Title: MATHEMATICAL KNOWLEDGE FOR TEACHING PROOF: COMPARING SECONDARY TEACHERS, PRE-SERVICE TEACHERS AND UNDERGRADUATE STEM MAJORS
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. more »« less
Butler, R.; Buchbinder, O.; McCrone, S.(
, Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education)
Karunakaran, S.; Higgins, A.
(Ed.)
Mathematical Knowledge for Teaching Proof (MKT-P) has been recognized as an important component of fostering student engagement with mathematical reasoning and proof. This study is one component of a larger study aimed at exploring the nature of MKT-P. The present study examines qualitative differences in feedback given by STEM majors, in-service and pre-service secondary mathematics teachers on hypothetical students’ arguments. The results explicate key distinctions in the feedback provided by these groups, indicating that this is a learnable skill. Feedback is cast as a component of MKT-P, making the results of this study significant empirical support for the construct of MKT-P as a type of knowledge that is unique to teachers.
Butler, R.; Buchbinder, O.; McCrone, S.(
, Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education)
Karunakaran, S.; Higgins, A.
(Ed.)
Mathematical Knowledge for Teaching Proof (MKT-P) has been recognized as an important component of fostering student engagement with mathematical reasoning and proof. This study is one component of a larger study aimed at exploring the nature of MKT-P. The present study examines qualitative differences in feedback given by STEM majors, in-service and pre-service secondary mathematics teachers on hypothetical students’ arguments. The results explicate key distinctions in the feedback provided by these groups, indicating that this is a learnable skill. Feedback is cast as a component of MKT-P, making the results of this study significant empirical support for the construct of MKT-P as a type of knowledge that is unique to teachers.
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.
Buchbinder, O.; McCrone, S.(
, The Journal of mathematical behavior)
Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.
Weingarden, M.; Buchbinder, O.(
, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)
Karunakaran, S. S.; Higgins, A.
(Ed.)
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.
Buchbinder, O., McCrone, S., Capozzoli, M., and Butler, R. MATHEMATICAL KNOWLEDGE FOR TEACHING PROOF: COMPARING SECONDARY TEACHERS, PRE-SERVICE TEACHERS AND UNDERGRADUATE STEM MAJORS. Retrieved from https://par.nsf.gov/biblio/10356259. Proceedings of the 43rd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education .
Buchbinder, O., McCrone, S., Capozzoli, M., & Butler, R. MATHEMATICAL KNOWLEDGE FOR TEACHING PROOF: COMPARING SECONDARY TEACHERS, PRE-SERVICE TEACHERS AND UNDERGRADUATE STEM MAJORS. Proceedings of the 43rd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (). Retrieved from https://par.nsf.gov/biblio/10356259.
Buchbinder, O., McCrone, S., Capozzoli, M., and Butler, R.
"MATHEMATICAL KNOWLEDGE FOR TEACHING PROOF: COMPARING SECONDARY TEACHERS, PRE-SERVICE TEACHERS AND UNDERGRADUATE STEM MAJORS". Proceedings of the 43rd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (). Country unknown/Code not available. https://par.nsf.gov/biblio/10356259.
@article{osti_10356259,
place = {Country unknown/Code not available},
title = {MATHEMATICAL KNOWLEDGE FOR TEACHING PROOF: COMPARING SECONDARY TEACHERS, PRE-SERVICE TEACHERS AND UNDERGRADUATE STEM MAJORS},
url = {https://par.nsf.gov/biblio/10356259},
abstractNote = {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.},
journal = {Proceedings of the 43rd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education},
author = {Buchbinder, O. and McCrone, S. and Capozzoli, M. and Butler, R.},
editor = {Olanoff, D. and Johnson, K. and Spitzer, S.}
}
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