The Teacher Response Coding Scheme: Disentangling Teacher Responses to Student Mathematical Thinking
Van Zoest, L. R., Bishop, J. P., Hardison, H., PrzybylaKuchek, J., Peterson, B. E., Stockero,
S. L., Leatham, K. R., Freeburn, B., & Kazemi, E. (2018, April). Analyzing teachers' responses to student mathematical contributions during wholeclass interactions: Goals, grain sizes, and coding schemes. Symposium presented at the annual meeting of the American Educational Research Association, New York, NY.
To contribute to the field's understanding of the teacher's 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' inthemoment responses to student thinking during instruction. The TRC differs from existing schemes in that it disentangles the teacher move from the actor (those who are publicly given the opportunity to consider the student thinking), the recognition (the extent to which the student recognizes their contribution in the teacher response), and the mathematics (the alignment of the mathematics in the teacher response to the mathematics in the student thinking). This disentanglement makes the TRC useful across a broad range of settings and allows researchers to interrogate their data in multiple ways without recoding.
The Nature of Student Thinking Available in a Secondary Mathematics Classroom
Ochieng, M. A., Ruk, J. M., Arslan, O., Van Zoest, L. R., Leatham, K. R., Peterson, B. E., & Stockero, S. L. (2018, February). The nature of student thinking available in a secondary mathematics classroom. Poster presented at the 22nd Annual Meeting of the Association of Mathematics Teachers Educators, Houston, TX.
Our investigation of a lesson revealed ways in which student thinking is not a uniform construct. We will illustrate the manner in which different types of student mathematical thinking provide different resources for instruction and require different responses from teachers to be used effectively.
Teachers' Orientations Around Using Student Mathematical Thinking as a Resource During Wholeclass Discussion
Leatham, K. R., Stockero, S. L., Ochieng, M. A., Van Zoest, L. R., & Peterson, B. E. (2018, February). Teachers' orientations around using student mathematical thinking as a resource during wholeclass discussion. Presentation at the 22nd Annual Meeting of the Association of Mathematics Teachers Educators, Houston, TX.
We characterize teachers' orientations related to using student mathematical thinking as a resource during wholeclass discussion. We consider the potential these orientations provide to either support or hinder the development of the practice of building on student mathematical thinking.
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, October). Teachers' responses to a common set of high potential instances of student mathematical thinking. Presentation at the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Indianapolis, IN.
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.
Learning to Become a Researcher in an Ongoing Research Project: A Communities of Practice Perspective
Arslan, O., Van Zoest, L. R., & Ruk, J. M. (2017, October). Learning to become a researcher in an ongoing research project: A communities of practice perspective. Presentation at the 39th Annual Meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education, Indianapolis, IN.
We apply Wenger's (1998) communities of practice ideas to the process of incorporating new researchers into an
ongoing mathematics education research project. We illustrate this application by describing how the Leveraging MOSTs
research project coding team can be viewed as a community of practice. We describe how we have used this particular
community of practice to bring new researchers into the project, and new researchers reflect on their experiences with
the coding team. Mutual engagement in project work with experienced researchers and having a rich shared repertoire
to draw on led to the new researchers developing a shared understanding of the project and being successfully
incorporated into the MOST community. This work speaks to the importance of deliberately creating communities of
practice for new mathematics researchers to participate within.
Barriers to Building on Student Mathematical Thinking
Stockero, S. L., Van Zoest, L. R., Leatham, K. R., & Peterson, B. E. (2017, February). Barriers to building on student thinking. Presentation at the 21st Annual Meeting of the Association of Mathematics Teachers Educators, Orlando, FL.
In our work with teachers, we have identified barriers that inhibit them from productively implementing the teaching practice of building on student thinking. We share examples of barriers and ways we have supported teachers to make progress toward overcoming them.
Beyond the Move: A Coding Scheme for Teacher Responses to Instances of Student Mathematical Thinking
Rougée, A. O. T., Peterson, B. E., Van Zoest, L. R., Freeburn, B., Stockero, S. L., Leatham, K. R., & Gunn, R. M. (2017, February). Beyond the move: A coding scheme for teacher responses to instances of student mathematical thinking. Poster presented at the 21st Annual Meeting of the Association of Mathematics Teachers Educators, Orlando, FL.
In our work with teachers, we have identified barriers that inhibit them from productively implementing the teaching practice of building on student thinking. We share examples of barriers and ways we have supported teachers to make progress toward overcoming them.
Imprecision in Classroom Mathematics Discourse
Leatham, K. R., Peterson, B. E., Merrill, L., Van Zoest, L. R., & Stockero, S. L. (2016, November). Imprecision in classroom mathematics discourse. Presentation at the 38th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ.
We theorize about ambiguity in mathematical communication and define a certain subset of ambiguous language usage as imprecise. For us, imprecision in classroom mathematics discourse hinders inthemoment communication because the instance of imprecision is likely to create inconsistent interpretations of the same statement among individuals. We argue for the importance of attending to such imprecision as a critical aspect of attending to precision. We illustrate various types of imprecision that occur in mathematics classrooms and the ramifications of not addressing this imprecision. Based on our conceptualization of these types and ramifications, we discuss implications for research on classroom mathematics discourse.
Conceptualizing the Teaching Practice of Building on Student Mathematical Thinking
Van Zoest, L. R., Peterson, B. E., Leatham, K. R., & Stockero, S. L. (2016, November). Conceptualizing the teaching practice of building on student mathematical thinking. Presentation at the 38th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ.
An important aspect of effective teaching is taking advantage of inthemoment expressions of student thinking that, by becoming the object of class discussion, can help students better understand important mathematical ideas. We call these highpotential 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 highpotential 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 subpractices.
We conclude by discussing the need for future research and our research agenda for studying the building prototype.
Theorizing the Mathematical Point of Building on Student Mathematical Thinking
Van Zoest, L. R., Stockero, S. L., Leatham, K. R., & Peterson, B. E. (2016, August). Theorizing the mathematical point of building on student mathematical thinking (handout). Presentation at the 40th Conference of the International Group for the Psychology of Mathematics Education, Szeged, Hungary.
Despite the fact that the mathematics education field recognizes the critical role that student thinking plays in high quality instruction, little is known about productive use of the inthemoment student thinking that emerges in the context of wholeclass discussion. We draw on and extend the work of others to theorize the mathematical understanding an instance of such student thinking can be used to build towards—the mathematical point (MP). An MP is a mathematical statement of what could be gained from considering a particular instance of student thinking. Examples and nonexamples are used to illustrate nuances in the MP construct. Articulating the MP for an instance of student thinking is requisite for determining whether there is instructional value in pursuing that thinking.
Leveraging MOSTs: Developing a Theory of Productive Use of Student Mathematical Thinking
Van Zoest, L. (2016, June). Leveraging MOSTs: Developing a theory of productive use of student mathematical thinking. Poster presented at the 2016 NSF Discovery Research PreK12 PI Meeting. Washington, DC.
This project focuses on improving the teaching of secondary school mathematics by improving teachers' abilities to use student thinking during instruction to develop mathematical concepts. The core research questions of the project are: (1) What is the nature of highleverage student thinking that teachers have available to them in their classrooms? (2) How do teachers use student thinking during instruction and what goals, orientations and resources underlie that use? (3) What is the learning trajectory for the teaching practice of productively using student thinking? and (4) What supports can be provided to move teachers along that learning trajectory?
Conceptualizing Teacher Discourse Moves Using Different Focal Lengths
Van Zoest, L. R., Stockero, S. L., Leatham, K. R., Peterson, B. E., Conner, A., Singletary, L. M., … O'Connor, C. (2016, April). Conceptualizing teacher discourse moves using different focal lengths. Presentation at the 2016 National Council of Teachers of Mathematics Research Conference, San Francisco, CA.
Using the metaphor of camera focal length, three research groups will share their conceptualizations of teacher moves to facilitate meaningful mathematical discourse. The approaches will be analyzed in relationship to each other to better understand teacher actions in response to student contributions during instruction.
How We Can "Attend to Precision" in Classroom Mathematics Discussions
Leatham, K. R., Peterson, B. E., & Merrill, L. (2016, April). How we can "attend to precision" in classroom mathematics discussions. Presentation at the 2016 National Council of Teachers of Mathematics Annual Conference, San Francisco, CA.
Explore examples of teacher and student imprecision in classroom mathematics discourse. Discuss types of imprecision that occur in classrooms, the ramifications of this imprecision, and strategies for addressing that imprecision. Learn how to minimize your own imprecision and to view student imprecision as an opportunity to learn mathematics.
I've got my Students Sharing Their Mathematical Thinking—Now What?
Stockero, S. L., Van Zoest, L. R., & Leatham, K. R. (2016, April). I've got my students sharing their mathematical thinking—Now what? Presentation at the 2016 National Council of Teachers of Mathematics Annual Conference, San Francisco, CA.
Once students share their ideas, creating meaningful mathematics discourse requires that teachers decide which ideas are worth pursuing and how to capitalize on those ideas. We will share a framework for determining which student ideas have significant potential to support mathematics learning, and we will discuss how teachers might productively use those ideas.
Engaging Teachers in Identifying the Point of Student Mathematical Thinking
Van Zoest, L. R., Fraser, E. H., & Ochieng, M. A. (2016, January). Engaging teachers in identifying the point of student mathematical thinking. Presentation at the 20th Annual Meeting of the Association of Mathematics Teachers Educators, Irvine, CA.
Explore productive use of student thinking through activities related to identifying the mathematical point an instance of student thinking can be used to build toward. Discuss the potential of such activities for supporting teachers to productively use student mathematical thinking.
Productive Use of Student Mathematical Thinking is More than a Single Move
Peterson, B. E., Van Zoest, L. R., Stockero, S. L., & Leatham, K. R. (2016, January). Productive use of student mathematical thinking is more than a single move. Presentation at the 20th Annual Meeting of the Association of Mathematics Teachers Educators, Irvine, CA.
We will introduce the teaching practice of building and its constituent components as the most productive use of worthwhile student mathematical thinking, analyze teaching examples for evidence of building, and consider how to support teachers' development of this practice.
Attributes of Student Mathematical Thinking That is Worth Building on in Whole Class Discussion
Van Zoest, L. R., Stockero, S. L., Atanga, N. A., Leatham, K. R., Peterson, B. E., & Ochieng, M. A. (2015, November). Attributes of student mathematical thinking that is worth building on in whole class discussion. Presentation at the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, East Lansing, MI.
This study investigated the attributes of 297 instances of student mathematical thinking during wholeclass interactions that were identified as having the potential to foster learners' understanding of important mathematical ideas (MOSTs). Attributes included the form of the thinking (e.g., question vs. declarative statement), whether the thinking was based on earlier work or generated inthemoment, the accuracy of the thinking, and the type of the thinking (e.g., sense making). Findings both illuminate the complexity of identifying student thinking worth building on during wholeclass discussion and provide insight into important attributes of MOSTs that teachers can use to better recognize them. For example, 96% of MOSTs were of three types, making these three particularly salient types of student mathematical thinking for teachers to develop skills in recognizing.
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, November). Uncovering teachers' goals, orientations, and resources related to the practice of using student thinking. Presentation at the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, East Lansing, MI.
Improving teachers' practice of using student mathematical thinking requires an understanding of why teachers 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 scenariobased interview tool developed to prompt teachers to discuss their decisions and rationales related to using student thinking. We examine cases of two individual teachers to illustrate how the tool contributes to (1) inferring individual teachers' goals, orientations and resources and (2) differentiating among teachers' uses of student thinking.
Toward a Theory of Productive Use of Student Mathematical Thinking
Van Zoest, L. R., Stockero, S. L., Peterson, B. E., Leatham, K. R., Atanga, N., Merrill, L., & Ochieng M. (2015, April). Toward a theory of productive use of student mathematical thinking. Presentation at the 2015 National Council of Teachers of Mathematics Research Conference, Boston, MA.
This research symposium consists of three presentations that consider: (1) the nature of student mathematical thinking (SMT) available to teachers during instruction, (2) teachers' perceptions of productive use of SMT, and (3) teachers' abilities to recognize and productively respond to SMT. The work will be discussed in the broader context of developing a theory of productive use of SMT.
Defining and Developing Teaching Practices Related to Responding to Students' Mathematical Thinking
Webel, C., DeLeeuw, W., Empson, S., Jacobs, V., Land, T., Leatham, K., … Conner, K. (2015, February). Defining and developing teaching practices related to responding to students' mathematical thinking. Presentation at the 19th Annual Meeting of the Association of Mathematics Teachers Educators, Orlando, FL.
This session builds on research on professional noticing of students' mathematical thinking by unpacking different ways of conceptualizing the teaching practice of responding to student thinking. Four projects focused on defining and developing this practice will be presented and discussed. The MOST project presented on Discerning Student Mathematical Thinking in Whole Class Discussion.
Teachers' Perceptions of "Use" of Student Mathematical Thinking in Whole Class Discussion
Ochieng, M. A., Leatham, K., Stockero, S. L., & Van Zoest, L. R. (2015, February). Teachers' perceptions of "use" of student mathematical thinking in whole class discussion. Presentation at the 19th Annual Meeting of the Association of Mathematics Teachers Educators, Orlando, FL.
What does it mean to productively "use" student mathematical
thinking in wholeclass discussion? The MOST project interviewed
mathematics teachers about their perceptions of such use. We
discuss our framework for categorizing teachers' perceptions of use
and implications for professional development.
What's the Point? Identifying Important Mathematics Underlying Student Thinking
Van Zoest, L. R. (2015, March). What's the point? Identifying important mathematics underlying student thinking. Presentations at the Teachers Development Group 2015 Leadership Seminar, Portland, OR.
This session focuses on understanding the instructional practice of building on student thinking and supporting teachers to enact this practice through activities related to identifying the mathematical point of an instance—that is, what the instance of student thinking can be used to build towards. In the MOST project (see LeveragingMOSTs.org), we define building as the teaching practice of making student thinking the object of consideration by the class in order to engage the class in making sense of that thinking to better understand an important mathematical idea. ...MORE...A critical aspect of building is recognizing what important mathematical idea the student thinking can be used to better understand; we call these important mathematical ideas mathematical points. As a result of our own learning through the process of identifying mathematical points, we hypothesized that teachers would also benefit from engaging with and developing skill in identifying mathematical points. To facilitate teacher learning, we defined mathematical point as a concise statement of a specific mathematical idea that students could learn and constructed activities that would develop teachers' knowledge of and skill with mathematical points. Session attendees will engage in these mathematical point activities and discuss the potential of identifying and conceptualizing mathematical points for an instance of student mathematical thinking as a "learning to teach" activity for preservice teachers and an "improving teaching" activity for practicing teachers. Research data from using the activities with preservice teachers will be used to inform the discussion. Focusing on the mathematical point of an instance of student thinking supports teachers in focusing on the mathematics in the student thinking and to recognize which student thinking can be pursued to meet CCSSM. This focus is seen as an antidote both for the temptation to see transmitting content to students as the only way to meet the CCSSM and for the idea that having students share their thinking is sufficient to meet the CCSSM. The research and examples are from work with secondary preservice teachers, but the session will be relevant to anyone involved in supporting teacher learning.
Teachers' Perceptions of Productive Use of Student
Mathematical Thinking
Leatham, K. R., Van Zoest, L. R., Stockero, S. L., & Peterson, B. E. (2014, July). Teachers' perceptions of productive use of student
mathematical thinking. Presentation at the Joint Meeting of PME 38 and PMENA
36, Vancouver, Canada.
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.
Recognizing Opportunities for Productive Use of
Student Thinking
Leatham, K. R., Peterson, B. E., Stockero, S. L., & Van Zoest,
L. R. (2014, April). Recognizing opportunities for productive
use of student thinking (handout #1, handout #2). Presentation at the 2014 National
Council of Teachers of Mathematics Research Conference, New Orleans,
LA.
Participants will be introduced to and use a framework that considers
the significance of student mathematical thinking and the pedagogical
opportunities that thinking might create. The affordances and
complexities of using the framework to analyze classroom discourse
and to support teachers in productively using student thinking
will be discussed.
Making the MOST of Student Mathematical Thinking
Van Zoest, L. R. (2014, February). Making the MOST of student mathematical thinking. Presentation at the Teachers Development
Group 2014 Leadership Seminar, Portland, Oregon.
This session draws on research about which student thinking during
mathematics lessons is most productive to pursue in the moment
to consider how to support teachers in productively using student
thinking to meet the CCSS for Mathematical Practice and Content.
The Mathematical Opportunities in Student Thinking (MOST) framework
will be considered as a way to identify student thinking that
has considerable potential to engage students in developing their
understanding of significant mathematics. One of the strengths
of the framework is the way in which it focuses teachers on mathematics.
This session will be relevant to anyone interested in thinking
about how to support teachers in productively using student thinking.
What Does it Mean to Build on Student Mathematical
Thinking?
Peterson, B. E., Leatham, K. R., & Van Zoest, L. R. (2014, February). What
does it mean to build on student mathematical thinking? Presentation
at the 18th Annual Meeting of the Association of Mathematics Teacher
Educators, Irvine, CA.
"Attend to," "respond to," "pursue," and "use" are terms often
used synonymously with "build on" student mathematical thinking.
This imprecision contributes to teachers' difficulty in implementing
the practice. Our discussion will work toward developing common
definitions among mathematics teacher educators.
Classroom Mathematics Discourse: Broadening Perspectives
by Integrating Tools for Analysis
Johnson, K. R., Steele, Michael D., HerbelEisenmann, B. A., Leatham,
K. R., & Peterson, B. E., Stockero, S. L., … Merrill, L. (2013, November). Classroom mathematics discourse: Broadening perspectives by integrating tools for analysis. Presentation at the 35th Annual Meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education, Chicago, IL.
This working group explores tools for analyzing mathematics classroom
discourse across two projects with different, but complementary
perspectives. The goals of the working group include generating
interaction about the theoretical lenses that we use to analyze
and discuss classroom mathematics discourse and the relationships
between these different theoretical frameworks. Participants
will engage with the individual frameworks in the first two sessions
and discuss interactions of the two frameworks in the third session. The attached presentation is from the MOST group on "Mathematically Significant Pedagogical Opportunities to Build on Student Thinking."
Conceptualizing Mathematically Significant Pedagogical
Opportunities to Build on Student Thinking
Van Zoest, L. R., Leatham, K. R., Peterson, B. E., & Stockero,
S. L. (2013, July). Conceptualizing mathematically significant pedagogical
openings to build on student thinking. Presentation at the 37th Conference of the International
Group for the Psychology of Mathematics Education, Kiel, Germany.
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
openings—what we call Mathematically Significant
Pedagogical Openings to Build on Student Thinking
(MOSTs)—and introduce a framework for determining when
they occur. We discuss how the MOST construct contributes to
facilitating and researching teachers' mathematicallyproductive
use of student thinking through providing a lens and generating
a common language for recognizing and agreeing upon highleverage
student mathematical thinking.
A Framework for Recognizing Teachable Moments in
Mathematics Classrooms
Leatham, K. R., Stockero, S. L., Peterson, B. E., & Van Zoest,
L. R. (2013, January). A framework for recognizing teachable
moments in mathematics classrooms. Presentation at the 17th
Annual Meeting of the Association of Mathematics Teacher Educators,
Orlando, FL.
We describe a tool for identifying when student thinking provides
a pedagogical opening for working towards a mathematical goal.
Attendees will discuss ideas for using the tool to analyze and
discuss instances of student mathematics thinking with teachers.
Mathematically Important Pedagogical Opportunities
(MIPO)
Leatham, K. R., Peterson, B. E., Stockero, S. L., & Van Zoest,
L. R. (2011, October). Mathematically important pedagogical opportunities. Presentation at the 33rd
Annual Meeting of the North American Chapter of the International
Group for the Psychology of Mathematics Education,
Reno, NV.
The mathematics education community values using student thinking
to develop mathematical concepts, but the nuances of this practice
are not clearly understood. For example, not all student thinking
provides the basis of productive discussions. In this paper we
describe a conceptualization of instances in a classroom lesson
that provide the teacher with opportunities to extend or change
the nature of students' mathematical understanding—what
we refer to as Mathematically Important Pedagogical Opportunities
(MIPOs). We analyze classroom dialogue to illustrate how this
lens can be used to make more tangible the often abstract but
fundamental goal of pursuing students’ mathematical thinking.
Investigating Mathematically Important Pedagogical
Opportunities
Leatham, K. R., Stockero, S. L., Van Zoest, L. R., & Peterson,
B. E. (2010, October). Investigating mathematically important pedagogical
opportunities. Presentation at the 32nd Annual Meeting of the North
American Chapter of the International Group for the Psychology
of Mathematics Education,
Columbus, OH.
Mathematically Important Pedagogical Opportunities (MIPOs) are
instances in a classroom lesson in which the teacher has an opportunity
to move the class forward in their development of significant
mathematics. Although this construct is widely recognized in
the literature as important to mathematics teaching and learning,
it is neither well defined nor clearly identified as a construct
that can be studied. This working group will build on the efforts
of two research groups, represented by the organizers, to define,
identify, and characterize MIPOs. Specifically, Session 1 will
focus on identifying MIPOs, including questioning and critiquing
working definitions and preliminary dimensions of MIPOs. Session
2 will explore subconstructs of MIPOs and the potential of subconstructs
to provide leverage in studying the broader construct. The first
two sessions will include examining instances of classroom practice
(written/video) that have been identified as containing MIPOs.
Session 3 will focus on issues around developing a research agenda
for investigating MIPOs and generating plans for continuing work
on MIPOs.
