CHARACTERIZING PIVOTAL TEACHING MOMENTS IN MATHEMATICS INSTRUCTION
Research identifying the use of student thinking as an instructional practice that enhances student learning (e.g., Fennema et al., 1996; Stein & Lane, 1996) has prompted the study of how to best support teachers in using this practice in their classroom (e.g., Santagata, Zannoni & Stigler, 2007; van Es & Sherin, 2008). Ball and Cohen (1999) discuss the notion of teachers learning in and from practice that involves teachers "siz[ing] up a situation from moment to moment" (p. 11) and using what they learn to modify their instruction. Van Es and Sherin (2008) use the term noticing to describe the way in which teachers attend to important elements of instruction that support student learning and then reason about them to make instructional decisions. Although the language used to describe the practice of keying in on important moments during instruction varies, the idea is the same—teachers need to learn to pay attention to students' ideas and improve their abilities to use these ideas to advance student learning.
Although skilled teachers and teacher educators often recognize important mathematical moments that occur during a lesson and act on them in ways that support student learning, such moments frequently either go unnoticed or are not acted upon by others in the profession, particularly novices (Peterson & Leatham, 2010). This raises the question of how teachers can learn to recognize important mathematical moments during their instruction and use them to support student learning. We focus on characterizing the circumstances surrounding such moments, as making them more visible is an important first step in supporting teacher learning.
We define a pivotal teaching moment (PTM) to be an instance in a classroom lesson in which a studentgenerated interruption in the flow of the lesson provides the teacher an opportunity to modify instruction in order to extend or change the nature of students' mathematical understanding. To better understand PTMs and teacher responses to them that support student learning, our exploratory study investigated these questions:
1. What are characteristics of PTMs faced by mathematics teachers during classroom instruction?
2. What types of decisions do mathematics teachers make when PTMs occur?
3. What relationships exist among PTM characteristics, teachers' decisions, and the likely impacts on student learning?
A better understanding of PTMs can improve students' learning of mathematics by helping teachers build on student ideas to enhance their mathematical understanding. PTMs can be used as a tool to improve the ways in which we educate teachers to use student thinking during instruction by providing a framework for helping teachers learn to focus on mathematically rich moments that occur during instruction.
Perspectives
The idea that there are important moments within a mathematics lesson that a teacher needs to notice and act upon is grounded in a particular vision of teaching—one in which teachers build on student thinking during instruction to develop mathematical understanding. Inherent in this vision is a need for teachers to reflectinaction (Schšn, 1983) in order to adapt instruction as it unfolds in response to students' current understandings (Ball & Cohen, 1999). In order to do so, teachers need to both notice (van Es & Sherin, 2008) important mathematical moments when they arise and have the orientations, resources and goals (Schoenfeld, 2011) necessary to act on them in ways that support student learning.
Data Collection and Analysis
As part of a project focused on examining teacher learning using practicebased materials, over 45 hours of classroom video was collected from six mathematics teachers of grades 812 in a variety of school settings. All were recent graduates of an NCTM (2000) Standardsbased teacher education program focused on teaching mathematics for student understanding.
The data analysis involved the reduction of the video to potential sites for PTMs, followed by the identification of 27 episodes containing 39 PTMs. These PTMs were then characterized using open coding, resulting in a framework that included the following components: (a) PTM type, (b) potential for advancing students' mathematical understanding, (c) teacher decision, (d) decision implementation, and (e) likely impact on student learning. We then used multiple comparison charts to determine whether there were patterns in the PTM triples (type, decision, likely impact) and to investigate relationships among different pairings of coded components.
Results
The study resulted in a preliminary framework for understanding PTMs (see Table 1). Analyzing the frequency that individual characteristics occurred in the data, as well as the frequency of groupings of characteristics, led to, among other things, five hypotheses about relationships among the framework components:
(1) There is no apparent relationship between the type of PTM and teachers' ability to respond effectively;
(2) Teachers' inability to notice PTMs negatively affects students' opportunities to learn;
(3) Improving teachers' abilities to implement productive decisions will reduce the number of negative learning outcomes;
(4) Focusing on mathematical trajectories and connections will produce more positive student learning outcomes; and
(5) Significant potential PTMs may improve student learning on their own, whereas moderate potential PTMs require a wellimplemented decision.
Table 1: PTM Framework
Pivotal Teaching Moment
 Teacher Decision
 Likely Impact on Student Learning
 
Type
 Potential
 Action
 Implementation
 
Extending
 Significant Moderate
 Extends math and/or makes connections
 Skillfully Moderately Poorly
 High positive

Incorrect math
 Pursues student thinking
 Medium positive
 
Sensemaking
 Emphasizes meaning of the mathematics
 Low positive
 
Contradiction
 Acknowledges, but continues as planned

 Neutral
 
Confusion
 Ignores or dismisses
 Negative

The results suggest that it is possible to categorize the circumstances that lead to PTMs—an important criterion for helping teachers learn to notice them during instruction. The work raises questions about which PTM components might be most productive to focus on in teacher education, as well as the types of activities that might help teachers learn to capitalize on PTMs; these questions will be the focus of the roundtable discussion.
PTMs, by their very nature, are highleverage moments that can significantly impact student learning, and are thus worthy of analysis. The PTM framework can be used to support teacher professional learning by focusing teachers on identifying and using student ideas during instruction to improve their students' mathematical understanding.
References
Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practicebased theory of professional education. In G. Sykes and L. DarlingHammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 332). San Francisco: Jossey Bass.
Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Epson, S. B. (1996). A longitudinal study of learning to use children's thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403434.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Peterson, B. E., & Leatham, K. R. (2010). Learning to use students' mathematical thinking. In L. Knott (Ed.), The role of mathematics discourse in producing leaders of discourse (pp. 99–128). Charlotte, NC: Information Age.
Santagata, R., Zannoni, C., & Stigler, J. W. (2007). The role of lesson analysis in preservice teacher education: An empirical investigation of teacher learning from a virtual videobased field experience. Journal of Mathematics Teacher Education, 10(2), 123140.
Schoenfeld, A. H. (2011). How we think: A theory of goaloriented decision making and its educational applications. New York: Routledge.
Schšn, D. A. (1983). The reflective practitioner: How professionals think in action. New York: Basic Books.
Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2(1), 5080.
van Es, E. A., & Sherin, M. G. (2008). Mathematics teachers' "learning to notice" in the context of a video club. Teaching and Teacher Education, 24, 244276.