Abstract
Although mathematical practice has traditionally valued two distinct kinds of mathematical work—referred to by Gowers (2000) as theory building and problem solving—activity in classrooms appears to be organized largely around the latter, rather than the former. This study takes up the question of whether there is a customary role for theory building in the secondary geometry course and to what extent teachers of geometry hold students accountable for that disposition. We analyze records from study groups composed of experienced geometry teachers in the USA, and ask to what extent the disposition towards theory building is visible in those records, and how teachers react to the suggestion that it could play a larger role in their practice. Among our findings, we report that participants consistently attached a higher value to work organized around problem solving, rejecting the idea that they might hold their students accountable for theory building.
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Notes
In the United States of America, it is customary for students to take a course in geometry during their first or second year of high school (that is, when they are 14–15 or 15–16 years old; see González & Herbst, 2006).
In this theorem and the next, the notation \( \left\langle XY\right\rangle \) denotes the signed distance from X to Y relative to a choice of a positive direction along the line \( \overleftrightarrow{XY} \). The choice of which direction is positive is arbitrary, so each of the signed distances is only defined up to a factor of ±1, but each of the ratios has an unambiguous sign regardless of which direction is chosen to be positive.
Study group sessions in the archive are named using the format XXXmmddyy, where the last six characters encode the session date, and the three-letter code XXX is either ITH, ESP, TMT, or TMW—codes used to distinguish the study groups from one another.
All personal and institutional names appearing in the data are pseudonyms.
In our report of these discussions, we often use first names to attribute the expression of dispositions because that helps direct the reader to the transcript evidence, which is indeed provided through the speech from individuals. The extent to which those expressions represent individual convictions or adaptations to the group is not decidable by this evidence alone. And our intent is not to adjudicate on that question but to bring out what is being expressed—the dispositions themselves.
In addition to the moderator, the discussion included up to two other project staff members whom we refer to as Researchers (see Nachlieli, 2011).
This story can be seen in LessonSketch (www.lessonsketch.org).
The teachers seemed to take for granted that special education students are less intellectually capable than others, a position that oversimplifies the wide range of conditions that special education students may have. As the goal here is to report teachers’ perceptions of what is and is not viable in the classroom, we defer to their own use of the term.
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Acknowledgments
The work done was supported by NSF grant ESI-0353285 to Patricio Herbst. Opinions expressed here are the sole responsibility of the authors and do not necessarily reflect the views of the Foundation.
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The research reported in this article is based in part on the first author’s doctoral dissertation directed by the second author. An earlier version was presented at the 2010 Annual Meeting of the American Educational Research Association in Denver.
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Weiss, M., Herbst, P. The role of theory building in the teaching of secondary geometry. Educ Stud Math 89, 205–229 (2015). https://doi.org/10.1007/s10649-015-9599-x
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DOI: https://doi.org/10.1007/s10649-015-9599-x