Source: *Journal of Mathematics Teacher Education, Vol. 16, No. 2*, April 2013, p. 125-147.

(Reviewed by the Portal Team)

This study focuses on understanding the types of instances that beginning teachers need to notice during instruction, how they currently respond to these instances, and how their responses potentially impact student learning.

The authors use the concept pivotal teaching moment (PTM) as an opportune mathematical instances during instruction.

PTM is defined as an instance in a classroom lesson in which an 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.

The authors used videos of beginning teachers’ classrooms to investigate the following questions:

a. What are characteristics of PTMs faced by beginning secondary school mathematics teachers during classroom instruction?

b. What types of decisions do beginning mathematics teachers make when a PTM occurs during their instruction?

c. What relationships exist among PTM characteristics, teachers’ decisions, and the likely impact on student learning?

**Methods**

Data collection included video of classroom mathematics instruction was collected from six teachers with fewer than four years teaching experience.

Each teacher was recorded for at least two class periods per day for three consecutive days in the middle of the fall 2008 semester.

All of the teachers were graduates of an NCTM (2000) Standards-based secondary mathematics teacher preparation program in the US that focused on teaching mathematics for student understanding.

At the time of the data collection, the participating teachers were teaching mathematics in grades 8–12 in a variety of school settings.

The topics taught included algebra, geometry, and trigonometry.

The data suggest that the decisions a teacher makes in response to PTMs significantly affect the way in which a PTM is likely to impact student learning.

For example, classrooms in which the teacher was focused on making student thinking public had higher numbers of PTMs, but PTMs existed in even the most transmission- based classrooms.

The results suggest that it is possible to categorize the circumstances that lead to PTMs—an important first step in helping teachers learn to notice them during instruction.

Therefore, five circumstances that led to PTMs were identified:

(a) when students made a comment or asked a question that was grounded in, but went beyond, the mathematics that the teacher had planned to discuss;

(b) when students were trying to make sense of the mathematics in the lesson;

(c) when students expressed incorrect mathematical thinking or an incorrect solution;

(d) when a mathematical contradiction occurred in the public space; and

(e) when students expressed mathematical confusion.

Similarly, the authors were able to classify a set of three productive teacher decisions:

(a) extend the mathematics and/or make connections among mathematical ideas,

(b) pursue student mathematical thinking, and

(c) emphasize the meaning of the mathematics.

These decisions provide a starting point for helping teachers learn to use student thinking in ways that support the development of students’ mathematical understanding.

In general, the data also suggest that focusing on mathematical trajectories and connections might produce more positive likely impacts on student learning.

In conclusion, the authors argue that the initial PTM framework that has resulted from this work has the potential to be used as a tool to help teachers focus on mathematically rich moments that occur during instruction and to inform teacher educators as they develop activities to support both teacher noticing and teacher decision-making.

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