Abstract
This article presents case studies of the classroom teaching practices of two algebra teachers. Our data consist of videotaped classroom observations during a single academic year. We identify and characterize specific teaching practices that establish the norm that the teacher is the sole arbiter of mathematical correctness in the classroom. We suggest that these practices are likely to promote the development of the authoritative proof scheme in students. Our results can provide a basis for future research investigating the prevalence of these teaching practices and their impact on student learning and can be used as parameters to investigate teacher change.
Résumé
Cet article présente deux études de cas sur les pratiques d’enseignement de deux enseignants d’algèbre. Nos données sont constituées d’observations de classe filmées au cours d’une même année scolaire. Nous définissons et analysons certaines pratiques d’enseignement qui imposent une norme selon laquelle l’enseignant est le seul arbitre de la véracité mathématique dans la salle de classe. Nous estimons que de telles pratiques sont susceptibles de promouvoir les arguments d’autorité dans les démonstrations formelles. Nos résultats pourraient servir de base pour des recherches ultérieures sur la prévalence de ces pratiques d’enseignement et leur impact sur l’apprentissage des étudiants, et pourraient également servir de paramètres pour analyser les possibilités de changement chez les enseignants.
Similar content being viewed by others
References
Anderson, J. R. (1980). Cognitive psychology and its implications. San Francisco: Freeman.
Brownlee, J. (2003). Paradigm shifts in preservice teacher education students: A case study of changes in epistemological beliefs for two teacher education students. Australian Journal of Educational & Developmental Psychology, 3, 1–6.
Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24, 359–387.
Cooney, T. J. (1995). On the notion of authority applied to teacher education. In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the Seventeenth Annual Meeting of the North American Chapter of the International Group for Psychology of Mathematics Education (pp. 91–96). Columbus, OH: ERIC/CSMEE Publications.
Dey, I. (1999). Grounding grounded theory: Guidelines for qualitative inquiry. San Diego: Academic Press.
Ellis, A. (2007). Connections between generalizing and justifying: students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
Etchberger, M. L., & Shaw, K. L. (1992). Teacher change as a progression of transitional images: A chronology of a developing constructivist teacher. School Science and Mathematics, 92(8), 411–417.
Fennema, E., & Nelson, B. S. (1997). Mathematics teachers in transition. Mahwah, NJ: Lawrence Erlbaum.
Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory; strategies for qualitative research. Chicago: Aldine.
Goldsmith, L., & Schifter, D. (1997). Understanding teachers in transition: Characteristics of a model for developing teachers. In E. Fennema & B. S. Nelson (Eds.), Mathematics teachers in transition (pp. 19–54). Mahwah, NJ: Erlbaum.
Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zaskis (Eds.), Learning and teaching number theory (pp. 185–212), Westport, CT: Ablex Publishing Corporation.
Harel, G., & Rabin, J. M. (2010). Teaching practices associated with the authoritative proof scheme. Journal for Research in Mathematics Education, 41, 14–19.
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education (Vol. 3, pp. 234–283). Providence, RI: American Mathematical Society.
Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). Greenwich, CT: Information Age Publishing.
Hart, E. W. (1994). A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory. In J. Kaput & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning (pp. 49–157). Washington, DC: Mathematical Association of America.
Healy, L., & Hoyles, C. (1998). Justifying and proving in school mathematics, executive summary. London: Institute of Education, University of London.
Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research on Mathematics Education, 31(4), 396–428.
Heinze, A. (2002)....Because a square is not a rectangle. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Annual Meeting of the International Group for the Psychology of Mathematics Education (pp. 81–88). Norwich, UK: University of East Anglia.
Herbst, P. G. (2002). Engaging students in proving: A double bind on the teacher. Journal for Research in Mathematics Education, 33(3), 176–203.
Housman, D., & Porter, M. (2003). Proof schemes and learning strategies of above-average mathematics students. Educational Studies in Mathematics, 53, 139–158.
Martin, T. S., McCrone, S. M. S., Bower, M. L. W., & Dindyal, J. (2005). The interplay of teacher and student actions in the teaching and learning of geometric proof. Educational Studies in Mathematics, 60, 95–124.
Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 10(1), 41–51.
Senk, S. L. (1985). How well do students write geometry proofs? Mathematics Teacher, 78(6), 448–456.
Sharon, M., & Martin, T. (2004). The impact of teacher actions on student proof schemes in geometry. In D. E. McDougal & J. A. Ross (Eds.), Proceedings of the Twenty-sixth Annual Meeting of the North American Chapter of the Psychology of Mathematics Education. Toronto: Ontario Institute for Studies in Education.
Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–132). New York: Erlbaum.
Sowder, L. (1988). Children’s solutions of story problems. Journal of Mathematical Behavior, 7, 227–238.
Sowder, J. (2007). The mathematical education and development of teachers. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 157–224). Greenwich, CT: Information Age Publishing.
Stigler, J. W., Gonzales, P., Kawanaka, T., Knoll, S., & Serrano, A. (1999). The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States (National Center for Education Statistics Report No. NCES 99-0974). Washington, DC: U.S. Government Printing Office.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: Free Press.
Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 127–146). New York, NY: Macmillan.
Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181–234). Albany, NY: SUNY Press.
Wilson, M., & Goldenberg, M. P. (1998). Some conceptions are difficult to change: One middle school mathematics teacher’s struggle. Journal of Mathematics Teacher Education, 1(3), 269–293.
Author information
Authors and Affiliations
Corresponding author
Additional information
A brief report on this work has appeared separately (Harel & Rabin, 2010).
This work is supported, in part, by the National Science Foundation (REC 0310128, G. Harel PI). Opinions expressed herein are those of the authors and are not necessarily those of the Foundation.
Rights and permissions
About this article
Cite this article
Harel, G., Rabin, J.M. Teaching Practices That Can Promote the Authoritative Proof Scheme. Can J Sci Math Techn 10, 139–159 (2010). https://doi.org/10.1080/14926151003778282
Published:
Issue Date:
DOI: https://doi.org/10.1080/14926151003778282