David Tall

Concept image and concept definition in mathematics with particular reference to limits and continuity

The concept image consists of all the cognitive structure in the individuals mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition.The development of limits and continuity, as taught in secondary school and university, are considered. Various investigations are reported which demonstrate individual concept images differing from the formal theory and containing factors which cause cognitive conflict.

The Psychology of Advanced Mathematical Thinking

Exponents of the two disciplines are likely to view the subject in different ways the psychologist to extend psychological theories to thinking processes in a more complex knowledge domain the mathematician to seek insight into the creative thinking process, perhaps with the hope of improving the quality of teaching or research. Although we will consider the cognitive side from a technical point of view as we discuss the most useful concepts in the psychology of advanced mathematical thinking, our main aim will be to seek insights of value to the mathematician in his professional work.

Functions and Calculus

This chapter is concerned with the changing perceptions of functions and calculus in recent years, both in terms of research into cognitive development of concepts and curriculum developments using computer technology.

Group representations, λ-rings and the J-homomorphism

THIS paper arose from a desire to apply the work of J. F. Adams ‘on the groups J(X)’ [2] to the case where Xis the classifying space B, of a finite group G. Since Adams’ calculations apply only to a finite complex X, and B, is infinite, the results could not be applied directly. Rather than quoting theorems and using limiting processes, the pure algebra has been isolated and independently reworked in such a way that it not only applies to the situation considered, but is also of general interest. This occurs in Parts I and III. The algebra used requires knowledge of special L-rings (which arise in K-theory and elsewhere). Part I is a self-contained study of these. (A special L-ring is a commutative ring together with operations {An} having the formal properties of exterior powers). Part III contains the main algebraic theorem, which readers familiar with the work of Adams [2] may recognise as essentially including a proof of his theorem ‘J’(X) = J”(X)‘.

The Transition to Formal Thinking in Mathematics.

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts towards a formal framework of axiomatic systems and mathematical proof. In this paper, the transition in thinking is formulated within a framework of ‘three worlds of mathematics’- the ‘conceptual-embodied’ world based on perception, action and thought experiment, the ‘proceptual-symbolic’ world of calculation and algebraic manipulation compressing processes such as counting into concepts such as number, and the ‘axiomatic-formal’ world of set-theoretic concept definitions and mathematical proof. Each ‘world’ has its own sequence of development and its own forms of proof that may be blended together to give a rich variety of ways of thinking mathematically. This reveals mathematical thinking as a blend of differing knowledge structures; for instance, the real numbers blend together the embodied number line, symbolic decimal arithmetic and the formal theory of a complete ordered field. Theoretical constructs are introduced to describe how genetic structures set before birth enable the development of mathematical thinking, and how experiences that the individual has met before affect their personal growth. These constructs are used to consider how students negotiate the transition from school to university mathematics as embodiment and symbolism are blended with formalism. At a higher level, structure theorems proved in axiomatic theories link back to more sophisticated forms of embodiment and symbolism, revealing the intimate relationship between the three worlds.

What is the object of the encapsulation of a process

Abstract Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?

Encouraging Versatile Thinking in Algebra Using the Computer

In this article we formulate and analyse some of the obstacles to understanding the notion of a variable, and the use and meaning of algebraic notation, and report empirical evidence to support the hypothesis that an approach using the computer will be more successful in overcoming these obstacles. The computer approach is formulated within a wider framework ofversatile thinking in which global, holistic processing complements local, sequential processing. This is done through a combination of programming in BASIC, physical activities which simulate computer storage and manipulation of variables, and specific software which evaluates expressions in standard mathematical notation. The software is designed to enable the user to explore examples and non-examples of a concept, in this case equivalent and non-equivalent expressions. We call such a piece of software ageneric organizer because if offers examples and non-examples which may be seen not just in specific terms, but as typical, or generic, examples of the algebraic processes, assisting the pupil in the difficult task of abstracting the more general concept which they represent. Empirical evidence from several related studies shows that such an approach significantly improves the understanding of higher order concepts in algebra, and that any initial loss in manipulative facility through lack of practice is more than made up at a later stage.

Symbols and the bifurcation between procedural and conceptual thinking

Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to do mathematical problems and to think about mathematical relationships. In this article we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra, and calculus, then on to the formalism of axiomatic mathematics. This evidence is taken from a number of research studies recently conducted for doctoral dissertations at the University of Warwick by students from the United States, Malaysia, Cyprus, and Brazil, with data collected in the United States, Malaysia, and the United Kingdom. All the studies form part of a broad investigation into why some students succeed, while others fail.RésuméLes symboles jouent un rôle central entre les processus à accomplir et les concepts auxquels on doit penser. Ils nous permettent à la fois de résoudre des problèmes mathématiques et de réfléchir aux relations mathématiques. Dans cet article, nous examinons les discontinuités qui apparaissent le long du chemin d’apprentissage emprunté par différents élèves et qui entraînent une divergence entre la pensée conceptuelle et la pensée procédurale. Nous fournirons des exemples tirés de plusieurs contextes différents illustrant l’acquisition des symboles par l’intermédiaire de l’arithmétique, de l’algèbre et du calcul différentiel et intégral, pour ensuite passer au formalisme des mathématiques axiomatiques. Ces exemples proviennent d’un certain nombre d’études menées récemment à l’Université de Warwick par des doctorants américains, malaisiens, chypriotes et brésiliens, et qui portent sur des données recueillies aux États‐Unis, en Malaisie et au Royaume‐Uni. Toutes ces études s’inscrivent dans le cadre d’une vaste recherche visant à déterminer les raisons pour lesquelles certains élèves réussissent alors que d’autres échouent.

Knowledge Construction and Diverging Thinking in Elementary & Advanced Mathematics

This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action upon it and reflection on both perception and action. We see an emphasis on one or more of these activities leading not only to different kinds of mathematics, but also to a spectrum of success and failure depending on the nature of the focus in the individual activity. For instance, geometry builds from the fundamental perception of figures and their shape, supported by action and reflection to move from practical measurement to theoretical deduction and euclidean proof. Arithmetic, on the other hand, initially focuses on the action of counting and later changes focus to the use of symbols for both the process of counting and the concept of number. The evidence that we draw together from a number of studies on childrens arithmetic shows a divergence in performance. The less successful seem to focus more on perceptions of their physical activities than on the flexible use of symbol as process and concept appropriate for a conceptual development in arithmetic and algebra. Advanced mathematical thinking introduces a new feature in which concept definitions are formulated and formal concepts are constructed by deduction. We show how students cope with the transition to advanced mathematical thinking in different ways leading once more to a diverging spectrum of success.

Students' Mental Prototypes for Functions and Graphs

This research study investigates the concept of function developed by students studying English A-level mathematics. It shows that, while students may be able to use functions in their practical mathematics, their grasp of the theoretical nature of the function concept may be tenuous and inconsistent. The hypothesis is that students develop prototypes for the function concept in much the same way as they develop prototypes for concepts in everyday life. The definition of the function concept, though given in the curriculum, is not stressed and proves to be inoperative, with their understanding of the concept reliant on properties of familiar prototype examples: those having regular shaped graphs, such as x2 or sin*, those often encountered (possibly erroneously), such as a circle, those in which y is defined as an explicit formula in x, and so on. Investigations reveal significant misconceptions. For example, threequarters of a sample of students starting a university mathematics course considered that a constant function was not a function in either its graphical or algebraic forms, and threequarters thought that a circle is a function. This reveals a wide gulf between the concepts as perceived to be taught and as actually learned by the students.