### Electronic Proceedings of the Tenth Annual Conference on
Technology in Collegiate Mathematics

*POSTER: 10-P8*
### "TangentField": A tool for "webbing"
the learning of differential equations

Margaret James, Phillip Kent and Phil Ramsden

Mathematics Department

Imperial College

University of London

Phone: +44(0) 171 594 8503

E-mail: p.kent@ic.ac.uk

#### ABSTRACT

Direction fields, or "tangent fields" as we have called
them, are a familiar visual representation for ordinary
differential equations, linking a first order ODE with
its solutions by distributing little "tangent stubs"
across the x-y plane, whose slopes are specified by the
ODE. Typically this representation is encountered in
learning about ODE's after a thorough treatment of
algebraic solution methods, especially for studying
ODE's that cannot, or cannot easily, be solved in closed
form. We wished to try a very different role for this
representation, as a *starting point* for undergraduate
students to learn about the concept of "differential
equation".
We designed a tool called "TangentField" in Mathematica,
and a set of activities for its use. In using
TangentField, anyone familiar with the concept of
differential equation recognises the families of
solutions which appear visually. However, learners do
not need the concept of *solutions* to see *curves* - we
hoped that the learners in constructing a meaning for
these families of curves would through this construct
the concepts of "solutions" and "differential equation".
And also that they would re-construct their
understanding of "derivative", which research suggests
is often strongly bound up with its meaning in
differentiation as derivative-at-a-point.

Noss & Hoyles ("Windows on Mathematical Meanings",
Kluwer Academic, 1996) have extensively researched the
idea of constructing computational tools in which the
expert perceives an embedded structure but which are
also intended to be in agreement with a learner's
initial conceptions, and responsive to the learner's
developing conceptions. They coined the term "webbing"
for such a structured, yet locally responsive, learning
environment.

We do not wish to use TangentField as a way of replacing
the formal meanings and algebraic language of
differential equations: on the contrary, we intend that
access to the formal meanings should be facilitated by
computational activity. For that to happen, the learners
must come to recognise explicitly the general structure
that they are relying on implicitly as they work with
TangentField. But in the meantime, they *can* rely on
it, and build a rich conceptual image before formal
definitions are needed. They must also forge links
between the representations of equations and families of
solutions as expressed in tangent fields, and those that
they meet in the algebraic language of ODE's.

We describe the theoretical basis for our design work,
and present an analysis of some student work on the
TangentField activities, and say how we see this
analysis feeding into the next cycle of design.