For Prospective Teachers, Less is More

An interview with John Layman, University of Maryland

Has the widespread use of computers for modeling and graphical analysis changed the kinds of mathematics (in school or college) that students need in order to pursue the study of science?

I would rephrase the question a bit: Has the widespread use of computers for modeling and graphical analysis in the study of science changed the kinds of mathematics that students need or can acquire in the act of doing their science?

With the use of microcomputer-based laboratories (MBL), students can obtain real-time graphs, provided by the computer, that represent many of the functions dealt with in mathematics. They can learn to use curve fitting techniques to recognize the kinds of functions that properly describe the physical system represented by a graph. This applies even to very complex systems such as motion or temperature vs. time relationships. Students can strengthen their understanding of functions when their actions produce a graph. In some cases, this may be their first introduction to certain functions (e.g., sinusoidal functions representing the motion of a pendulum).

In science, observations lead to hypotheses, whereas in mathematics courses applications seem to flow from decontextualized theory. How can teachers best reconcile these rather different approaches to the mathematics education of students whose primary interest is in science?

In mathematics classes that do hands-on work, each experiment is chosen to clearly illustrate a particular function that is known in advance. Thus students have no need to review a multitude of possible mathematical functions in order to choose one or two to test in support of the prediction or hypothesis.

In the MBL approach used in our introductory physics course for elementary teachers-to-be, the students are given a challenge problem and are asked to (a) predict the behavior of a related physical system; (b) experiment with the system; (c) obtain a graph of the actual behavior (e.g., how the period of the pendulum depends on the amplitude of swing); and (d) describe the observed behavior and its relationship to their predictions. In this approach, there is no advance clue as to what mathematical function will contribute to their work, although the groups often turn to mathematics for help with their final explanations. This provides a more open context for their use of mathematics. (We are doing research into this aspect of the common ground of mathematics and science in our introductory physics class.)

This is a fundamental difference in the approach to using mathematics and might be considered by our mathematics colleagues who use science as a mechanism for illuminating the mathematics. It is a challenge worth pursuing.

Mathematics teachers often worry that the distinctive nature of mathematics will be lost unless it is taught as a separate subject rather than as part of an integrated interdisciplinary program. Do you share this concern?

Not at all. Evidence from my experience with pre-service teachers in physics classes is that the abstract context of mathematics is not shared by many of these students. This is especially true for their experience in the "mathematics for elementary teachers" courses offered by the mathematics department. These students have a very poor image of their own prowess in mathematics even though it will be part of their responsibilities as elementary teachers.

Mathematics teachers also worry that the context-rich environment of an interdisciplinary course will impede learning rather than enhance it-- since it will be harder for students to sort out the mathematics principles from the surrounding context. What's been your experience as a physicist in this regard?

Just the opposite. It is the richer context of science experiences that provides prospective teachers with a context that they can understand. These contexts relate mathematics to the kinds of things that may be part of their teaching. Pendulums play a role in almost all elementary school science; the study of motion is often there as well. Thermodynamics with water, ice, and steam may also be used.

Given all these concerns about interdisciplinary teaching, and about teaching mathematics in the context of other subjects, is integrated or interdisciplinary teaching worth the effort?

I think it is very much worth the effort. The key is the "less is more" theme and actually honoring it so students experience a very rich inquiry environment that naturally involves both mathematics and science. I do feel that this theme is substantiated by insights we have gained related to a model of teaching and learning that involves experiments, stories, graphs, and tabular representation of data

John Layman is Professor of Physics and Physics Education at the University of Maryland. he can be reached by e-mail at jl15@umail.umd.edu