We investigated how US secondary mathematics teachers expect students to present geometric proofs at the board. We analyzed video records of geometry classrooms and found students to be engaged in a practice that we call proof transcription—i.e., mark-for-mark reproductions of previously completed proofs that were not reasoned reconstructions of arguments. To investigate whether or not US secondary mathematics teachers recognize transcriptions as routine occurrences when students presented proofs, we conducted a survey experiment. Participants (n = 60) viewed episodes of instruction and answered questions that elicited their reactions to those episodes. The analysis of open- and closed-ended responses to the survey indicated that participants recognized transcriptions as routine. Our study contributes a fine-grained description of what teachers expect from students when students are called to present their work at the board.
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Mathematics teachers are also obligated to recognize and serve students as individual learners; to oversee the interpersonal dynamics of the students in the class (e.g., how students talk to one another or share communal classroom resources); and to enforce school rules, work within specific curricular frameworks, or comply with policies that pertain to schooling as an institution (see Herbst & Chazan, 2012).
Dimmel and Herbst (2015) describe a variety of these markings, including hash marks on strokes to associate segments which are congruent, and others.
We focus here on student presentations of proofs. For a multimodal analysis of a disciplinary expert’s presentation of a proof, see Núñez, 2009 (p. 321–324).
The data analyzed in this report are a subset of data that were gathered as part of a dissertation study (see Dimmel, 2015). In addition to the sequence norm, the larger study investigated the kinds of details that teachers expected in students’ written proofs (Dimmel & Herbst, 2018). There were five experimental groups in total for the data collection for the larger study, but only four of those groups were included in the experimental design for the sequence norm portion of the study that is reported here. Groups A, B, C, and D of this study correspond to Groups 1, 3, 4, and 5 (respectively) of the larger study.
The choices for the closed-ended rating questions were: 1 (very inappropriate), 2 (inappropriate), 3 (somewhat inappropriate), 4 (somewhat appropriate), 5 (appropriate), 6 (very appropriate).
We acknowledge the support of Nicolas Boileau for his assistance with the reliability study.
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This research was supported in part by Rackham Merit Fellowship from the University of Michigan (first author). The video archive was collected with the support of NSF grant REC-0133619 and the experiment data with NSF grant DRL-0918425 (P. Herbst, PI). The Opinions expressed here are the authors’ and do not reflect the views of the University of Michigan or the National Science Foundation.
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The research reported here is based on the first author’s doctoral dissertation at the University of Michigan, for which the second author served as the dissertation committee chair. Please address correspondence concerning this article to Justin K. Dimmel, School of Learning and Teaching, University of Maine, 330 Shibles Hall, Orono, ME 04469.
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Dimmel, J.K., Herbst, P.G. Proof transcription in high school geometry: a study of what teachers recognize as normative when students present proofs at the board. Educ Stud Math 105, 71–89 (2020). https://doi.org/10.1007/s10649-020-09975-y
- Instructional practices
- Instructional situations
- Breaching experiments
- Mathematical communication