Conceptualising a university teaching practice in an activity theory perspective
DOI:
https://doi.org/10.7146/nomad.v20i2.148679Abstract
In this article I present a theorisation of a university mathematics teaching practice, based on a research study into the teaching of linear algebra in a first year mathematics undergraduate course. The research was largely qualitative and consisted of data collected in interviews with the lecturer and in observations of his lectures. Using Leontiev’s (1981) activity theory framework I categorised the teaching of linear algebra on three levels: activity-motive, actions-goals and operations-conditions. Each level of analysis provided insights into the lecturer’s teaching approach, his motivation, his intentions and his strategies in relation to his teaching. I developed a model of the teaching process that relates goals as expressed by the lecturer in interviews to the strategies that he designed for his teaching.
References
Alsina, C. (2001). Why the professor must be a stimulating teacher. In D. Holton (Ed.), The teaching and learning of mathematics at university level: an ICMI study (pp. 3-12). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47231-7_1
Barnard, T. & Morgan, C. (1996). Theory and practice in undergraduate mathematics teaching: a case study. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 43-50). Valencia: PME.
Bergsten, C. (2007). Investigating quality of undergraduate mathematics lectures. Mathematics Education Research Journal, 19 (3), 48-72. https://doi.org/10.1007/BF03217462
Burton, L. (2004). Mathematicians as enquirers: learning about learning mathematics. Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-1-4020-7908-5
Carlson, D. (1993). Teaching linear algebra: must the fog always roll in? The College Mathematics Journal, 24 (1), 29-40. https://doi.org/10.1080/07468342.1993.12345737
Carlson, D., Johnson, C. R., Lay, D. & Porter, D. (1993). The Linear Algebra Curriculum Study Group recommendations for the first course in linear algebra. The College Mathematics Journal, 24 (1), 41-46. https://doi.org/10.1080/07468342.1993.11973504
Daniels, H. (2008). Vygotsky and research. London: Routledge. https://doi.org/10.4324/9780203891797
Dorier, J.-L. (1998). The role of formalism in the teaching of the theory of vector spaces. Linear algebra and its applications, 275-276, 141-160. https://doi.org/10.1016/S0024-3795(97)10061-1
Dorier, J.-L. (Ed.). (2000). On the teaching of linear algebra. Dordrecht: Kluwer Academic Publishers.
Dorier, J.-L., Robert, A., Robinet, J. & Rogalski, M. (2000a). On a research programme concerning the teaching and learning of linear algebra in the first-year of a French science university. International Journal of Mathematical Education in Science and Technology, 31 (1), 27-35. https://doi.org/10.1080/002073900287354
Dorier, J.-L., Robert, A., Robinet, J. & Rogalski, M. (2000b). The meta lever. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 151-176). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47224-4_5
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. 225-273). Dordrecht: Kluwer Academic Publishers.
Fischbein, E. (1987). Intuition in science and mathematics: an educational approach. Dordrecht: D. Reidel Publishing Company.
Glaser, B. G. (1992). Basics of grounded theory analysis: emergence vs forcing. Mill Valley: Sociology Press.
Glaser, B. G. & Strauss A. L. (1967). The discovery of grounded theory. Piscataway: Aldine Transactions.
Goodchild, S. (1997). An exploratory study of year ten students' goals in the mathematics classroom (Unpublished doctoral dissertation). University of Exeter.
Hadamard, J. (1945, 1973). The mathematician's mind. Princetown University Press.
Hannah, J., Stewart, S. & Thomas, M. (2013). Emphazising language and visualization in teaching linear algebra. International Journal of Mathematical Education in Science and Technology, 44 (4), 475-489. https://doi.org/10.1080/0020739X.2012.756545
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 (1-2), 139-148.
Harel, G. (1997). The Linear Algebra Curriculum Study Group recommendations: moving beyond concept definition. In D. Carlson, C. R. Johnson, D. Lay, D. Porter, A. Watkins & W. Watkins (Eds.), Resources for teaching linear algebra (MAA Notes, Vol. 42, pp. 107-126). Washington: Mathematical Association of America.
Hemmi, K. (2010). Three styles characterising mathematicians' pedagogical perspectives on proof. Educational Studies in Mathematics, 75, 271-291. https://doi.org/10.1007/s10649-010-9256-3
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: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47224-4_7
Hillel, J. & Sierpinska, A. (1994). On one persistent mistake in linear algebra. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 65-72). Lisbon: PME.
Holton, D. (2005). Tertiary mathematics education for 2024. International Journal of Mathematical Education in Science and Technology, 36 (2-3), 303-313. https://doi.org/10.1080/00207390412331317031
Ioannou, M. & Nardi, E. (2009). Engagement, abstraction and visualisation: cognitive and emotional aspects of year 2 mathematics undergraduates' learning experience in abstract algebra. Proceedings of the British Society for Research into Learning Mathematics, 29 (2), 35-40.
Jaworski, B. (2002). Sensitivity and challenge in university mathematics tutorial teaching. Educational Studies in Mathematics, 51, 71-94. https://doi.org/10.1023/A:1022491404298
Jaworski, B., Treffert-Thomas, S. & Bartsch, T. (2011). Linear algebra with a didactical focus. Nieuw Archief voor Wiskunde, 5 (4), 265-270.
Jaworski, B., Treffert-Thomas, S. & Bartsch, T. (2009). Characterising the teaching of university mathematics: a case of linear algebra. In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of the 33rd conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 249-256). Thessaloniki: PME.
Leontiev, A. N. (1978). Activity, consciousness and personality. Englewood Cliffs: Prentice Hall.
Leontiev, A. N. (1981). The problem of activity in psychology. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 37-71). New York: M. E. Sharpe.
Love, B., Hodge, A., Grandgenett, N. & Swift, A. W. (2014). Student learning and perceptions in a flipped linear algebra course. International Journal of Mathematical Education in Science and Technology, 45 (3), 317-324. https://doi.org/10.1080/0020739X.2013.822582
Millett, K. C. (2001). Making large lectures effective: an effort to increase student success. In D. Holton (Ed.), The teaching and learning of mathematics at university level: an ICMI study (pp. 137-152). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47231-7_13
Nardi, E. (2008). Amongst mathematicians: teaching and learning mathematics at the university level. New York: Springer. https://doi.org/10.1007/978-0-387-37143-6
Nardi, E., Jaworski, B. & Hegedus, S. (2005). A spectrum of pedagogical awareness for undergraduate mathematics: from "tricks" to "techniques". Journal for Research in Mathematics Education, 36 (4), 284-316.
Nanes, K. M. (2014). A modified approach to team-based learning in linear algebra courses. International Journal of Mathematical Education in Science and Technology, 45 (8), 1208-1219. https://doi.org/10.1080/0020739X.2014.920558
Pedersen, T. V. (2007). Design of a didactical situation - mathematical experiments in linear algebra. Nordic Studies in Mathematics Education, 12 (4), 27-52.
Petropoulou, G., Potari, D. & Zachariades, T. (2011). Inquiring mathematics teaching at the university level. In B. Ubuz (Ed.), Proceedings of the 35th conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 385-392). Ankara: PME.
Pritchard, D. (2010). Where learning starts? A framework for thinking about lectures in university mathematics. International Journal of Mathematical Education in Science and Technology, 41 (5), 609-623. https://doi.org/10.1080/00207391003605254
Rasmussen, C., Wawro, M. & Zandieh, M. (2015). Examining individual and collective level mathematical progress. Educational Studies in Mathematics, 88, 259-281. https://doi.org/10.1007/s10649-014-9583-x
Sfard, A. (2008). Thinking as communicating: human development, the growth of discourses, and mathematizing. Cambridge University Press. https://doi.org/10.1017/CBO9780511499944
Sierpinska, A. (2005). Innovations in the teaching of linear algebra: why you never hear the end of it. Retrieved from http://atlas-conferences.com/c/a/o/w/54.htm
Simon, M. A. (2009). Amidst multiple theories of learning in mathematics education. Journal for Research in Mathematics Education, 40 (5), 477-490. https://doi.org/10.5951/jresematheduc.40.5.0477
Speer, N. M., Smith III, J. P. & Horvath, A. (2010). Collegiate mathematics teaching: an unexamined practice. Journal of Mathematical Behavior, 29 (2), 99-114. https://doi.org/10.1016/j.jmathb.2010.02.001
Stewart S. (2009). Understanding linear algebra concepts through APOS and the three worlds of mathematical thinking theories. In M. Tzekaki, M. Kaldrimidou & H. Sakonidis (Eds.), Proceedings of the 33rd conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 169-176). Thessaloniki: PME.
Stewart, S. & Thomas, M. O. J. (2009). A framework for mathematical thinking: the case of linear algebra. International Journal of Mathematical Education in Science and Technology, 40 (7), 951-961. https://doi.org/10.1080/00207390903200984
Thomas, S. (2012). An activity theory analysis of linear algebra teaching within university mathematics (Unpublished doctoral dissertation). Loughborough University.
Treffert-Thomas, S. (2013a). IMA education grant: teaching linear algebra. Mathematics Today, 49 (2), 70-72.
Treffert-Thomas, S. (2013b). Enculturation into mathematical practice: an explorative study into the student perspective. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the International Group for the Psychology of Mathematics Education (Vol. 5, p. 181). Kiel: PME.
Treffert-Thomas, S. & Jaworski, B. (2015). Developing mathematics teaching: What can we learn from the literature? In M. Grove, A. C. Croft, J. Kyle & D. Lawson (Eds.), Transitions in undergraduate mathematics education (pp. 259-276). University of Birmingham & Higher Education Academy.
Uhl, J. J. (1999). Why (and how) I teach without long lectures. In S. G. Krantz (Ed.), How to teach mathematics (pp. 253-259). Providence: The American Mathematical Society.
Uhlig, F. (2002). The role of proof in comprehending and teaching elementary linear algebra. Educational Studies in Mathematics, 50 (3), 335-346. https://doi.org/10.1023/A:1021245213997
Uhlig, F. (2003). A new unified, balanced, and conceptual approach to teaching linear algebra. Linear Algebra and Its Applications, 361, 147-159. https://doi.org/10.1016/S0024-3795(02)00319-1
Viirman, O. (2014). The function concept and university mathematics teaching (Doctoral dissertation). Department of mathematics and computer science, Karlstad University.
Vygotsky, L. S. (1978). Mind in society: the development of higher psychological processes. Cambridge: Harvard University Press.
Vygotsky, L. S. (1981). The genesis of higher mental functions. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 144-188). New York: M. E. Sharpe.
Weber, K. (2004). Traditional instruction in advanced mathematics courses: a case study of one professor's lectures and proofs in an introductory real analysis course. Journal of Mathematical Behavior, 23, 115-133. https://doi.org/10.1016/j.jmathb.2004.03.001
Wertsch, J. V. (1991). Voices of the mind: a sociocultural approach to mediated action. Cambridge: Harvard University Press.
Wu, H. (1999). The joy of lecturing. In S. G. Krantz (Ed.), How to teach mathematics (pp. 261-271). Providence: The American Mathematical Society.
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