# Reducing Abstraction Level When Learning Abstract Algebra Concepts

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## Abstract

How do undergraduate students cope with abstract algebra concepts? How should we go about researching this question? Based on interviews with undergraduate students and on written questionnaires, a theoretical framework evolved which could coherently account for most of the data. According to this theoretical framework, students' responses can be interpreted as a result of reducing the level of abstraction. In this paper, the theme of reducing abstraction is examined, based on three interpretations for *levels of abstraction* discussed in mathematics education research literature. From these three perspectives on abstraction, ways in which students reduce abstraction level are analyzed and exemplified.

## Keywords

Undergraduate Student Mathematics Education Education Research Research Literature Abstraction Level## Preview

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## References

- 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*, The American Mathematics Society, pp. 1–32.Google Scholar - 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.Google Scholar - 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.Google Scholar - Beth, E. W. and Piaget, J.: 1966,
*Mathematical Epistemology and Psychology*, D. Reidel Publishing Company, Dordrecht, The Netherlands.Google Scholar - Brown, A., DeVries, D. J., Dubinsky, E. and Thomas, K.: 1997, ‘Learning binary operations, groups and subgroups’,
*Journal of Mathematical Behavior*16(3), 187–239.Google Scholar - Buchthal, D. C.: 1977, ‘Operative and nonoperative group definitions’,
*Mathematics Teacher*70(3), 262–263.Google Scholar - Confrey, J.: 1990, ‘Chapter 8: What constructivism implies for teaching’, in R. B. Davis, C. A. Maher and N. Noddings (eds.),
*Journal for Research in Mathematics Education, Monograph Number 4, Constructivist Views on the Teaching and Learning of Mathematics*, The National Council of Teachers of Mathematics, Inc., pp. 107–122.Google Scholar - Davis, R. B., Maher, C. A. and Noddings, N.: 1990, ‘Chapter 12: Suggestions for the improvement of mathematics education’, in R. B. Davis, C. A. Maher and N. Noddings (eds.),
*Journal for Research in Mathematics Education, Monograph Number 4, Constructivist Views on the Teaching and Learning of Mathematics*, The National Council of Teachers of Mathematics, Inc., pp. 187–191.Google Scholar - Dreyfus, T.: 1990, ‘Advanced mathematical thinking’, in P. Nesher and J. Kilpatrick (eds.),
*Mathematics and Cognition: A research Synthesis by the International Group for the Psychology of Mathematical Education*, Cambridge University Press, pp. 113–134.Google Scholar - Dreyfus, T.: 1995, ‘Guest editorial, special issue — Advanced mathematical thinking’,
*Educational Studies in Mathematics*29, 93–95.Google Scholar - Dubinsky, E., Elterman, F. and Gong, C.: 1988, ‘The student's construction of quantification’,
*For the Learning of Mathematics*8(2), 44–51.Google Scholar - Dubinsky, E.: 1991, ‘Reflective abstraction in advanced mathematical thinking’, in D. Tall (ed.),
*Advanced Mathematical Thinking*, Kluwer Academic Press, pp. 95–123.Google Scholar - Dubinsky, E.: 1997, ‘On learning quantification’,
*Journal of Computers in Mathematics and Science Teaching*16(2/3), 335–362.Google Scholar - Dubinsky, E., Dautermann, J., Leron, U. and Zazkis, R.: 1994, ‘On learning fundamental concepts of group theory’,
*Educational Studies in Mathematics*27, 267–305.Google Scholar - Dubinsky, E. and Leron, U.: 1994,
*Learning Abstract Algebra with Isetl,*Springer-Verlag.Google Scholar - Freedman, H.: 1983, ‘A way of teaching abstract algebra’,
*American Mathematical Monthly*90(9), 641–644.Google Scholar - Frorer, P., Hazzan, O. and Manes, M.: 1997, ‘Revealing the faces of abstraction’,
*The International Journal of Computers for Mathematical Learning*2(3), 217–228.Google Scholar - Gallian, J. A.: 1990,
*Contemporary Abstract Algebra, second edition*, D. C. Heath and Company.Google Scholar - Glaser, B. and Strauss, A. L.: 1967,
*The Discovery of Grounded Theory: Strategies for Qualitative Research*, Chicago, Aldine.Google Scholar - Harnik, V.: 1986, ‘Infinitesimals from Leibniz to Robinson, Time to bring them to school’,
*The Mathematical Intelligencer*8(2), 41–47, 63.Google Scholar - Hart, E.: 1994, ‘A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory’, in J. Kaput and E. Dubinsky (eds.),
*Research Issues in Mathematics Learning: Preliminary Analyses and Results*, MAA notes, No. 33.Google Scholar - Hazzan, O.: 1994, ‘A students' belief about the solutions of the equation x=x
^{−1}in a group’,*Proceedings of the 18th international conference for the Psychology of Mathematics Education*, Lisbon, Portugal, Vol. III, pp. 49–56.Google Scholar - Hazzan, O.: 1995,
*Undergraduate Students' Understanding of Abstract Algebra Concepts*. Ph.D. thesis, Technion — Israel Institute of Technology. Unpublished (Hebrew).Google Scholar - 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, 23–26.Google Scholar - Herstein, I. N.: 1986,
*Abstract Algebra,*Macmillan Publishing Company.Google Scholar - Kilpatrick, J.: 1987, ‘What constructivism might be in mathematics education’,
*Proceedings of the Eleventh International Conference for the Psychology of Mathematical Education*, Volume I, 3–27.Google Scholar - Kleiner, I.: 1991, ‘Rigor and proof in mathematics: A historical perspective’,
*Mathematics Magazine*64(5), 291–314.Google Scholar - Leron, U.: 1987, ‘Abstraction barriers in mathematics and computer science’,
*Proceedings of the Third International Conference for the Psychology of Mathematical Education,*Montreal.Google Scholar - Leron, U. and Dubinsky, E.: 1995, ‘An abstract algebra story’,
*The American Mathematical Monthly*102(3), 227–242.Google Scholar - Leron, U., Hazzan, O. and Zazkis, R.: 1994, ‘Students constructions of group isomorphism’,
*Proceedings of the Eleventh International Conference for the Psychology of Mathematical Education*3, 152–159.Google Scholar - Leron, U., Hazzan, O. and Zazkis, R.: 1995, ‘Learning group isomorphism: A crossroads of many concepts’,
*Educational Studies in Mathematics*29, 153–174.Google Scholar - Lesh, R. A.: 1976, ‘The influence of two types of advanced organizers on an instructional unit about finite groups’,
*Journal for Research in Mathematics Education*7(2), 87–91.Google Scholar - Lichtenberg, D. R.: 1981, ‘A group whose elements are functions’,
*Mathematics Teacher*74(7), 521–523.Google Scholar - Macdonald, I. D.: 1976, ‘Modern algebra in the nineteenth century’,
*Australian Mathematics Teacher*32(1), 33–38.Google Scholar - Nesher, P. and Teubal, E.: 1975, ‘Verbal cues as an interfering factor in verbal problem solving’,
*Educational Studies in Mathematics*6, 41–51.Google Scholar - Nesher, P.: 1980, ‘The stereotyped nature of school word problems’,
*For the Learning of Mathematics*1(1), 41–48.Google Scholar - Noss, R. and Hoyles, C.: 1996,
*Windows on Mathematical Meanings — Learning Cultures and Computers*, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar - Papert, S.: 1980,
*Mindstorms — Children, Computers and Powerful Ideas*, Basic Books, Inc., Publishers / New York.Google Scholar - Pedersen, J. J.: 1972, ‘Sneaking up on a group’,
*Two Year College Mathematics Journal*3(1), 9–12.Google Scholar - Petricig, M.: 1988, ‘Combining individualized instruction with the traditional lecture method in a college algebra course’,
*Mathematics Teacher*81, 385–387.Google Scholar - Piaget, J.: 1977, ‘Problems of Equilibration’, in M. H. Appel and L. S. Goldberg (eds.),
*Topics in Cognitive Development, Volume 1: Equilibration: Theory, Research and Application*, Plenum Press, NY, pp. 3–13.Google Scholar - Polya, G.: 1973,
*How to Solve it?*, (Second edition), Princeton University Press, Princeton, New Jersey.Google Scholar - Quadling, D.: 1978, ‘A contorted isomorphism’,
*Mathematics Teaching*85, 48–49.Google Scholar - Rumelhart, D. E.: 1989, ‘Toward a microstructural account of human reasoning’, in S. Vosniadou and A. Ortony (eds.),
*Similarity and Analogical Reasoning*, Cambridge University Press, pp. 298–312.Google Scholar - Schoenfeld, A. H.: 1985,
*Mathematical Problem Solving*, Academic press, Inc.Google Scholar - Selden, A. and Selden, J.: 1987, ‘Errors and misconceptions in college level theorem proving’,
*Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics*, Cornell University, Vol. III, pp. 456–471.Google Scholar - Selden, A. and Selden, J.: 1993, ‘Collegiate mathematics education research: What would that be like?’
*The College Mathematics Journal*24(5), 431–445.Google Scholar - 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.Google Scholar - Sfard, A.: 1992, ‘Operational origins of mathematical objects and the quandary of reification — The case of function’, in E. Dubinsky and G. Harel (eds.),
*The Concept of Function — Aspects of Epistemology and Pedagogy*, MAA Notes.Google Scholar - Simmonds, G.: 1982, ‘Computer discovery of a theorem of modern algebra’,
*Mathematics and Computer Education*16(1), 58–61.Google Scholar - Sinclair, H.: 1987, ‘Constructivism and the psychology of mathematics’,
*Proceedings of the Eleventh International Conference for the Psychology of Mathematical Education,*Volume I, pp. 28–41.Google Scholar - Smith, J. P., diSessa, A. A., Roschelle, J.: 1993, ‘Misconceptions reconceived: A constructivist analysis of knowledge in transition’,
*The Journal of the Learning Sciences*3(2), 115–163.Google Scholar - Staub, F. C. and Stern, E.: 1997, ‘Abstract reasoning with mathematical constructs’,
*International Journal of Educational Research*27(1), 63–75.Google Scholar - Tall, D.: 1991 (ed.),
*Advanced Mathematical Thinking*, Kluwer Academic Press, Dordrecht, The Netherlands.Google Scholar - Thompson, P. W.: 1985, ‘Experience, problem solving, and learning mathematics: Considerations in developing mathematics curricula’, in E. A. Silver (ed.),
*Teaching and Learning Mathematical Problem Solving: Multiple Research Perspective*, Hillsdale, NJ, pp. 189–236.Google Scholar - Thompson, P. W.: 1993, ‘Yes, Virginia, some children do grow up to be mathematicians’,
*Journal for Research in Mathematics Education*24(3), 279–284.Google Scholar - Thrash, K. R. and Walls, G. L.: 1991, ‘A classroom note on understanding the concept of group isomorphism’,
*Mathematics and Computer Education*25(1), 53–55.Google Scholar - Wilensky, U.: 1991, ‘Abstract meditations on the concrete and concrete implications for mathematical education’, in I. Harel and S. Papert (eds.),
*Constructionism*, Ablex Publishing Corporation, Norwood, NJ, pp. 193–203.Google Scholar