# A Qualitative Research on Example Generation Capabilities of University Students

## Abstract

Examples which are used in exploring a procedure or comprehending/concretizing a mathematical concept are powerful teaching tools. Generating examples other than conventional ones is both a means for research and a pedagogical method. The aim of this study is to determine the transition process between example generation strategies, and the factors affecting success of the students in generating examples in a Real Analysis course. The participants of the study consisted of 27 undergraduate mathematics students. At the end of the study, it was observed that some of the participants used especially the trial and error strategy as an effective step in the transition to the transformation strategy. Definitions were used by participants as a trigger for example generation and to reflect on concepts during this process in order to reduce cognitive demand.

## Keywords

Example generation Example space University students Real Analysis## References

- Antonini, S. (2006). Graduate students’ processes in generating examples of mathematical objects. In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková (Eds.),
*Proceedings of the 30th Annual Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 57-64). Prague: Czech Republic.Google Scholar - Arzerello, A., Ascari, M. & Sabena, C. (2011). A model for developing students’ example spaces: the key role of the teacher.
*Zentralblatt für Didaktik der Mathematik, 43*, 295–306.CrossRefGoogle Scholar - Bills, L., Dreyfus, T. Mason, J. Tsamir, P. Watson, A. & Zaslavsky, O. (2006). Example generating in mathematics education, In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková (Eds.),
*Proceedings of the 30th Annual Conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 126-154). Prague: Czech Republic.Google Scholar - Bills, L. & Tall, D. (1998). Operable definitions in advanced mathematics: The case of the least upper bound. In A. Olivier & K. Newstead (Eds.),
*Proceedings of the 22nd Annual Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 104–111). Stellenbosch, South Africa: University of Stellenbosch.Google Scholar - Dahlberg, R. P. & Housman, D. L. (1997). Facilitating learning events through example generation.
*Educational Studies in Mathematics, 33*, 283–299.CrossRefGoogle Scholar - Edwards, A. & Alcock L. (2010). How do undergraduate students navigate their example spaces?
*In Proceedings of the 13th annual conference on research in undergraduate mathematics education*. Retrieved from http://sigmaa.maa.org/rume/crume2010/Archive/Edwards.pdf - Furinghetti, F., Morselli, F. & Antonini, S. (2011). To exist or not to exist: example generation in real analysis.
*Zentralblatt für Didaktik der Mathematik, 43*, 219–232.CrossRefGoogle Scholar - Goldenberg, P. & Mason, J. (2008). Shedding light on and with example spaces.
*Educational Studies in Mathematics, 69*, 183–194.CrossRefGoogle Scholar - Iannone, P., Inglis, M., Mejia-Ramos, J. P., Siemons, J. & Weber, K. (2009). How do undergraduate students generate examples of mathematical concepts? 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. 217–224). Thessaloniki, Greece: PME.Google Scholar - Iannone, P., Inglis, M., Mejia-Ramos, J. P., Siemons, J. & Weber, K. (2011). Does generating examples aid proof production?
*Educational studies in Mathematics, 77*, 1–14.CrossRefGoogle Scholar - Leung, I. K. C. & Lew, H. (2012). The ability of students and teachers to use counter-examples to justify mathematical propositions: a pilot study in South Korea and Hong Kong.
*ZDM Mathematics Education, 45*(1), 91–105.CrossRefGoogle Scholar - Moore, R. C. (1994). Making the transition to formal proof.
*Educational Studies in Mathematics, 27*, 249–266.CrossRefGoogle Scholar - Morselli, F. (2006). Use of examples in conjecturing and proving: an exploratory study. In J. Novotná, H. Moraová, M. Krátká, & N. Stehliková (Eds.)
*Proceedings of the 30th Annual Conference of the International Group for the Psychology of Mathematics Education*, (Vol. 4, pp.185-192). Prague: Czech Republic.Google Scholar - Strauss, A. & Corbin, J. (1998).
*Basics of qualitative research: Grounded theory procedures and technique*. London, England: Sage Publications.Google Scholar - Watson, A. & Mason, J. (2002). Student‐generated examples in the learning of mathematics.
*Canadian Journal of Science, Mathematics and Technology Education, 2*(2), 237–249.CrossRefGoogle Scholar - Watson, A. & Mason, J. (2005).
*Mathematics as a constructive activity: Learners generating examples*. London, England: Lawrence Erlbaum Associates.Google Scholar - Watson, A. & Shipman, S. (2008). Using learner generated examples to introduce new concepts.
*Educational Studies in Mathematics, 69*(2), 97–109.CrossRefGoogle Scholar - Zaslavsky, O. & Peled, I. (1996). Inhibiting factors in generating examples by mathematics teachers and student teachers: The case of binary operation.
*Journal for Research in Mathematics Education, 27*, 67–78.Google Scholar - Zazkis, R. & Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool.
*For the Learning of Mathematics, 27*, 11–17.Google Scholar - Zodik, I. & Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for the mathematics classroom.
*Educational Studies in Mathematics, 69*, 165–18.CrossRefGoogle Scholar