Examining Mathematics Classroom Interactions: Elevating Student Roles in Teaching and Learning

Abstract

This article introduces a model entitled, “Responsive Teaching through Problem Posing” or RTPP, that addresses a type of reform oriented mathematics teaching based on posing relevant problems, positioning students as experts of mathematics, and facilitating discourse.  RTPP incorporates decades of research on students’ thinking in mathematics and more recent research on responsive teaching practices.  Two classroom case studies are presented.  A high school unit on functions is explored utilizing individual research on the part of the teacher to enact RTPP lessons.  A middle school teacher enacts a RTPP lesson on proportions and utilizes this model to bridge students’ incorrect additive reasoning strategies with correct multiplicative reasoning strategies.  The results showed that both teachers were able to elevate students’ roles in classroom discussions through implementation of RTPP.  Individual research conducted by the high school teacher informed his RTPP approach while participation in professional development sessions with a classroom embedded component influenced the middle school teacher’s enactment of RTPP lessons.  Both teachers used specific teacher moves within RTPP to relinquish their role as mathematics experts in order to elevate their students’ roles in classroom discussions.  The RTPP cycle is offered as a potential model for studying mathematics teaching and learning across a variety of secondary mathematics classrooms.

Keywords

• Problem posing, responsive teaching, identity, mathematics teaching and learning

References

1. Ball, D. L., & McDiarmid, G. W. (1990). The subject-matter preparation of teachers. In W. R. Houston (Ed.), Handbook of research on teacher education (pp. 437-448). New York: MacMillan Publishing Company.
2. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for research in mathematics education, 29 (1), 41-62.
3. Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge of children's mathematical thinking in classroom teaching: An experimental study. American educational research journal, 26 (4), 499-531.
4. Cobb, P., Boufi, A., McClain, K., & Whitenack (1997). Reflective discourse and collective reflection. Journal for research in mathematics education 28 (3), 258-277.
5. Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for research in mathematics education, 40 (1), 40-68.
6. Dreyfus, T., & Eisenberg, T. (1982). Intuitive functional concepts: A baseline study on intuitions. Journal for research in mathematics education, 13, 360-380.
7. Dyer, E. B., & Sherin, M. G. (2016). Instructional reasoning about interpretations of student thinking that supports responsive teaching in secondary mathematics. ZDM, 48(1-2), 69-82.
8. Empson, S. B. (2014). Responsive teaching from the inside out: teaching base ten to young children. Investigations in mathematics learning, 7(1), 23-53.

9. Empson, S. B. (2003). Low-performing students and teaching fractions for understanding: An interactional analysis. Journal for research in mathematics education, 34 (4), 305-343.
10. 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.
11. Freudenthal, H. (1994). Thoughts on teaching mechanics: Didactical phenomenology of the concept of force. In L. Streefland,(ed.), The legacy of Hans Freudenthal. Springer: Dordrecht, The Netherlands: Kluwer Academic Publishers.
12. Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and instruction, 26(4), 430-511.
13. Howard, T. C. (2010). Why race and culture matter in schools. New York: Teachers College Press.
14. Jacobs, V. R., Empson, S. B., Krause, G. H., & Pynes, D. (2015). Responsive teaching with fractions. In Annual meeting of the research conference of the national council of teachers of mathematics, Boston.
15. Kent, L. B. (2015). Change in the era of common core standards: A mathematics teacher’s journey. International journal of learning, teaching and educational research, 12(2), 48-63.
16. Kiefer, S. M., Ellerbrock, C., & Alley, K. (2014). The role of responsive teacher practices in supporting academic motivation at the middle level. RMLE Online, 38(1), 1-16.
17. Lamon, S. 1. (1993). Ratio and proportion: Connecting content and children's thinking. Journal for research in mathematics education, 24 (1), 41-61.
18. Leinwand, S. (2014). Principles to actions: Ensuring mathematical success for all. National Council of Teachers of Mathematics.
19. McClain, K., & Cobb, P. (2001). An analysis of development of sociomathematical norms in one first-grade classroom. Journal for research in mathematics education, 32(3), 236-266.
20. Nathan, M. J., & Petrosino, A. (2003). Expert blind spot among preservice teachers. American educational research journal, 40(4), 905-928.
21. National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.
22. National Governors Association. (2014). Common core state standards initiative. Washington, DC: National Governors Association.
23. Nielsen, L., Steinthorsdottir, O. B., & Kent, L. B. (2016). Responding to student thinking: Enhancing mathematics instruction through classroom based professional development. Middle school journal, 47(3), 17-24.
24. Robertson, A. D., Scherr, R., & Hammer, D. (Eds.). (2015). Responsive teaching in science and mathematics. Routledge.
25. Sfard, A. (1995). Symbolizing mathematical reality into being. Paper presented for the symposium Symbolizing, communicating, and mathematizing, Nashville, September, 1995.
26. Simon, M. (1997). Developing new models of mathematics teaching: An imperative for research on mathematics teacher development. In E. Fennema & B. S. Nelson (Eds.), Mathematics teachers in transition. New Jersey: Lawrence Erlbaum Associates.
27. Stake, R. E. (1994). Case studies. In N. K. Denzin & Y. S. Lincon (Eds.), Handbook of qualitative research. Thousand Oaks, CA: Sage Publications.
28. Staples, M. (2007). Supporting whole-class collaborative inquiry in a secondary mathematics classroom. Cognition and instruction, 25(2-3), 161-217.
29. Steinthorsdottir, O. B. (2006, July). Proportional reasoning: Variable influencing the problems difficulty level and one’s use of problem solving strategies. In Proceedings of the 30th conference of the international group of Psychology of Mathematics Education (Vol. 5, pp. 169-176).
30. Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht, The Netherlands: Kluwer Academic Publishers.
31. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning, NCTM, 495-511.
32. Treffers, A. (1991). Realistic mathematics education in the Netherlands 1980-1990. Realistic mathematics education in primary school, 11-20.
33. Tjoe, H. (2015). Giftedness and aesthetics: Perspectives of expert mathematicians and mathematically gifted students. Gifted child quarterly, 59(3), 165-176.