# Analyzing the teaching of advanced mathematics courses via the enacted example space

- 793 Downloads
- 7 Citations

## Abstract

Examples are believed to be very important in developing conceptual understanding of mathematical ideas, useful both in mathematics research and instruction (Bills & Watson in *Educational Studies in Mathematics* 69:77–79, 2008; Mason & Watson, 2008; Bills & Tall, 1998; Tall & Vinner, 1981). In this study, we draw on the concept of an example space (Mason & Watson, 2008) and variation theory (Runesson in *Scandinavian Journal of Educational Research* 50:397–410, 2006) to create a lens to study how examples are used for pedagogical purposes in undergraduate proof-based instruction. We adapted the construct of an example space and extended its application to the constructs of example neighborhood, methods of example construction, and the functions of examples. We explained how to use our new lens to analyze the collection of examples and non-examples that the students had access to. We then demonstrate our method by analyzing the collection of examples and non-examples of a mathematical group the professor of an abstract algebra class presented during lectures or assigned to students in problem sets or exams.

## Keywords

Undergraduate teaching Examples Abstract algebra Variation theory## References

- Alcock, L. (2010). Mathematicians’ perspectives on the teaching and learning of proof. In F. Hitt, D. Holton, & P. W. Thompson (Eds.),
*Research in collegiate mathematics education VII*(pp. 63–91). Washington, DC: MAA.Google Scholar - Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures.
*Educational Studies in Mathematics, 69*, 111–129.CrossRefGoogle Scholar - Barab, S. A., Cherkes-Julkowski, M., Swenson, R., Garrett, S., Shaw, R. E., & Young, M. (1999). Principles of self-organization: Learning as participation in autocatakinetic systems.
*The Journal of the Learning Sciences, 8*(3/4), 349–390.CrossRefGoogle Scholar - Bills, L., & Tall, D. O. (1998). Operable definitions in advanced mathematics: The case of the least upper bound.
*Proceedings of PME 22*, Stellenbosch, South Africa, 2, 104–111.Google Scholar - Bills, L., & Watson, A. (2008). Special issue: Role and use of exemplification in mathematics education [Editorial introduction].
*Educational Studies in Mathematics, 69*, 77–79.CrossRefGoogle Scholar - Cuoco, A., Goldenberg, E. P., & Mark, J. (1997). Habits of mind: An organizing principle for mathematics curriculum.
*Journal of Mathematical Behavior, 15*(4), 375–402.CrossRefGoogle Scholar - Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation.
*Educational Studies in Mathematics, 33*, 283–299. doi: 10.1023/A:1002999415887.CrossRefGoogle Scholar - Dienes, Z. (1963).
*An experimental study of mathematics-learning*. London: Hutchinson Educational.Google Scholar - Ernst, P. (2006). Reflections on theories of learning.
*Zentralblatt für Didaktik der Mathematik, 38*(1), 3–8.CrossRefGoogle Scholar - Fraleigh, J. (1999).
*A first course in abstract algebra*(6th ed.). Reading: Addison-Wesley.Google Scholar - Gibson, J. J. (1979).
*The ecological approach to visual perception*. Boston: Houghton Mifflin.Google Scholar - Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces.
*Educational Studies in Mathematics, 69*, 183–194. doi: 10.1007/s10649-008-9143-3.CrossRefGoogle Scholar - Greeno, J. G., & Gresalfi, M. S. (2008). Opportunities to learn in practice and identity. In P. A. Moss, D. C. Pullin, J. P. Gee, E. H. Haertel, & L. J. Young (Eds.),
*Assessment, equity, and opportunity to learn*(pp. 170–199). New York: Cambridge University Press.CrossRefGoogle Scholar - Greeno, J. G., & MMAP. (1998). The situativity of knowing, learning, and research.
*American Psychologist, 53*, 5–26.CrossRefGoogle Scholar - Gresalfi, M. S. (2009). Taking up opportunities to learn: Constructing dispositions in mathematics classrooms.
*Journal of the Learning Sciences, 18*, 327–369.CrossRefGoogle Scholar - Gresalfi, M., Barnes, J. Cross, D. (2011, December 16). When does an opportunity become an opportunity? Unpacking classroom practice through the lens of ecological psychology.
*Educational Studies in Mathematics*, 1–19. doi: 10.1007/s10649-011-9367-5 - Harel, G., & Fuller, E. (2009). Contributions toward perspectives on learning and teaching proof. In D. Stylianou, M. Blanton, & E. Knuth (Eds.),
*Teaching and learning proof across the grades: A K-16 perspective*(pp. 355–370). New York: Routledge.Google Scholar - Harel, G., & Sowder, L. (2007). Towards a comprehensive perspective on proof. In F. Lester (Ed.),
*Second handbook of research on mathematical teaching and learning*(pp. 805–842). Washington, DC: NCTM.Google Scholar - Hazzan, O. (1999). Reducing abstraction level when learning abstract algebra concepts.
*Educational Studies in Mathematics, 40*(1), 71–90.CrossRefGoogle 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*(pp. 371–404). Charlotte, NC: Information Age Publishers.Google Scholar - Lai, Y., Weber, K., & Mejia-Ramos, J. P. (2012). Mathematicians’ perspectives on features of a good pedagogical proof.
*Cognition and Instruction, 30*(2), 146–169.CrossRefGoogle Scholar - Lakatos, I. (1976).
*Proofs and refutations*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Larsen, S., & Zandieh, M. (2008). Proofs and refutations in the undergraduate mathematics classroom.
*Educational Studies in Mathematics, 67*, 205–216.CrossRefGoogle Scholar - Mason, J., & Watson, A. (2008). Mathematics as a constructive activity: Exploiting dimensions of possible variation. In M. Carlson & C. Rasmussen (Eds.),
*Making the connection: Research and practice in undergraduate mathematics*(pp. 189–202). Washington, DC: MAA.Google Scholar - Mejia-Ramos, J. P., & Inglis, M. (2009). Argumentative and proving activities in mathematics education research. In F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Eds.),
*Proceedings of the ICMI study 19 conference: Proof and proving in mathematics education*(Vol. 2, pp. 88–93). Taipei, Taiwan: ICMIGoogle Scholar - Mills, M. (2014). A framework for example usage in proof presentations.
*The Journal of Mathematical Behavior, 33*, 106–118.Google Scholar - Mills, M. (2012). Investigating the teaching practices of professors when presenting proofs: The use of examples. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.),
*Proceedings of the 15th Conference for Research in Undergraduate Mathematics Education*(pp. 512–516). Portland, OR.Google Scholar - National Research Council. (2001).
*Adding it up: Helping children learn mathematics*. Washington, DC: The National Academies Press.Google Scholar - Rasmussen, C., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly & R. Lesh (Eds.),
*Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching*(pp. 195–215). Mawah, NJ: Erlbaum.Google Scholar - Rokeach, M. (1968).
*Beliefs, attitudes, and values*. San Francisco, CA: Jossey-Bass.Google Scholar - Runesson, U. (2006). What is it possible to learn? On variation as a necessary condition for learning.
*Scandinavian Journal of Educational Research, 50*, 397–410.CrossRefGoogle Scholar - Schoenfeld, A. (1998). Toward a theory of teaching-in-context.
*Issues in Education, 4*(1), 1–78.CrossRefGoogle Scholar - Shaw, R. E., Effken, J. A., Fajen, B. R., Garrett, S. R., & Morris, A. (1997). An ecological approach to the online assessment of problem-solving paths.
*Instructional Science, 25*, 151–166.CrossRefGoogle Scholar - Sousa, D. A. (2008).
*How the brain learns mathematics*. Thousand Oaks, CA: Corwin Press.Google Scholar - Speer, N., Smith, J., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice.
*The Journal of Mathematical Behavior, 29*, 99–114.CrossRefGoogle Scholar - Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference to limits and continuity.
*Educational Studies in Mathematics, 12*(2), 151–169.Google Scholar - Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 25–41). Dordecht: Kluwer.Google Scholar - Watson, A., & Mason, J. (2005).
*Mathematics as a constructive activity: Learners generating examples*. Mahwah, NJ: Erlbaum.Google Scholar - Weber, K. (2002). Beyond proving and explaining: Proofs that justify the use of definitions and axiomatic structures and proofs that illustrate technique.
*For the Learning of Mathematics, 22*(3), 14–17.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.
*Journal of Mathematical Behavior, 23*, 115–133.CrossRefGoogle Scholar - Weber, K. (2009). How syntactic reasoners can develop understanding, evaluate conjectures, and construct counterexamples in advanced mathematics.
*Journal of Mathematical Behavior, 28*, 200–208.CrossRefGoogle Scholar - Weber, K., & Alcock, L. J. (2005). Using warranted implications to understand and validate proofs.
*For the Learning of Mathematics, 25*(1), 34–38. 51.Google Scholar - Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square.
*Educational Studies in Mathematics, 69*(2), 131–148.CrossRefGoogle Scholar - Zodik, I., & Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for the mathematics classroom.
*Educational Studies in Mathematics, 69*, 165–182. doi: 10.1007/s10649-008-9140-6.CrossRefGoogle Scholar