Studying the Practical Rationality of Mathematics Teaching: What Goes Into “Installing” a Theorem in Geometry?

Abstract

This article presents a way of studying the rationality that mathematics teachers utilize in managing the teaching of theorems in high-school geometry. More generally, the study illustrates how to elicit the rationality that guides teachers in handling the demands of teaching practice. In particular, it illustrates how problematic classroom scenarios represented through animations of cartoon characters can facilitate thought experiments among groups of practitioners. Relying on video records from four study group sessions with experienced teachers of geometry, the study shows how these records can be parsed and inspected to identify categories of perception and appreciation with which experienced practitioners relate to an instance of an instructional situation. The study provides initial evidence that supports a theoretically derived hypothesis, namely that teachers of geometry as a group recognize as normative the expectation that a teacher will sanction or endorse those propositions that are to be remembered as theorems for later use. In interacting with a story in which students had proven a proposition that the teacher had not identified as a theorem, the study also shows the kind of tactical resources that teachers of geometry could use to make it feasible for students to reuse such a proposition.

ACKNOWLEDGMENTS

The research reported in this article is supported by NSF, grant ESI-0353285 to the first and third authors. Opinions expressed here are the sole responsibility of the authors and do not reflect the views of the Foundation.

Notes

Note that installation does not mean the same as learning since students might be held accountable for knowing something, while at the same time some individual students might not actually be able to use that knowledge.

The reader may also watch an animation of this scenario in ThEMaT Online, http://grip.umich.edu/themat, selecting the Isosceles Triangles story.

This thought overlooks what Davis and Hersh (1981) argue—that the social shapes what is installed as a theorem in the discipline.

Philosopher of education Thomas Green (1976) also alluded to “practical rationality” but without a definition. And Immanuel Kant (see Reath, 2006) used “practical rationality” also but while his interest seems to have been on the structure and logic of practical reason, our interest is in its content.

At various points in time there have been calls for laboratory geometry courses that follow inductive or informal approaches to the subject. We consider those as different contracts that are not addressed in this study.

The making of such a hypothesis is grounded in a corpus of geometry lessons in four intact classrooms where most times that a theorem is installed the teachers sanction it as such.

We note also that in making this hypothesis we mean neither to subscribe nor to object to the norm. This is a norm that (we hypothesize) underlies practice as it exists regardless of our opinion of it.

Study groups eventually met for two years and generated a large data corpus of over 120 hours of group discussion.

Megan had moved from the “Wednesday” to the “Tuesday” group after Christmas, so she participated in all four sessions. Some references were carried over from one set of discussions to the other because of this.

Note. *Excludes the intervals spent watching the movie.

The acronym and numbers refer to group (ITH or ESP) and session date, interval number, and turn.