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Undergraduate Mathematics Teaching in First Year Lectures: Can it be Responsive to Student Learning Needs?

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Abstract

This paper investigates mathematics teaching to first year students of two mathematics departments in the context of large-group lectures. It goes beyond what may be seen as implicit in this context by focussing on the links between two lecturers’ thinking about students’ learning of advanced mathematics and their practice. Data includes lecture observations and reflective discussions with the lecturers. Three layers of data analysis are drawn on Activity Theory, the Teaching Triad, and, grounded analytical approaches. Findings suggest that the formation of teaching goals is motivated by aspects of sensitivity to student’s participation and that the complementarities among these aspects are important qualitative characteristics of teaching in this context.

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  1. Please note all citations were originally in Greek and have been translated into English for this paper by the first author.

References

  • Artemeva, N., & Fox, J. (2011). The writing's on the board: The global and the local in teaching undergraduate mathematics through chalk talk. Written Communication, 28(4), 345–379.

    Google Scholar 

  • Artigue, M., Batanero, C., & Kent, P. (2007). Mathematics thinking and learning at post-secondary level. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: National Council of teachers of mathematics (pp. 1011–1049). Charlotte, NC: Information Age Publishing.

    Google Scholar 

  • Axiak, C. (2004). Being sensitive to Students' mathematical needs: What does it take? In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th International Group for the Psychology of Mathematics Education (Vol. V2, pp. 79–86). Bergen, Norway: PME.

    Google Scholar 

  • Barton, B., Oates, G., Paterson, J., & Thomas, M. (2015). A marriage of continuance: Professional development for mathematics lecturers. Mathematics Education Research Journal, 27(2), 147–164.

    Google Scholar 

  • Biggs, J. (2003). Teaching for quality learning at university (2nd ed.). Maidenhead, England: Open University Press.

  • Biza, I., Jaworski, B., & Hemmi, K. (2014). Communities in university mathematics. Research in Mathematics Education, 16(2), 161–176.

    Google Scholar 

  • Charmaz, K. (2006). Constructing grounded theory: A practical guide through qualitative analysis. London, England: Sage.

  • Cretchley, P. (2005). Mathematics and dumping lectures?: Another perspective on the shift towards learner pragmatism. In Bulmer, M., MacGillivray, H. & Varsavsky, C. (Eds.), Proceedings of the fifth southern hemisphere conference on undergraduate mathematics and statistics teaching and learning, 42–48. Queensland, Australia.

  • Dawkins, P. C., Oehrtman, M., & Mahavier, W. T. (2019). Professor goals and student experiences in traditional IBL real analysis: A case study. International Journal of Research in Undergraduate Mathematics Education, 5, 315–336.

    Google Scholar 

  • Evans, J. (2006). Affect and emotion in mathematical thinking and learning. In J. Maasz & W.Schloeglmann (Eds.), New mathematics education research and practice (pp. 233–255). Rotterdam: Sense Publishers.

  • Freeman, T. M., Anderman, L. H., & Jensen, J. M. (2007). Sense of belonging in college freshmen at the classroom and campus levels. The Journal of Experimental Education, 75(3), 203–220.

    Google Scholar 

  • Frymier, A. B. (2002). Making content relevant to students. In J.L. Chesebro & J.C. McCroskey (Eds.), Communication for Teachers, (pp. 83–92).

  • Fukawa-Conelly, P., & T. (2012). A case study of one instructor’s lecture-based teaching of proof in abstract algebra: Making sense of her pedagogical moves. Educational Studies in Mathematics, 81, 325–345.

    Google Scholar 

  • Gabel, M., & Dreyfus, T. (2017). Affecting the flow of a proof by creating presence — A case study in number theory. Educational Studies in Mathematics, 96, 187–205.

    Google Scholar 

  • Hannah, J., Stewart, S., & Thomas, M. (2011). Analysing lecturer practice: The role of orientations and goals. International Journal of Mathematical Education in Science and Technology, 42(7), 975–984.

    Google Scholar 

  • Hannula, M. S. (2006). Motivation in mathematics: Goals reflected in emotions. Educational Studies in Mathematics, 63(2), 165–178.

  • Hemmi, K. (2010). Three styles characterising mathematicians' pedagogical perspectives on proof. Educational Studies in Mathematics, 75(3), 271–291.

    Google Scholar 

  • Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning, 1 (pp. 371–404). Greenwich, CT: Information Age.

    Google Scholar 

  • Jaworski, B. (1994). Investigating mathematics teaching: A constructivist enquiry. London, England: Falmer.

  • Jaworski, B. (2002). Sensitivity and challenge in university mathematics teaching. Educational Studies in Mathematics, 51, 71–94.

    Google Scholar 

  • Jaworski, B., & Didis, M. G. (2014). Relating student meaning-making in mathematics to the aims for and design of teaching in small group tutorials at university level. In P. Liljedahl, S. Oesterle, C. Nicol, & D. Allan (Eds.), Proceedings of the 38th conference of the International Group for the Psychology of mathematics education (Vol. 4, pp. 377–384). Vancouver, Canada: PME.

    Google Scholar 

  • Jaworski, B., & Potari, D. (2009). Bridging the macro- and micro-divide: Using an activity theory model to capture sociocultural complexity in mathematics teaching and its development. Educational Studies in Mathematics, 72, 219–236.

    Google Scholar 

  • Jaworski, B., Mali, A., & Petropoulou, G. (2016). Critical Theorising from studies of undergraduate mathematics teaching for students’ meaning making in mathematics. International Journal of Research in Undergraduate Mathematics Education 3(1), 168–197.

  • Jaworski, B., Potari, D. & Petropoulou, G. (2017). Theorising university mathematics teaching: The teaching triad within an activity theory perspective. In T. Dooley, & Gueudet, G. (Eds.). Proceedings of the 10th Congress of the European Society for Research in Mathematics Education (CERME10), (pp. 2105–2112). Dublin, Ireland: DCU Institute of education and ERME.

  • Johnson, E., Caughman, J., Fredericks, J., & Gibson, L. (2013). Implementing inquiry -oriented curriculum: From the mathematicians’ perspective. The Journal of Mathematical Behavior, 32(4), 743–760.

    Google Scholar 

  • Laursen, S. L., & Rasmussen, C. (2019). I on the prize: Inquiry approaches in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 5(1), 129–146.

  • Leont’ev, A. N. (1978). Activity, consciousness and personality. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Lew, K., Fukawa-Connelly, T. P., Mejía-Ramos, J. P., & Weber, K. (2016). Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education, 47(2), 162–198.

    Google Scholar 

  • Lobato, J., Clarke, D., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 36(2), 101–136.

    Google Scholar 

  • Mali, A., & Petropoulou, G. (2017). Characterising undergraduate mathematics teaching across settings and countries: An analytical framework. Nordic Studies in Mathematics Education, 22(4), 23–42.

    Google Scholar 

  • McClendon, D. M., Weller, K., Trouba, J., Brigance, S., & Dekker, M. (2019). Transforming tradition: From one instructor to an entire department. Problems, Resources, and Issues in Mathematics Undergraduate Studies (PRIMUS), 1–29.

  • Mills, M. (2014). A framework for example usage in proof presentations. The Journal of Mathematical Behavior, 33, 106–118.

    Google Scholar 

  • Nardi, E. (2008). Amongst mathematicians: Teaching and learning mathematics at the university level. New York, NY: Springer.

  • 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.

    Google Scholar 

  • Olson, J. C., & Knott, L. (2013). When a problem is more than a teacher’s question. Educational Studies in Mathematics, 83, 27–36.

    Google Scholar 

  • Paoletti, T., Krupnik, V., Papadopoulos, D., Olsen, J., Fukawa-Connelly, T., & Weber, K. (2018). Teacher questioning and invitations to participate in advanced mathematics lectures. Educational Studies in Mathematics, 98, 1–17.

    Google Scholar 

  • Paterson, J., Thomas, M., & Taylor, S. (2011). Decisions, decisions, decisions: What determines the path taken in lectures? International Journal of Mathematical Education in Science and Technology, 42(7), 985–995.

    Google Scholar 

  • 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, 3 (pp. 386–392). Ankara, Turkey: PME.

  • Petropoulou, G., Jaworski, B., Potari, D., & Zachariades, T. (2013). Mathematical challenge and sensitivity to students in university lecturing: An uneasy balance. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, 5, (p. 144) . PME: Kiel, Germany.

  • Petropoulou, G., Jaworski, B., Potari, D., & Zachariades, T. (2015). How do research mathematicians teach Calculus? In K. Krainer & N. Vondrová (Eds.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (pp. 2221–2227). Prague, Czech Republic: CERME.

  • Petropoulou, G., Jaworski, B., Potari, D., & Zachariades, T. (2016). Addressing large cohorts of first year mathematics students in lectures. In E. Nardi, C. Winslow, & T. Hausberger (Eds.), Proceedings of 1st Conference of International Network for Didactic Research in University Mathematics (pp. 390–399). Montpellier, France: INDRUM.

  • Pinto, A. (2013). Revisiting university mathematics teaching: A tale of two instructors. In Paper presented in the eighth congress of European research in mathematics education (CERME 8). Antalya: Turkey.

  • Polya, G. (1971). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton, New Jersey: Princeton University Press.

  • Potari, D., & Jaworski, B. (2002). Tackling complexity in mathematics teaching development: Using the teaching triad as a tool for reflection and analysis. Journal of Mathematics Teacher Education, 5, 251–280.

    Google Scholar 

  • Pring, R. (2000). Philosophy of educational research. London, England: Continuum.

  • 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.

    Google Scholar 

  • Roth, W. M., & Radford, L. (2011). A cultural-historical perspective on mathematics teaching and learning. Rotterdam, The Netherlands: Sense Publishers.

    Google Scholar 

  • Schoenfeld, A. H. (2011). How we think: A theory of goal-oriented decision making and its educational applications. New York, New York: Routledge.

  • Speer, N. M., & Wagner, J. F. (2009). Knowledge needed by a teacher to provide analytic scaffolding during undergraduate mathematics classroom discussions. Journal for Research in Mathematics Education, 40(5), 530–562.

    Google Scholar 

  • Speer, N. M., Smith, J. P., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. The Journal of Mathematical Behavior, 29, 99–114.

    Google Scholar 

  • Stouraitis, K. (2016). Decision making in the context of enacting a new curriculum: An activity theoretical perspective. In C. Csíkos, A. Rausch, & J. Szitányi (Eds.), Proceedings of the 40th conference of the International Group for the Psychology of mathematics education (Vol. 4, pp. 235–242). Szeged, Hungary: PME.

    Google Scholar 

  • Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24.

    Google Scholar 

  • Van Oers, B. (2001). Educational forms of initiation in mathematical culture. Educational Studies in Mathematics, 46, 59–85.

    Google Scholar 

  • Viirman, O. (2014). The functions of function discourse: University mathematics teaching from a commognitive standpoint. International Journal of Mathematical Education in Science and Technology, 45(4), 512–527.

    Google Scholar 

  • Viirman, O. (2015). Explanation, motivation and question posing routines in university mathematics teachers' pedagogical discourse: A commognitive analysis. International Journal of Mathematical Education in Science and Technology, 46(8), 1165–1181.

    Google Scholar 

  • von Cranach, M., & Valach, L. (1984). 14-the social dimension of goal-directed action. In Tajfel, T. (Ed.). The Social Dimension: European Developments in Social Psychology, 1, 285–299. UK: Cambridge University Press.

  • Vygotsky, L. S. (1978). Mind and society: The development of higher mental processes. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Wagner, J. F., & Keene, K. A. (2014). Exploring differences in teaching practice when two mathematics instructors enact the same lesson. In T. Fukawa-Connolly, G. Karakok, K. Keene, & M. Zandieh (Eds.), Proceedings of the Seventeenth Annual Conference on Research in Undergraduate Mathematics Education (pp. 322–335). Denver, CO.

  • Wagner, J., Speer, N., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician's knowledge needed for teaching an inquiry-oriented differential equations course. The Journal of Mathematical Behavior, 26(3), 247–266.

    Google Scholar 

  • Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.

    Google Scholar 

  • 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. The Journal of Mathematical Behavior, 23, 115–133.

    Google Scholar 

  • Weinberg, A., Wiesner, E., & Fukawa-Connelly, T. (2016). Mathematics lectures as narratives: Insights from network graph methodology. Educational Studies in Mathematics, 91, 203–226.

    Google Scholar 

  • Wertsch, J. V. (1991). Voices of the mind: A sociocultural approach to mediated action. Cambridge, MA: Harvard University Press.

  • Winsløw, C., & Grønbæk, N. (2014). Klein's double discontinuity revisited. Recherches en Didactique des Mathématiques, 34(1), 59–86.

    Google Scholar 

  • Winsløw, C., Barquero, B., De Vleeschouwer, M., & Hardy, N. (2014). An institutional approach to university mathematics education: From dual vector spaces to questioning the world. Research in Mathematics Education, 16(2), 95–111.

    Google Scholar 

  • Winsløw, C., Gueudet, G., Hochmuth, R., & Nardi, E. (2018). Research on university mathematics education. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education - twenty years of communication, cooperation and collaboration in Europe (pp. 60–74). London and New York: Routledge.

    Google Scholar 

  • Yin, R. K. (2017). Case study research and applications: Design and methods. (6th edition), Los Angeles, United States: Sage Publications.

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Petropoulou, G., Jaworski, B., Potari, D. et al. Undergraduate Mathematics Teaching in First Year Lectures: Can it be Responsive to Student Learning Needs?. Int. J. Res. Undergrad. Math. Ed. 6, 347–374 (2020). https://doi.org/10.1007/s40753-020-00111-y

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