Proof transcription in high school geometry: a study of what teachers recognize as normative when students present proof
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Proof transcription in high school geometry: a study of what teachers recognize as normative when students present proofs at the board Justin K. Dimmel 1
& Patricio G. Herbst
2
# Springer Nature B.V. 2020
Abstract
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-formark 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. Keywords Instructional practices . Instructional situations . Norms . Breaching experiments . Mathematical communication . Geometry . Proof
1 Introduction A mathematical expert’s presentation of a proof in real-time usually regenerates the proof in a way that is accountable; that is, it can be explained as a valid way of establishing what has to 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.
* Justin K. Dimmel [email protected]
1
School of Learning and Teaching and the Research in STEM Education (RiSE) Center, University of Maine, 330 Shibles Hall, Orono, ME 04469, USA
2
School of Education and Department of Mathematics, University of Michigan, Ann Arbor, USA
Dimmel J.K., Herbst P.G.
be proved (Livingston, 1986). To make their proof presentations accountable, mathematicians write words or symbols, draw pictures, and make other inscriptions while simultaneously talking and gesturing (Greiffenhagen, 2014; Núñez, 2009). The talk that accompanies such a presentation includes both verbalizing the steps of the proof as they are being written and also providing commentary about the process of the proof as it unfolds. It would be unreasonable to expect students that are just learning the rudiments of doing proofs to be capable presenters that skillfully use multimodal resources to recreate arguments, as mathematical experts do. However, as mathematics teachers are obligated to represent the discipline of mathematics faithfully (Herbst & Chazan, 2012), it is feasible that teachers might expect students to approximate these multimodal communication practices.1 One circumstance wher
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