Exploiting Mental Imagery with Computers in Mathematics Education

Solving mathematical problems, as well as understanding mathematical concepts, very often involves building and handling representations. Representations are a powerful means for a person to communicate with and reflect on his environment. They can be a tool for dialogue both with oneself and with others.

This paper is divided into three main sections. In the first one, Sect. 2, there is a collection of contributions from recent literature about the processes implied by terms such as to visualize, to imagine, etc. reported by researchers in mathematics education. It is clear that the objects of such processes are mental images, but the debate about the nature of mental images, though interesting, will not be considered here. Some statements on the role of graphical representations in mathematics activities will also be reported.

Formal mathematics involves definitions and deductions in a manner which is quite different from the mental processes of school mathematics. Formal definitions of function, limit, continuity, differentiation and integration (both Riemann and Lebesgue) involve possibilities that often conflict with the students’ previous experience, leading to confusion and alienation. Examples given to ,motivate - definitions invariably have specific properties that do not follow logically from the definition itself. For instance, examples of sequences are usually given by formulae so that the sequence ,gets closer and closer - to the limit, without actually reaching it. Consequently, many students believe that this is an essential property of the limit concept. Functions are nearly always given by formulae whose graphs look ,smooth - so that students have difficulty imagining anything different. When discontinuities are exemplified by drawing a graph, the picture is often represented as a number of curved pieces with a ,jump - at the point under consideration. The result is a widespread belief that a typical function is given by a formula and is continuous except at occasional isolated points. In this way the student builds up a personal concept image of the concepts at variance with the theory.

If I wake in the night I may lie awake thinking and this thinking makes use of mental images. If I think about a person I may create a visual image of the person. If I think about a recent meeting with people I may imagine the language of the discussion. If I think about how to develop a part of my garden I may juxtapose the plant shapes and not the plant names. If I think about a mathematical problem I may imagine my recent work on paper or with a computer. These imaginings come from previous experiences and the richer the experiences the easier it is to symbolise, recreate and create in the imagination.

In this essay I try to investigate the potential role of imagery and representations for learning and understanding pure and more advanced mathematics. To do this it is necessary to analyse the discourse of pure mathematics and how it makes use of all kinds of representations. In my view, we will find that there is a rather wide gap between imagery and representation on the one hand and the usual talk in pure mathematics on the other. For bridging this gap specific attitudes, agreements, and convictions, within a process of socialization are necessary. This will be illustrated by various examples. The references point out work to which I owe some of the basic ideas and notions used in this article as well as publications related to my theme. I feel especially dependent on Johnson, Lakoff, Vaihinger and Wittgenstein. In Dörfler [1] information can be found about the theory of image schemata as developed by Johnson and Lakoff (see [4,5]). The term representation is used here in the standard way, e.g. as in the contributions in Janvier [3]. The main idea which I borrow (and use implicitly) from Wittgenstein [7] is his notion of Sprachspiel. Finally, Vaihinger [6] has developed a comprehensive theory of fictitious thinking which highlights the importance of the role of adopting an as-if attitude.

Geometry is a school subject, but also and primarily geometry is a mathematical domain. As mathematics educators we are interested in geometry from both points of view. That is the reason why a first discussion is devoted to highlighting some characteristics of geometry as a mathematical domain.

Modern software for geometry like Cabri-géomètre [2], GEOLOG [4] or Felix [6] allows a drawing to be varied on the screen by direct manipulation with the mouse. The user can move certain parts of a drawing, and the software maintains all existing geometrical relationships. For instance, Cabri-Géomètre provides this basic feature by its ‘drag mode’: Fig. 1 shows the cursor dragging the vertex of a square. While the vertex is being moved around, the computer continues to draw a square.

Over the last thirty years, a considerable body of practice has been built up in schools in the U.K. on developing students’ geometric awareness. These practices involve the use of physical objects, verbal descriptions, diagrams and film. More recently, these have been augmented by computer software — sometimes devised in imitation of these practices, but also in new ways suggested by the medium. If we are to be clearer about how the computer might be used as a stimulus for learners’ developing control of their imaginations, we need to consider the purposes of the earlier stimuli and how their features are utilised. In this paper, I examine these features by situating them in a wider context of visual phenomena, and offer suggestions for further enquiry. Owing to difficulty in the notion of ‘mental imagery’, it is necessary to begin by considering the various meanings of geometric images.

It seems obvious to contrast physical representation (a drawing on paper or on a screen) of an object with mental images of this same object. However, there is an important contrast which attracts less attention, between a drawing of a physical object (a house, for example, outlined by a square with a triangle at the top and rectangles for the door…) and a drawing of a geometrical figure (square, circle, triangle…).

This chapter explores the visual and imagistic aspects of cybernetic manipulatives, how they may be designed to improve upon physical manipulatives, what their potentials and pitfalls may be, and how they fit into the larger evolution of technology use in support of mathematics learning. We will examine questions of ,physicality - of representations, dynamic connections between ,natural - and formal representations, and, especially, ways that new forms of records of actions may alter these connections and elevate levels of thinking involved in the doing of mathematics, from low-level computation to higher level planning, strategic and structural thinking. This paper extends earlier work [11, 12, 13], which also includes references to the wider literature relating to these topics, references not repeated here. It also relates closely to other papers on another aspect of the representational use of new technologies, dealing with dynamic linkages between formal mathematical notations and authentic human experience, particularly as instantiated in realistic simulations [14, 15, 16].

In his systematic attempt to discuss mathematicians’ thinking, Hadamard [2] shows that although there are differences concerning the manner in which they rely on mental images, mathematicians frequently exploit visual reasoning and often the images used are of a geometric nature. He recounts that during problem-solving, many mathematicians avoid not only the use of words but also that of algebraic or other symbols; instead, they use visual reasoning incorporating geometrical and other images to provide the basis for their intuitions, and only subsequently code them in symbolic terms.

When I first agreed to join the NATO Advanced Workshop on ‘Exploiting Mental Imagery with Computers in Mathematics Education,’ it was clear to me that the mental imagery I was most interested in was what I call ‘dynamic imagery’: moving pictures in the mind. My colleagues and I were beginning a new curriculum development project at Education Development Center that would exploit dynamic imagery on the screen: tools like Cabri-géomètre, and Sketchpad in geometry, our own DynaGraphs¹(a brief description will follow), and perhaps Apple’s QuickTime capabilities for animation

What is the relationship between screen images and mental images?

Suppose you want students to acquire a set of mental images to help them think about a topic. What is the best way to use images on a computer or video screen to achieve this? As a designer of screen images, how do you increase the chances that someone will acquire the mental images you wish to convey?

The position advanced in this paper is derived in part from the work of Varela, Thompson and Rosch [24] as described in their 1991 book, The Embodied Mind: Cognitive science and human experience, which in turn draws heavily from Varela’s earlier work with Maturana [19] as found in their 1987 book, The Tree of Knowledge: The biological roots of human understanding. We begin with a brief perspective of the philosophical background which gave rise to and inspired Varela et al.’s enactive view of cognition. The major issues addressed revolve around the notion of representation and what is commonly referred to as the mindbody problem. Attempts to resolve this problem essentially define and motivate developments in cognitive science, thus affecting our understanding of mental imagery as well. It will be seen, as we subsequently present Varela et al.’s formulation, of their enactive view of cognition as embodied action, that their theory is no exception in this regard. Implications of this view will eventually require a complete reconsideration of the notion of representation. Theories of mental imagery presupposing a representationalism must then, in some manner, be recast in terms of the immediate experiential presentations of consciousness in all of its modalities. In the final section of the paper, some of the manifest implications of this view for teaching and learning and the environments in which these occur will be discussed.

More easily than written scientific communication, pictures created during scientific research (when they are models of real objects) give access to the way in which researchers look at the real world.

Imagery has an important role to play in the teaching and learning of geometry since content of an image can be the very subject content which is being studied. I will not consider this fruitful area here since there are several chapters in this book which deal with this. Instead, I will consider how images can be used as a tool by a teacher to assist students in their learning of algebra. My initial consideration is to present reasons why a teacher may be interested in using images. To help with this, I invite the reader to carry out the following exercise:

Imagine walking from your house to the nearest shop. If you are like me, with a shop across the road from your house, you may take this walk slowly; if your shop is some distance away, then you may speed it up. As you make your walk, look either side of you and consider what you see, both in the foreground and in the background.

Mathematical thinking and teaching has been liberated by the graphics computer screen. Mandelbrot’s fractal geometry revivified theoretical work by Julia and Fatou, and the screen images of structure in’ chaos’ have entranced people, particularly artists, mathematicians, and entrepreneurs eager to find an effective image for marketing purposes. Spreadsheets enable people to lay out their calculations and see the results all at once (depending on screen size). LaBorde’s Cabri-géomètre and Jackiw’s Geometer’s Sketchpad, which enable users to make Euclidean constructions on general rather than specific geometrical objects, have offered a new lease of life to Euclidean geometry and we can expect to see the same construction mechanism employed for other forms of geometry and in other domains. Modelling tools like STELLA provide a graphic language in which to express complex inter-relationships based on the single image of controlled flow through pipes. Symbolic manipulators like Maple and Mathematica have the potential for providing many more people with access to symbolic generality than ever before. ISETL enables the language of sets and relations to be as used by mathematicians to be deployed and tested out on a wide range of particular (finite) examples. LOGO and Boxer present exploratory and expressive environments which depend upon and exploit the graphics screen.

When asked to participate in a conference on exploiting mental images in mathematics education I thought uitecarefully efore accepting; since, by all measures, I am a mathematical novice. I am by training a fine artist who, via postgraduate research, moved into computing where I have been teaching and researching design related computing for some considerable time. Just recently, I moved back into art and design education where I hope to exploit my research and computing experience in advancing art and design education and practice.