Pedagogical Content Knowledge (PCK)

Ten-minute talks

Ten-Minute Talks are similar to Number Talks in that they are to be used routinely as brief openers to lessons and that they offer opportunities for students to share their thinking and for teachers to explore that thinking without guiding, approving or correcting it. For example, students might be asked to share all the thoughts they have about a particular term, phrase, or mathematical situation. Ten-Minute Talks provide students with opportunities to articulate their thinking in a safe environment and teachers with opportunities to gain insight into student thinking which could include sound understandings,  connections, and misconceptions. Ten-minute talks provide teachers and students with opportunities to explore and improve SMP 2: Reason abstractly and quantitatively, SMP 3: Construct viable arguments and critique the reasoning of others, and SMP 7: Look for and make use of structure.



Ten-Minute_Talk_Number Sense


Error analysis

These error analysis tasks could be used with either teachers or students. Teachers or students examine student work on several tasks to identify mathematical thinking and misconceptions. Teachers may then brainstorm activities that will help students revise their conceptions. The content focuses on equivalent expressions (factoring and exponents) and involves SMP 2, 3, and 7.

Error Analysis Tasks


Examining student work

Teachers examine student work on the Intersections Task and discuss ways to improve discourse while implementing the task.

student work 1

student work 2

Task dialogue_IntersectionTask1&2

Examining student work

The goals of this activity are to have teachers engage in a process for examining student work with attention to learning from student thinking and to strengthen teachers’ ability to use student thinking in teaching by fostering intentional noticing of student thinking and an inquiry stance in their PLCs. The first part provides an example and the second part provides a protocol for teachers to analyze their own students’ work.


Examining students’ work – example

Examining a lesson plan

This lesson plan was written by a PLC to use the Growing Rectangles Task on the first day of Algebra 1. The facilitation notes provide a guide to help PLCs critically examine the lesson plan. PLCs should first do the Rectangles Task, anticipating what their students might do on the task on the first day of Algebra 1.

Growing Rectangles


Growing rectangles Lesson Plan

Planning a lesson using the 5 Practices

This activity is intended for a group of at least 15 teachers. The goals are to enact the 5 Practices (Smith & Stein) of planning and teaching a lesson using the Staircase task, which participants solve in multiple ways. They examine the solutions to consider ways to make connections and teach for coherence. We also focus on noticing and discussing the SMP used. Mathematically, the goal of the task is to understand structure of quadratic functions, and explore equivalent expressions; distinguish and relate expressions, equations, and functions, recursively defined functions, second common differences. The CCSS standards addressed are:

A-SSE.1, A-SSE.3, F-IF.3, F-IF.5, F-IF.7a, F-IF.8, F-BF (also, to some extent: F-IF.2, F-IF.4, F-IF.6)





Observing a lesson together

Teachers observe a video of a lesson called Staircase (Annenberg Learner) together for questioning, content, and engagement. They focus on observable evidence, and make connections between the teachers’ strategies and the purpose of the lesson. Learning to notice objectively supports deepening awareness of what is happening in the classroom and makes us aware of our own practices. The mathematics in the lesson involves explicit vs. recursive descriptions of sequences, seeing structure, and creating equations, while teachers are also trying to notice questioning, engagement, and how these questioning and engagement support the content purposes of the lesson.



Cognitive Complexity

In the first activity teachers identify and discuss the cognitive demand of several tasks involving exponential expressions. They also identify the SMP students may use. The second activity involves identifying the Cognitive Demand of a variety of Algebra 1 tasks. Directions are included on the first page of each activity and the tasks follow.

Cognitive demand activityExponents