Eddie Gray

Duality, ambiguity, and flexibility: A `proceptual' view of simple arithmetic

In this paper we consider the duality between process and concept in mathematics, in particular using the same symbolism to represent both a process (such as the addition of two numbers 3+2) and the product of that process (the sum 3+2). The ambiguity of notation allows the successful thinker the flexibility in thought to move between the process to carry out a mathematical task and the concept to be mentally manipulated as part of a wider mental schema. We hypothesise that the successful mathematical thinker uses a mental structure which is an amalgam of process and concept which we call a procept. We give empirical evidence from simple arithmetic to show that this leads to a qualitatively different kind of mathematical thought between the more able and the less able, in which the less able are actually doing a more difficult form of mathematics, causing a divergence in performance between success and failure.

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?

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.

An Analysis of Diverging Approaches to Simple Arithmetic: Preference and Its Consequences.

Earlier research by the author indicated that many below average attainers do not remember number facts and use alternative strategies to obtain solutions to basic arithmetical problems. These alternatives were frequently seen as the ‘best way’ of finding a solution.This paper considers the relationship between the various strategies used by mixed ability children aged 7 to 12. An analysis of alternatives suggests that the selection is not underpinned by regression through the learning sequence, but by regression dominated by the childs preference for certain strategies over others. Through the evaluation of a hierarchy of preferences, divergence between the strategies available to the less able and the more able child is revealed. The alternative strategies used are based either on counting — procedural strategies, or on the use of selected known knowledge — deductive strategies. Above average children have both available as alternatives; evidence of deduction is rare amongst below average children. The more able child appears to build up a growing body of known facts from which new known facts are deduced. Less able children — relying mainly on procedural strategies — do not appear to have this feedback loop available to them.The paper contends that, for some children, procedural methods do not encourage the need to remember; the procedure provides security. On the other hand, deductive methods initially enhance the ability to remember other basic facts and eventually help children make extensive use of facts that are known to remove the need to remember new ones. More able children appear to be doing a qualitatively different sort of mathematics than the less able.

Abstraction as a Natural Process of Mental Compression

This paper considers mathematical abstraction as arising through a natural mechanism of the biological brain in which complicated phenomena are compressed into thinkable concepts. The neurons in the brain continually fire in parallel and the brain copes with the saturation of information by the simple expedient of suppressing irrelevant data and focusing only on a few important aspects at any given time. Language enables important phenomena to be named as thinkable concepts that can then be refined in meaning and connected together into coherent frameworks. Gray and Tall (1994) noted how this happened with the symbols of arithmetic, yielding a spectrum of performance between the more successful who used the symbols as thinkable concepts operating dually as process and concept (procept) and those who focused more on the step-by-step procedures and could perform simple arithmetic but failed to cope with more sophisticated problems. In this paper, we broaden the discussion to the full range of mathematics from the young child to the mature mathematician, and we support our analysis by reviewing a range of recent research studies carried out internationally by research students at the University of Warwick.

Objects, Actions, and Images: A Perspective on Early Number Development

It is the purpose of this article to present a review of research evidence that indicates the existence of qualitatively different thinking in elementary number development. In doing so, the article summarizes empirical evidence obtained over a period of 10 years. This evidence first signaled qualitative differences in numerical processing, and was seminal in the development of the notion of procept. More recently, it examines the role of imagery in elementary number processing. Its conclusions indicate that in the abstraction of numerical concepts from numerical processes qualitatively different outcomes may arise because children concentrate on different objects or different aspects of the objects, which are components of numerical processing.