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Conceptualizing important facets of teacher responses to student mathematical thinking
International Journal of Mathematical Education In Science & Technology
Abstract
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.
... These decisions are crucial to promoting a collaborative student-centered learning environment and depend on the teacher's interpretation of students' thinking (Jacobs et al., 2010). Their interpretation of students' mathematical contributions and how they respond to them can shape and direct students' mathematical thinking significantly by enhancing cognitive opportunities for sense-making (Boaler & Brodie, 2004;Van Zoest et al., 2022). Given the importance of this decision, teachers must become adept at developing effective questioning strategies that help to engage students, deepen understanding, and encourage productive discussion in the classroom. ...
... Encouraging students to participate in whole-class discussions is consistent with the NCTM's (2014) goal of having teachers support their students' mathematical thinking. For example, Van Zoest et al. (2022) observed in their study that asking a student to evaluate another student's contribution creates a different learning opportunity from having a teacher evaluate it. It gives students the opportunity to share their thinking as well as help other students to have a better understanding of the contribution. ...
... Researchers (Stockero et al., 2017a;Van Zoest et al., 2022) have uncovered subtle but significant variations in how teachers act as a facilitator and engage students with student contributions. ...
In the student-centered classroom, a teacher’s interpretation and response to student mathematical contributions plays an important role to shape and direct students’ opportunities for sense-making. This research used a scenario-based survey questionnaire to examine what types of questions middle and high school mathematics teachers indicate they would ask to engage students in making sense of a high-leverage student mathematical contribution and their reasoning about why particular questions are or are not productive. From the results, it could be concluded that teachers asked more productive questions after seeing a set of possible questions. Their beliefs about the productivity of the questions related to a variety of factors, including the specificity of the question, student participation, student ability and whether incorrect solutions should be discussed. The results could inform future work with teachers to productively use student thinking in their teaching.
... The coding was both inductive and deductive. We first created a code list by primarily drawing on the comprehensive list of moves mathematics teachers use to elicit and work with students' mathematics thinking drawn up by Van Zoest et al. (2021). We chose this list because it is comprehensive and could be matched with the accountable talk moves we supplied to the PSTs. ...
... • Re-stating question (Tytler & Aranda, 2015) • Collect (Van Zoest et al., 2021) Invite ...
... • Canvassing opinion (Tytler & Aranda, 2015) • Prompts others to elaborate • Allow (Van Zoest et al., 2021) • Asking authentic questions which invite students to take a position (Bansal, 2018) • Handling agreements/ disagreements (Bansal, 2018) Restate ...
As more and more science teacher educators are subscribing to a practice‐based teacher education curriculum, it is becoming increasingly necessary to identify and articulate smaller grain‐sized teaching practices nested within a core practice in important instructional contexts in order to facilitate preservice science teachers' (PSTs') learning of core practices. This paper illustrates how to achieve this goal using data from a qualitative study that aimed to characterize how PSTs enact the core practice of eliciting and working with student thinking in the three key stages of collaborative group work in rehearsals: whole‐class warm‐up, small group work, and whole‐class share‐out. Based on existing teaching practices synthesized from prior studies, as well as empirical data from the study, we identify the intermediate‐level practices (teaching moves) and technique‐level practices (talk moves) enacted by PSTs when orchestrating group work as well as the PSTs' strengths and missed opportunities in enacting these different grain‐sized practices. Findings reveal that, although the PSTs were able to enact a variety of talk moves when orchestrating group work, they missed opportunities to engage in intermediate‐level practices. For example, there were only a few instances in which the PSTs could explicitly work with the student thinking they noticed while circulating among the groups during whole‐class share‐out. We also identify three ways in which the PSTs used this intermediate‐level practice, reflective of their differing orientations toward using student thinking. We propose an integrative framework that decomposes the focal core practice into a nested set of practices of varying grain sizes, including medium grain‐sized teaching moves and small grain‐sized talk moves, during and across the key stages of group work. We discuss the implications of our findings for science teacher preparation and methodological approaches for capturing PSTs' enactment of the focal core practice, as well as the possible contributions of the framework.
... Even more critical to the understanding learners develop in a mathematics classroom is the way teachers respond to the thinking students' exhibit during enactment [18,19]. Van Zoest et al. [20] and Van Zoest et al. [21] identified that different teachers can respond differently to the same student thinking making engagement different and productive in different ways. According to Van Zoest et al. [20], the difference they found could be as a result of the fact that some teachers engage with student response, providing deeper insights to the learners and developing it further while another teacher might mobilize other students in the class to respond to the student response under consideration. ...
... Van Zoest et al. [20] and Van Zoest et al. [21] identified that different teachers can respond differently to the same student thinking making engagement different and productive in different ways. According to Van Zoest et al. [20], the difference they found could be as a result of the fact that some teachers engage with student response, providing deeper insights to the learners and developing it further while another teacher might mobilize other students in the class to respond to the student response under consideration. ...
This study investigated an experienced teacher’s moves in preparing high school students for an end of year certificate examination. Six consecutive lessons taught by this teacher were observed and videotaped in the second term of the 2020/2021 academic year. These lessons were transcribed and coded for teacher moves as well as teacher knowledge deployed in any of the moves. First, the two researchers coded teacher moves independently and agreed on them. Second, interviews were used to obtain justification for teacher moves used. Third, Ball, Thames and Phelps’s (2008) categorizations of mathematical knowledge for teaching was used to determine which teacher knowledge motivated the move. Results revealed that this teacher, Ngwa, a pseudonym, deployed the following moves: reading through all the topics to identify key concepts and determine connections between and among them, connecting ideas from one topic to another, returning to previous chapters or units for review of mathematical ideas, reinforce understanding by modifying problems, adoption/adaptations and extension of problems to generate the use of more ideas, withdrawal of the teacher and creation of space for interaction, announcing or advertising what will be learned in the next lesson, and learners designing their own questions for the class to engage with. The findings also revealed that understanding the structure of an examination might be a form of specialized content knowledge as it drives teaching. In addition, teaching and learning was found to be forward and backward to review concepts and reinforce learning in learners. Furthermore, teachers must intentionally create space for student-student interaction which provide opportunities for learners to make sense of what is being taught. The findings of this study might be beneficial to teacher educators and professional development experts to focus training on productive teaching moves. Classroom teachers can also use the outcome of this study for the improvement of their practice of teaching.
This study explores how novice mathematics teachers (NMTs) respond to Mathematically Significant Pedagogical Opportunities to Build on Student Thinking (MOSTs)—critical classroom moments with high potential to enhance mathematical understanding—in middle school classrooms. Addressing the challenges faced by early-career teachers and understanding their instructional decisions and interactions with student mathematical thinking are crucial. Data were collected from five NMTs with less than five years of teaching experience through classroom observations and video recordings. The data were analyzed using the Teacher Response Coding Scheme (TRC), a framework that categorizes teacher responses according to three facets: who engages with the student thinking (actor), what instructional action is taken (move), and how the response connects to the student’s mathematical idea (recognition). The findings reveal that teachers predominantly positioned themselves as the main actors during interactions with student mathematical thinking, limiting student involvement and frequently relying on brief explanations such as literal, validating, or correct moves. However, the findings also indicate that more than half of the teacher responses incorporated student actions and ideas explicitly, showing an effort to honor and engage with student thinking.
The purpose of this study was to investigate whether a TMSSR (Teacher Moves for Supporting Student Reasoning)-based intervention could enhance prospective middle school teachers' abilities for supporting student reasoning on quadratic functions. Participants (N = 17) engaged with quadratic functions, practiced the TMSSR framework through scriptwriting assignments, received feedback and revised their scripts, compared and contrasted scenarios, and discussed ways to achieve high-potential moves during the intervention. Data sources included participants' pre/post scriptwriting tasks along with take-home assignments across the intervention. Results indicated the TMSSR-based intervention was effective in improving prospective teachers' repertoire of high-leverage moves for eliciting, responding to, facilitating, and extending student mathematical reasoning. Specific areas of improvement included eliciting understanding, promoting error correction, providing guidance, encouraging reflection, and pressing for generalizations. This study suggests a framework that mathematics teacher educators can use to improve prospective teachers' abilities for supporting student reasoning.
Supporting students' mathematical reasoning is an important goal of mathematics instruction, but can be challenging for many teachers .We report the results of a study aimed at better understanding and identifying the ways in which teachers support student reasoning when provided with conceptually rich tasks. This study resulted in the Teacher Moves for Supporting Student Reasoning (TMSSR) framework, which organizes moves vis-à-vis their function and their potential for fostering student thinking. We describe the TMSSR framework, illustrate its affordances for studying teacher practices, and highlight its utility for teachers, teacher educators, and researchers.
Responsive teaching occurs when teachers take up and respond to their students’ ideas during instruction (J. L. Pierson, 2008). Although responsive teaching is gaining recognition as an effective strategy for encouraging student learning, few methods of analysis are capable of characterizing the different ways in which teachers take up their students’ ideas in the moment. This article presents and exemplifies a new methodological construct, the redirection, which provides researchers with a means of detecting nuanced differences in how teachers respond to their students’ thinking. The redirection construct emerged via systematic discourse analysis of 1 science teacher’s classroom discussions during 3 implementations of an inquiry-based module on the water cycle. Redirections are defined as instances when a teacher invites students to shift or redirect their attention to a new locus. Such shifts reflect different types of teacher responsiveness and, as such, can be used to capture the different ways in which teachers take up their students’ ideas. This article presents a comprehensive coding scheme for the redirection, in addition to segments of classroom discourse to exemplify each redirection coding category. A comparison of 3 5th-grade teachers using the construct provides an illustrative example of the type of analysis such a coding scheme affords learning sciences researchers.
Teachers who attempt to use inquiry-based, student-centered instructional tasks face challenges that go beyond identifying well-designed tasks and setting them up appropriately in the classroom. Because solution paths are usually not specified for these kinds of tasks, students tend to approach them in unique and sometimes unanticipated ways. Teachers must not only strive to understand how students are making sense of the task but also begin to align students' disparate ideas and approaches with canonical understandings about the nature of mathematics. Research suggests that this is difficult for most teachers (Ball, 1993, 2001; Leinhardt & Steele, 2005; Schoenfeld, 1998; Sherin, 2002). In this article, we present a pedagogical model that specifies five key practices teachers can learn to use student responses to such tasks more effectively in discussions: anticipating, monitoring, selecting, sequencing, and making connections between student responses. We first define each practice, showing how a typical discussion based on a cognitively challenging task could be improved through their use. We then explain how the five practices embody current theory about how to support students' productive disciplinary engagement. Finally, we close by discussing how these practices can make discussion-based pedagogy manageable for more teachers.
To understand how teacher education might build the knowledge it needs, the editors of this issue investigate how organizations outside of education create and maintain "self-improving" systems that enable them to learn how to get better at what they do. To learn about knowledge building in teacher education, the research reported here also looks outside-outside of mathematics teacher education and outside the structure of professional preparation for teaching as it occurs in universities and colleges. This article explores features of a system of collective knowledge building in and for teaching and teacher education that are in place in an international school-based program for teachers of Italian as a foreign language. We use our study of this unusual teacher education program to investigate how a carefully chosen set of instructional activities, built out of the essential social and intellectual routines of ambitious teaching can make it possible for novices to teach ambitiously and for teacher educators to buildknowledge.
This paper examines teachers’ classroom talk as teachers respond to, interact with and take forward learners’ contributions. Teachers’ responses to learner contributions provide a useful lens in understanding teachers’ hybrid practices as they take up aspects of reform practice. A set of codes for teacher moves is developed to describe teacher responses, which build on previous work in describing classroom talk and provide the beginnings of an elaborated language of description for changes in teaching practice. The codes illuminate the similarities and differences across four secondary school mathematics teachers as they shift their practices to take account of learners’ thinking.
Informed by theory and research in inquiry-based mathematics, this study examined how classroom practices create a press for conceptual learning. Using videotapes of a lesson on the addition of fractions in 4 primarily low-income classrooms from 3 schools, we analyzed conversations that create a high or lower press for conceptual thinking. We use examples of interactions from these fourth- and fifth-grade lessons to propose that a high press for conceptual thinking is characterized by the following sociomathematical norms: (a) an explanation consists of a mathematical argument, not simply a procedural description; (b) mathematical thinking involves understanding relations among multiple strategies; (c) errors provide opportunities to reconceptualize a problem, explore contradictions in solutions, and pursue alternative strategies; and (d) collaborative work involves individual accountability and reaching consensus through mathematical argumentation.
Pressing for more than curricular change, the current mathematics reforms place an unprecedented emphasis on students' mathematical communication and discourse. These visions of mathematics classrooms imply substantial changes for the teacher's role. Unlike much of their own past experience and practice, teachers are urged to open the classroom to student talk. In small and large groups, students are to present their ideas and solutions, explain their reasoning, and question one another. Accustomed to being the source of knowledge, and doing most of the talking, teachers now must refashion their role in ways that responsibly and effectively shift more authority and autonomy to students. The challenge to the teacher has not gone unrecognized by reformers, 1 and yet, we argue, that the teacher's role in this new vision of classroom learning has been underconsidered. Some people seem to believe that teachers should never tell students anything or that students will learn appropriate mathematics on their own if they are engaged in worthwhile tasks with useful tools and materials. We argue that such a view underestimates the role of the teacher. Few opportunities have existed for careful examination of the complicated decisions and moves that teachers must make in classrooms, and even when such opportunities arise—for example in discussions of videotaped classroom sessions—the linguistic tools available for parsing and analyzing practice often seem too blunt as discussions focus on what the teacher should do, whether they should "tell," "show," "go over," or "let students figure it out on their own." 2
If teacher education is to prepare novices to engage successfully in the complex work of ambitious instruction, it must somehow
prepare them to teach within the continuity of the challenging moment-by-moment interactions with students and content over
time. With Leinhardt, we would argue that teaching novices to do routines that structure teacher–student–content relationships
over time to accomplish ambitious goals could both maintain and reduce the complexity of what they need to learn to do to
carry out this work successfully. These routines would embody the regular “participation structures” that specify what teachers
and students do with one another and with the mathematical content. But teaching routines are not practiced by ambitious teachers
in a vacuum and they cannot be learned by novices in a vacuum. In Lampert’s classroom, the use of exchange routines occurred
inside of instructional activities with particular mathematical learning goals like successive approximation of the quotient
in a long division problem, charting and graphing functions, and drawing arrays to represent multi-digit multiplications.
To imagine how instructional activities using exchange routines could be designed as tools for mathematics teacher education,
we have drawn on two models from outside of mathematics education. One is a teacher education program for language teachers
in Rome and the other is a program that prepares elementary school teachers at the University of Chicago. Both programs use
instructional activities built around routines as the focus of a practice-oriented approach to teacher preparation.
KeywordsExchange routines–ambitious teaching–instructional activities–teacher preparation–deliberate practice
Try these tools to engage your students with one another's thinking and to reflect on mathematically responsive classroom interactions.
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.
Teachers play a critical role in supporting students’ mathematical engagement. There is evidence that meaningful student engagement occurs more often in student-centered classrooms, in which the teacher and the students mutually share mathematical authority. However, teacher-centered instruction continues to dominate classroom discourse, and teachers struggle to effectively support student inquiry. This paper presents a framework of teacher moves specific to inquiry-oriented instruction, the Teacher Moves for Supporting Student Reasoning (TMSSR) framework. Based on the analysis of four instructors’ implementations of a middle grades (ages 12–14) research-based unit on ratio and linear functions, the TMSSR framework organizes pedagogical moves into four categories, eliciting, responding, facilitating, and extending, and then places individual moves within each category on a continuum according to their potential for supporting student reasoning. In this manner, the TMSSR framework characterizes how multiple teacher moves can work together to foster an inquiry-oriented environment. We detail the framework with data examples and then present a classroom episode exemplifying the framework’s operation.
This study investigated attributes of 278 instances of student mathematical thinking during whole-class interactions that were identified as having high potential, if made the object of discussion, to foster learners’ understanding of important mathematical ideas. Attributes included the form of the thinking (e.g., question vs. declarative statement), whether the thinking was based on earlier work or generated in the moment, the accuracy of the thinking, and the type of thinking (e.g., sense-making). Findings illuminate the complexity of identifying student thinking worth building on during whole-class discussion and provide insight into important attributes of these high potential instances that could be used to help teachers more easily recognize them. Implications for researching, learning, and enacting the teaching practice of building on student mathematical thinking are discussed.
To learn mathematical practices, students need opportunities to engage in them. But simply providing such opportunities may not be sufficient to support all students. Simultaneously, explicitly teaching mathematical practices could be problematic if instruction becomes prescriptive. This study investigated how teachers might make mathematical practices explicit in classroom discourse. Analyses of 26 discussions from 3 mathematics classes revealed that teachers made mathematical practices explicit primarily after students had participated in them. I present a framework of 8 types of teacher moves that made mathematical practices explicit and argue that they did so without turning practices into prescriptions or reducing students’ opportunities to engage in them. This suggests a need to expand conceptions of explicitness to promote access to mathematical practices.
Background/Context
Discussion is central to mathematics teaching and learning, as well as to mathematics as an academic discipline. Studies have shown that facilitating discussions is complex work that is not easily done or learned. To make such complex aspects of the work of teaching learnable by beginners, recent research has focused on approaches to teacher education that decompose practice into smaller tasks or routines that can be articulated, unpacked, studied, and rehearsed.
Purpose/Objective/Research Question/Focus of Study
Drawing on the elements of Grossman et al.'s (2009) framework for pedagogies of practice in professional education, this article describes an approach to analyzing teaching practice that supports the design of teacher education in which novices to learn how to engage in high-leverage teaching practices and simultaneously develop a sense of why the work is done.
Setting
The context for the study is a mathematics methods course for prospective elementary teachers.
Research Design
The study involved qualitative analysis of both the design of the course and its implementation.
Conclusions/Recommendations
Results from this study suggest that: (1) Decomposing teaching into nested practices of varying grain sizes that maintain the connection between techniques and domains can support beginning teachers in attending simultaneously to the how and the why of practice, as well as provide a map of teaching that teacher educators can use to support the learning of teaching. (2) Nesting early approximations of practice inside subsequent ones is a way to support beginning teachers in building toward recom-posed teaching practice. As practices are nested, teacher educators can increase the complexity of the approximations by adjusting the content and authenticity of the context of practice as well as by decreasing the amount scaffolding provided. (3) Assessment is vital to pedagogies of practice. Decompositions, approximations, and representations of practice need to be developed in ways that support assessment for the array of teacher education purposes. (4) Because subject matter is at the heart of teaching, it is crucial to attend to the ways in which the content interacts with pedagogies of practice.
The mathematics education community values using student thinking to develop mathematical concepts, but the nuances of this practice are not clearly understood. We conceptualize an important group of instances in classroom lessons that occur at the intersection of student thinking, significant mathematics, and pedagogical opportunities-what we call Mathematically Significant Pedagogical Opportunities to Build on Student Thinking. We analyze dialogue to illustrate a process for determining whether a classroom instance offers such an opportunity and to demonstrate the usefulness of the construct in examining classroom discourse. This construct contributes to research and professional development related to teachers' mathematically productive use of student thinking by providing a lens and generating a common language for recognizing and agreeing on a critical core of student mathematical thinking that researchers can attend to as they study classroom practice and that teachers can aspire to notice and build upon when it occurs in their classrooms.
In this paper five classrooms at grades 5-7 (students aged 11-14 years) are studied for one week each during their mathematics lessons. The aim is to study the students’ comments in order to develop categories describing the comments’ different contributions to the mathematical discourse. The main categories developed are student initiatives, explanations, partial answers, teacher-led responses and unexplained answers. The practices analysed are all dominated by the IRE pattern (Initiation-Response-Evaluation), and the different categories of student comments can be seen as a description of the different types of ‘R’ (student response) from the IRE pattern. This also illustrates that different patterns can be hidden behind the IRE-label. The categories can be used to study student comments on a turn-by-turn basis, describing different types of student contribution to the mathematical discourse.
Evidence shows that class discussion is important in students' development of mathematical conceptions. Theoretically, the process of contradiction and resolution is central to the transformation of thought. This article is a report of an 18-month investigation of a teacher's actions during class discussions in a 2nd-grade classroom in which students' disagreement was resolved by argumentation. Although the teacher valued children's reports of their reasoning, the context of argument in discussion was characterized by the high priority she afforded their roles as critical listeners. Her sensitivity in communicating her expectations for students' participation was evident during both discussion and disagreement. Moreover, the teacher participated with the students to create patterns of interaction and discourse that enabled children to shift their cognitive attention from making social sense to making sense of their mathematical experiences.
We propose a framework for examining how teachers may support collective argumentation in secondary mathematics classrooms, including teachers’ direct contributions to arguments, the kinds of questions teachers ask, and teachers’ other supportive actions. We illustrate our framework with examples from episodes of collective argumentation occurring across 2 days in a teacher’s classroom. Following from these examples, we discuss how the framework can be used to examine mathematical aspects of conversations in mathematics classrooms. We propose that the framework is useful for investigating and possibly enhancing how teachers support students’ reasoning and argumentation as fundamentally mathematical activities.
In order to describe and analyze teachers’ orchestrating of classroom discourse, detailed descriptions of teachers’ comments and questions are critical. The purpose of this article is to suggest new concepts that enable us to describe in detail how teachers use or do not use students’ comments to work with the mathematical content. Five teachers from upper primary school (grades five to seven, students aged 10 to 13) were studied. Beginning with the analysis of a pattern where the teacher gives a confirmation followed by a question that indicates a rejection, their practices form the basis for the development of 13 categories of teacher comments. These categories are then grouped into redirecting, progressing, and focusing actions. The categories and their groupings shed light on tools and techniques which these teachers use to make student strategies visible, to make students justify, apply and assess, to ensure progress towards a conclusion, or to redirect the students into alternative approaches. These findings can help us develop in the direction of a more profound understanding of how communication affects learning.
This article examines how a coding scheme for mathematics classroom discussion that was created to highlight how teachers negotiate student responses during whole-class discussion around high-level, cognitively demanding tasks was used to help teachers shift what they notice when analyzing classroom discourse. Data from an intervention that trained teachers how to use the coding scheme and then provided them opportunities to use the scheme to code transcripts of classroom discussion are presented. Results suggest that teachers’ ability to notice interactions between teacher and students when analyzing classroom discussion (as opposed to focusing on one actor or the other) can be increased and that teachers can learn to identify specific discourse moves teachers use to negotiate student responses. However, teachers’ capacity to identify how students’ opportunities to learn are related to teacher discourse moves did not change as a result of the intervention. The article goes on to examine how discussion during the intervention itself may have contributed to what teachers learned to notice. This research contributes to the body of work on teachers’ noticing by examining the feasibility and efficacy of using transcripts and a coding scheme to foster teachers’ ability to notice how they can increase their students’ opportunities to learn through mathematics discourse.
This is a case study of a highly regarded high-school mathematics teacher in Israel. It examines the kinds of responses to
students’ talk used repeatedly by the teacher, directing and shaping the classroom discourse, during different parts of the
lesson. The main data source included 21 h of observations in two of this teacher’s classrooms. Analysis of the video-taped
lessons showed that almost the entire whole-class work comprised of mathematical activity that was triggered by, built or
followed on, students’ talk. This was mainly due to the teacher’s responsiveness to students. The most common teacher response
was elaborating. Accompanying talk occurred considerably less, and the teacher rarely expressed puzzlement or opposition when
responding to students’ talk. The chapter demonstrates how the teacher combined her attention to students’ talk, with the
goal of making progress on the main topic.
Cognitively Guided Instruction (CGI) researchers have found that while teachers readily ask initial questions to elicit students’ mathematical thinking, they struggle with how to follow up on student ideas. This study examines the classrooms of three teachers who had engaged in algebraic reasoning CGI professional development. We detail teachers’ questions and how they relate to students’ making explicit their complete and correct explanations. We found that after the initial “How did you get that?” question, a great deal of variability existed among teachers’ questions and students’ responses.
The treatment of errors in mathematics classrooms has gained attention in recent years, with many researchers suggesting that errors should be used as starting points for student inquiry into mathematics. In the study reported in this article, we examined how teachers used discourse around errors to generate inquiry by looking at the treatment of mistakes in U.S. and Chinese elementary mathematics lessons. To do so, we videotaped 44 lessons from Chinese and U.S. first-grade (n = 15), and fourth- and fifth-grade (n = 29) classrooms and also interviewed the teachers of the lessons. We separated the lessons by the topic taught (place value or fractions) and analyzed them for frequency of students' errors and the types of teachers' responses to these errors. Results indicated that U.S. and Chinese students made errors at similar frequencies. However, the teachers in the 2 countries responded to errors differently. In particular, the U.S. teachers made more statements about errors than the Chinese teachers, who instead asked more follow-up questions about errors. Relying on qualitative analysis of teacher interview and in-class statements about errors, we shed light on both how teachers used errors for inquiry and what teachers believed about errors.
The current reform movement in mathematics education urges teachers to support students as they make sense of mathematics,
while also ensuring that they gain specific mathematical skills and knowledge. The tension between these two expectations
gives rise to what we call the dilemma of telling: how to ensure that students come to certain mathematical understandings,
without directly telling them what they need to know or do. Our study focused on how two middle school mathematics teachers
who were incorporating many aspects of reform mathematics into their instruction responded to this dilemma. Data sources include
classroom observations and videotapes of lessons over a three-year period. We found that both teachers devoted the majority
of class time to student conversations, both small group and whole class; however, the teachers strategically entered the
student-dominated conversations by “telling” to meet specific curricular goals.
KeywordsDiscourse-Constructivist teaching-Scaffolding
A case study of teacher moves in mathematically responsive interactions
- J P Bishop
- J Przybyla-Kuchek
- H Hardison
Bishop, J. P., Przybyla-Kuchek, J., & Hardison, H. (2018, April 13-17). A case study of teacher moves
in mathematically responsive interactions. Paper presented at the 2018 Annual Meeting of the
American Educational Research Association. Retrieved April 16, 2018, from the AERA Online
Paper Repository.
Teachers' initial responses to high leverage instances of student mathematical thinking
- S L Stockero
- B E Peterson
- M A Ochieng
- J R Ruk
- L R Van Zoest
- K R Leatham
Stockero, S. L., Peterson, B. E., Ochieng, M. A., Ruk, J. R., Van Zoest, L. R., & Leatham, K. R. (2019).
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Theorizing the mathematical point of building on student mathematical thinking
- L R Van Zoest
- S L Stockero
- K R Leatham
- B E Peterson
Van Zoest, L. R., Stockero, S. L., Leatham, K. R., & Peterson, B. E. (2016). Theorizing the mathematical point of building on student mathematical thinking. In C. Csíkos, A. Rausch, & J.
Szitányi (Eds.), Proceedings of the 40th Conference of the International Group for the Psychology
of Mathematics Education (Vol. 4, pp. 323-330). PME.
Beyond the “move”: A scheme for coding teachers' responses to student mathematical thinking
- B E Peterson
- L R Van Zoest
- A O T Rougée
- B Freeburn
- S L Stockero
- K R Leatham
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 Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 17-24).
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Teachers' responses to a common set of high potential instances of student mathematical thinking
- S L Stockero
- L R Van Zoest
- B E Peterson
- K R Leatham
- A O Rougée
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).
Conceptualizing the teaching practice of building on student mathematical thinking
- L R Van Zoest
- B E Peterson
- K R Leatham
- S L Stockero
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). University of
Arizona.
Profiles of responsiveness in middle grades mathematics classrooms
- J Bishop
- H Hardison
- J Przybyla-Kuchek
Bishop, J., Hardison, H., & Przybyla-Kuchek, J. (2016). Profiles of responsiveness in middle grades
mathematics classrooms. 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. 1173-1180). The University of Arizona.
Classroom discussions: Using math talk to help students learn, grades K-6
- S H Chapin
- C O'connor
- N C Anderson
Chapin, S. H., O'Connor, C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to
help students learn, grades K-6. Math Solutions.
Routines for reasoning: Fostering the mathematical practices in all students
- G Kelemanik
- A Lucenta
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