Abstract
Abstract algebra courses tend to take one of two pedagogical routes: from examples of mathematics structures through definitions to general theorems, or directly from definitions to general theorems. The former route seems to be based on the implicit pedagogical intention that students will use their understanding of particular examples of an algebraic structure to get a sense of those properties which form the basis of the fundamental definitions. We will explain the transition from examples to abstract algebra as a series of shifts of attention and in this paper we will use a case study to examine the initial shift, which we will call apprehending a structure, and examine how one student came to apprehend the structure of the commutative ring Z 99.
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Alibert, D. and Thomas, M.: 1991, ‘Research on mathematical proof’, in D. Tall (ed.), Advanced Mathematical Thinking, Kluwer Academic Publishers, The Netherlands, pp. 215–230.
Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D. and Thomas, K.: 1996, ‘A framework for research and curriculum development in undergraduate mathematics education’, Research in Collegiate Mathematics Education II, pp. 1–35.
Asiala, M., Brown, A., Kleiman, J. and Mathews, D.: 1998, ‘The development of students’ understanding of permutations and symmetries’, International Journal of Computers for Mathematical Learning 3(1), 13–43.
Asiala, M., Dubinsky, E., Mathews, D.M., Morics, S. and Oktaç, A.: 1997, ‘Development of students’ understanding of cosets, normality, and quotient groups’, Journal of Mathematical Behavior 16(3), 241–309.
Cosmides, L. and Tooby, J.: 1994, ‘Origins of domain specificity: The evolution of functional organization’, in L. A. Hirschfeld and S. A. Gelman (eds.), Mapping the Mind: Domain Specificity in Cognition and Culture, Cambridge University Press, New York, pp. 85– 116.
Deacon, T.: 1997, The Symbolic Species, Allen Lane, The Penguin Press, London.
Dorier, J.L.: 1995, ‘Meta level in the teaching of unifying and generalizing concepts in mathematics’, Educational Studies in Mathematics 29(2), 175–197.
Dreyfus, T.: 1991, ‘Advanced mathematical thinking processes’, in D. Tall (ed.), Advanced Mathematical Thinking, Kluwer Academic Publishers, The Netherlands, pp. 25–41.
Dubinsky, E.: 1991, ‘Reflective abstraction in mathematical thinking’, in D.O. Tall (ed.) Advanced Mathematical Thinking, Kluwer, Dordrecht, pp. 95–123.
Dubinsky, E., Dautermann, J., Leron, U. and Zazkis, R.: 1994, ‘On learning fundamental concepts of group theory’, Educational Studies in Mathematics 27(3), 267–305.
Ginsburg, H.: 1981, ‘The clinical interview in psychological research on mathematical thinking: Aims, rationales, techniques’, For the Learning of Mathematics 1, 4–11.
Glaser, B. and Strauss, A.: 1967, Discovery of Grounded Theory, Aldine, Chicago.
Gray, E.M. and Tall, D.O.: 1994, ‘Duality, ambiguity and flexibility: A proceptual view of simple arithmetic’, Journal for Research in Mathematics Education 25, 115–141.
Harel, G. and Sowder, L.: 1998, ‘Students proof schemes’, in A. Schoenfeld, J. Kaput and E. Dubinsky (eds.), Research in Collegiate Mathematics Education III, American Mathematical Society, Washington, DC, pp. 234–282.
Hart, E.W.: 1994, ‘A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory’, in J.J. Kaput and E. Dubinsky (eds.), Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results, MAA, Washington, DC.
Hazzan, O.: 1994, ‘A student’s belief about the solutions of the equation x = x − 1 in a group’, in J.P. da Ponte and J.F. Matos (eds.), Proceedings of the Eighteenth International Conference for the Psychology of Mathematics Education, Vol. 3, PME, Lisbon, pp. 49–56.
Hazzan, O.: 1999, ‘Reducing abstraction level when learning abstract algebra concepts’, Educational Studies in Mathematics 40(1), 71–90.
Hazzan, O. and Leron, U.: 1996, ‘Students’ use and misuse of mathematical theorems: The case of Lagrange’s theorem’, For the Learning of Mathematics 16(1), 23–26.
Hejný, M.: 1992, ‘Analysis of student’s solution of the equation x 2 = a 2 and x 2 − a 2 = 0’, Acta Didactica Universitatis Comenianae 1, 65–82.
Kaput, J.: 1994, ‘Democratizing access to calculus: New routes to old roots’, in A. Schoenfeld (ed.), Mathematical Thinking and Problem Solving, Erlbaum, Hillsdale, NJ, pp. 77–156.
Karmiloff-Smith, A.: 1992, Beyond Modularity: A Developmental Perspective on Cognitive Science, MIT Press, Cambridge, Massachusetts.
Leron, U. and Dubinsky, E.: 1995, ‘An abstract algebra story’, American Mathematical Monthly 102, 227–242.
Mason, J. and Pimm, D.: 1984, ‘Generic examples: Seeing the general in the particular’, Educational Studies in Mathematics 15(3), 277–290.
Mason, J.: 1989, ‘Mathematical abstraction seen as a delicate shift of attention’, For the Learning of Mathematics 9(2), 2–8.
Moore, R.C.: 1994, ‘Making the transition to formal proof’, Educational Studies in Mathematics 27, 249–266.
Morgan, C.: 1998, Writing Mathematically. The Discourse of Investigation, Falmer Press, London.
Novak, J.D.: 1990, ‘Concept maps and Vee diagrams: Two metacognitive tools for science and mathematics education’, Instructional Science 19, 29–52.
Piaget, J. and Garcia, R.: 1989, Psychogenesis and the History of Science, Columbia University Press, New York.
Selden, A.A. and Selden, J.: 1978, ‘Errors students make in mathematical reasoning’, Bogazici Üniversitesi Dergis 6, 67–87.
Selden, A.A. and Selden, J.: 2002, ‘Validations of proofs written as texts: Can undergraduates tell whether an argument proves a theorem?’, Journal for Research in Mathematics Education 34(1), 4–36.
Sfard, A.: 1991, ‘On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin’, Educational Studies in Mathematics 22, 1–36.
Sierpinska, A.: 1994, Understanding in Mathematics, The Falmer Press, London.
Skemp, R.R.: 1971, The Psychology of Learning Mathematics, Penguin, London.
Stehlíková, N.: 2004, Structural Understanding in Advanced Mathematical Thinking, Charles University in Prague, Faculty of Education, Prague.
Tomasello, M.: 1999, The Cultural Origins of Human Cognition, Harvard University Press.
van Dormolen, J.: 1986, ‘Textual analysis’, in B. Christiansen, A.G. Howson and M. Otte (eds.), Perspectives on Mathematics Education, Reidel Publishing Company, The Netherlands, pp. 141–171.
Zazkis, R., Dubinsky, E. and Dautermann, J.: 1996, ‘Coordinating visual and analytic strategies: A study of students’ understanding of the group D4’, Journal for Research in Mathematics Education
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Simpson, A., Stehlíková, N. Apprehending Mathematical Structure: A Case Study of Coming to Understand a Commutative Ring. Educ Stud Math 61, 347–371 (2006). https://doi.org/10.1007/s10649-006-1300-y
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DOI: https://doi.org/10.1007/s10649-006-1300-y