Abstract
We elaborate on the notion of the instructional triangle, to address the question of how the nature of instructional activity can help justify actions in mathematics teaching. We propose a practical rationality of mathematics teaching composed of norms for the relationships between elements of the instructional system and obligations that a person in the position of the mathematics teacher needs to satisfy. We propose such constructs as articulations of a rationality that can help explain the instructional actions a teacher takes in promoting and recognizing learning, supporting work, and making decisions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We use constraint in the sense of bind or dependence, and without assuming that constraints are necessarily impediments. They can impede some things and facilitate others, just like the rules of a game do.
The content of studies is not the same as disciplinary knowledge, but the result of the process of didactic transposition (Chevallard, 1991) or alchemy of school subjects (Popkewitz, 2004). The two characteristics listed here are minimal in order to develop the instructional triangle so as to account for the work of the teacher. In an analysis of the knowledge itself, it would be desirable to make finer distinctions among these various versions of knowledge.
By norm we mean statements of behaviors that are unmarked or unremarked upon when participants do them but that call for elaborations or repairs by participants when those behaviors are missing. The word norm has thus an objective sense, as most frequent behavior in a recurrent social encounter, and a subjective sense, as behavior expected by actors of a recurrent social encounter. Herbst, Nachlieli, and Chazan (2011) show how norms can be empirically confirmed using an adaptation of the ethnomethodological practice of breaching experiments (Garfinkel & Sacks, 1970).
This use of situation and framing comes from interpretive sociology, dating back to the work of Goffman (1964/1997).
Examples of situations include “solving for x” in algebra I, “doing proofs” in high school Geometry; also calculating a measure and exploring a figure in middle and high school geometry, to mention a few.
Conceptions of task have also been proposed to examine teacher education (e.g., Zaslavsky & Sullivan, 2011) and teacher learning from teaching (Leikin, 2010) suggesting that task might be a more central notion than what we provide here. An examination of that issue is beyond the scope of this paper.
When Brousseau introduced the subject–milieu system, his main interest was in showing how adidactical situations could embody mathematical ideas. Thus, Brousseau’s emphasis was on the adidacticity of the milieu—the possibility that the student perceive the milieu as devoid of didactical (instructional) intention. In our account of “task” we continue to find helpful to speak about a milieu, though the milieu may or may not be adidactical. Clearly not all tasks are adidactical situations.
Needless to say, whatever individual students may actually do in response to whatever task is chosen is not determined by the norms of the task chosen in a situation.
References
Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373–397.
Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques 1970–1990 (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. and Trans.). Dordrecht: Kluwer.
Buchmann, M. (1987). Role over person: Morality and authenticity in teaching. Teachers’ College Record, 87(4), 529–543.
Chazan, D., & Herbst, P. (2012). Animations of classroom interaction: Expanding the boundaries of video records of practice. Teachers’ College Record, 114(3). http://www.tcrecord.org.proxy.lib.umich.edu/library. Accessed 8 June 2012.
Chazan, D., & Sandow, D. (2011). “Why did you do that?” Reasoning in algebra classrooms. The Mathematics Teacher, 104(6), 460–464.
Chazan, D., & Yerushalmy, M. (2003). On appreciating the cognitive complexity of school algebra: Research on algebra learning and directions of curricular change. In J. Kilpatrick, D. Schifter, & G. Martin (Eds.), A research companion to the principles and standards for school mathematics. Reston: NCTM.
Chazan, D., Yerushalmy, M., & Leikin, R. (2008). An analytic conception of equation and teachers’ views of school algebra. Journal of Mathematical Behavior, 27(2), 87–100.
Chevallard, Y. (1991). La transposition didactique: Du savoir savant au savoir enseignée. Grenoble: La Pensée Sauvage.
Cohen, D. (2011). Teaching and its predicaments. Cambridge, MA: Harvard University Press.
Cohen, D., Raudenbush, S., & Ball, D. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25(2), 119–142.
Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23(2), 167–180.
Garfinkel, H., & Sacks, H. (1970). On Formal Structures of Practical Action. In J. McKinney & E. Tiryakian (Eds.), Theoretical Sociology (pp. 337–366). New York: Appleton-Century-Crofts.
Goffman, E. (1997). The neglected situation. In C. Lemert & A. Branaman (Eds.), The Goffman reader (pp. 229–233). Oxford: Blackwell (Original work published 1964).
Hawkins, D. (2002). I, thou, and it. In D. Hawkins (Ed.), The informed vision: Essays on learning and human nature (pp. 52–64). New York: Agathon (Original work published in 1967).
Henderson, K. (1963). Research on teaching secondary school mathematics. In N. L. Gage (Ed.), Handbook of research on teaching. Chicago: Rand McNally.
Herbst, P. (2003). Using novel tasks to teach mathematics: Three tensions affecting the work of the teacher. American Educational Research Journal, 40, 197–238.
Herbst, P. (2006). Teaching geometry with problems: Negotiating instructional situations and mathematical tasks. Journal for Research in Mathematics Education, 37, 313–347.
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.
Herbst, P., & Chazan, D. (2011). Research on practical rationality: Studying the justification of actions in mathematics teaching. The Mathematics Enthusiast, 8(3), 405–462.
Herbst, P., & Miyakawa, T. (2008). When, how, and why prove theorems: A methodology to study the perspective of geometry teachers. ZDM—The International Journal on Mathematics Education, 40(3), 469–486.
Herbst, P., Nachlieli, T., & Chazan, D. (2011). Studying the practical rationality of mathematics teaching: What goes into “installing” a theorem in geometry? Cognition and Instruction, 29(2), 1–38.
Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems in practice. Harvard Educational Review, 55(2), 178–194.
Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press.
Leikin, R. (2010). Learning through teaching through the lens of multiple solution tasks. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics (pp. 69–85). New York: Springer.
Lemke, J. (2000). Across the scales of time: Artifacts, activities, and meanings in ecosocial systems. Mind, Culture, and Activity, 7(4), 273–290.
Marcus, R., & Chazan, D. (2010). What experienced teachers have learned from helping students think about solving equations in the one-variable-first algebra curriculum. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics (pp. 169–187). Berlin: Springer.
Margolinas, C. (1995). La structuration du milieu et ses apports dans l’analyse a posteriori des situations. In C. Margolinas (Ed.), Les débats de didactique des mathématiques. Grenoble: La Pensée Sauvage.
Popkewitz, T. (2004). The Alchemy of the mathematics curriculum: Inscriptions and the fabrication of the child. American Educational Research Journal, 41(1), 3–34.
Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge.
Simon, M., & Tzur, R. (1999). Explicating the teacher’s perspective from the researchers’ perspectives: Generating accounts of mathematics teachers’ practice. Journal for Research in Mathematics Education, 30(3), 252–264.
Skott, J. (2009). Contextualising the notion of ‘belief enactment’. Journal of Mathematics Teacher Education, 12, 27–46.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: Free Press.
Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing. London: Falmer.
Zaslavsky, O., & Sullivan, P. (2011). Setting the stage: A conceptual framework for examining and developing tasks for mathematics teacher education. In O. Zaslavsky & P. Sullivan (Eds.), Constructing knowledge for teaching secondary mathematics (pp. 1–19). New York: Springer.
Acknowledgments
The ideas reported in this paper have been developed in part with the support of National Science Foundation Grants ESI-0353285 and DRL-0918425 to the authors. All opinions are those of the authors and do not necessarily represent the views of the Foundation. The authors thank Ander Erickson and three anonymous reviewers for valuable comments on an earlier version.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Herbst, P., Chazan, D. On the instructional triangle and sources of justification for actions in mathematics teaching. ZDM Mathematics Education 44, 601–612 (2012). https://doi.org/10.1007/s11858-012-0438-6
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11858-012-0438-6