Number 2, June 1997.
Number 2, June 1997.
Phillip Kent, Phil Ramsden and Margaret James
The METRIC Project, Imperial College, University of London, United Kingdom.
Email: metric-proj@ic.ac.uk
WWW: http://metric.ma.ic.ac.uk/
Many mathematics educators have advocated computer algebra software as a medium for exploration and experiment in the learning of mathematics. In this article we outline our own work in this area over the past four years, developing exploratory learning environments using the computer algebra system Mathematica.
During 1993-6 (working under the name ``Transitional Mathematics Project'', TMP) we developed, through a cyclical process of design and evaluation (for a history of which see [1]), a set of learning materials for mathematical topics at the transition between school and university. These materials were ultimately published as the book Experiments in Undergraduate Mathematics [2]. (Note: John Wood was the third project member of TMP, replaced by Margaret James in the METRIC project.)
The book contains both (a) text explaining mathematical theory and (b) descriptions of computer-based activities for exploring mathematical concepts. Students have access to special functions, written by us, to support their explorations, but no text appears on the screen apart from what they themselves put there. This use of Mathematica is quite different from is often known as ``courseware'', in which the student works with a pre-prepared on-screen document containing text, graphics and code. Although the courseware concept informed our early designs, we later moved away from this model and towards an approach that involved students building their own documents (the reasons for this move are given in [1][3]).
Currently, we are developing microworlds for use by undergraduate science students. These owe their origins to the research tradition that began with microworlds based on the programming language Logo (see, for example [5][4]). We use ``microworld'' to mean a computational environment that represents a particular knowledge domain and that is constructed for the purpose of learning about that domain. It contains:
Our first attempts at Mathematica-based microworlds (currently under design and testing) have at their core a tool-kit of Mathematica functions. Paper documents contain activities related to the development of mathematical concepts, and scientific contexts in which the mathematics will be used. By making the common toolkit the central structure of the microworld, we have tried to provide many access routes: it should be possible to begin using the tools within the scientific context, and to move freely from there to the exploration of the mathematical concepts (and back again).
In designing the elements of the toolkits we took account not only of where we wanted students to end up mathematically, but also of where they were likely to be starting. The tools were built to embody the fresh mathematical concepts we wanted the students to learn, but to be meaningful to them from the beginning.
For example, in a microworld on ``Ordinary Differential Equations'', we have designed a tool called TangentField, our aim being to focus a learner's attention onto the meanings of ``differential equation'' and of ``solutions'' to one. These meanings are by no means transparent, but attention is rarely given to them in the standard mathematics curriculum (an important aspect of our microworld design is to focus on what the computer might be particularly useful for, and not to seek to somehow replace standard pen-and-paper work). Learners may interpret the equation as a relation which enables them to calculate the derivative at any point: it is not immediately obvious from this that a function exists which satisfies the relation at every point along its curve, and it is even less obvious that a ``family'' of such functions exists.
TangentField produces ``tangent fields'' for first-order ordinary differential equations (ODEs)-loosely, pictures of the plane peppered with tangent stubs. Someone who already knew about ODEs would be able to see a tangent field as a visual representation linking a first order ODE and its solutions. For the learner, we propose quite a different role for this tool-as a starting point for learning.
TangentField can plot a tangent stub at any point in the plane, as specified by an input list of coordinate points. The important part of the last sentence is ``at any point'': the students using the tool may have a good pointwise conception of the differential relation and this is in harmony with the tool. However, the tool has (in a sense) the structure of the more general, global, conception embedded in it. Our hypothesis is that as students use the tool they may come to perceive this structure. They will implicitly make use of the ``at any point'' facility and as they do this, solution curves will appear:
TangentField for the equation dy/dx = -y.
Our hope is that students will begin to construct a meaning for solution and for family of solutions, and will also develop a formal meaning for ``differential equation'' itself. That is, they will become explicitly aware of the embedded structure in the tool.
At the time of writing, we are evaluating TangentField and the ODE microworld with students at Imperial College. We do not yet know if the microworld ``works''; it represents a major step away from our past, proven design, and we expect major revisions may be necessary. Hence, for the evaluation, we are interviewing students to assess prior knowledge, observing them closely whilst they work in the microworld, and interviewing them again after that work to dissect their experiences and possible changes in understanding.
We are committed to a programme of research in which theoretical work develops hand-in-hand with design. Design, we believe, is such a rich source of good theory, and theory such an aid to good design, that unique opportunities are offered by an approach which unites the two. The ODE microworld, as it stands, can be seen both as a design prototype and as the embodiment of a set of educational hypotheses. We hope to be able to report developments in both aspects over the next couple of years.
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