Cognitive Functions in Mathematical Thinking. Basic Issues for Learning

This paper focuses on some main distinctions that are necessary to analyze mathematical knowledge from a learning point of view and explain how many students come up against difficulties at each level of the curriculum. From a learning point of view, visualization cannot be used as an immediate and obvious support for understanding, even though it is the only relevant cognitive modality in mathematics. (Contains 21 references.) (DDR) Reproductions supplied by EDRS are the best that can be made from the original document. PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL HAS BEEN GRANTED BY TO TIN HE EDUCATIONAL R TER ESOURCES FORMATION CEN (ERIC) U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) This document has been reproduced as received from the person or organization originating it. Minor changes have been made to improve reproduction quality. Points of view or opinions stated in this document do not necessarily represent official OERI position or policy. REPRESENTATION, VISION AND VISUALIZATION: COGNITIVE FUNCTIONS IN MATHEMATICAL THINKING. BASIC ISSUES FOR LEARNING Raymond Duval Universite du Littoral Cote-d'Opale, Boulogne, et Centre IUFM Nord Pas-de Calais, Lille [email protected] Mathematics education has been very sensitive to needed changes over the past fifty years. Researches in developmental psychology, new technologies, andnew requirements in assessment have supported them. But their impact has been more effective on mathematics curriculum and on means of teaching than on the explanations of the deep processes of understanding and learning in mathematics. Difficulties of such research stem from the necessity of defining a framework within the epistemological constraints specific to mathematical activity and the cognitive functions of thought which it involves are not separated. That requires going beyond local studies of concept acquiring at each level of the curriculum and beyond mere reference to very general theories of learning and even beyond global description of student's activity in classroom. Representation and visualization are at the core of understanding in mathematics. But in which framework can their role in mathematical thinking and in learning of mathematics be analyzed? Already in 1961, Piaget admitted the difficulty to understand what mathematicians call "intuition", a way of understanding which has close links with representation and visualization: "rien n'est plus difficile a comprendre pour un psychologue que ce que les mathematiciens entendent par intuition". He distinguished "many forms of mathematical intuition" (1961, pp. 223-241) from the empirical ones to the symbolizing ones. From a cognitive viewpoint, the question is not easier. Representation refers to a large range of meaning activities: steady and holistic beliefs about something, various ways to evoke and to denote objects, how information is coded. On the contrary, visualization seems to emphasize images and empirical intuition of physical objects and actions. Which ones are relevant to analyze the understanding in mathematics in order to bring out conditions of learning? Our purpose in this panel is to focus on some main distinctions which are necessary to analyze the mathematical knowledge from a learning point of view and to explain how many students come up against difficulties at each level of curriculum and very often cannot go beyond. Studies about reasoning, proving, using geometrical figures in problem solving, reading