Title: A Sociocultural Perspective on Beginning Teachers Enacting Reasoning and Proving Practices
The critical role of teachers in supporting student engagement with reasoning and proving has long been recognized (Nardi & Knuth, 2017; NCTM, 2014). While some studies examined how prospective secondary teachers (PSTs) develop dispositions and teaching practices that promote student engagement with reasoning and proving (e.g., Buchbinder & McCrone, 2020; Conner, 2007), very little is known about long-term development of proof-related practices of beginning teachers and what factors affect this development (Stylianides et al., 2017). During the supervised teaching experiences, interns often encounter tensions between balancing their commitments to the university and cooperating teacher, while also developing their own teaching styles (Bieda et al., 2015; Smagorinsky et al., 2004; Wang et al., 2008).
Our study examines how sociocultural contexts of the teacher preparation program and of the internship school, supported or inhibited proof-related teaching practices of beginning secondary mathematics teachers. In particular, this study aims to understand the observed gap between proof-related teaching practices of one such teacher, Olive, in two settings: as a PST in a capstone course Mathematical Reasoning and Proving for Secondary Teachers (Buchbinder & McCrone, 2020) and as an intern in a high-school classroom. We utilize activity theory (Leont’ev, 1979) and Engeström’s (1987) model of an activity system to examine how the various components of the system: teacher (subject), teaching (object), the tasks (tools), the curriculum and the expected teaching style (rules), the cooperating teacher (community) and their involvement during the teaching (division of labor) interact with each other and affect the opportunities provided to students to engage with reasoning and proving (outcome).
The analysis of four lessons from each setting, lesson plans, reflections and interviews, showed that as a PST, Olive engaged students with reasoning and proving through productive proof-related teaching practices and rich tasks that involved conjecturing, justifying, proving and evaluating arguments. In a sharp contrast, as an intern, Olive had to follow her school’s rigid curriculum and expectations, and to adhere to her cooperating teacher’s teaching style. As a result, in her lessons as an intern students received limited opportunities for reasoning and proving. Olive expressed dissatisfaction with this type of teaching and her desire to enact more proof-oriented practices. Our results show that the sociocultural components of the activity system (rules, community and division of labor), which were backgrounded in Olive’s teaching experience as a PST but prominent in her internship experience, influenced the outcome of engaging students with reasoning and proving. We discuss the importance of these sociocultural aspects as we examine how Olive navigated the tensions between the proof-related teaching practices she adopted in the capstone course and her teaching style during the internship. We highlight the importance of teacher educators considering the sociocultural aspects of teaching in supporting beginning teachers developing proof-related teaching practices. more »« less
Liu, J.; Buchbinder, O.(
, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)
Karunakaran, S. S.; Higgins, A.
(Ed.)
Preparing prospective secondary teachers (PSTs) to teach mathematics with a focus on
reasoning and proving is an important goal for teacher education programs. A capstone course, Mathematical Reasoning and Proving for Secondary Teachers, was designed to address this goal. One component of the course was a school-based experience in which the PSTs designed and taught four proof-oriented lessons in local schools, video recorded these lessons, and reflected on them. In this paper, we focus on one PST – Nancy, who took the course in Fall 2020 during the pandemic, when the school-based experience moved online. We analyzed how Nancy’s Mathematical Knowledge for Teaching Proof (MKT-P) evolved through her attempts to teach proof online and through repeated cycles of reflection.
This research paper describes a study of elementary teacher learning in an online graduate program in engineering education for in-service teachers. While the existing research on teachers in engineering focuses on their disciplinary understandings and beliefs (Hsu, Cardella, & Purzer, 2011; Martin, et al., 2015; Nadelson, et al., 2015; Van Haneghan, et al., 2015), there is increasing attention to teachers' pedagogy in engineering (Capobianco, Delisi, & Radloff, 2018). In our work, we study teachers' pedagogical sense-making and reflection, which, we argue, is critical for teaching engineering design.
This study takes place in [blinded] program, in which teachers take four graduate courses over fifteen months. The program was designed to help teachers not only learn engineering content, but also shift their thinking and practice to be more responsive to their students. Two courses focus on pedagogy, including what it means to learn engineering and instructional approaches to support this learning. These courses consist of four main elements, in which teachers:
1) Read data-rich engineering education articles to reflect on learning engineering; 2) Participate in online video clubs, looking at classroom videos of students’ engineering and commenting on what they notice; 3) Conduct interviews with learners about the mechanism of a pull-back car; and 4) Plan and teach engineering lessons, collecting and analyzing video from their classrooms.
In the context of this program, we ask: what stances do teachers take toward learning and teaching engineering design? What shifts do we observe in their stances?
We interviewed teachers at the start of the program and after each course. In addition to reflecting on their learning and teaching, teachers watched videos of students’ engineering and discussed what they saw as relevant for teaching engineering. We informally compared summaries from previous interviews to get a sense of changes in how participants talked about engineering, how they approached teaching engineering, and what they noticed in classroom videos. Through this process, we identified one teacher to focus on for this paper: Alma is a veteran 3rd-5th grade science teacher in a rural, racially-diverse public school in the southeastern region of the US. We then developed content logs of Alma's interviews and identified emergent themes. To refine these themes, we looked for confirming and disconfirming evidence in the interviews and in her coursework in the program. We coded each interview for these themes and developed analytic memos, highlighting where we saw variability and stability in her stances and comparing across interviews to describe shifts in Alma's reasoning. It was at this stage that we narrowed our focus to her stances toward the engineering design process (EDP).
In this paper, we describe and illustrate shifts we observed in Alma's reasoning, arguing that she exhibited dramatic shifts in her stances toward teaching and learning the EDP. At the start of the program, she was stable in treating the EDP as a series of linear steps that students and engineers progress through. After engaging and reflecting on her own engineering in the first course, she started to express a more fluid stance when talking more abstractly about the EDP but continued to take it up as a linear process in her classroom teaching. By the end of the program, Alma exhibited a growing stability across contexts in her stance toward the EDP as a fluid set of overlapping practices that students and engineers could engage in.
Buchbinder, O.; Brisard, S.; Butler, R.; McCrone, S.(
, Journal of technology and teacher education)
The benefits of using video in teacher education as a tool for reflection and for developing professional expertise have long been recognized. Recent introduction of 360 video technology holds promise to extend these benefits as it allows prospective teachers to reflect on their own performance by considering the classroom from multiple perspectives. This study examined nine prospective secondary teachers’ (PSTs) noticing and self-reflection on the 360 recordings of their own teaching. The PSTs, enrolled in a capstone course Mathematical Reasoning and Proving for Secondary Teachers, taught a proof-oriented lesson to small groups of students in local schools while capturing their teaching with 360 video cameras. We analyzed the PSTs’ written comments on their video and reflection reports to identify the categories of noticing afforded by the 360 technology as well as the instances of PSTs’ learning. The results point to the powerful potential of 360 videos for supporting PSTs’ self-reflection and professional growth.
Buchbinder, O.; McCrone, S.; Capozzoli, M.; Butler, R.(
, Proceedings of the 43rd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education)
Olanoff, D.; Johnson, K.; Spitzer, S.
(Ed.)
It has been suggested that integrating reasoning and proof in mathematics teaching requires a special type of teacher knowledge - Mathematical Knowledge for Teaching Proof (MKT-P). Yet, several important questions about the nature of MKT-P remain open, specifically, whether MKT-P is a type of knowledge specific to teachers, and whether MKT-P can be improved through intervention. We explored these questions by comparing performance on an MKT-P questionnaire of in-service secondary mathematics teachers, undergraduate STEM majors, and pre-service secondary mathematics teachers. The latter group completed the questionnaire twice- before and after participating in a capstone course, Mathematical Reasoning and Proving for Secondary Teachers. Our data suggest that MKT-P is indeed a special kind of knowledge specific to teachers and it can be improved through interventions.
Mathematics teacher education programs in the United States are charged with preparing prospective secondary teachers (PSTs) to teach reasoning and proving across grade levels and mathematical topics. Although most programs require a course on proof, PSTs often perceive it as disconnected from their future classroom practice. Our design research project developed a capstone courseMathematical Reasoning and Proving for Secondary Teachersand systematically studied its effect on PSTs’ content and pedagogical knowledge specific to proof. This paper focuses on one course module—Quantification and the Role of Examples in Proving,a topic which poses persistent difficulties to students and teachers alike. The analysis suggests that after the course, PSTs’ content and pedagogical knowledge of the role of examples in proving increased. We provide evidence from multiple data sources: pre-and post-questionnaires, PSTs’ responses to the in-class activities, their lesson plans, reflections on lesson enactment, and self-report. We discuss design principles that supported PSTs’ learning and their applicability beyond the study context.
Weingarden, M., and Buchbinder, O. A Sociocultural Perspective on Beginning Teachers Enacting Reasoning and Proving Practices. Retrieved from https://par.nsf.gov/biblio/10406078. Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education .
Weingarden, M., & Buchbinder, O. A Sociocultural Perspective on Beginning Teachers Enacting Reasoning and Proving Practices. Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education, (). Retrieved from https://par.nsf.gov/biblio/10406078.
Weingarden, M., and Buchbinder, O.
"A Sociocultural Perspective on Beginning Teachers Enacting Reasoning and Proving Practices". Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education (). Country unknown/Code not available. https://par.nsf.gov/biblio/10406078.
@article{osti_10406078,
place = {Country unknown/Code not available},
title = {A Sociocultural Perspective on Beginning Teachers Enacting Reasoning and Proving Practices},
url = {https://par.nsf.gov/biblio/10406078},
abstractNote = {The critical role of teachers in supporting student engagement with reasoning and proving has long been recognized (Nardi & Knuth, 2017; NCTM, 2014). While some studies examined how prospective secondary teachers (PSTs) develop dispositions and teaching practices that promote student engagement with reasoning and proving (e.g., Buchbinder & McCrone, 2020; Conner, 2007), very little is known about long-term development of proof-related practices of beginning teachers and what factors affect this development (Stylianides et al., 2017). During the supervised teaching experiences, interns often encounter tensions between balancing their commitments to the university and cooperating teacher, while also developing their own teaching styles (Bieda et al., 2015; Smagorinsky et al., 2004; Wang et al., 2008). Our study examines how sociocultural contexts of the teacher preparation program and of the internship school, supported or inhibited proof-related teaching practices of beginning secondary mathematics teachers. In particular, this study aims to understand the observed gap between proof-related teaching practices of one such teacher, Olive, in two settings: as a PST in a capstone course Mathematical Reasoning and Proving for Secondary Teachers (Buchbinder & McCrone, 2020) and as an intern in a high-school classroom. We utilize activity theory (Leont’ev, 1979) and Engeström’s (1987) model of an activity system to examine how the various components of the system: teacher (subject), teaching (object), the tasks (tools), the curriculum and the expected teaching style (rules), the cooperating teacher (community) and their involvement during the teaching (division of labor) interact with each other and affect the opportunities provided to students to engage with reasoning and proving (outcome). The analysis of four lessons from each setting, lesson plans, reflections and interviews, showed that as a PST, Olive engaged students with reasoning and proving through productive proof-related teaching practices and rich tasks that involved conjecturing, justifying, proving and evaluating arguments. In a sharp contrast, as an intern, Olive had to follow her school’s rigid curriculum and expectations, and to adhere to her cooperating teacher’s teaching style. As a result, in her lessons as an intern students received limited opportunities for reasoning and proving. Olive expressed dissatisfaction with this type of teaching and her desire to enact more proof-oriented practices. Our results show that the sociocultural components of the activity system (rules, community and division of labor), which were backgrounded in Olive’s teaching experience as a PST but prominent in her internship experience, influenced the outcome of engaging students with reasoning and proving. We discuss the importance of these sociocultural aspects as we examine how Olive navigated the tensions between the proof-related teaching practices she adopted in the capstone course and her teaching style during the internship. We highlight the importance of teacher educators considering the sociocultural aspects of teaching in supporting beginning teachers developing proof-related teaching practices.},
journal = {Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education},
author = {Weingarden, M. and Buchbinder, O.},
editor = {Karunakaran, S. S. and Higgins, A.}
}
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