Twenty years ago, Seymour Papert put forth a vision for how computer
software could transform the learning of mathematics: “The idea of ‘talking
mathematics’ to a computer can be generalised to a view of learning mathematics
in ‘Mathland’; that is to say, in a context which is to learning mathematics
what living in France is to learning French.” (Mindstorms, 1980, p. 6)
When I first read this (in 1994), it connected very powerfully with
my work at that time, which was developing new mathematics curriculum for
undergraduate students based on the computer algebra system (CAS), Mathematica.
Surely (I thought) a CAS is the closest one can get to “talking mathematics”
with a computer? Was I standing on the borders of Mathland? Well, six years
later I am still excited by Papert’s vision, but, of course, still a long
distance from a realisation of Mathland.
In my presentation, I will describe two curriculum developments undertaken
by myself and colleagues at Imperial College in London during the past
five years, and the two-way flow of ideas that has taken place between
the practice of curriculum design and the theoretical principles that have
influenced the design work. Inspired by ideas developed within the Logo
research community, we have conceived a CAS as a computational, expressive
medium: a medium where actions are carried out by means of programming
in a syntactically precise language. Expressiveness means that it is possible
to express ideas (mental objects) in concrete form (visible, public objects):
it is possible to build mathematical structures “…into the fabric of the
medium, thus shaping the types of action that are possible: they do not
only exist in the mind of the learner. The level of what can be thought
about, talked about, is notched up a rung or two” (Noss & Hoyles, Windows
on Mathematical Meanings, 1996, p. 126).
I will illustrate our uses of expressiveness (and the related idea
of “webbing”, also due to Noss & Hoyles) by means of a curriculum example
that concerns first-year university science students at the very beginning
of their learning about ordinary differential equations (ODEs). We developed
a Mathematica “toolkit” (i.e. a set of programmed functions) to enable
students to produce graphs of “direction fields”—these are a familiar visual
representation for ODEs, linking a first-order ODE with its solutions by
distributing small “tangent segments” across the x-y plane, whose gradients
are specified by the ODE. In paper-and-pencil mathematics, a direction
field is only really usable by someone with the substantial experience
to know the shapes and behaviours of typical solutions. Given an expressive
medium, however, our hope was that the working with direction fields could
become an entry point into understanding the essential concept of differential
equation.
My second example is concerned with a course on dynamics taken by third-year
mathematics students. In this work, I’ve tried to make Mathematica “expressive”
for dynamics by exploiting a feature of its programming language that allows
for the creation of new data structures (objects), and functions that can
operate on them. As the students program (create and manipulate) these
objects, they are dealing with a concrete implementation of abstract mathematical
concepts. In principle, meaning can be given to mathematics through this
constructive, explicit interplay of concrete and abstract; but, of course,
this is easier said than done. On the theoretical side, this work has shown
up some interesting comparisons between “abstraction” as it is conventionally
conceived in mathematics education, and the quite different ideas about
abstraction that have been developed by computer scientists.