"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**

- 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. - Noss, R. and Hoyles, C. (1996).
*Windows on Mathematical Meanings*. Dordrecht: Kluwer Academic.