On the instructional triangle and sources of justification for actions in mathematics teaching
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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.
KeywordsMathematics Teaching Mathematics Teacher Mathematical Task Practical Rationality Mathematical Work
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.
- 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
- Buchmann, M. (1987). Role over person: Morality and authenticity in teaching. Teachers’ College Record, 87(4), 529–543.Google Scholar
- 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.Google Scholar
- 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
- Chevallard, Y. (1991). La transposition didactique: Du savoir savant au savoir enseignée. Grenoble: La Pensée Sauvage.Google Scholar
- 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
- 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
- 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
- Henderson, K. (1963). Research on teaching secondary school mathematics. In N. L. Gage (Ed.), Handbook of research on teaching. Chicago: Rand McNally.Google Scholar
- Herbst, P. (2006). Teaching geometry with problems: Negotiating instructional situations and mathematical tasks. Journal for Research in Mathematics Education, 37, 313–347.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
- 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
- Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems in practice. Harvard Educational Review, 55(2), 178–194.Google Scholar
- Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press.Google Scholar
- 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
- Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. New York: Routledge.Google Scholar
- 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