Conceptualizing Important Facets of Teacher Responses to Student Mathematical Thinking

Van Zoest, L. R., Peterson, B. E., Rougée, A. O. T., Stockero, S. L., Leatham, K. R., & Freeburn, B. (in press). Conceptualizing important facets of teacher responses to student mathematical thinking. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2021.1895341

We argue that progress in the area of research on mathematics teacher responses to student thinking could be enhanced were the field to attend more explicitly to important facets of those responses, as well as to related units of analysis. We describe the Teacher Response Coding Scheme (TRC) to illustrate how such attention might play out, and then apply the TRC to an excerpt of classroom mathematics discourse to demonstrate the affordances of this approach. We conclude by making several further observations about the potential versatility and power in articulating units of analysis and developing and applying tools that attend to these facets when conducting research on teacher responses.

 

Articulating the Student Mathematics in Student Contributions

Van Zoest, L. R., Stockero, S. L., Leatham, K. R., Peterson, B. E., & Ruk, J. M. (2020). Articulating the student mathematics in student contributions. In A. I. Sacrista´n, J. C. Corte´s-Zavala, & P. M. Ruiz-Arias (Eds.), Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mexico (pp. 2105–2109). Cinvestav / AMIUTEM / PME-NA. https:/doi.org/10.51272/pmena.42.2020

We draw on our experiences researching teachers' use of student thinking to theoretically unpack the work of attending to student contributions in order to articulate the student mathematics (SM) of those contributions. We propose four articulation-related categories of student contributions that occur in mathematics classrooms and require different teacher actions:(a) Stand Alone, which requires no inference to determine the SM; (b) Inference-Needed, which requires inferring from the context to determine the SM; (c) Clarification-Needed, which requires student clarification to determine the SM; and (d) Non-Mathematical, which has no SM. Experience articulating the SM of student contributions has the potential to increase teachers' abilities to notice and productively use student mathematical thinking during instruction.

 

Teachers' Orientations Toward Using Student Mathematical Thinking As a Resource During Whole-Class Discussion

Stockero, S. L., Leatham, K. R., Ochieng, M. A., Van Zoest, L. R., & Peterson, B. E. (2020). Teachers' orientations toward using student mathematical thinking as a resource during whole-class discussion. Journal of Mathematics Teacher Education, 23(3), 237-267. https://doi.org/10.1007/s10857-018-09421-0

Using student mathematical thinking during instruction is valued by the mathematics education community, yet practices surrounding such use remain difficult for teachers to enact well, particularly in the moment during whole-class instruction. Teachers' orientations—their beliefs, values, and preferences—influence their actions, so one important aspect of understanding teachers' use of student thinking as a resource is understanding their related orientations. To that end, the purpose of this study is to characterize teachers' orientations toward using student mathematical thinking as a resource during whole-class instruction. We analyzed a collection of 173 thinking-as-a-resource orientations inferred from scenario-based interviews conducted with 13 teachers. The potential of each orientation to support the development of the practice of productively using student mathematical thinking was classified by considering each orientation's relationship to three frameworks related to recognizing and leveraging high-potential instances of student mathematical thinking. After discussing orientations with different levels of potential, we consider the cases of two teachers to illustrate how a particular collection of thinking-as-a-resource orientations could support or hinder a teacher's development of the practice of building on student thinking. The work contributes to the field's understanding of why particular orientations might have more or less potential to support teachers' development of particular teaching practices. It could also be used as a model for analyzing different collections of orientations and could support mathematics teacher educators by allowing them to better tailor their work to meet teachers' specific needs.

 

Teachers' Responses to Instances of Student Mathematical Thinking with Varied Potential to Support Student Learning

Stockero, S. L., Van Zoest, L. R., Freeburn, B., Peterson, B. E., & Leatham, K. R. (2020). Teachers' responses to instances of student mathematical thinking with varied potential to support student learning. Mathematics Education Research Journal, 1-23. https://doi.org/10.1007/s13394-020-00334-x

Teacher responses to student mathematical thinking (SMT) matter because the way in which teachers respond affects student learning. Although studies have provided important insights into the nature of teacher responses, little is known about the extent to which these responses take into account the potential of the instance of SMT to support learning. This study investigated teachers' responses to a common set of instances of SMT with varied potential to support students' mathematical learning, as well as the productivity of such responses. To examine variations in responses in relation to the mathematical potential of the SMT to which they are responding, we coded teacher responses to instances of SMT in a scenario-based interview. We did so using a scheme that analyzes who interacts with the thinking (Actor), what they are given the opportunity to do in those interactions (Action), and how the teacher response relates to the actions and ideas in the contributed SMT (Recognition). The study found that teachers tended to direct responses to the student who had shared the thinking, use a small subset of actions, and explicitly incorporate students' actions and ideas. To assess the productivity of teacher responses, we first theorized the alignment of different aspects of teacher responses with our vision of responsive teaching. We then used the data to analyze the extent to which specific aspects of teacher responses were more or less productive in particular circumstances. We discuss these circumstances and the implications of the findings for teachers, professional developers, and researchers.

 

Clarifiable Ambiguity in Classroom Mathematics Discourse, Investigations in Mathematics Learning

Peterson, B. E., Leatham, K. R., Merrill, L. M., Van Zoest, L. R., & Stockero, S. L. (2020). Clarifiable Ambiguity in Classroom Mathematics Discourse, Investigations in Mathematics Learning. Investigations in Mathematics Learning, 12(1), 28-37. https://doi.org/10.1080/19477503.2019.1619148

Ambiguity is a natural part of communication in a mathematics classroom. In this paper, a particular subset of ambiguity is characterized as clarifiable. Clarifiable ambiguity in classroom mathematics discourse is common, frequently goes unaddressed, and unnecessarily hinders in-the-moment communication because it likely could be made more clear in a relatively straightforward way if it were attended to. We argue for deliberate attention to clarifiable ambiguity as a critical aspect of attending to meaning and as a necessary precursor to productive use of student mathematical thinking. We illustrate clarifiable ambiguity that occurs in mathematics classrooms and consider ramifications of not addressing it. We conclude the paper with a discussion about addressing clarifiable ambiguity through seeking focused clarification.

 

The Complexity of Interpreting Student Thinking and Inferring its Potential to Foster Learning

Ochieng, M. A., Ruk, J. M., Leatham, K. R., Peterson, B. E., Stockero, S. L., & Van Zoest, L. R. (2019). The Complexity of Interpreting Student Thinking and Inferring its Potential to Foster Learning. In Graven, M., Venkat, H., Eissen, A. A., & Vale, P (Eds.), Proceedings of the 43rd Annual Conference of the International Group for the Psychology of Mathematics Education (pp. 161-168). University of Pretoria: Pretoria, South Africa.

Research has shown that listening to and interpreting student thinking is challenging, yet critical for effective incorporation of student mathematical thinking (SMT) into instruction. We examine an exemplary teacher's interpretations of SMT, his inference of the potential of the SMT to foster learning, and the rationale for his responses to that thinking. Our findings reveal some reasons why teachers may fail to successfully act on SMT that emerges during whole class discussion. This study confirms previous research findings, that in order to incorporate SMT into instruction in a way that fosters learning, teachers must correctly interpret that SMT. The study also shows that even good teachers may need support in developing skills that will enable them accurately interpret SMT and its potential to foster learning.

 

Teachers' Initial Responses to High-Leverage Instances of Student Mathematical Thinking

Stockero, S. L., Peterson, B. E., Ochieng, M. A., Ruk, J. R., Van Zoest, L. R., & Leatham, K. R. (2019). Teachers' Initial Responses to High Leverage Instances of Student Mathematical Thinking. In Graven, M., Venkat, H., Eissen, A. A., & Vale, P (Eds.), Proceedings of the 43rd Annual Conference of the International Group for the Psychology of Mathematics Education (pp. 335-342). University of Pretoria: Pretoria, South Africa.

We investigate teachers' initial in-the-moment responses to instances of high-potential student mathematical thinking (SMT) during whole class discussion to understand what it means to productively incorporate SMT into instruction. Teachers' initial responses were coded using the Teacher Response Coding scheme, which disentangles the teacher action, who the response is directed to, and the degree to which the SMT is honored. We found that teachers incorporated students' actions and ideas in their response, but tended to address the SMT themselves and did not fully take advantage of the SMT. We consider the productivity of teachers' initial responses in relation to principles of productive use of SMT and compare the results to those of a previous study of teachers' hypothetical initial responses to SMT in an interview setting.

 

Teachers' Responses to Instances of Student Mathematical Thinking with Varied Potential to Support Student Learning

Stockero, S. L., Freeburn, B., Van Zoest, L. R., Peterson, B. E., & Leatham, K. R. (2018). Teachers' responses to instances of student mathematical thinking with varied potential to support student learning. In T. E. Hodges, G. J. Roy, & A. M. Tyminski (Eds.), Proceedings of the 40th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1076-1083). Greenville, SC: University of South Carolina & Clemson University.

We investigated teachers' responses to a common set of varied-potential instances of student mathematical thinking to better understand how a teacher can shape meaningful mathematical discourse. Teacher responses were coded using a scheme that both disentangles and coordinates the teacher move, who it is directed to, and the degree to which student thinking is honored. Teachers tended to direct responses to the same student, use a limited number of moves, and explicitly incorporate students' thinking. We consider the productivity of teacher responses in relation to frameworks related to the productive use of student mathematical thinking.

 

A Characterization of Student Mathematical Thinking That Emerges During Whole-Class Instruction: An Exploratory Study

Van Zoest, L. R., Leatham, K. R., Arslan, O., Ochieng, M. A., Ruk, J. M., Peterson, B. E., & Stockero, S. L. (2018). A characterization of student mathematical thinking that emerges during whole-class instruction: An exploratory study. In T. E. Hodges, G. J. Roy, & A. M. Tyminski (Eds.), Proceedings of the 40th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1126-1129). Greenville, SC: University of South Carolina & Clemson University.

This exploratory study investigated 164 instances of student mathematical thinking that emerged during whole-class instruction in a high-school geometry course. The MOST Analytic Framework provided a way to categorize these instances according to their Building Potential—that is, the potential for learning to occur if the student thinking of the instance were made the object of consideration by the class. We discuss variations in Building Potential found in these instances by examining the subcategorizations that emerged from our additional analyses. The variations in the building potential of student thinking revealed in the study highlight the complexity of teaching, and the need to support teachers in identifying and appropriately responding to instances with different levels of Building Potential.

 

Teachers' Responses To a Common Set of High Potential Instances of Student Mathematical Thinking

Stockero, S. L., Van Zoest, L. R., Peterson, B. E., Leatham, K. R., & Rougée, A. O. T. (2017). Teachers' responses to a common set of high potential instances of student mathematical thinking. In E. Galindo, & J. Newton (Eds.), Proceedings of the 39th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1178–1185). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators.

This study investigates teacher responses to a common set of high potential instances of student mathematical thinking to better understand the role of the teacher in shaping meaningful mathematical discourse in their classrooms. Teacher responses were coded using a scheme that disentangles the teacher move from other aspects of the teacher response, including who the response is directed to and the degree to which the student thinking is honored. Teachers tended to direct their response to the student who had shared their thinking and to explicitly incorporate ideas core to the student thinking in their response. We consider the nature of these responses in relation to principles of productive use of student mathematical thinking.

 

Beyond the "Move": A Scheme for Coding Teachers' Responses To Student Mathematical Thinking

Peterson, B. E., Van Zoest, L. R., Rougée, A. O. T., Freeburn, B., Stockero, S. L., & Leatham, K. R. (2017). Beyond the "move": A scheme for coding teachers' responses to student mathematical thinking. In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy. (Eds.), Proceedings of the 41st annual meeting of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 17–24). Singapore: International Group for the Psychology of Mathematics Education.

To contribute to the field's understanding of the teachers' role in using student thinking to shape classroom mathematical discourse, we developed the Teacher Response Coding Scheme (TRC). The TRC provides a means to analyze teachers' in-the-moment responses to student thinking during instruction. The TRC differs from existing schemes in that it disentangles the teacher move from the actor (the person publically asked to consider the student thinking), the recognition (the extent to which the student recognizes their idea in the teacher move), and the mathematics (the alignment of the mathematics in the teacher move to the mathematics in the student thinking). This disentanglement makes the TRC less value-laden and more useful across a broad range of settings.

 

Conceptualizing the Teaching Practice of Building On Student Mathematical Thinking

Van Zoest, L. R., Peterson, B. E., Leatham, K. R., & Stockero, S. L. (2016). Conceptualizing the teaching practice of building on student mathematical thinking. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Proceedings of the 38th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1281–1288). Tucson, AZ: University of Arizona.

An important aspect of effective teaching is taking advantage of in-the-moment expressions of student thinking that, by becoming the object of class discussion, can help students better understand important mathematical ideas. We call these high-potential instances of student thinking MOSTs and the productive use of them building. The purpose of this paper is to conceptualize the teaching practice of building on MOSTs as a first step toward developing a common language for and an understanding of productive use of high-potential instances of student thinking. We situate this work in the existing literature, introduce core principles that underlie our conception of building, and present a prototype of the teaching practice of building on MOSTs that includes four sub-practices.

We conclude by discussing the need for future research and our research agenda for studying the building prototype.

 

Uncovering Teachers' Goals, Orientations, and Resources Related to the Practice of Using Student Thinking

Stockero, S. L., Van Zoest, L. R., Rougee, A., Fraser, E. H., Leatham, K. R. & Peterson, B. E. (2015). Uncovering teachers' goals, orientations, and resources related to the practice of using student thinking. Proceedings of the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. East Lansing, MI: Michigan State University, MI.

Despite years of efforts to promote using student thinking during instruction, the practice of productively using student thinking is still difficult for teachers to enact well. Improving teachers' use of this practice requires an understanding of why teachers currently respond to student thinking as they do; that is, an understanding of the goals, orientations and resources (Schoenfeld, 2011) that underlie their enactment of this practice. We describe a scenario-based interview tool developed to prompt teachers to discuss their decisions and rationales related to using student thinking. We examine cases of three individual teachers to illustrate how the tool allows us (1) to infer individual teachers' goals, orientations and resources and (2) to differentiate among teachers' uses of student thinking. Understanding reasons behind teachers' use of student thinking will help us to match professional development to teachers' needs.

 

Teachers' Perceptions of Productive Use of Student Mathematical Thinking

Leatham, K. R., Van Zoest, L. R., Stockero, S. L., & Peterson, B. E. (2014). Teachers' perceptions of productive use of student mathematical thinking. In P. Liljedahl, S. Oesterle, C. Nicol, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 4, pp. 73–80). Vancouver, Canada: PME.

We argue that the teaching practice of productively using student mathematical thinking [PUMT] needs to be better conceptualized for the construct to gain greater traction in the classroom and in research. We report the results of a study wherein we explored teachers' perceptions of PUMT. We interviewed mathematics teachers and analyzed these interviews using and refining initial conjectures about the process teachers might go through in learning PUMT. We found that teachers' perceptions of PUMT ranged from valuing student participation, to valuing student mathematical thinking, to using that thinking in a variety of ways related to eliciting, interpreting and building on that thinking.