a public forum for parents to hear a balanced discussion about these new programs and the alternatives.

Lastly, we would like to thank you for providing this opportunity to correct the public record with respect to our assessment of the District 2 Mathematics programs.


Professor Alan Siegel
Former Director for Industrial Relations
Former Deputy Chairman
Department of Computer Science
New York University

Professor Jonathan Goodman
Department of Mathematics
New York University

Reproduced with permission from Town and Village.

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