, Volume 44, Issue 5, pp 601–612 | Cite as

On the instructional triangle and sources of justification for actions in mathematics teaching

  • P. Herbst
  • D. Chazan
Original Article


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.


Mathematics Teaching Mathematics Teacher Mathematical Task Practical Rationality Mathematical Work 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. 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.CrossRefGoogle Scholar
  2. 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.Google Scholar
  3. Buchmann, M. (1987). Role over person: Morality and authenticity in teaching. Teachers’ College Record, 87(4), 529–543.Google Scholar
  4. Chazan, D., & Herbst, P. (2012). Animations of classroom interaction: Expanding the boundaries of video records of practice. Teachers’ College Record, 114(3). Accessed 8 June 2012.
  5. Chazan, D., & Sandow, D. (2011). “Why did you do that?” Reasoning in algebra classrooms. The Mathematics Teacher, 104(6), 460–464.Google Scholar
  6. 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.Google Scholar
  7. 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.CrossRefGoogle Scholar
  8. Chevallard, Y. (1991). La transposition didactique: Du savoir savant au savoir enseignée. Grenoble: La Pensée Sauvage.Google Scholar
  9. Cohen, D. (2011). Teaching and its predicaments. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar
  10. Cohen, D., Raudenbush, S., & Ball, D. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25(2), 119–142.CrossRefGoogle Scholar
  11. Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23(2), 167–180.CrossRefGoogle Scholar
  12. 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.Google Scholar
  13. Goffman, E. (1997). The neglected situation. In C. Lemert & A. Branaman (Eds.), The Goffman reader (pp. 229–233). Oxford: Blackwell (Original work published 1964).Google Scholar
  14. 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).Google Scholar
  15. Henderson, K. (1963). Research on teaching secondary school mathematics. In N. L. Gage (Ed.), Handbook of research on teaching. Chicago: Rand McNally.Google Scholar
  16. Herbst, P. (2003). Using novel tasks to teach mathematics: Three tensions affecting the work of the teacher. American Educational Research Journal, 40, 197–238.CrossRefGoogle Scholar
  17. Herbst, P. (2006). Teaching geometry with problems: Negotiating instructional situations and mathematical tasks. Journal for Research in Mathematics Education, 37, 313–347.Google Scholar
  18. 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
  19. Herbst, P., & Chazan, D. (2011). Research on practical rationality: Studying the justification of actions in mathematics teaching. The Mathematics Enthusiast, 8(3), 405–462.Google Scholar
  20. 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.CrossRefGoogle Scholar
  21. 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.CrossRefGoogle Scholar
  22. Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems in practice. Harvard Educational Review, 55(2), 178–194.Google Scholar
  23. Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press.Google Scholar
  24. 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.CrossRefGoogle Scholar
  25. Lemke, J. (2000). Across the scales of time: Artifacts, activities, and meanings in ecosocial systems. Mind, Culture, and Activity, 7(4), 273–290.CrossRefGoogle Scholar
  26. 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.CrossRefGoogle Scholar
  27. 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.Google Scholar
  28. Popkewitz, T. (2004). The Alchemy of the mathematics curriculum: Inscriptions and the fabrication of the child. American Educational Research Journal, 41(1), 3–34.CrossRefGoogle Scholar
  29. Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge.Google Scholar
  30. 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.CrossRefGoogle Scholar
  31. Skott, J. (2009). Contextualising the notion of ‘belief enactment’. Journal of Mathematics Teacher Education, 12, 27–46.CrossRefGoogle Scholar
  32. 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.Google Scholar
  33. Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing. London: Falmer.CrossRefGoogle Scholar
  34. 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.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2012

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.University of MarylandCollege ParkUSA

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