CPM Glossary

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Says-Means-Matters Protocol

A reading strategy used with a longer reading passage to aid in understanding where a reader reads a passage/article, describes what it says, interprets what they think it means and writes why they think it matters.

Second CPM Principle of Assessment

Teachers need to read and work through all assessment items carefully before giving them to students, making sure it is clear what kind of response is expected and that there are no errors.


  • CRUCIAL STEP - What did you want to highlight?
  • Purposefully select those that will advance mathematical ideas

Selecting is Step 3 of the 5 Practices for Orchestrating Productive Math Discussions

Sentence-Phrase-Word Protocol

A reading strategy where group members read the text silently selecting a meaningful sentence that captures a core idea; a moving, engaging or provoking phrase; and a powerful word or one that captures attention.  Discuss and record group choices. Looking at the groups’ choices of words, phrases, and sentences, reflect on the conversation by identifying emerging themes, implications and/or aspects of the text not yet captured.


  • In what order do you want to present the student work samples?
  • Do you want the most common? Present misconceptions first?
  • How will students share their work? Draw on board? Put under doc cam?

Sequencing is Step 4 of the 5 Practices for Orchestrating Productive Math Discussions

Share Math Authority

The idea that authority should be “shared” between the teacher and the students—that authority should be openly co-constructed between all the individuals involved in the classroom.  This is an important step in getting students to take ownership in team collaboration and in their own learning.

Shareable Content Object Reference Model

Also known as a SCORM, is a collection of standards and specifications for web-based electronic educational technology. It defines communications between client side content and a host system, which is commonly supported by a learning management system.

Shared Authority

The idea that authority should be “shared” between the teacher and the students—that authority should be openly co-constructed between all the individuals involved in the classroom.  This is an important step in getting students to take ownership in team collaboration and in their own learning.

Silent Debate

Mode of Instruction: Partner           Purpose: Present logical arguments

Objective: To use and connect mathematical representations, two students write about concepts and strategies while critiquing the understanding of others. The teacher monitors through circulation.

To improve writing and communications skills, students are prompted to write clear and concise statements about topics. The process is similar to oral debates, except that it is silent. Partners are assigned a topic and one partner writes pro statements while the other responds with con statements. One paper and pencil is shared by the partners. The pro partner begins and writes a statement in favor of the prompt. The con partner reads the statement and writes a statement against it or against the original prompt. The process continues.

  • Students work in pairs.

  • Partner (1) is assigned the pro (for) position, Partner (2) takes the con (against) position.

  • Partners share a pencil and one sheet of paper. A prompt or topic is given by the teacher.

  • Partner (1) makes a pro, or supportive statement in writing.

  • Partner (2) reads the statement, and writes a comment against.

  • Process continues—three or four times.

Six Word Synthesis Protocol

A reading strategy where text is read and marked to gain an understanding of the ideas and applications. Ideas about the reading are synthesized into only six words which could be a sentence, phrase, connection, personal learning, or an Aha. Each member then shares his/her words with the group along with an explanation.  The group could then create a six word synthesis with all of the words.

Standards for Mathematical Practice

Also known as SMPs, are enumerated in the Common Core State Standards and describe varieties of expertise that math teachers should strive to develop in their students. CPM lessons are aligned to the SMPs, which can be found in the Mathematical Practices section of the Teacher Notes.


Status is the perception of students’ academic capability and social desirability.  Status will play a role in all classrooms and in all teams.  To support Collaborative Learning, teachers must continually monitor status and take action to raise a student's status using strategies such as Team Roles and STTS.

Stoplight Problems

Ip5p-8whUW-hfnbVQPYsywoibpnKieXHEowXeW6Ole8u0PD38cQwgrvQceFmL7zmUOAkynL1QJe0T0q-wxuA8FbwD4hHgvEEsHiYgCfyzAwMUF9mQGalNbL_vZrhXXkssoQMMDsy Problems identified with a Stoplight icon. The Stoplight signifies that the problem contains an error, in reasoning or procedure. Stoplight problems often contain multiple subproblems, not all of which contain an error. The question often asks the students to find the error and explain why it is wrong or to solve it correctly.

Strength in Numbers

A book on collaborative learning in secondary mathematics by Ilana Seidel Horn.

Study Team and Teaching Strategies

Sometimes referred to as STTS, these strategies help structure effective collaboration among students. They are set up with particular ways for students to interact. Some are useful for brainstorming, for creating individual think time before team discussion, or for ensuring that all students have an opportunity to be vocal in a discussion. 

Success Criteria

Success Criteria explain how students can demonstrate a Learning Goal.  Success criteria often use words such as explain, describe, model, show, write, justify, or create.  In instances where hinge questions are used, success criteria may designate a particular part of the lesson.

Students will be able to explain the rule and growth after question 4-14b.

Suggested Assessment Plan

The Suggested Assessment Plan is in the Teacher Notes of each Chapter Opening.  It provides suggestions for Team Assessments, Participation Quizzes, and Individual Assessments.  The problems listed in this plan can be shared with students via a Learning Management System in order for teachers to be transparent about the connections between Review & Preview and Summative Assessments.

Support Productive Struggle in Learning Mathematics

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 consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.


Mode of Instruction: Teamwork           Purpose: Share ideas

Objective: To facilitate meaningful mathematical discourse, teams swap two members with another team to compare and analyze approaches to problem solving. The teacher monitors through circulation and makes ongoing instructional decisions.

Teams work on problems where ideas are generated, strategies are formed, and solutions are developed. At times, teams reach a point of impasse. When this happens, this strategy can be used to allow team members to swap ideas with other teams. Two team members rotate to the next team to exchange strategies and ideas with that team. After sharing, students return to their original teams to share new understanding.

  • When a team task is partially finished, one pair from each team rotates to the next team.

  • Pairs from the two teams exchange ideas, solutions, thinking, etc.

  • Pairs return to their original teams and each pair shares what they learned from other teams.

  • Teams continue to work on problems.

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