CPM Glossary


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P

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.



Planning

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.


Players-Coach

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.


Principles of Assessment

Guidance for teachers and all stakeholders regarding assessment practices and suggest teachers create their own tests, work through all assessments, only assess material students have had ample time to engage with, formatively assess as a learning experience for both students and themselves, and be flexible in grading to allow differences in reaching mastery.


Principles to Actions

Also known as PtA, strategies for teachers to engage students in mathematical thinking, reasoning, and sense making to significantly strengthen teaching and learning’ put forth by the National Council Teachers of Mathematics to offer guidance to teachers, mathematics coaches, administrators, parents, and policymakers.


Principles to Actions Executive Summary

An introduction which sets out the Mathematics Teaching Practices - consistent components research states are needed for every mathematics lesson; productive and unproductive beliefs for facing obstacles as well as suggestions for combating the issues; and a call to action to recognize the critical need in education to develop understanding in math education and confidence for all students.




Problem-Based Learning

Research says students learn ideas more usefully for other arenas when they learn by attacking problems.

Problems-Based Learning is evident in a classroom when

  • Students and teachers share math authority as they value and engage in productive struggle
  • Teachers guide without taking over the thinking.

Process Goals

Process goals are designed to build habits, stick to consistent routines, and define success as growth in one's skills and abilities.  For example, process goals might refer to how students complete their work using the Standards for Mathematical Practice.  Teachers may have particular success criteria attached to these Learning Goals that explain how and when students may demonstrate their proficiency level.


Product Goals

Product goals are project-oriented, stick to firm deadlines, and define success by the completion of great work.  For example, product goals might refer to which math problems that students should complete by the end of the day or class period. Teachers may have particular success criteria attached to these Learning Goals that explain how and when students may demonstrate their proficiency level.


Productive Struggle

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.

“…productive struggle comprises the work that students do to make sense of a situation and determine a course of action when a solution strategy is not stated, implied, or immediately obvious. From an equity perspective, this implies that each and every student must have the opportunity to struggle with challenging mathematics and to receive support that encourages their persistence without removing the challenge.”

Boston, Melissa D., Fredrick Dillon, Margaret S. Smith, and Stephen Miller. Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 9-12. Reston, VA: National Council of Teachers of Mathematics, 2017. [p.208]


Professional Noticing

Professional Noticing requires that the teacher be able to: identify relevant aspects of the teaching situation; use knowledge to interpret the events, and establish connections between specific aspects of teaching and learning situations and more general principles and ideas about teaching and learning. Professional noticing is a crucial component of CPM math teacher competency and requires not only knowledge and expertise with mathematics, but also knowledge of the pedagogy associated with using the curriculum as intended.


Professional Outcome 2

Establish professional relationships and learning communities that support lifelong professional growth and a commitment to mathematics education


Professional Outcome 3

Reflect on the efficacy of their instructional practices and share instructional challenges and successes with colleagues


Professional Outcome 4

Recognize the importance of being transparent with their instructional practices with students, colleagues, and administration


Professional Outcome 5

Hold themselves accountable to be prepared to teach their course and commit to purposefully planning each chapter and lesson


Professional Outcome 6

Understand that professional growth develops over time and requires an ongoing commitment to engage in professional learning


Professionalism Outcome 1

Recognize that effective teaching requires implementation of research-based instructional strategies that advance student learning


Proximity Partner

Mode of Instruction: Partner Work           Purpose: Build shared understanding

Objective: To facilitate meaningful mathematical discourse, students build shared understanding through comparing approaches and arguments. Teacher monitors approaches through circulation.

To find proximity partners, students stand up, push in chairs, touch 2 tables/desks, three walls, and a chair. The two students closest together are proximity partners. Partners discuss a topic or concept for a period of time—one or two minutes. Partners thank each other and return to the teams.

Stand-Up Hand-Up Partner-Up (Proximity Partners variation)

To find a partner, students stand up, push in chairs, put a hand up, make eye contact, and put hand down. After all hands are down, partners move to meet and discuss a topic or concept.

  • Students stand up and push in chairs.

  • Students touch 3 walls, 2 tables/desks, and 1 chair.

  • The two students closest to one another are proximity partners.

  • Proximity partners briefly discuss a topic or concept.

  • Proximity Partners return to teams.




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