Abstract
This chapter reports on some current educational issues related to the teaching and learning of tertiary algebra—in particular, abstract algebra, discrete mathematics, linear algebra, and number theory. The causes of conceptual difficulties experienced by many students are identified and possible ways of overcoming them, sometimes using a specific pedagogical framework, are discussed. Issues related to students’ motivation are explored and pedagogical possibilities for overcoming some of the problems in both these areas are also explored. This report also addresses issues associated with the dissemination of educational work to tertiary instructors who are typically mathematicians rather than mathematics educators. Furthermore, the role of computers in tertiary algebra courses is considered, focusing on the use of Computer Algebra Systems (at the tertiary level) and the use of the programming language ISETL that helps students construct and work with algebraic objects. This chapter makes recommendations for improving practices for teaching tertiary algebra and proposes areas for further research.
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References
Artigue, M. (2001). What can we learn from educational research at the university level? In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 207–220). Dordrecht, The Netherlands: Kluwer Academic.
Asiala, A., Dubinsky, E., Mathews, D., Morics, S., & Oktaç, A. (1997). Development of student understanding of cosets, normality, and quotient groups. Journal of Mathematical Behavior, 16, 241–309.
Brown, A., DeVries, D., Dubinsky, E., & Thomas, K. (1997). Learning binary operations, groups, and subgroups. Journal of Mathematical Behavior, 16, 187–239.
Campbell, S., & Zazkis, R. (Eds.). (2001). The learning and teaching of number theory. New York: Ablex Publishers.
Carlson, D., Johnson, C., Lay, D., & Porter, A. D. (1993). The linear algebra curriculum study group recommendations for the first course in linear algebra. College Mathematics Journal, 24, 41–46.
Carlson, D., Johnson, C., Lay, D., & Porter, A. D. (Eds.) (2002). Linear algebra gems: Assets for undergraduate mathematics. Washington, D.C.: Mathematical Association of America.
Carlson, D., Johnson, C., Lay, D., Porter, A. D., Watkins, A., & Watkins, W. (Eds.). (1997). Resources for teaching linear algebra. (MAA Notes Vol. 42). Washington, D.C.: Mathematical Association of America.
Clark, J., Hemenway, C., St. John, D., Tolias, G., & Vakil, R. (1999). Student attitudes toward abstract algebra. Primus, 9, 76–96.
Dorier, J.-L. (Ed.). (2000). On the teaching of linear algebra. Dordrecht, The Netherlands: Kluwer Academic.
Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (1994). The teaching of linear algebra in first year of French science university. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 137–144). Lisbon, Portugal: Program Committee.
Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacle of formalism in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 85–124). Dordrecht, The Netherlands: Kluwer Academic.
Dorier, J.-L., & Sierpinska, A. (2001). Research into the teaching and learning of linear algebra. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 255–273). Dordrecht, The Netherlands: Kluwer Academic.
Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Dordrecht, The Netherlands: Kluwer Academic.
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht, The Netherlands: Kluwer.
Dubinsky, E. (1997). On learning quantification. Journal of Computers in Mathematics and Science Teaching, 16(1), 335–362.
Dubinsky, E., Elterman, F., & Gong, C. (1988). The student’s construction of quantification I. For the Learning of Mathematics, 8(2), 44–51.
Dubinsky, E., Mathews, D., & Reynolds, B. (Eds.). (1997). Readings in cooperative learning for undergraduate mathematics. Washington, D.C.: Mathematical Association of America.
Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 275–282). Dordrecht, The Netherlands: Kluwer Academic.
Dubinsky, E., & Yiparaki, O. (2000). On student understanding of AE and EA quantification. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education IV (pp. 293–289). Providence, RI: American Mathematical Society.
Fenton, W., & Dubinsky, E. (1996). Introduction to discrete mathematics with ISETL. New York: Springer.
Forgasz, H. J., & Leder, G. C. (1998). Affective dimensions and tertiary mathematics students. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd annual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, 296–303). Stellenbosch, South Africa: Program Committee.
Hagelgans, N., Reynolds, B., Schwingendorf, K., Vidakovic, D., Dubinsky, E., Shanin, M., & Wimbish, G. (Eds.). (1995). A practical guide to cooperative learning in collegiate mathematics. (MAA Notes Vol. 37). Washington, D.C.: MAA.
Harel, G. (1989). Learning and teaching linear algebra: Difficulties and an alternative approach to visualizing concepts and processes. Focus on Learning Problems in Mathematics, 11, 139–148.
Harel, G. (1991). Using geometric models and vector arithmetic to teach high school students basic notions in linear algebra. International Journal for Mathematics Education in Science and Technology, 21, 387–392.
Harel, G. (1999). Students’ understanding of proofs: A historical analysis and implications for the teaching of geometry and linear algebra. Linear Algebra and its Applications, 302–303, 601–613.
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234–283). Providence, RI: American Mathematical Society.
Hart, E. (1994). Analysis of the proof-writing performances of expert and novice students in elementary group theory. In J. Kaput & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning (MAA Notes Vol. 33, pp. 49–62). Washington, D.C.: Mathematical Association of America.
Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts. Educational Studies in Mathematics, 40(1), 71–90.
Hibbard, A., & Maycock, E. (Eds.). (2002). Innovations in teaching abstract algebra. Washington, D.C.: Mathematical Association of America.
Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 191–207). Dordrecht, The Netherlands: Kluwer Academic.
Holton D. (Ed.) (2001). The teaching and learning of mathematics at university level: An ICM1 study. Dordrecht, The Netherlands: Kluwer Academic.
Katz, V. J. (2001). Using the history of algebra in teaching algebra. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 353–359). Melbourne, Australia: The University of Melbourne.
Kleiner, I. (1998). A historically focused course in abstract algebra. Mathematics Magazine, 71, 105–111.
Lajoie, C. (2001). Students’ difficulties with the concepts of group, subgroup, and group isomorphism. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 384–391). Melbourne, Australia: The University of Melbourne.
Leon, S., Herman, E., & Faulkenberry, R. (1997). ATLAST computer exercises for linear algebra. Upper Saddle River, NJ: Prentice-Hall.
Leron, U., & Dubinsky, E. (1995). An abstract algebra story. American Mathematical Monthly, 102(3), 227–242.
Leron, U., & Hazzan, O. (1997). The world according to Johnny: A coping perspective in mathematics education. Educational Studies in Mathematics, 32, 265–292.
Oktaç, A. (2001). The teaching and learning of linear algebra: Is it the same at a distance? In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 501–506). Melbourne, Australia: The University of Melbourne.
Rogers, E., Reynolds, B., Davidson, N., & Thomas, A. (Eds.). (2001). Cooperative learning in undergraduate mathematics: Issues that matter and strategies that work. Washington, D.C.: Mathematical Association of America.
SEFI Mathematics Working Group (2002). Core curriculum 2002. Retrieved May 16, 2003, from http://learn.lboro.ac.uk/mwg/core.html
Selden, A., & Selden, J. (1987). Errors and misconceptions in college level theorem proving. In J. Proceedings of the second international seminar on misconceptions and educational strategies in science and mathematics (Vol. III, pp. 457–470). New York: Cornell University.
Selden, A., & Selden, J. (2001). Research into the teaching and learning of linear algebra. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 237–254). Dordrecht, The Netherlands: Kluwer Academic.
Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 209–246). Dordrecht, The Netherlands: Kluwer Academic.
Sierpinska, A., Nnadozie, A., & Oktaç, A. (2002). A study of the relationship between theoretical thinking and high achievement in linear algebra. Montreal, Canada: Concordia University. Retrieved May 16, 2003, from http://alcor.Concordia.ca/~sierp/downloadpapers.html
Sierpinska, A., Trgalova, J., Hillel, J., & Dreyfus, T. (1999). Teaching and learning linear algebra with Cabri. In Zaslavsky, O. (Ed.), Proceedings of the 23rd annual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 119–134). Haifa, Israel: Program Committee.
Siu, M-K. (2001). Why is it difficult to teach abstract algebra? In H. Chick, K., Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 541–547). Melbourne, Australia: The University of Melbourne.
Sowder, L., & Harel, G. (2003). Case studies of mathematics majors’ proof understanding, production and appreciation. Canadian Journal of Science, Mathematics and Technology Education, 3(2), 251–267.
Tall, D. O., & Vinner, S. (1981). Concept images and concept definition in mathematics, with particular reference to limit and continuity. Educational Studies in Mathematics, 22(2), 125–147.
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall, D. (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht, The Netherlands: Kluwer Academic.
Weller, K., Montgomery, A., Clark, J., Cottrill, J., Trigueros, M., Arnon, I., & Dubinsky, E. (2002). Learning linear algebra with ISETL. Retrieved May 16, 2003 from http://www.ilstu.edu/~jfcottr/linear-alg/
Wood, L. N. (1999). Teaching definitions in undergraduate mathematics. Talum Newsletter, No. 10, April. May 16, 2003, http://www.bham.ac.uk/ctimath/talum/newsletter/wood.htm
Wood, L., & Smith, G. (2001). Assessment in linear algebra. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 663–667). Melbourne, Australia: The University of Melbourne.
Zazkis, R., Dautermann, J., & Dubinsky, E. (1996). Coordinating visual and analytic strategies: A study of students’ understandings of the group D4. Journal for Research in Mathematics Education, 27, 435–457
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Carlson, D. (2004). The Teaching and Learning of Tertiary Algebra. In: Stacey, K., Chick, H., Kendal, M. (eds) The Future of the Teaching and Learning of Algebra The 12th ICMI Study. New ICMI Study Series, vol 8. Springer, Dordrecht. https://doi.org/10.1007/1-4020-8131-6_11
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DOI: https://doi.org/10.1007/1-4020-8131-6_11
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