‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation Article

First Online: 06 August 2011 DOI :
10.1007/s10649-011-9345-y

Cite this article as: Nardi, E., Biza, I. & Zachariades, T. Educ Stud Math (2012) 79: 157. doi:10.1007/s10649-011-9345-y
Abstract In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori , empirical , institutional and evaluative ). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal.

Keywords Teacher knowledge and beliefs Freeman’s classification of warrants Toulmin’s model for argumentation Practical rationality of teaching Visualisation Example use Tangent line

References Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching. What makes it special?

Journal of Teacher Education, 59 (5), 389–407.

CrossRef Google Scholar Bingolbali, E., & Monaghan, J. (2008). Concept image revisited.

Educational Studies in Mathematics, 68 , 19–35.

CrossRef Google 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.

CrossRef Google Scholar Biza, I., Nardi, E., & Zachariades, T. (2007). Using tasks to explore teacher knowledge in situation-specific contexts.

Journal of Mathematics Teacher Education, 10 , 301–309.

CrossRef Google Scholar Biza, I., Nardi, E., & Zachariades, T. (2009). Teacher beliefs and the didactic contract on visualisation.

For the Learning of Mathematics, 29 (3), 31–36.

Google Scholar Brousseau, G. (1997).

Theory of didactical situations in mathematics . Dordrecht: Kluwer.

Google Scholar Castela, C. (1995). Apprendre avec et contre ses connaissances antérieures: Un example concret, celui de la tangente [Learning with and in contradiction to previous knowledge: A concrete example, that of tangent].

Recherches en Didactiques des mathématiques, 15 (1), 7–47.

Google Scholar Chevallard, Y. (1985).

La transposition didactique [The didactic transposition] . Grenoble: La Pensée Sauvage Éditions.

Google Scholar Cook, S. D., & Brown, J. S. (1999). Bridging epistemologies: The generative dance between organizational knowledge and organizational knowing.

Organization Science, 10 (4), 381–400.

CrossRef Google Scholar Evens, H., & Houssart, J. (2004). Categorizing pupils' written answers to a mathematics test question: 'I know but I can't explain'.

Educational Research, 46 (3), 269–282.

CrossRef Google Scholar Freeman, J. B. (2005a). Systematizing Toulmin’s warrants: An epistemic approach.

Argumentation, 19 (3), 331–346.

CrossRef Google Scholar Freeman, J. B. (2005b).

Acceptable premises: An epistemic approach to an informal logic problem . Cambridge, UK: Cambridge University Press.

Google Scholar Giannakoulias, E., Mastoridis, E., Potari, D., & Zachariades, T. (2010). Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims.

The Journal of Mathematical Behavior, 29 , 160–168.

CrossRef Google Scholar Harel, G., & Sowder, L. (2007). Towards comprehensive perspectives on the learning and teaching of proof. In F. K. Lester (Ed.),

The second handbook of research on mathematics teaching and learning (pp. 805–842). USA: NCTM.

Google Scholar Herbst, P., & Chazan, D. (2003). Exploring the practical rationality of mathematics teaching through conversations about videotaped episodes: The case of engaging students in proving.

For the Learning of Mathematics, 23 (1), 2–14.

Google Scholar Hill, H. C., & Ball, D. L. (2004). Learning mathematics for teaching: Results from California's mathematics professional development institutes.

Journal for Research in Mathematics Education, 35 (5), 330–351.

CrossRef Google Scholar Hoyles, C., & Küchemann, D. (2002). Students' understanding of logical implication.

Educational Studies in Mathematics, 51 , 193–223.

CrossRef Google Scholar Inglis, M., & Mejia-Ramos, J. P. (2008). How persuaded are you? A typology of responses.

Research in Mathematics Education, 10 (2), 119–133.

CrossRef Google Scholar Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification.

Educational Studies in Mathematics, 66 , 3–21.

CrossRef Google Scholar Jacobs, J. K., & Morita, E. (2002). Japanese and American teachers' evaluations of videotaped mathematics lessons.

Journal for Research in Mathematics Education, 33 (3), 154–175.

CrossRef Google Scholar Kennedy, M. M. (2002). Knowledge and teaching.

Teachers & Teaching, 8 (3/4), 355–370.

CrossRef Google Scholar Knuth, E. J. (2002). Secondary school mathematics teachers' conceptions of proof.

Journal for Research in Mathematics Education, 33 (5), 379–405.

CrossRef Google Scholar Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.),

The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229–269). Hillsdale, NJ: Lawrence Erlbaum.

Google Scholar Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions.

The Journal of Mathematical Behavior, 26 (1), 60–82.

CrossRef Google Scholar Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems in practice.

Harvard Educational Review, 55 (2), 178–194.

Google Scholar Leatham, K. R. (2006). Viewing mathematics teachers' beliefs as sensible systems.

Journal of Mathematics Teacher Education, 9 , 91–102.

CrossRef Google Scholar Leder, G., Pehkonen, E., & Törner, G. (Eds.). (2002).

Beliefs: A hidden variable in mathematics education? Dordrecht: Kluwer.

Google Scholar Miyakawa, T., & Herbst, P. (2007). Geometry teachers' perspectives on convincing and proving when installing a theorem in class. In T. Lamberg & L. R. Wiest (Eds.),

29th Annual Meetings of PME-NA (pp. 366–373). Lake Tahoe, NV: University of Nevada, Reno.

Google Scholar Pedemonte, B. (2005). Quelques outils pour l’analyse cognitive du rapport entre argumentation et démonstration [Some tools for the cognitive analysis of the relation between argumentation and demonstration].

Recherches en Didactique des Mathématiques, 25 , 313–348.

Google Scholar Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed?

Educational Studies in Mathematics, 66 , 23–41.

CrossRef Google Scholar Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.

Educational Researcher, 15 (2), 4–14.

Google Scholar Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform.

Harvard Educational Review, 57 (1), 1–22.

Google Scholar Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations.

The Journal of Mathematical Behavior, 21 (4), 459–490.

CrossRef Google 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 , 151–169.

CrossRef Google Scholar Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.),

Handbook of research on mathematics teaching and learning (pp. 122–127). New York: Macmillan.

Google Scholar Toulmin, S. (1958).

The uses of argument . Cambridge, UK: Cambridge University Press.

Google Scholar Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs.

For the Learning of Mathematics, 25 , 34–38.

Google Scholar Whitenack, J. W., & Knipping, N. (2002). Argumentation, instructional design theory and students' mathematical learning: A case for coordinating interpretive lenses.

The Journal of Mathematical Behavior, 21 (4), 441–457.

CrossRef Google Scholar Yackel, E. (2001). Explanation, justification, and argumentation in A mathematics classroom. In M. v. d. Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 9–24). Utrecht, The Netherlands.

Yackel, E. (2002). What we can learn from analyzing the teacher's role in collective argumentation.

The Journal of Mathematical Behavior, 21 (4), 423–440.

CrossRef Google Scholar © Springer Science+Business Media B.V. 2011

Authors and Affiliations 1. School of Education University of East Anglia Norwich UK 2. Mathematics Education Centre Loughborough University Loughborough UK 3. Department of Mathematics University of Athens Athens Greece