There is a disconnect between how mathematical proof is taught and what K-12 students need to develop critical thinking and reasoning skills. In the mathematics discipline, rich proving processes include generalizing patterns, formulating conjectures, generating arguments, evaluating others’ arguments, and communicating mathematical knowledge. These processes are key habits of mind that equip students to think critically within and outside mathematics classrooms. K-12 students typically learn proof, however, by verifying given statements using a two-column format, not by creating and justifying their own conjectures (Ball, Hoyles, Jahnke, & Movshovitz-Hadar, 2002; Harel & Sowder, 1998; Herbst, 2002; Usiskin, 1980). This restrictive emphasis on statement-and-reason proofs is contrary to proving processes in mathematics and contributes to K-12 students’ and their teachers’ pervasive difficulties and limited views of proof (Balacheff, 1988; Martin & Harel, 1989). Furthermore, many current and future elementary teachers have not learned about proof in process-oriented ways, but are increasingly expected to teach about proof using activity-based instruction (NCTM, 2000). It is crucial, then, for prospective teachers of elementary grades (PTEs) to develop substantial understandings of proof in order to provide meaningful opportunities for their students to engage with proof. How can college educators better equip PTEs for this vital endeavor?

Research Focus

Math content courses specifically designed for PTEs are one means for addressing this concern. Often, these courses are taught solely by graduate teaching assistants (TAs). As the instructors, TAs represent a central feature of PTEs’ experiences in learning about proof, and teachers’ beliefs have been implicated as a major influence on teachers’ practices (Putnam & Borko, 1997; Thompson, 1992). Since fostering richer conceptions of proof is one goal of these courses, I used classroom observations to examine how TAs and PTEs interacted when exploring proof, and in-depth interviews to study TAs’ views of proof. First, I examined how proof-related tasks were enacted in a mathematics content course for PTEs. I focused on how TAs and PTEs engaged in reasoning and proving (RP) in order to understand how TAs provided PTEs with opportunities to experience RP. Second, I studied TAs’ views of the nature and purposes of RP in mathematics and in teaching. Since teachers’ beliefs influence their practices, I studied TAs’ beliefs, as expressed during interviews, to illuminate their instructional decisions.

Method

In the spring 2011 semester, I conducted a qualitative, multiple-case study of six TAs who were teaching a geometry content course (referred to as Geom) at a large Mid-Western University. I chose this setting and semester because at this university all PTEs are required to take Geom, and the curriculum used (Beckmann, 2008) is considered to be student-centered; it is organized around activities intended to promote engagement, exploration, and discussion. Therefore, the TAs were more likely to have opportunities to use innovative proof-related tasks in their teaching. In such a scenario, understanding TAs’ teaching practices and views of proof can provide insights about how TAs use the curriculum and suggest future professional development strategies to help improve the teaching of proof to PTEs.

All six mathematics and mathematics-education graduate students assigned to teach Geom agreed to participate. The six participants included Laura, completing an Applied Mathematics Master’s degree; Peter, Isaac, and Ace, pursuing Mathematics PhDs; and Kelly and Evan, Mathematics Education doctoral students.

Measures

This study relied on qualitative data about TAs’ teaching practices and beliefs gathered from interviews and observations. To examine TAs’ teaching, I observed each TA’s class at least five times. Each observation coincided with sections of the curriculum that contained proof-related tasks. Conducting at least five observations helped me observe lessons on multiple days and across a variety of contexts. I video-recorded and took field notes focused on TA and PTEs’ actions and interactions around reasoning and proving. To examine TAs’ views about proof, I audio-recorded two types of interviews: (a) follow-up interviews conducted after each observation, and (b) interviews not associated with a specific observation. The first type of interview was conducted the same day as each observation. These interviews aimed to understand how TAs’ beliefs influenced their in-class decisions. The second type of interview consisted of three stages:  First, at the beginning of 2011, I asked TAs about their views of the purposes of proof in mathematics. Second, before the Geom text covered proof, I examined how TAs viewed the teaching and learning of proof in Geom. Third, at the end of Spring 2011, I aimed to provide additional measures to enhance the data from their each TA’s earlier responses.

Analysis

I used G. J. Stylianides’s (2009) analytic framework for analyzing RP activities in textbooks to identify proof-related tasks in Geom’s curriculum (another trained rater double-coded 31% of the tasks and we agreed on 81% of the codes). To analyze TAs’ classroom teaching, I coded observational data by identifying how TAs and PTEs were involved in proving processes and how proof-related tasks were set up and enacted (Bieda, 2010; Henningsen & Stein, 1997). To analyze the measures of TAs’ beliefs, I transcribed and coded the interviews for prevalent themes related to TAs’ views of the purposes of RP in mathematics and in teaching. I used a constant comparative method (Glaser & Strauss, 1967) to identify factors that may have influenced TAs’ teaching decisions, and I coded for the ways TAs emphasized characteristics of proof (Hanna, 2000; Hanna & Barbeau, 2008; A. J. Stylianides, 2007; de Villiers, 1999).

Key Findings

During my 42 classroom observations, 82 mathematical tasks were enacted that were coded as having the potential to engage students in RP-processes (i.e., generalize patterns, formulate conjectures, generate arguments, and/or evaluate others’ arguments). Preliminary analyses of how TAs set up these tasks indicated RP-opportunities were maintained (64.63%) or decreased (35.37%). Maintaining RP-opportunities means that when TAs introduced tasks they did not modify the tasks in such a way that changed PTEs’ opportunities to engage in RP-processes. TAs decreased RP-opportunities when introducing tasks by excluding RP-related questions or providing PTEs with conjectures that tasks asked PTEs to generate. Considering the enactment of these tasks, I also focused on RP-opportunities as PTEs worked (individually or in groups) and when the class discussed solutions to tasks. For the 82 tasks, during the enactment of a task, the RP-opportunities were maintained (31.71%), decreased (54.88%), or increased (13.41%). RP-opportunities increased during enactment primarily when TAs pressed PTEs for justifications when the written tasks did not. RP-opportunities decreased during enactment when, for example, TAs provided examples of justifications instead of eliciting justifications from PTEs.

Initial analyses of interview data suggest that a primary factor influencing TAs’ instructional decisions was TAs’ goals for PTEs as undergraduate students and as future teachers:  some participants (Ace and Laura) focused primarily on providing PTEs with mathematics content necessary to teach their future elementary students; other participants (Evan, Kelly, and Peter) endeavored to help PTEs experience RP processes in Geom to be able to engage their future elementary students in RP; and one participant (Isaac) articulated aspects of both views. I will also identify additional factors through my on-going analyses.

Discussion

The processes of proving are key habits of mind that should be fostered in K-12 and college mathematics classes. TAs assigned to teach PTEs, then, are positioned to significantly impact the quality of instruction for college students and PTEs’ future elementary students. Despite this positioning, research on TAs’ mathematics teaching and beliefs is virtually non-existent (Speer, Gutmann, & Murphy, 2005). This dissertation, therefore, contributes important first steps for understanding TAs’ mathematics teaching practices and beliefs. By examining each TA’s teaching practices and beliefs when teaching PTEs about RP, I reveal how proving was emphasized in these math-content classes, thereby indicating the ways in which particular teaching practices do or do not reinforce PTEs’ limited proof-related conceptions and experiences.

By considering TAs’ instructional moves restricting or increasing RP-opportunities for PTEs, I uncovered patterns and factors that can illuminate these instructional decisions and inform professional development for collegiate mathematics instructors. In particular, the differences in TAs’ goals likely contributed to if/how TAs engaged PTEs in proving processes in the classroom. Since these TAs were teaching undergraduates enrolled in a mathematics content course who were also potentially future elementary teachers, this dissertation indicates ways in which these mathematics instructors helped PTEs understand proof-related content (Ace and Laura), experience proving processes (Evan, Peter, and Kelly), or some combination of the two (Isaac). Investigating these different cases, therefore, suggests some factors that inhibited or helped these TAs focus their courses around mathematical processes. These factors could help inform professional development for TAs and other college mathematics instructors as well as potentially inform instructional goals and teaching practices for other mathematics courses.

Presentation-Related Notes

Since learning to teach is a lifelong process (Feiman-Nemser, 2001), this session addresses the “professional learning” research area by focusing on teaching practices and beliefs of TAs who are mathematics instructors, graduate students, and potential future faculty. The implications also include suggestions for professional development for TAs and other college mathematics instructors.

During the interactive paper session, I will briefly outline my study and focus on major findings and implications. I look forward to the opportunity to engage in focused discussions with participants around the ways in which understanding these cases could help mathematics departments make sense of what to do in other courses that contain some prospective elementary or secondary mathematics teachers.

References

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