"TangentField":
A tool for "webbing" the learning of differential equations

Margaret James, Phillip Kent
Phil Ramsden
Mathematics Department,
Imperial College, University of London.

In the field of mathematics education, Noss and Hoyles (1996) have extensively researched the idea of constructing computational tools in which the mathematical expert perceives an embedded structure but which are also intended to be syntonic with the learner’s developing conceptions. They coined the term "webbing" for such a structured, yet locally responsive, learning environment. We are using the theoretical framework of webbing in the development of a microworld for the learning of differential equations by undergraduate students. Our work is mostly with the computer algebra system Mathematica, though recently we have begun to work with StarLogo, exploring the rather different approaches to differential equations which this software makes possible.

Our poster will concentrate on the design of one particular computational tool in our microworld, "TangentField", and the activities for learners we designed around it. It has been implemented and evaluated using Mathematica, though we also have a StarLogo version: we will compare some of the relative advantages and disadvantages of using these two computational media as the basis for a differential equations microworld.

Research suggests that learners of calculus often begin with a point-by-point conception of derivative (Dreyfus 1990). "TangentField" produces pictures of the plane peppered with tangent stubs whose gradients are given by a differential equation. It can be used to plot a tangent stub at a single point: this encapsulates an epistemological continuity with the notion of the derivative giving the gradient "at a point". We hope that through carefully designed activity the students in making use of the "at any set of points" facility of the tool will develop their concept of derivative. But also, as they do this, solution curves to the related differential equation (as an expert recognises them) appear and we suggest that students may begin to construct meanings for differential equations.

We will present an analysis of some biographies of student meaning-making with "TangentField" and say how we see this analysis feeding into the next cycle of microworld design and theorizing.

References

1. Dreyfus, T. (1990). "Advanced Mathematical Thinking". In P. Nesher and J. Kilpatrick (Eds.), Mathematics and Cognition (ICMI Study Series). Cambridge University Press. pp. 113-134.
2. Noss, R. and Hoyles, C. (1996). Windows on Mathematical Meanings. Dordrecht: Kluwer Academic.