#### Date of Award

Spring 5-5-2023

#### Level of Access Assigned by Author

Open-Access Thesis

#### Degree Name

Master's of Science in Teaching (MST)

#### Department

Teaching

#### Advisor

Justin Dimmel

#### Second Committee Member

Eric Pandiscio

#### Third Committee Member

Timothy Boester

#### Additional Committee Members

Patricio Herbst

Craig Mason

#### Abstract

Expert mathematical communication is a synthesis of speaking, writing, diagramming, and gesturing. What opportunities are there for secondary mathematics students to learn the discipline-specific ways in which these modalities can be combined to communicate proofs? In an initial effort to investigate this question, Dimmel and Herbst (2020) conducted a multimedia survey experiment to test a hypothesis about how secondary mathematics teachers expect students to communicate when presenting proofs at the board in secondary geometry classrooms. Their hypothesis, based on an analysis of episodes of instruction from a small sample of secondary geometry classrooms, was that teachers expected student presentations of proofs to default to mark-for-mark reproductions of previously completed proofs. Dimmel and Herbst (2020) referred to this practice as proof transcription. A key aspect of this practice is that proofs were not expected to be evaluated as mathematical arguments until the transcription was completed. They tested their hypothesis with a survey experiment that used storyboard representations of episodes of geometry instruction to gauge teacher’s reactions to instructional practices that endorsed or challenged proof transcription. They found participants reacted more positively to instructional practices that endorsed proof transcription. While this study provided insight into secondary classroom mathematical communication practices, the result of the study is representative of only a regional population (Midwestern) of secondary mathematics teachers (N=60) all of whom were not geometry teachers. Therefore, I extended this research work by investigating whether the communication practices of secondary school students observed by Dimmel & Herbst (2020) are generally recognized by a larger, nationally representative sample of specifically secondary geometry teachers. My research work is both descriptive—investigating the normative ways teachers expect students to present their proofs—and expansive—investigating teachers’ reaction to alternative communication practices thereby drawing attention to ways in which teachers could create opportunities for students to develop multimodal communication skills during their proof presentations.

I analyzed data from a multimedia survey experiment with a nationally representative sample (N = 405) of secondary geometry teachers to investigate instructional communication practices during proof presentation. Participants were shown storyboard depictions of instructional episodes and asked to rate the appropriateness of the (hypothetical) teacher’s actions using a Likert-like response format. I analyzed participants' responses using ANOVA. The purpose of the experiment was to investigate how secondary geometry teachers expect students to communicate when presenting proofs during class. My study (1) replicated findings from an investigation of what teachers expect when students present proofs Dimmel & Herbst (2020), (2) investigated how geometry teachers reacted to instructional practices that attempted to steer student presentations of proofs toward disciplinary communication practices, and (3) investigated how geometry teachers justify or criticize hypothetical instructional scenarios where proof transcription were either allowed or disrupted. The findings from this study support findings from existing literature that teachers expect students’ presentation of proofs to default to proof transcription. Similarly, it provides evidence that teachers can create opportunities for students to develop and practice disciplinary approximate communication practices during their proof presentations. The warrant for this claim is evidenced by teachers' generally positive reactions to episodes that steered students toward engaging in multimodal and disciplinary-approximate presentations of their proofs.

#### Recommended Citation

Ake, Bukola T., "An Investigation of the Instructional Norms of Mathematical Communication when Students Present Geometry Proofs at the Board" (2023). *Electronic Theses and Dissertations*. 3773.

https://digitalcommons.library.umaine.edu/etd/3773