# Prospective Middle School Mathematics Teachers’ Covariational Reasoning for Interpreting Dynamic Events During Peer Interactions

## Abstract

This study investigated the covariational reasoning abilities of prospective middle school mathematics teachers in a task about dynamic functional events involving two simultaneously changing quantities in an individual process and also in a peer interaction process. The focus was the ways in which prospective teachers’ covariational reasoning abilities re-emerge in the peer interaction process in excess of their covariational reasoning. The data sources were taken from the individual written responses of prospective teachers, transcripts of individual comments, and transcripts of conversations in pairs. The data were analyzed for prospective teachers in terms of the cognitive and interactive aspects of individual behavior and also interaction. The findings revealed that prospective teachers at different levels working in pairs benefited from the process in terms of developing an awareness of their own individual and also a pair’s understanding of covarying quantities. Furthermore, the prospective teachers had opportunities to develop their knowledge on the connection between variables, rate of change, and slope. The prospective teachers’ work in pairs provided salient explanations for their reasoning about the task superior to their individual responses.

### Keywords

Covariational reasoning Dynamic functional events Peer learning### References

- Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division.
*Journal of Research in Mathematics Education, 21*, 132–144.CrossRefGoogle Scholar - Brasell, H. M. & Rowe, M. B. (1993). Graphing skills among high school physics students.
*School Science and Mathematics, 93*(2), 63–70.CrossRefGoogle Scholar - Carlson, M. (1998). A cross-sectional investigation of the development of the function concept. In J. J. Kaput, E. Dubinsky & A. H. Schoenfeld (Eds.),
*Research in collegiate mathematics education, III. Issues in Mathematics Education*(Vol. 7, pp. 115–162). Washington, DC, American Mathematical Society.Google Scholar - Carlson, M., Jacobs, S., Coe, E., Larsen, S. & Hsu, E. (2002). Applying covariational R-reasoning while modeling dynamic events.
*Journal for Research in Mathematics Education, 33*(5), 352–378.CrossRefGoogle Scholar - Carlson, M., Larsen, S. & Lesh, R. (2003). Integrating models and modeling perspective with existing research and practice. In R. Lesh & H. Doerr (Eds.),
*Beyond constructivism: A models and modeling perspective*(pp. 465–478). Mahwah, NJ: Erlbaum.Google Scholar - Carlson, M., Oehrtman, M. & Engelke, N. (2010). The precalculus concept assessment: a tool for assessing students’ reasoning abilities and understandings.
*Cognition and Instruction, 28*(2), 113–145. doi:10.1080/07370001003676587.CrossRefGoogle Scholar - Common Core State Standards Initiative (CCSSI) (2010).
*Common core state standards for mathematics.*Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Retrieved from http://www.corestandards.org/Math/ - Confrey, J. & Smith, E. (1991). A framework for functions: prototypes, multiple representations, and transformations. In R. G. Underhill (Ed.),
*Proceedings of the Thirteenth Annual Meeting North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 57–63). Atlanta, GA: Georgia State University.Google Scholar - Confrey, J. & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit.
*Educational Studies in Mathematics, 26*(2), 135–164.CrossRefGoogle Scholar - Confrey, J. & Smith, E. (1995). Splitting, covariation and their role in the development of exponential functions.
*Journal for Research in Mathematics Education, 26*(1), 66–86.CrossRefGoogle Scholar - Cooper, T. (2003). Open ended realistic division problems, generalisation and early algebra.
*International Group for the Psychology of Mathematics Education, 2*, 245–252.Google Scholar - Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K. & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme.
*The Journal of Mathematical Behavior, 15*(2), 167–192.CrossRefGoogle Scholar - Creswell, J. W. (2007).
*Qualitative inquiry and research design: Choosing among five approaches*(3rd ed.). Thousand Oaks, CA: Sage Publications.Google Scholar - De Lisi, R. & Golbeck, S. L. (1999). Implications of Piagetian theory for peer learning. In A. M. O’Donnell & A. King (Eds.),
*Cognitive perspectives on peer learning*(pp. 3–37). Mahwah, NJ: Erlbaum.Google Scholar - Dreyfus, T., Hershkowitz, R. & Schwarz, B. B. (2001). Abstraction in context: The case of peer interaction.
*Cognitive Science Quarterly, 1*, 307–358.Google Scholar - Eizenberg, M. M. & Zaslavsky, O. (2003). Cooperative problem solving in combinatorics: The inter-relations between control processes and successful solutions.
*Journal of Mathematical Behavior, 22*(4), 389–403.CrossRefGoogle Scholar - Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept.
*Journal for Research in Mathematics Education, 24*(2), 94–116.CrossRefGoogle Scholar - Fujii, T. (2003). Probing students’ understanding of variables through cognitive conflict problems: Is the concept of variable so difficult for students to understand? In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.),
*Proceedings of the 27th PME International Conference*(Vol. 1, pp.49-65). Honolulu: University of Hawaii.Google Scholar - Goos, M., Galbraith, P. & Renshaw, P. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving.
*Educational Studies in Mathematics, 49*(2), 193–223.CrossRefGoogle Scholar - Johnson, H. L. (2012). Reasoning about variation in the intensity of change in covarying quantities involved in rate of change.
*The Journal of Mathematical Behavior, 31*(3), 313–330. doi:10.1016/j.jmathb.2012.01.001.CrossRefGoogle Scholar - Houssart, J. & Evens, H. (2011). Conducting task-based interviews with pairs of children: Consensus, conflict, knowledge construction and turn taking.
*International Journal of Research and Method in Education, 34*(1), 63–79. doi:10.1080/1743727X.2011.552337.CrossRefGoogle Scholar - Kaput, J. J. (1992). Patterns in students’ formalization of quantitative patterns. In G. Harel & E. Dubinsky (Eds.),
*The concept of function: Aspects of epistemology and pedagogy, MAA Notes*(Vol. 25, pp. 290–318). Washington, DC: Mathematical Association of America.Google Scholar - Karlsson, M. (2013). Emotional identification with teacher identities in student teachers' narrative interaction.
*European Journal of Teacher Education, 36*(2), 133–146.CrossRefGoogle Scholar - Keene, K. A. (2007). A characterization of dynamic reasoning: Reasoning with time as parameter.
*The Journal of Mathematical Behavior, 26*(3), 230–246.CrossRefGoogle Scholar - Kertil, M. (2014).
*Pre-service elementary mathematics teachers' understanding of derivative through a model development unit*(Unpublished doctoral dissertation). Middle East Technical University, Ankara, Turkey.Google Scholar - Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 390–419). New York, NY: Macmillan.Google Scholar - Koklu, O. (2007).
*An investigation of college students’ covariational reasonings*(Phd dissertation). Retrieved from Florida State University database http://search.proquest.com/docview/304869139?accountid=13014. (Order No. 3282631). - Koklu, O. & Jakubowski, E. (2010). From interpretations to graphical representations: A Case study investigation of covariational reasoning.
*Eurasian Journal of Educational Research, 40*, 151–170.Google Scholar - Kruger, A. C. (1993). Peer collaboration: conflict, cooperation, or both?
*Social Development, 23*, 165–182.CrossRefGoogle Scholar - Leikin, R. (2004). The wholes that are greater than the sum of their parts: Employing cooperative learning in mathematics teachers’ education.
*Journal of Mathematical Behavior, 23*, 223–256.CrossRefGoogle Scholar - Milli Eğitim Bakanlığı [Ministry of National Education] (MoNE) (2013).
*Ortaokul matematik dersi (5, 6, 7, ve 8. Sınıflar) öğretim programı*[Middle school mathematics curriculum grades 5 to 8]. Ankara, Turkey: Author.Google Scholar - Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. Harel & E. Dubinsky (Eds.),
*The concept of function: Aspects of epistemology and pedagogy, MAA Notes*(Vol. 25, pp. 175–193). Washington, DC: Mathematical Association of America.Google Scholar - Monk, S. & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. In J. Dubinsky, E., Schoenfeld A. H, Kaput (Ed.),
*Research in Collegiate Mathematics Education I*(Vol. 4, pp. 139–168). Washington, DC, American Mathematical Society.Google Scholar - Moshman, D. & Geil, M. (1998). Collaborative reasoning: Evidence for collective reality.
*Thinking and Reasoning, 4*, 231–248.CrossRefGoogle Scholar - National Council of Teachers of Mathematics [NCTM]. (2000).
*Principles and Standards for School mathematics*. Reston, VA: Author.Google Scholar - Patchan, M. M., Hawk, B., Stevens, C. A. & Schunn, C. D. (2013). The effects of skill diversity on commenting and revisions.
*Instructional Science, 41*(2), 381–405.CrossRefGoogle Scholar - Portnoy, N., Heid, M. K., Lunt, J. & Zembat, I. O. (2005). Prospective secondary mathematics teachers unravel the complexity of covariation through structural and operational perspectives. In G. M. Lloyd, M. R. Wilson, J. L. M. Wilkins & S. L. Behm (Eds.),
*Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*[CD-ROM]. Eugene, OR: All Academic.Google Scholar - Psaltis, C. & Duveen, G. (2007). Conservation and conversation types: Forms of recognition and cognitive development.
*British Journal of Developmental Psychology, 25*(1), 79–102. doi:10.1348/026151005X91415.CrossRefGoogle Scholar - Saldanha, L. & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective- Simultaneous continuous variation. In S. B. Berensen, K. R. Dawkins, M. Blanton, W. N. Coulombe, J. Kolb, K. Norwood & L. Stiff (Eds.),
*Proceedings of the 20th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 298–303). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar - Sela, H. & Zaslavsky, O. (2007). Resolving cognitive conflicts with peers: Is there a difference between two and four? In J-H. Woo, H-C. Lew, K-S. Park, & D-Y. Seo (Eds.),
*Proceedings of the 31*^{st}*Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp.169-176). Atlanta, GA: Georgia State University.Google Scholar - Şen-Zeytun, A., Çetinkaya, B. & Erbaş, A. K. (2010). Mathematics teachers’ covariational reasoning levels and predictions about students’ covariational reasoning.
*Educational Sciences: Theory and Practice, 10*(3), 1601–1612.Google Scholar - Strom, A. D. (2006). The role of covariational reasoning in learning and understanding exponential functions. In S. Alatorre, J. L. Cortina, M. Sáiz & A. Méndez (Eds.),
*Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education 2*(pp. 624–630). Mérida, México: Universidad Pedagógica Nacional.Google Scholar - Swan M., Bell A., Burkhardt H., Janvier C., with the Shell Center team. (1985).
*The language of functions and graphs: An examination module for secondary school*. Manchester, England: Joint Matriculation Board. Retrieved February 20th, 2013, from http://www.mathshell.com/publications/tss/lfg/lfg_teacher.pdf - Tabach, M., Hershkowitz, R. & Schwarz, B. (2006). Constructing and consolidating algebraic knowledge within dyadic processes: A case study.
*Educational Studies in Mathematics, 63*, 238–258. doi:10.1007/s10649-005-9012-2.CrossRefGoogle Scholar - Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. H. Schoenfeld & J. J. Kaput (Eds.),
*Research in Collegiate Mathematics Education 1*(pp. 21–44). Providence, RI: American Mathematical Society.CrossRefGoogle Scholar - Tutty, J. I. & Klein, J. D. (2008). Computer-mediated instruction: A comparison of online and face-to-face collaboration.
*Educational Technology, Research & Development, 56*(2), 101–124.CrossRefGoogle Scholar - Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning.
*Educational Studies in Mathematics, 34*, 97–129.CrossRefGoogle Scholar - Webb, N. M. (1991). Task related verbal interaction and mathematics learning in small groups.
*Journal for Research in Mathematics Education, 22*(5), 366–389.CrossRefGoogle Scholar - Webb, N. M. & Farivar, S. (1999). Developing productive group interaction in middle school mathematics. In A. M. O’Donnell & A. King (Eds.),
*Cognitive perspectives on peer learning*(pp. 117–149). Hillsdale, NJ: Erlbaum.Google Scholar - Webb, N. M., Troper, J. D. & Fall, R. (1995). Constructive activity and learning in collaborative small groups.
*Journal of Educational Psychology, 87*(3), 406–423.CrossRefGoogle Scholar - Yin, R. K. (2009).
*Case study research: Design and methods*(4th Ed.). Thousand Oaks, CA: Sage.Google Scholar - Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. Schoenfeld & J. Kaput (Eds.),
*Research in Collegiate Mathematics Education IV*(pp. 103–127). Providence, RI: American Mathematical Society.Google Scholar - Zeidler, D. L., Walker, K. A., Ackett, W. A. & Simmons, M. L. (2002). Tangled up in views: Beliefs in the nature of science and responses to socioscientific dilemmas.
*Science Education, 86*(3), 343–367.CrossRefGoogle Scholar