**Click title to explore thematic outlines on this page below:**

**Cognitive Development in Mathematics | Concept Image and Concept Definition
Cognitive Units | Cognitive Roots | Generic Organizers
Procepts | Algebra | Limits, infinity and infinitesimals | Visualization
Calculus (& Computers) | Computers in School and College Mathematics
Problem Solving | Advanced Mathematical Thinking | Proof
Three Worlds of Mathematics
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The following themes in my research, developed with students and colleagues, are outlined here. By clicking on a heading, of a section, greater detail of specific papers will be given, including the opportunity to download them for further study.

My interests in cognitive development in mathematics have matured over the years, starting from the viewpoint of a mathematician and building a cognitive theory of growth from child to adult. Initially, as a mathematics lecturer at university, I used mathematical theories (such as catastrophe theory) to formulate cognitive theory. After researching concepts of limits and infinity, I designed computer software to visualize calculus ideas using the idea of 'local straightness' rather than formal limits, and expanded my interest in visualization. Working first with Michael Thomas on algebra, then Eddie Gray on arithmetic, I was able to work in cooperation to develop insight into symbolic development in arithmetic, algebra and calculus. This led directly to the theory of procepts with Eddie Gray, which is concerned essentially with symbols that represent both process and concept. Eddie and I were then able to see what we termed ‘the proceptual divide’ - the bifurcation between those children who remain focused on more specific procedures rather than develop the flexibility of manipulating symbols as concepts or processes as appropriate. From here the cognitive theory developed to integrate visual and symbolic aspects and their eventual transformation into axiomatic definition and proof in advanced mathematical thinking.The framework of *Three Worlds of Mathematics * is a simple theory based on perception, action and reflection, which is capable of giving insight into why some individuals are successful in mathematics but many others find it increasingly difficult. It formulates three distinct developments in embodiment, symbolism and formalism. A new construct termed a *crystalline concept* brings all three developments to a single overall insight. This is a concept which has rich internal relationships constrained by its context. For more information, on papers in cognitive development click here.

CONCEPT IMAGE AND CONCEPT DEFINITION

The terms ‘concept image’ and ‘concept definition’ were formulated in 1980 by Shlomo Vinner. When he visited me in Warwick that year, I had a huge quantity of data gleaned from undergraduate mathematicians that I could not analyse from a mathematical viewpoint. Shlomo's idea immediately clarified the issues and resulted in the joint paper on Concept Image and Concept Definition published in 1981. Here I should declare that this has led to two different meanings given to 'concept image' in the literature. Shlomo's definition was philosophically based and was a thought experiment to analyse what happens when students focus in different ways on images and definitions. My perception was more humanly based, so that where Shlomo talked about 'the mind' and thought about it as separate from 'the brain' in a cartesian sense, I always thought of it as a physical phenomenon in the brain. For me the mind is the way the brain works, and is an indivisible part of the structure of the brain. Shlomo has always written about 'concept image' and 'concept definition' as being 'two distinct cells' whereas I see the concept definition as a form of words that can be written or spoken that is part and parcel of the total concept image in the mind/brain. It is up to you to choose which version you want. It is even easy to accommodate both. However, when I say 'concept image', I mean the definition given in Tall and Vinner 1981. When I think about concept image, I think of the conundrum raised by the composer Mendelssohn, when he was told that music is too vague to be represented by musical notation. He replied, on the contrary, that music is too precise every to be captured by notation. Speaking of concept image can sometimes be vague, but as Pat Thompson once told me, this is precisely what makes it so useful. It helps us to grasp that there are subtleties in mathematical thinking that cannot be precisely conveyed by the apparent precision of mathematics. For more information, on concept image, click here.

A cognitive unit is a piece of information we can hold consciously in our focus of attention, together with all the links (many of which are unconscious) that can be made to other parts of our cognitive structure. For more information on cognitive units, click here.

A cognitive root is a very special kind of cognitive unit. It is one which links to the (intuitive) experiences of the individual, yet also contains the potential to be transformed to the future formal theory. For more information on cognitive roots, click here.

A generic organiser is an environment designed for the learner to interact with in a manner that can be focused on examples of specific mathematical concepts and/or processes. These include such equipment as Dienes blocks (to focus on the notion of place value and its role in arithmetic) or the software in Graphic Calculus, which enables the student to build up the notion of gradient of a graph by seeing it as ‘locally straight‘ under high magnification to ‘see’ the gradient over a small piece of the curve, then to have a routine to trace a moving straight line along the curve to see the changing gradient of the graph. My definition of generic organiser requires an ‘organizational agent’ to support the student playing with the organiser and focus on salient ideas. It also demanded that the limitations of the particular representation would become apparent as the use of the organiser was surpassed by more subtle theory. For more information on generic organisers, click here.

The notion of 'procept' began its life as an idea generated by looking at a symbol such as 3+2 both as a process (of addition) and a concept (of sum). It was extended (Gray & Tall, 1994) to include different symbols and different processes that give rise to the same mental object in the mind of a particular individual. Thus 3+2, 4+1, 5, 7-2 can all represent the same procept, involved in composing and decomposing arithmetic processes that give 5. Gray & Tall distinguished between the specific *procedure* as an explicit sequence of steps and the input-output *process* where different procedures can have the same input-output effect. Thus 'count-all' and 'count-on' as procedures for evaluating 3+2 are different procedures to give the same arithmetic process. In various papers we have studied the procedure-process-procept spectrum of performance. For more information on procepts, click here.

My collaborative research in algebra began with Michael Thomas, using the computer to give a meaning to an expression such as A+3 by the process of evaluation, so that, if A is 2, then A+3 is 5. By using the language BASIC to program expressions such as 3*A+6 and 3*(A+2) the student could have the experience that the two expressions always give the same answer. In procept theory terminology, this means that the two distinct procedures are actually the same process. Later research began to consider the different kind of cognitive activities that arise in arithmetic, algebra and calculus. For more information on algebra, click here.

LIMITS, INFINITESIMALS AND INFINITY

My earliest research began with calculus and limits, leading to the discovery of differences between mathematical theories and cognitive beliefs in many individuals. (For instance, the limit 'nought point nine repeating' has mathematical limit equal to one, but cognitively there is a tendency to view the concept as getting closer and closer to one, without actually ever reaching it. This has a procept theory interpretation. The process of tending to a limit is a *potential* process than may never reach its limit (it may not even have an explicit finite procedure to carry out the limit process). This gives a new way of describing the way in which the mind contemplates potentially infinite processes as procepts. For more information on limits,infinitesimals, infinity, click here.

My interest in visualisation began in the calculus, but then extended beyond this to the wider role of visual and spatial ideas in mathematics. For more information on visualization, click here.

Having found inherent difficulties limit concept, I sought a method of introducing ideas in the calculus that used the limit concept implicitly, rather than making it the explicit foundation of the theory for beginners. I determined that ‘local straightness’ of the graph (which generalises to the notion of locally euclidean in manifold theory) is a good cognitive root on which to build the calculus. To achieve this purpose, I designed the Graphic Calculus software to enable the student to interact with examples of the ideas. These include magnifying a portion of the graph to see it look straight under high magnification, and tracing the gradient numerically along the graph. This allows the concept of differential equations to be seen as determining locally straight graphs which have known gradient at each point (given by the equation). Software (the solution sketcher) allows the student to point the mouse at a place in the plane and the software to draw a short line segment of the given gradient. By enactively sticking such segments end to end, an approximate solution can be constructed physically and visually. The locally straight approach has been filled out in my papers to cover a wide range of ideas including visual representations of differentials, partial derivatives, tangent planes, parametric curves, composit functions, implicit functions, area and integral, numerical methods of solving equations etc. For more information on calculus, click here.

COMPUTERS IN SCHOOL AND COLLEGE

These papers look at a wider range of uses of computers in mathematics education. For more information, click here.

This work, developed by Yudariah Yusof for her PhD thesis, considers the attitudes of students to learning mathematics and solving problems. It shows how exposure to a problem-solving course increases positive attitudes as desired by the students' teachers, but when the students return to standard mathematics courses, some of the effect is reversed. For more information on problem solving, click here.

ADVANCED MATHEMATICAL THINKING

Advanced mathematical thinking is concerned with the introduction of formal definitions and logical deduction in formal axiomatic theories. Of particular interest is the transition from elementary school mathematics (geometry, arithmetic, algebra) to advanced mathematical thinking at university. This includes the full cycle of formal mathematics, including the creation of new theories using problem-solving techniques of conjecturing, testing, modifying, and proving theorems to build a formal theory. For more information on advanced mathematical thinking, click here.

Interests include the cognitive development of proof concepts as the individual matures and the relationship between the mental representations used and the kinds of proof which are possible, in problem-solving in school and university and on to formal proof in university mathematics. This includes work with Marcia Pinto on *natural learners* who develop their concept imagery to embrace and illuminate the concept definition and *formal learners* who use the concept definition to contruct the formal mathematics directly. For more information on proof, click here.

As I worked on various aspects of mathematics through the years I began to realise that there were three very different developments that occur in mathematics based on our *perceptions* of **objects**, *actions* on objects which are represented by **symbols** that can be manipulated as mental objects, and on the **properties** of physical and mental objects that we *reflect* upon and use as a basis for deduction. This led me to categorise mathematical thinking into three different worlds: a ** conceptual-embodied world** of objects perceived and conceived, a

Brief explanations of terms used in my papers, both those defined by me and my colleagues and also other terminology which I may use in a particular way. Click here to go to the Glossary.

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