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Paying Attention to Mathematics

An interview with Judith Sowder, San Diego State University

Like many mathematicians, you worry that in most interdisciplinary programs mathematics exists to serve science and that the mathematics itself gets lost. How important is it that students see mathematics as a separate subject rather than just as a powerful tool in the service of other subjects?

I don't really think students should see mathematics as something completely separate from other subjects. However, when people design interdisciplinary curricula, there is danger that the only mathematics that gets into the curriculum is that which is based on the immediate needs of the science or other subjects being developed. When such an approach is taken, a great deal of mathematics that I think should be part of anyone's mathematical literacy falls through the cracks

For example, I was on the advisory board for a very innovative NSF-funded middle school mathematics curriculum development project. It used many real life situations such as building a research station in Antarctica and tracking the wolf population in Alaska. Originally, it was intended as supplementary to a regular curriculum. But then the developers received additional funding to develop a complete grade 7-9 curriculum. Suddenly, they became aware of a lot of gaps--many important topics were missing that they themselves believed should be part of the curriculum. So long as it was viewed as a supplement, they had not paid much attention to these gaps during the project's initial years. How to address these gaps became a real problem for them once the nature of the project changed.

When people sit down to develop an interdisciplinary curriculum, they do not think of what questions about mathematics really need to be addressed. Do the students know when different arithmetic operations are appropriate? Have they thought about why multiplication does not always "make bigger"? When is the median a better measure of central tendency than the mean? These kinds of questions rarely arise naturally, but need to be part of the curriculum.

Mathematics teachers also worry that for many students the context-rich environment of an interdisciplinary course will impede rather than enhance learning. Mathematics itself is hard enough, without the confusing context of another subject. Does research or experience have anything to say about how context affects mathematical learning?

Certainly the whole psychological theory of situated cognition would argue that mathematics is best learned in an appropriate situational environment. Of course, the problem of transfer is an old one in psychology.

I would never be one to argue that mathematics should be taught in a context-free environment. But really good curricula must provide situations in which the mathematics can be conceptualized by the students and in which adequate attention is paid to mathematical topics and their connectedness. I strongly believe that development of such curricula is extremely time consuming and all too often is undertaken without the needed time or expertise. Some of the new middle school materials from NSF funded projects look interdisciplinary to me, and are context-rich, but mathematics has always been their primary focus.

The problem of "gaps" in integrated programs is an ever-present concern--not only for mathematics teachers, but also for teachers of other subjects. Does this problem take on a different perspective if instead of focusing on what is covered or not, we look at what is still remembered and used five or ten years later?

I find this question phrased in a way that makes it difficult to answer. It sounds as though I am concerned about "content coverage" and that may be the impression I left. That term suggests "scope and sequence charts" to check off each small skill that, of course, students may not remember in five or ten years.

If children in the lower grades do not have sufficient experiences with place value so that all the algorithms they learn are without meaning, I would not expect them to be remembered. However, if children learn place value in a problem setting where they are encouraged to develop a deep understanding of our base ten system and to use their place value knowledge to invent their own algorithms, then I suspect that in five or ten years they would still be able to invent procedures for operations that they needed.

In the middle grades, I would hope that children develop good number sense about rational numbers. Number sense is developed in computational situations that lend themselves to good questioning--by teachers and by peers. Similarly, if children in the middle grades are given a broad range of opportunities that help them distinguish situations that call for additive reasoning from those that call for multiplicative reasoning (which includes proportional reasoning), this distinction will not be forgotten. This can, of course, be done in an integrated program if there is a focus on the tasks and situations that allow for this kind of conceptual development. What I am concerned about is that there be a conceptual focus. It has to be attended to. But is it, in the typical integrated program?

You speak of selecting contexts that enable students to conceptualize mathematics, whereas integrated projects usually select the mathematics (and the science, and history...) that enables students to deal with the problem context. Which takes priority--the context or the mathematics? Is there a danger that in fitting the context to the mathematics one is finding problems to fit a solution, rather than solutions to fit a problem?

Yes, there is this danger. I am more inclined to deliberately chose some contexts that will allow conceptualization of mathematics. There can, of course, also be the second kind--contexts where the problem is the focus and answers are sought that fit the problem (where the mathematics may or may not be attended to and understood by students). But this comes right down to what I fear--that in an integrated program there may only be the second kind of context, without also the first kind. I don't believe that children learn the mathematics they need accidently, without any attention being paid to it by the teacher and the curriculum.

Judith Sowder is professor of Mathematics Education at the Center for Research in Mathematics and Science Education at San Diego State University. She can be reached by e-mail at jsowder@sciences.sdsu.edu.

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Last Update: June 17, 1997