Abstract
The reform mathematics curricula present students with problems that cannot be solved using routine methods. Ms. Latterell and Mr. Copes wonder about the best ways to teach students to solve such problems -- a research question that leads them into a classroom where a teacher is refereeing an animated discussion about algebra. WE ARE hearing a lot of noise these days about wars being waged over the content and pedagogy of new math curricula. These disagreements echo similar discussions in other fields: between whole language and phonics in reading, between prophetic versus priestly in the social sciences, between positivists and antipositivists in general. Can we ever decide which of these approaches is better? One way in which the so-called reform curricula in mathematics differ from the more traditional curricula is in their approach to teaching problem solving. Many who have taken mathematics courses remember solving lots of problems. is 3 + 2? Solve the equation 3x + 17 = 2 for x. The National Council of Teachers of Mathematics (NCTM), however, defines problem solving as engaging in a task for which the solution path is not routine.1 Problems in symbol manipulation can be solved by routine methods, and so would not constitute problem solving in the NCTM sense. Those notorious word problems, problems with practical applications, or puzzles more nearly match the NCTM criterion. There are no routine methods by which a student (or a computer) could solve these problems. Certainly, the traditional curricula do not eschew problem solving in the NCTM sense. Indeed, almost all curricula claim to be satisfying the new math standards as they are spelled out in Principles and Standards for School Mathematics. But the differences in how to teach problem solving are very real. What are those differences, and how should problem solving be taught? Let us begin to address these questions by considering one seventh-grade teacher's first-person description of an attempt to teach problem solving. Classroom Episode One I wrote the problem on a transparency as students were coming into the room, and I put it on the overhead projector as they got into their base groups and signed in. The problem was: Represent as an equation the claim that there are six times as many students as teachers in this school. Use S for the number of students and T for the number of teachers. I said, Please work on this for a few minutes in your groups. As I circulated, I found that all but one group had written 6S = T. Then I overheard Chris saying, Wait a minute -- if S was 100, then T would be 600, and that's not right. The group began to argue heatedly. I gave Chris a thumbs-up sign to keep the argument going and sent representatives from other groups to eavesdrop on the discussion. Soon the entire class was in turmoil. Different Approaches to Teaching Problem Solving Is using a problem that provokes controversy like this a good way to teach problem solving? Before we consider that question, let us look at the differences between reform-oriented and traditional approaches to teaching problem solving in mathematics. The restrictions of space demand that we oversimplify. Traditional mathematics curricula are based on the walk before you run philosophy. Until you know how to add 3 + 2, there is no way you can solve a problem such as You have 3 apples and are given 2 more; how many do you have? In these curricula, students learn the and then see how to apply them. That is, students must learn mathematics skills before they learn problem solving. Reform curricula, in contrast, are based on the idea that students retain basic mathematics best while trying to use them to solve engaging problems. Students who are set to work on interesting problems they do not know how to solve can be quite receptive to learning and problem-solving techniques on a need-to-know basis. …
Published Version
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