CPM Glossary

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Numbered Heads

Mode of Instruction: Teamwork           Purpose: Individual accountability

Objective: To support productive struggle in learning mathematics, students collectively and individually support each other through building mathematical ideas and relationships. Teacher monitors the equitable participation of team members and poses purposeful questions.

Numbered Heads is an effective strategy for keeping all students involved and accountable during teamwork. Students number off in teams. Then they are given a problem to solve, a question to answer, or any task to complete. The team members work together making sure that each student in the team understands what they are doing, knows the answers, and can explain the team's work.

  • Students number themselves 1 - 4 in their team.

  • Each team member is assigned a team role, task, or problem number that they are responsible for.

  • Teacher circulates, stops at a group, and asks a random number to answer a checking for understanding question.

  • The Teacher can use a random number to have students share out at the end of class.


Pairs Check

Mode of Instruction: Partner Work           Purpose: Check for understanding

Objective: To promote productive struggle with a topic before mastery is expected, two students share mathematical ideas with each other. Teacher monitors through circulation.

Use when practicing a new skill or procedure, or to pre-assess a topic that will soon be taught. Within each team of four, students work in pairs to solve problems and then check solutions with the other pair. Each set of partners has one sheet of paper and one pencil. While one student writes, the other student explains. If the student writing disagrees with the explanation, then a discussion happens before the step is recorded on the paper. When finished the role of writer is rotated to the other student, and the process continues. After problems are complete, the partners check the explanations of the other team members. If both pairs agree, a checkmark is added to the paper. If pairs disagree, teams conduct error analysis.

  • Team Member (1) writes while Team Member (2) explains the first problem.

  • Team Member (1) asks clarifying questions to Team Member (2).

  • The partners check with the other partners from the team—if they agree, put a ✅, if they disagree, find mistakes.

  • Team Member (1) rotates the paper to Team Member (2), and roles are reversed for the next problem.

Parent Guide with Extra Practice

The Parent Guide with Extra Practice provides an explanation for selected topics, with worked examples, and additional practice problems with answer key. The resource is accessible via free PDF file download by topic, by chapter, or for the entire course. For most courses the Parent Guide with Extra Practice is also available for purchase in softbound format.

Participation Quiz

Mode of Instruction: Teamwork/Teacher-led           Purpose: Promote equitable team routines

Objective: To establish mathematics goals to focus learning, students practice equitable routines and team roles that progress throughout tasks and problem solving. Teacher monitors and records participation in expected routines and roles.

During a team worthy task, the teacher picks a class norm on which to focus and provide students with feedback. The teacher explains to students how the feedback will be presented (rubric, laminated feedback cards, overhead, posters, chalkboard, electronic, etc.). The teacher may want to pause to debrief in the middle and end of class, or just debrief at the end of the class period.

  • Teacher displays the team norm that will be the focus of the lesson.

  • Teacher explains to teams how they will monitor teamwork.

  • Teacher records comments while students are working.

  • Debrief together as a class.

Peer Edit

Mode of Instruction: Partner Work           Purpose: Collaboration

Objective: To elicit and use evidence of student thinking, pairs of students edit each other's work, by asking questions and improving evidence. Teacher adjusts instruction based on the edits.

This strategy allows peers to give each other feedback either orally or in writing. It can be used when writing Toolkit or Learning Log entries, or any problem that asks for an explanation or justification. Peers should be positive with their comments and specific with their feedback. They should highlight the things that they like about what was written. They should share ideas to improve their partner’s writing.

  • Students complete a rough draft of their writing entry.

  • Students trade papers with a partner and read their partner’s work.

  • Students use another color to make edits, provide comments and suggestions, ask clarifying questions, or provide praise.

  • Student (1) shares out to Student (2), what they like about the writing, and any additional notes or feedback.

  • Student (2) shares out to Student (1), what they like about the writing, and any additional notes or feedback.

  • Students make changes or additions to their rough draft thinking.

Pick Three

Mode of Instruction: Independent/Teamwork           Purpose: Collaboration

Objective: To establish mathematics goals to focus learning, students highlight strengths that will make the team stronger. The teacher monitors the equitable status of all team members.

The purpose of this activity is to provide a quick reminder at the beginning of class about the importance of having every team member contribute to the team's work, and of what is involved in good mathematics work. Examples of what the list of strengths may include are: Looking for patterns, Asking questions, Understanding vocabulary, Making a drawing or model, Acting out the problem, Helping others, Explaining my thinking and justifying answers, Noticing details, Organizing, Predicting, Writing equations from patterns, Looking at things in different ways, Reading aloud, Keeping people on task, Following directions, Learning from our mistakes, Remembering a similar problem, Encouraging your team members to persevere.

  • Teacher posts a list of strengths.

  • Each student selects and writes down three strengths they can contribute to their team.

  • Students take turns sharing their strengths with their team.

  • Students use strengths as they work on the lesson.


Thinking about and organizing the activities, sequence, manner of presentation, study team teaching strategies and materials needed to implement a lesson.  The article on Purposefully Planning a Lesson can be found here.


Mode of Instruction: Teamwork           Purpose: Articulate understanding through mathematical discourse

Objective: To build procedural fluency from conceptual understanding, students share self-generated, flexible strategies to team members. Teacher monitors the development of strategies over an extended period of time.

Team Member (1) acts as a coach to teach the other members about a topic or concept. Team member (1) shares self-generated strategies to build team comprehension. The team members acting in the role of player may ask clarifying questions.

  • Team Member (1) assumes the role of coach.

  • Team Members (2), (3), and (4) assume the role of player.

  • Team Member explains a topic or concept.

  • Team Members (2), (3), and (4) ask clarifying questions.

Pose Purposeful Questions

One of the eight Mathematics Teaching Practices from Principles to Actions that needs to be a consistent component of every mathematics lesson. Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships.

Position Papers

Articles written that present a position about a topic to an audience that the opinion presented is valid and worth listening to.

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