Abstract
In this chapter, we present a study that investigated the nature of the task solving practices developed by students in a first Analysis course at a North American university, and how these practices may be shaped by the evaluations (assignments and exams) given in the course. Task-based interviews with 15 students after their successful completion of the course revealed that students’ practices could vary in nature, being more or less “mathematical,” i.e., more or less reflective of mathematicians’ practices. As suggested by previous research on Calculus courses, we also found that the practices students develop in this Analysis course are likely shaped by the minimal requirements for success. To try to make sense of this, we introduce the theoretical notion of “path to a practice” and a characterization of three ways in which students’ practices may reveal themselves to be “non-mathematical.”
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Both in those studies and in ours, successful completion of a course means obtaining a passing grade.
- 2.
- 3.
One should pause and reflect on the relationship between the terms “practice” and “knowledge” from an ATD perspective. We let this hang in the subtext of our chapter, to be addressed in further discussion and subsequent theoretical research.
- 4.
It is possible that an individual will not explicitly engage in each of these actions when solving a task. Following the example of Chevallard (1999), we take the position that “having a practice” means being able to engage in four actions reflecting the four components of a praxeology. For example, if an individual has a practice, they would be able to give some description of why their chosen technique works. This description need not be “mathematical”: e.g., “I know the technique works because my teacher told me to do it that way.”
- 5.
We are assuming that there are some uniform, implicit ideas among mathematicians of what is (or is not) acceptable and worthwhile. We also acknowledge that there could be pertinent differences between mathematicians’ judgements depending, for example, on the specific area of mathematics in which they work (mathematical physics, numerical analysis, algebraic topology, …), which could warrant a definition of mathematical practice that depends on a specified area of mathematics. We did not consider such differences in this research.
- 6.
A student who enacts a mathematical practice could simply be mimicking behaviour. But the focus of our work is the development of non-mathematical practices. Given our task-based interview approach (see Sect. 7.3), we are convinced that the students we interviewed who enacted non-mathematical practices had not developed mathematical practices; they had developed non-mathematical practices.
- 7.
We consider the important distinction between the action of solving a task for which one has developed a practice and the action of solving a task for which one has not developed a practice. For instance, an individual may engage in the tasks of cooking a meal or hammering a nail without having developed practices for doing so. In contrast, professional chefs or carpenters are typically required to develop practices to ensure the regular and suitable accomplishment of those tasks.
- 8.
At the time of our study, students were evaluated by taking the best of two possible distributions: 10% assignments, 30% midterm, 60% final exam or 10% assignments, 90% final exam.
- 9.
We use “(non-)mathematical” to mean “non-mathematical or mathematical.”
- 10.
To be clear, we are not referring to the expectations that the institution or the teacher may have, which may well be that students learn mathematical practices. We are referring to our expectations as researchers critical of the tasks being proposed. Based on previous research, we expected students to develop some non-mathematical practices (e.g., focusing on superficial, non-mathematically relevant features of highly frequent tasks).
- 11.
The time available for solving a given task was constrained by the planned duration of the interview (2 h) and the priority of observing a participant formulate at least one approach, and a reason for the approach, for each of the six interview tasks.
- 12.
See Sect. 7.3 for the meaning of suggested practice in the context of this study.
- 13.
Of the four participants who did not immediately speak of using the IVT, one (S15) spoke about needing to use a “theorem” but could not remember which one, one (S2) immediately took the derivative of f, and the other two were S9 and S3 mentioned below.
- 14.
Our judgement is that their identification of the task exclusively on the fact that it is about the zeros of a function is not worthwhile from the perspective of scholarly mathematics.
- 15.
Some participants found negative values for f(x) due to calculation errors.
- 16.
This is not something we had anticipated based on our model of a suggested practice and so we do not know where this first step came from. Since S1 was not the only one to do it, perhaps it was shown to students in lectures.
- 17.
S11 made the specification that “it’s not this function, it’s another function” when giving the example of the interval [0, 5] because he had already tried plugging in x = 0, 1, 2, 3, 4, 5 and had not found the two zeros. This said, the zeros for f(x) do indeed occur on [0, 5].
- 18.
This may be an example where students were exhibiting “practices in development” (see the last paragraph of Sect. 7.3).
- 19.
As in the use of the adjective “mathematical” in this study, “mathematically” here refers to a way of reasoning that is acceptable and worthwhile by the institution of scholarly mathematics.
- 20.
S1 was one of six participants who did not mention the continuity condition required for applying the IVT during his solving of T2.
References
Artigue, M., & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM Mathematics Education, 45, 797–810. https://doi.org/10.1007/s11858-013-0506-6
Artigue, M., Menigaux, J., & Viennot, L. (1990). Some aspects of students’ conceptions and difficulties about differentials. European Journal of Physics, 11, 262–267.
Bergé, A. (2008). The completeness property of the set of real numbers in the transition from Calculus to Analysis. Educational Studies in Mathematics, 67, 217–235. https://doi.org/10.1007/s10649-007-9101-5
Bergqvist, E. (2007). Types of reasoning required in university exams in mathematics. Journal of Mathematical Behavior, 26, 348–370. https://doi.org/10.1016/j.jmathb.2007.11.001
Bosch, M., Chevallard, Y., García, F. J., & Monaghan, J. (Eds.). (2020). Working with the anthropological theory of the didactic in mathematics education: A comprehensive casebook. Routledge.
Brandes, H., & Hardy, N. (2018). From single to multi-variable calculus: A transition? In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N. M. Hogstad (Eds.), Proceedings of the second conference of the International Network for Didactic Research in University Mathematics (pp. 477–486). University of Agder and INDRUM.
Broley, L. (2020). The development of (non-)mathematical practices through paths of activities and students’ positioning: The case of Real Analysis [Unpublished doctoral dissertation]. Concordia University.
Broley, L. (2021). The development of (non-)mathematical practices through paths of activities and students’ positionings. The case of real analysis. In Canadian mathematics education study group proceedings of the 2021 annual meeting (pp. 109–116). https://www.cmesg.org/wp-content/uploads/2022/05/CMESG-2021.pdf
Broley, L., & Hardy, N. (2018). A study of transitions in an undergraduate mathematics program. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N. M. Hogstad (Eds.), Proceedings of the second conference of the International Network for Didactic Research in University Mathematics (pp. 487–496). University of Agder and INDRUM.
Chevallard, Y. (1985). La transposition didactique: Du savoir savant au savoir enseigné (1st ed.). La Pensée Sauvage.
Chevallard, Y. (1991). La transposition didactique: Du savoir savant au savoir enseigné (2nd ed.). La Pensée Sauvage.
Chevallard, Y. (1992). Concepts fondamentaux de la didactique: Perspectives apportées par une approche anthropologique. Recherches en didactique des mathématiques, 12(1), 73–111.
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en didactique des mathématiques, 19(2), 221–265.
Chevallard, Y. (2015). Teaching mathematics in tomorrow’s society: A case for an oncoming counter paradigm. In S. J. Cho (Ed.), Proceedings of the 12th international congress on mathematical education (pp. 173–187). Springer.
Cox, W. (1994). Strategic learning in A-level mathematics? Teaching Mathematics and its Applications, 13(1), 11–21. https://doi.org/10.1093/teamat/13.1.11
Darlington, E. (2014). Contrasts in mathematical challenges in A-level mathematics and further mathematics, and undergraduate mathematics examinations. Teaching Mathematics and its Applications, 33, 213–229. https://doi.org/10.1093/teamat/hru021
Florensa, I., Barquero, B., Bosch, M., & Gascón, J. (2019). Study and research paths at university level: Managing, analysing and institutionalizing knowledge. In U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the eleventh congress of the European Society for Research in Mathematics Education (pp. 2484–2491). Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.
Goldin, G. (1997). Chapter 4: Observing mathematical problem solving through task-based interviews. Journal for Research in Mathematics Education, 9, 40–62. https://doi.org/10.2307/749946
Goldin, G. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). Lawrence Erlbaum Associates, Inc.
Hardy, N. (2009). Students’ perceptions of institutional practices: The case of limits of functions in college level calculus courses. Educational Studies in Mathematics, 72, 341–358. https://doi.org/10.1007/s10649-009-9199-8
Kondratieva, M., & Winsløw, C. (2018). Klein’s plan B in the early teaching of analysis: Two theoretical cases of exploring mathematical links. International Journal of Research in Undergraduate Mathematics Education, 4, 119–138. https://doi.org/10.1007/s40753-017-0065-2
Lithner, J. (2000). Mathematical reasoning in task solving. Educational Studies in Mathematics, 41, 165–190.
Lithner, J. (2003). Students’ mathematical reasoning in university textbook exercises. Educational Studies in Mathematics, 52, 29–55. https://doi.org/10.1023/A:1023683716659
Lithner, J. (2004). Mathematical reasoning in Calculus textbook exercises. Journal of Mathematical Behavior, 23, 405–472. https://doi.org/10.1016/j.jmathb.2004.09.003
Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67, 255–276. https://doi.org/10.1007/s10649-007-9104-2
Maciejewski, W., & Merchant, S. (2016). Mathematical tasks, study approaches, and course grades in undergraduate mathematics: A year-by-year analysis. International Journal of Mathematical Education in Science and Technology, 47(3), 373–387. https://doi.org/10.1080/0020739X.2015.1072881
Orton, A. (1983). Students’ understanding of integration. Educational Studies in Mathematics, 14, 1–18. https://doi.org/10.1007/BF00704699
Raman, M. (2002). Coordinating informal and formal aspects of mathematics: Student behavior and textbook messages. Journal of Mathematical Behavior, 21, 135–150. https://doi.org/10.1016/S0732-3123(02)00119-0
Raman, M. (2004). Epistemological messages conveyed by three high-school and college mathematics textbooks. Journal of Mathematical Behavior, 23, 389–404. https://doi.org/10.1016/j.jmathb.2004.09.002
Selden, J., Selden, A., & Mason, A. (1994). Even good Calculus students can’t solve nonroutine problems. Research Issues in Undergraduate Mathematics Learning, MAA Notes, 33, 19–16.
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. https://doi.org/10.1007/BF00302715
Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 495–511). Macmillan.
Tallman, M. A., Carlson, M. P., Bressoud, D. M., & Pearson, M. (2016). A characterization of Calculus I final exams in U.S. colleges and universities. International Journal of Research in Undergraduate Mathematics Education, 2, 105–133.
Timmermann, S. (2005). Undergraduates’ solving strategies and teachers’ practice. In C. Winsløw (Ed.), Didactics of mathematics – The French way (pp. 71–82). Unpublished manuscript, Center for Naturfagenes Didaktik, University of Copenhagen.
Weber, K. (2005a). A procedural route toward understanding aspects of proof: Case studies from Real Analysis. Canadian Journal of Science, Mathematics and Technology Education, 5(4), 469–483. https://doi.org/10.1080/14926150509556676
Weber, K. (2005b). Problem-solving, proving, and learning: The relationship between problem-solving processes and learning opportunities in the activity of proof construction. Journal of Mathematical Behavior, 24, 351–360. https://doi.org/10.1016/j.jmathb.2005.09.005
White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory Calculus. Journal for Research in Mathematics Education, 27(1), 79–95. https://doi.org/10.2307/749199
Winsløw, C. (2006). Transformer la théorie en tâches: La transition du concret à l’abstrait en Analyse Réelle. In A. Rouchier & I. Bloch (Eds.), Actes de la XIIIe École d’été de didactique des mathématiques (pp. 1–12). Éditions La Pensée Sauvage.
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. https://doi.org/10.1080/14794802.2014.918345
Acknowledgements
We thank the reviewers, the editors, and our colleague Dr. Ildiko Pelczer, whose thoughtful comments and constructive criticism have greatly improved the content of this chapter. This chapter draws on research supported by the Social Sciences and Humanities Research Council of Canada.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Broley, L., Hardy, N. (2022). University Students’ Development of (Non-)Mathematical Practices: The Case of a First Analysis Course. In: Biehler, R., Liebendörfer, M., Gueudet, G., Rasmussen, C., Winsløw, C. (eds) Practice-Oriented Research in Tertiary Mathematics Education. Advances in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-14175-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-14175-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-14174-4
Online ISBN: 978-3-031-14175-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)