# Interactions between defining, explaining and classifying: the case of increasing and decreasing sequences

## Abstract

This paper describes a study in which we investigated relationships between defining mathematical concepts — increasing and decreasing infinite sequences — explaining their meanings and classifying consistently with formal definitions. We explored the effect of defining, explaining or studying a definition on subsequent classification, and the effect of classifying on subsequent explaining and defining. We report that (1) student-generated definitions and explanations were highly variable in content and quality; (2) explicitly considering the meaning of the concept facilitated subsequent classification, and giving a personal definition or explanation had a greater effect than studying a given definition; (3) classifying before defining or explaining resulted in significantly poorer definitions and explanations. We discuss the implications of these results for the teaching of abstract pure mathematics, relating our discussion to existing work on the concept image/concept definition distinction and on working with examples.

### Keywords

Sequences Definitions Examples Real analysis Classification### References

- Alcock, L., & Simpson, A (2004). Convergence of sequences and series: Interactions between visual reasoning and the learner’s beliefs about their own role.
*Educational Studies in Mathematics*,*57*(1), 1–32.CrossRefGoogle Scholar - Alcock, L., & Simpson, A. (2011). Classification and concept consistency.
*Canadian Journal of Science, Mathematics and Technology Education*,*11*(2), 91–106.CrossRefGoogle Scholar - Bergé, A. (2008). The completeness property of the set of real numbers in the transition from calculus to analysis.
*Educational Studies in Mathematics*,*67*(3), 217–235.CrossRefGoogle Scholar - Bergqvist, E. (2007). Types of reasoning required in university exams in mathematics.
*The Journal of Mathematical Behavior*,*26*(4), 348–370.CrossRefGoogle Scholar - Bingolbali, E., & Monaghan, J. (2008). Concept image revisited.
*Educational Studies in Mathematics*,*68*(1), 19–35.CrossRefGoogle Scholar - Biza, I., Christou, C., & Zachariades, T. (2008). Student perspectives on the relationship between a curve and its tangent in the transition from euclidean geometry to analysis.
*Research in Mathematics Education*,*10*(1), 53–70.CrossRefGoogle Scholar - Brown, J. R (1998). What is a definition?
*Foundations of Science*,*3*(1), 111–132.CrossRefGoogle Scholar - Dahlberg, R. P, & Housman, D. L (1997). Facilitating learning events through example generation.
*Educational Studies in Mathematics*,*33*(3), 283–299.CrossRefGoogle Scholar - Dawkins, P. C (2014). How students interpret and enact inquiry-oriented defining practices in undergraduate real analysis.
*The Journal of Mathematical Behavior*,*33*, 88–105.CrossRefGoogle Scholar - Dubinsky, E., & Yiparaki, O. (2000). On student understanding of AE and EA quantification.
*Research in Collegiate Mathematics*,*4*, 239–289.Google Scholar - Dubinsky, E., Elterman, F., & Gong, C. (1988). The student’s construction of quantification.
*For the Learning of Mathematics*,*8*(2), 44–51.Google Scholar - Fujita, T. (2012). Learners’ level of understanding of the inclusion relations of quadrilaterals and prototype phenomenon.
*The Journal of Mathematical Behavior*,*31*(1), 60–72.CrossRefGoogle Scholar - Fukawa-Connelly, T. P, & Newton, C. (2014). Analyzing the teaching of advanced mathematics courses via the enacted example space.
*Educational Studies in Mathematics*,*87*(3), 323–349.CrossRefGoogle Scholar - Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces.
*Educational Studies in Mathematics*,*69*(2), 183–194.CrossRefGoogle Scholar - Heinze, A., & Kwak, J. Y (2002). Informal prerequisites for informal proofs.
*Zentralblatt für Didaktik der Mathematik*,*34*(1), 9–16.CrossRefGoogle Scholar - Iannone, P., Inglis, M., Mejía-Ramos, J. P, Simpson, A., & Weber, K. (2011). Does generating examples aid proof production?
*Educational studies in Mathematics*,*77*(1), 1–14.CrossRefGoogle Scholar - Inglis, M., & Simpson, A. (2008). Reasoning from features or exemplars. In O. Figueras, J. Cortina, S. Alatorre, T. Rojana, & A. Sepúlveda (Eds.),
*Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3 pp. 217–224). Morelia: PME.Google Scholar - Kruschke, J. K (2005). Category learning. In K. Lamberts & R. Goldstone (Eds.),
*The handbook of cognition*(pp. 183–201). London: Sage.Google Scholar - Lakatos, I. (1976).
*Proofs and refutations: The logic of mathematical discovery*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Larsen, S., & Zandieh, M. (2008). Proofs and refutations in the undergraduate mathematics classroom.
*Educational Studies in Mathematics*,*67*(3), 205–216.CrossRefGoogle Scholar - Moore, R. C (1994). Making the transition to formal proof.
*Educational Studies in Mathematics*,*27*(3), 249–266.CrossRefGoogle Scholar - Moore-Russo, D., Conner, A., & Rugg, K. I (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction.
*Educational Studies in Mathematics*,*76*(1), 3–21.CrossRefGoogle Scholar - Oehrtman, M., Swinyard, C., & Martin, J. (2014). Problems and solutions in students’ reinvention of a definition for sequence convergence.
*The Journal of Mathematical Behavior*,*33*, 131–148.CrossRefGoogle Scholar - Ouvrier-Buffet, C. (2011). A mathematical experience involving defining processes: In-action definitions and zero-definitions.
*Educational Studies in Mathematics*,*76*(2), 165–182.CrossRefGoogle Scholar - Pollard, P. (1982). Human reasoning: Some possible effects of availability.
*Cognition*,*12*(1), 65–96.CrossRefGoogle Scholar - Raman, M. (2004). Epistemological messages conveyed by three high-school and college mathematics textbooks.
*The Journal of Mathematical Behavior*,*23*(4), 389–404.CrossRefGoogle Scholar - Roh, K. H (2008). Students’ images and their understanding of definitions of the limit of a sequence.
*Educational Studies in Mathematics*,*69*(3), 217–233.CrossRefGoogle Scholar - Sandefur, J., Mason, J., Stylianides, G., & Watson, A. (2013). Generating and using examples in the proving process.
*Educational Studies in Mathematics*,*83*(3), 323–340.CrossRefGoogle Scholar - Swinyard, C. (2011). Reinventing the formal definition of limit: The case of Amy and Mike.
*The Journal of Mathematical Behavior*,*30*(2), 93–114.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.CrossRefGoogle Scholar - Vargha, A., & Delaney, H. D (2000). A critique and improvement of the CL common language effect size statistics of McGraw and Wong.
*Journal of Educational and Behavioral Statistics*,*25*(2), 101–132.Google Scholar - Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 65–81). Dordrecht: Kluwer Academic Publishers.Google Scholar - Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function.
*Journal for Research in Mathematics Education*,*20*(4), 356–366.CrossRefGoogle Scholar - Watson, A., & Mason, J. (2005).
*Mathematics as a constructive activity: The role of learner generated examples*. Mahwah: Erlbaum.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*(2), 115–133.CrossRefGoogle Scholar - Yopp, D. A (2014). Undergraduates’ use of examples in online discussions.
*The Journal of Mathematical Behavior*,*33*, 180–191.CrossRefGoogle Scholar - Zandieh, M., & Rasmussen, C. (2010). Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning.
*The Journal of Mathematical Behavior*,*29*(2), 57–75.CrossRefGoogle Scholar - Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition.
*Journal for Research in Mathematics Education*, 317–346.Google Scholar