The role of abstraction and generalisation in the development of mathematical knowledge

Although mathematics deals with generalizations relating abstract ideas, very little attention has been given in the mathematics education literature to the role of abstraction and generalization in the development of mathematical knowledge. In this paper, the meanings of "abstraction" and "generalization" are first explored by examining various definitions given by mathematics educators and others. These concepts are illustrated by reference to the author's research on children's understanding of angle concepts. The question of how mathematics teaching would be affected if more explicit attention were given to abstraction and generalization in the classroom is then considered. Finally, some recent experiments in "teaching for abstraction" are presented and discussed. (Contains 29 references.) (Author/MM) Reproductions supplied by EDRS are the best that can be made from the original document. THE ROLE OF ABSTRACTION AND GENERALISATION IN THE DEVELOPMENT OF MATHEMATICAL KNOWLEDGE Michael C. Mitchelmore Macquarie University, Sydney, Australia E-mail: mmitch@ted. educ.mq. edu. au Although mathematics deals with generalisations relating abstract ideas, very little attention has been given in the mathematics education literature to the role of abstraction and generalisation in the development of mathematical knowledge. In this paper, the meanings of "abstraction" and "generalisation" are first explored by examining various definitions given by mathematics educators and others. These concepts are illustrated by reference to the author's research on children's understanding of angle concepts. The question of how mathematics teaching would be affected if more explicit attention were given to abstraction and generalisation in the classroom is then considered. Finally, some recent experiments in "teaching for abstraction" are presented and discussed. Children's Knowledge of Angles Many of the ideas to be presented in this paper derive from research on children's understanding of angles conducted over the past decade by Paul White and myself. So this first section summarises our research on angles. Those who are familiar with our research may confidently skip to the next section. Australian students (and they are probably not alone in this) find angles difficult to understand. For example, Grade 7 teachers always despair at teaching students to use a protractor to measure angles. Our research suggests that the source of students' difficulties is their inability to link angles in different contexts. Learning to use a protractor is difficult because they have been taught in Grade 4 that "angle is an amount of turning", whereas the angles they are confronted with in Grade 7 textbooks do not turn at all and neither does the protractor. To learn how to use a protractor to measure angles, they have to link two very different contexts: the dynamic turning context and the static diagram context. On the other hand, we have found that young children can easily link physically similar angle contexts. For example, almost all Grade 2 children could show how the angles in two turning contexts (a toy ballerina and a regulator knob) are related (Mitchelmore, 1998). A similar success rate was found among Grade 2 children in relating the angles in scissors and road bends (Mitchelmore & White, 1998) but not the angles of opening in doors and tilting windows (White & Mitchelmore, 2001). Two questions arose from our earlier research: (1) Among all angle contexts, which do students find most similar and why? (2) How does their recognition of angular similarity change as children grow older? Do distinct clusters of angle concepts develop (e.g., turns and corners) or only one angle concept? To investigate these questions, we presented 192 students in Grades 2, 4, 6, and 8 with examples or models of 9 different angle contexts and asked them to show us how the angles were similarfor example, by using a bent straw to indicate the angles in each context (Mitchelmore & White, 2000a). The variation in the similarity matrices at each grade level was then explored using cluster analysis. Figure 1 summarises the results. At each grade level, there was one main cluster. In Grade 2, this consisted of junction, tile, and wall; scissors was nearby and joined the main cluster in Grade 4. By Grade 6, the main cluster consisted of junction, tile, wall, scissors, fan, and signpost. Even by Grade 8, the remaining contexts (door, hill, and wheel) had not joined the main cluster. The results suggest that the angle concept develops from a single core concept which gradually expands to include other contexts. They also suggest that the major factor affecting children's recognition of angular similarity is . the presence of visible lines and vertices: the angle arms in the scissors, fan, and signpost contexts are rather indefinite; the doors and the hill only have one arm of the angle visible; and a wheel has no visible arm. PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIALHAS BEEN GRANTED BY TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement DUCATIONAL RESOURCES INFORMATION CENTER (ERIC) cument 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. EARCOME 2002 Proceedings Volume 1, Plenary & Regular Lectures, 157-167