CPM Glossary


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1

1-2-3 Reading Protocol

This is a reading strategy where teachers select 1 practice or challenge to work on in the classroom, list 2 reasons for working on it, and list 3 tasks needed in order to be successful.  


5

5 Practices for Orchestrating Productive Mathematics Discussions

The 5 Practices for Orchestrating Productive Mathematics Discussions provides teachers with concrete guidance for engaging students in discussions that make the mathematics in classroom lessons transparent to all.

  1. Anticipating
  2. Monitoring
  3. Selecting
  4. Sequencing
  5. Connecting


6

6 Word Synthesis

1. Read and mark up the text to gain an understanding of the ideas and applications. 

2. Synthesize your ideas about the reading into only six words. Your six words could be a sentence, phrase, connection, personal learning, or an Aha. 

3. Record your six words for presentation to the group. 

4. Be prepared to connect your six words to content in the text.


A

Additional Strategies

Attached is a list of STTS.  Some teachers make a poster of these for their classrooms and place a checkmark next to those that they have tried so that their students become familiar with these strategies.


Ambassador

Mode of Instruction:  Teamwork           Purpose: To have students share strategies

Objective: To support productive struggle in the classroom, students share math authority in their learning by sharing strategies and multiple methods. The teacher can monitor the procedural development of conceptual understanding.


Students are eligible to be Ambassadors once the team has finished problem solving and the teacher has assessed for understanding. An Ambassador is sent to work with other teams to support productive struggle. The Ambassador asks the team questions to guide understanding during problem solving.


  • Teacher appoints Ambassador

  • Ambassadors help other teams



Anticipating

  • Do the problem yourself.
  • What are students likely to produce?
  • Which problems will most likely be the most useful in addressing the mathematics?

Anticipating is Step 1 of the 5 Practices for Orchestrating Productive Math Discussions.


Assessment Practices Outcome 1

Understand CPM and NCTM assessment documents and connect them to instructional practice and assessment decisions


Assessment Practices Outcome 2

Reflect on and make connections between formative assessment and instructional strategies


Assessment Practices Outcome 3

Utilize CPM’s assessment tools and resources


Assessment Practices Outcome 4

Identify appropriate formative and summative assessment topics and strategies for each chapter based on the learning progression


Assessment Practices Outcome 5

Understand the purpose and value of team tests


Assessment Practices Outcome 6

Develop feedback and expectations for all forms of assessments


Attend to precision

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently and express numerical answers with a degree of precision appropriate for the problem context.


B

Board Report

Mode of Instruction: Teacher-led           Purpose: Self-assessment, Collaboration, Discourse

Objective: To facilitate meaningful mathematical discourse, students share and view others' solutions at the board. Teacher monitors learning through circulation and analysis of student work on board.


Teacher monitors student progress while students self-assess work and increase mathematical discourse. This is recommended for questions with short solutions, not for all questions from a lesson. If a problem requires choosing a tool and setting up an equation with many steps to solve, it is best if teams only report the end solution, or part of the solution.


  • Teacher creates a space in the classroom to write a row of problem numbers from the lesson.

  • When teams get to the problem listed on the board report, the team writes their answer on a sticky note.

  • A student from the team goes to the board to place the sticky note and compare to other teams.

  • Teacher monitors student work on board and through circulation. Based on work, teachers may ask specific teams to do a Swapmeet, or I Spy.

  • Repeat this process for each problem listed on the board, with a new student placing the sticky note each time.



Build Procedural Fluency from Conceptual Understanding

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 builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.


C

Carousel: Around the World

Mode of Instruction: Teamwork             Purpose: Brainstorm

Objective: To facilitate meaningful mathematical discourse, students share thinking and generate ideas by viewing multiple rounds of presentations. The teacher monitors learning through posing purposeful questions. 

Teams explore topics or questions displayed on poster paper around the classroom. After a brief discussion—two or three minutes, teams agree on a written statement to add to the poster. Teams rotate several times to discuss additional topics or questions. Teams read the previous written statements before adding to the list. Teacher monitors and determines when to conclude the activity. A Gallery Walk closure provides students time to read all of the written statements.


  • Display topics or questions around the classroom.

  • Provide a different colored marker for each team.

  • Assign one team to each topic or question to start.

  • Teams discuss and agree on a written statement to include about the topic.

  • Teams rotate to the next topic or question and repeat the process every few minutes.

  • For closure, facilitate a Gallery Walk to view all topics or questions.

Carousel: Index Card

Mode of Instruction: Teamwork           Purpose: Brainstorm 

Objective: To facilitate meaningful mathematical discourse, students share thinking and generate ideas by viewing multiple rounds of presentations. The teacher monitors learning through posing purposeful questions.


Teachers or students write one struggle about learning mathematics including time management, Review and Preview, partner work, teamwork, etc., on separate index cards. The index card rotates to other students that offer suggestions to support the struggles.


  • Students record one struggle/question/comment/concern on an index card.

  • Index card rotates within a team of students or to the next team.

  • Students or teams write suggestions on the index card.

  • Rotate the index card several times.

  • Index card is returned to the original student or can be displayed in class for all to benefit from.



Carousel: Station Rotation

Mode of Instruction: Teamwork           Purpose: Review

Objective: To facilitate meaningful mathematical discourse, students share thinking and generate ideas by viewing multiple rounds of presentations. Teacher monitors learning through posing purposeful questions.


Stations include review problems—possibly four to six—placed into a sheet protector. There should be more stations than teams. Teams record written explanations on a prepared sheet—in numerical order—to manage teacher review of work. After teams have completed a written explanation for a station, the paper is submitted to the teacher. Teams rotate to an available station.


  • Stations include several review problems.

  • Set up more stations than teams.

  • Teams record written explanations on a prepared record sheet.

  • Teams check in with the teacher.

  • Teams rotate to an available station.

  • Repeat until time is up or stations are complete.



Checkpoint Problem

LiwTtlrpE6cc1EX3sZA5xKdQEeKCgxLkNeUmx0eQKAL1iFVclzFha2YCV7axRoVmkn_2WRIN26S8GFjEXzWnq9l7qKsKsiXkAWIpg4B7M7OK6iaqphli58DpEWsC0B0HeZLz6aB4 These problems have been identified for determining if students are building skills at the expected level. Checkpoint problems are designed to support students in taking responsibility for the development of their own skills. When students find that they need help with these problems, worked examples and practice problems are available in the Checkpoint Problems section at the back of their book.



Checkpoint Problems

LiwTtlrpE6cc1EX3sZA5xKdQEeKCgxLkNeUmx0eQKAL1iFVclzFha2YCV7axRoVmkn_2WRIN26S8GFjEXzWnq9l7qKsKsiXkAWIpg4B7M7OK6iaqphli58DpEWsC0B0HeZLz6aB4 These problems have been identified for determining if students are building skills at the expected level. Checkpoint problems are designed to support students in taking responsibility for the development of their own skills. When students find that they need help with these problems, worked examples and practice problems are available in the Checkpoint Problems section at the back of their book.


Cognitive Delays in Processing

Students who have the ability to engage with the mathematics but need more time and supports may struggle to keep up.

Supporting Question to Ask:
Can assignments be extended or modified to allow more time? Are there opportunities for extra help available?


Collaborative Learning

Research says students learn ideas more deeply when they discuss ideas with classmates.

Collaborative learning is evident in a classroom when

  • Students and teachers are aware of the purpose for and value of working in teams. 
  • Students and teachers are familiar with team norms and roles.

Concept Goals

Concept Goals are those focused on the mathematics students learn.  Teachers may have particular success criteria attached to these Learning Goals that explain how and when students may demonstrate their proficiency level.


Connect-Extend-Challenge protocol

This is a reading strategy used for longer passages where students are asked to make connections to things they already know, extend their thinking by finding new ideas in the material, identify ideas that challenge them, and then share these with their group or the class.




Connecting

  • Craft questions to make the math visible.
  • Compare and contrast 2 or 3 students' work. What are the mathematical relationships?
  • What do parts of a student's work represent in the original problem? The solution? Work done in the past?

Connecting is Step 5 of the 5 Practices for Orchestrating Productive Math Discussions


Construct viable arguments and critique the reasoning of others

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.


Content Goals

A content goal is linked to a specific content standard or objective.  A content goal example might be "creating equations and inequalities in one variable and using them to solve problems."


Core Problems

If time is limited, use these problems to meet the lesson objectives and the Common Core State Standards.  A problem that is not listed as part of the core is either an extension, an opportunity for deeper understanding, or further practice. Core Problems for each lesson are listed in the Teacher Notes.


Course Notebook

Your course notebook is the place where you record solutions to all of your classwork and Review & Preview problems.  Some teachers ensure that their written solutions are complete with the intent of sharing them with their students.  Some teachers use this area to take notes about formative and summative assessments, including questioning.  You will also want to think about how your students should organize their own Course Notebook. How will you support your students with their notebook organization throughout the school year?


CPM Principles of Assessment

Teachers understand that students learn at different rates and through different experiences. The CPM materials have been designed to support mastery over time through a student-centered, problem-based course, and this approach supports students’ different learning styles. But when changing the materials and changing the methodology, teachers must also change their assessment practices. Teachers cannot tell students they want them to explain their thinking during class and then assess them with only a multiple choice test. Students will quickly realize that “explaining” is not valued enough to be given the time to be assessed.


CPM Workshops

CPM Workshops are a partnership created with teachers and site administration to improve instruction through specialized workshops and coaching.




CPM's Position Paper on Assessment

To ensure all students are afforded the same opportunities for appreciation and success, CPM researches the best practices to support learning. It is on this research that CPM has based its philosophy and methodology for the position paper on assessment.  


CPM's Position Paper on Homework

CPM's philosophy and methodology surrounding homework.  The Review & Preview portion of each lesson is CPM's opportunity for independent practice.  


CPM's Three Pillars of Research

In the seven years since the original CPM Research Report was posted, the new research has continued to validate the efficacy of the three pillars of CPM pedagogy:

  1. Students learn ideas more deeply when they discuss ideas with classmates.
  2. Students learn ideas more usefully for other arenas when they learn by attacking problems—ideally from the real world.
  3. Students learn ideas more permanently when they are required to engage and re-engage with the ideas for months or even years.

These three principles (termed respectively as collaborative learning, problem-based learning and mixed, spaced practice) have driven the development of the CPM textbooks from the beginning, and each year these principles are validated by more research to prove their effectiveness.


D

Dashboard

The dashboard is a virtual location where CPM users can interact with the Learning Management System. It provides at-a-glance views of instructional element outcomes progress.


Deficit Mindset

Deficit thinking refers to the notion that students (particularly low income, minority students) fail in school because such students and their families experience deficiencies that obstruct the learning process (e.g. limited intelligence, lack of motivation and inadequate home socialization). Deficit Thinking does not Support Productive Struggle in Learning Mathematics.

A deficit mindset often results in Educators rescuing students from difficult tasks, and removing the opportunity for productive struggle

  • “Teachers sometimes perceive student frustration or lack of immediate success as indicators that they have somehow failed their students. As a result, they jump in to “rescue” students by breaking down the task and guiding students step by step through the difficulties. Although well intentioned, such “rescuing” undermines the efforts of students, lowers the cognitive demand of the task, and deprives students of opportunities to engage fully in making sense of the mathematics” Principles to Action, pg. 48

Mindsets must shift about what it means to be “successful” in mathematics. Productive struggle should be considered as a valuable part of the learning process. 

  • “Mathematics classrooms that embrace productive struggle necessitate rethinking on the part of both students and teachers. Students must rethink what it means to be a successful learner of mathematics, and teachers must rethink what it means to be an effective teacher of mathematics. “Principles to Action, pg. 49

“Teachers greatly influence how students perceive and approach struggle in the mathematics classroom. Even young students can learn to value struggle as an expected and natural part of learning, as demonstrated by the class motto of one first-grade math class: “If you are not struggling, you are not learning” Principles to Action, pg. 50


Depth of Knowledge

Depth of Knowledge or DoK is another type of framework used to identify the level of rigor for an assessment. In 1997, Dr. Norman Webb developed the DoK to categorize activities according to the level of complexity in thinking.  CPM Classwork problems are often DoK level 3 or 4, while Review & Preview problems are often level 1 or 2.


Descriptive, Effective Feedback

Good feedback improves student learning. It has the following qualities:

Specific: It is a tool for future change.  Ask yourself, "What worked?" or "What does the student understand?" Then ask, "What needs improvement?"  

Actionable: Emphasize what could be done differently rather than what is wrong.  Actionable feedback is often in the form of a question.  "How could you have justified this differently?"

Timely: The most effective feedback is immediate and frequent.  How can your feedback be timely for both formative and summative work?

Respectful:  Make an effort to look for the good while still focusing on future changes. How can this work be an asset for future learning not just for this student but other students?


Discussion Points

Some lessons include questions embedded in the task that study teams should use to guide their discussions, investigations, and problem-solving processes.


Dyad

Mode of Instruction: Partner work           Purpose: Making connections

Objective: To implement tasks that promote reasoning and problem solving, students share without interruption for a short period of time. Teacher monitors through circulation.


Student think-alouds are used in a variety of ways. For example, partners take turns talking about feelings of returning to school. Or, students express concern about math topics on upcoming assessments. A Dyad allows students time to talk without interruption. Each student receives equal time. The listener does not talk; a Dyad is not a conversation. Students maintain eye contact and positive body language.


  • Students share—without interruption—with a partner.

  • Each partner shares for an equal amount of time.

  • Listening partner remains quiet and uses positive body language.




E

Elevator Talk

Mode of Instruction: Partner/Teamwork/Teacher-led           Purpose: Summarize a topic

Objective: To facilitate meaningful mathematical discourse, students summarize understanding of topics through brief conversations. Teachers monitor through circulation techniques.


Students summarize a concept or topic in a 30-second, relatively short, talk. Basically, if someone gets onto an elevator and asks about a problem, you have until the elevator gets to your floor to explain it. Use Proximity Partners or a Whiparound to share out the summaries.


  • Each individual/partner/team is given a topic or concept.

  • Students summarize the topic or concept into a brief presentation.

  • Teacher facilitates a share out of presentations.



Elicit and Use Evidence of Student Thinking

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 evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.


Error Analysis

Error analysis is a method commonly used to identify the cause of student errors when they make consistent mistakes. In this context, it is the students who are analyzing either the teacher's justification or another student's justification.  It is difficult for students to judge the quality of their own math justification unless they have first spent time analyzing someone else's justification.  Having a discussion with students about how the justification could be improved for a particular solution can be used in conjunction with Review & Preview at the beginning of the class period.


Establish Math Goals

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 establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.


Establish Mathematics Goals to Focus Learning

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 establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.


eTool

Technology tools that provide a way to interact with lessons electronically. The Teacher Notes offer suggestions for incorporating this technology into the classroom experience. Some eTools are intended to be used as part of a whole-class demonstration, while others are meant for individuals or pairs of students to use while exploring.


Executive Function

Executive function is a set of mental skills that include working memory, flexible thinking, and self-control. We use these skills every day to learn, work, and manage daily life. Executive functions have very limited capacity due to working memory.


Executive Summary of Research


F

Facilitate Meaningful Mathematical Discourse

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 facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.


Facilitator

The Facilitator helps the teams get started by having someone in the team read the task aloud. Facilitators also make sure each person understands the task and knows how to get started. Before anyone moves on to a new problem, the facilitator makes sure that each team member can explain the team’s answer. 

Typically, a teacher could expect to hear a Facilitator asking:

“Who wants to read?”

“Does anyone know how to get started?”

“What does the first question mean?”

“I’m not sure. What are we supposed to do?”

“Do we all agree?”

“I’m not sure I get it yet—can someone explain?”



Final Say Protocol

A reading strategy where team members read a selection, highlight important information, and select 3 points/quotes that stood out as most important. In small teams, the first person leads by reading their quote aloud. Others take turns responding individually to the quote shared. First person finally explains why they chose the quote, what it meant to them, and any new connections or new thinking that stemmed from the responses of the other team members. Sharing moves to the other group members in the same manner.




Fishbowl

Mode of Instruction: Teacher-led           Purpose: Modeling

Objective: To establish mathematics goals to focus learning, teams are tasked with modeling an activity for other teams. Teacher monitors the goals of the lesson to ensure that there is an appropriate learning progression.


Teacher facilitates one team modeling an activity for the whole class. Students watch the team collaborate on a task while the teacher highlights positive behaviors and team norms. Questions may be asked about the goals for the lesson.


  • Classmates stand in a circle around one team that models an activity.

  • Teacher highlights positive behaviors and team norms.

  • Questions may be asked.

  • Students return to teams and complete activity.


Formative Assessment

All of those activities undertaken by teachers and/or by their students, which provide information to be used as feedback to modify the teaching and learning activities in which they are engaged (Black and Wiliam 1998).  Formative assessment can be viewed as having two parts: checking for understanding and an action taken based on that check.  Both parts can be accomplished by both teachers and students. However, the teacher usually orchestrates effective formative assessment.


Fortune Cookie

Mode of Instruction: Teamwork           Purpose: Elicit responses to a prompt before discussion

Objective: To facilitate meaningful mathematical discourse, students share understanding of a topic to analyze and compare other student approaches to the same topic. Teacher monitors through effective circulation.

Teams receive five or six sentence starters (fortunes) related to topics/concepts/norms in an envelope. Team Member (1) draws a sentence starter from the envelope, reads it aloud, and shares a brief—30 second—explanation. Team Member (2) receives the same sentence starter, makes one comment about Team Member (1)'s explanation, and rotates to continue the process for each member.

Team member (2) draws a sentence starter from the envelope, reads it aloud, and shares. Team member (3) receives the same sentence starter, makes one comment about Team Member (2)'s explanation. Continue this rotation for each of the sentence starters in the envelope.

Teacher circulates to make instructional decisions about team discussions. For closure, each sentence starter (fortune) is included on a separate poster—not previously mentioned to the students. Teams rotate through each poster location to add one comment/explanation about the sentence starter.


  • Teams receive sentence starters (fortunes) in an envelope.

  • Team Member (1) reads one sentence starter and shares a brief explanation.

  • Team Member (2) receives the same sentence starter and comments on Team Member (1)'s explanation.

  • Rotate to each team member.

  • When complete, Team Member (2) reads one sentence starter and shares.

  • Team Member (3) comments on Team Member (2)'s explanation.

  • Continue the rotation through all sentence starters (fortunes).

  • Posters with each fortune are displayed around the room after the activity is complete.

  • Each team adds one comment to each poster for closure.



Four “A’s” Protocol

A reading strategy where the group reads the text silently, highlighting and writing notes in the margin on post-it notes in answer to these four questions: What assumptions does the author of the text hold? What do you agree with in the text? What do you want to argue with in the text? What parts of the text do you want to aspire to? Discussion then occurs within the group to talk about the text in light of each of the “A’s”, taking them one at a time. What do people want to argue with, agree with, and aspire to in the text? 




Four Corners Jigsaw

Purpose: Build understanding of a large quantity of information

A study team and teaching strategy where teams work collaboratively to understand a large quantity of information. There are various ways to organize the jigsaw activity, but the central concept is that teams of people are assigned or select topics that they teach to others.  The teams decide collectively how they are going to share what they know. One way to acquire knowledge from a large amount of material is to break it into smaller pieces.  Each student becomes an expert for part of the material and then shares their knowledge with their team.  


Fourth CPM Principle of Assessment

Formative assessment is a learning experience for both the student and the teacher.


Further Guidance

Some lessons include additional support for students immediately following the task statement. This section of the lesson has step-by-step instructions for students to follow. Several large investigations in this course will have this structure. This design allows the teacher either to have teams attack the problem using their own strategies and available tools or to have students follow a more directed approach using Further Guidance. The beginning and end of each Further Guidance section is clearly marked.


G

Gallery Walk

Mode of Instruction: Teamwork/Teacher-Led           Purpose: To share ideas

Objective: To elicit and use evidence of student thinking, students share math authority by explaining understanding and critiquing the work of others. Teacher monitors through circulation and questioning.

Teams display posters or presentations. Students explain and critique as individuals/teams rotate about the classroom. Rotations are completed quietly—Museum Walk—or through discussion—Gallery Walk. If the teacher decides to allow feedback, students provide positive feedback Two Stars and a Wish or Glow and Grow.


  • Teams display posters or presentations.

  • Students explain and critique displayed work.

  • Students rotate to each location.

  • Feedback is given with Two Stars and a Wish or Glow and Grow.


Gaps in Understanding/Learning

Students who understand the bigger picture of the mathematics but have gaps in the skills necessary to complete the task, or students who possess discrete skills but do not understand how to put their skills to use, will struggle to progress.

Supporting Question to Ask:

Does the student need additional learning opportunities to fill in learning gaps?


Give One - Get One

Mode of Instruction: Partner Work           Purpose: Share ideas

Objective: To facilitate meaningful mathematical discourse, students give and receive information about a concept to build shared understanding. Teacher monitors through circulation.

Students explain and critique ideas with members of the class. For example, students write three ideas on separate note cards for creating positive team norms. Students circulate to give one idea to a classmate, while they get one idea. Student names are recorded next to the idea. For closure, a volunteer reads an idea from a classmate, and then the named person continues to share another idea. Allow many to share.

  • Students record three ideas to share about a given topic.

  • Students circulate and share ideas.

  • For each idea the student gives, they get one in return to record on paper - including the name of the student who gives the idea.

  • After many ideas are gathered, the teacher asks a volunteer to read an idea from a classmate and their name.

  • Named classmate then shares the idea of another classmate and the sharing process continues.



Glow and Grow

Mode of Instruction: Independent/Teamwork           Purpose: Self/Peer assessment

Objective: To establish goals to focus learning, students self-assess strengths and areas for growth. Teacher monitors the shared understanding to guide instructional decisions.

Students use think time to write one topic, team norm, or idea that is a strength and one topic, team norm, or idea where improvement is needed. Students use Glow and Grow to provide feedback team posters, topics, team roles, team norms, assessment, goals.

  • Students share one topic, team norm, or idea that is a strength—Glow.

  • Students share one topic, team norm, or idea where improvement is needed—Grow.



Go-Around One Protocol

A reading strategy where persons in the group read the text silently, highlighting or using Post-it notes to identify those parts in the text that raise questions, confirm beliefs, cause “aha” thoughts, conflict with beliefs, cause reconsideration of prior assumptions and/or show constraints of the problem or topic.  One person reports one idea that he or she recorded while other group members listen, but do not question. The next person does the same until all group members have reported. The group discusses ideas that were reported. 




Golden Line Protocol

A reading strategy where persons in the group read the text silently, highlighting or using Post-it notes to identify those parts in the text that raise questions, confirm beliefs, cause “aha” thoughts, conflict with beliefs, cause reconsideration of prior assumptions and/or show constraints of the problem or topic.  Each person then chooses two different “Golden Lines” that they want to share with the group.  Taking turns, members direct others to their line, reading it and explaining the significance. Once everyone has shared, the whole group discusses together.




GPS

Mode of Instruction: Teamwork/Teacher-led           Purpose: Task Completion

Objective: To implement tasks that promote reasoning and problem solving, students move throughout the problem set to ensure multiple opportunities for learning conceptually. Teacher monitors through circulation and questioning.

To help teams navigate through a problem set, the GPS strategy is a visual road map of the tasks. Students know the final destination to achieve the goals for the lesson. One member of the team reports the progress on a one-quadrant grid displayed in the room, where the team travels along one axis and the problem numbers travel the other axis. Teachers adjust instruction throughout the lesson monitoring where each team is located. Teachers may send Ambassadors from teams that are further along in the lesson to deepen understanding, or they may utilize a Swapmeet with teams that are similarly located on the grid. If all teams land at one-point on the grid—unable to continue—the teacher may conduct a Huddle to progress teams past that point.

  • A one-quadrant grid is displayed for the class—on whiteboard, reusable laminated grid, paper or electronic.

  • Teams are informed of the goals of the lesson included within each problem.

  • One team member checks off progress on the grid as each problem is completed.

  • Teacher makes instructional decisions about STTS used to support teams and/or closure.



Guiding Principles

These beliefs guide CPM throughout course implementation which are rooted in research about collaborative learning, problem based learning, and mastery over time.


  1. Students' involvement in effective study teams increases their ability to learn mathematics.
  2. Students have significantly better retention of mathematics when concepts are grounded in context.
  3. Students deepen their mathematical understanding when they engage with concepts over time.
  4. Effective study teams are guided, supported, and summarized by a reflective, knowledgeable teacher.
  5. Assessing what students understand requires more than one method and more than one opportunity.
  6. When students and stakeholders embrace a growth mindset, they understand that mastery takes time, effort, and support.

H

Homework Help

This online tool links each homework problem in the eBooks to the same problem at homework.cpm.org.   Homework help works on most computers and mobile devices.  Many of the problems include interactive eTools to help students visualize and understand the problem.


Hosted Gallery Walk

A Gallery Walk is a STTS where teams display posters or presentations. Students explain and critique as individuals/teams rotate about the classroom. Rotations are completed quietly—Museum Walk—or through discussion—Gallery Walk. If the teacher decides to allow feedback, students provide positive feedback Two Stars and a Wish or Glow and Grow

  • Teams display posters or presentations. 
  • Students explain and critique displayed work. 
  • Students rotate to each location. 
  • Feedback is given with Two Stars and a Wish or Glow and Grow.


Hot Potato

Mode of Instruction: Teamwork           Purpose: Collaboration

Objective: To support productive struggle in learning multiple step procedures, teams grapple with mathematical problems and explain their thinking. Teacher monitors through circulation and purposeful questioning.

Students practice concepts before mastery or review multiple step problems. Teams receive one sheet of paper. Each team member is provided a different colored writing utensil. Team Member (1) records the first step of the strategy, and rotates the paper to Team Member (2). Team Member (2) corrects mistakes, explains the step aloud, and records the second step. The rotation continues until the problem is completed. Teacher decides if each team member signs the paper agreeing to everything that was written down.


  • Team uses one sheet of paper to solve a problem or practice a procedure.

  • Each team member uses a different colored writing utensil.

  • Team member (1) records the first step of the strategy.

  • The paper rotates to the next team member to correct mistakes and explain and write the next step.

  • Continue the rotation until the problem is completed.


Hot Seat

Mode of Instruction: Teamwork           Purpose: Individual accountability during teamwork

Objective: To build procedural fluency from conceptual understanding, students use opportunities to develop strategies for similar mathematical problems. Teacher monitors strategies for problems.

Students demonstrate individual accountability by working on a problem individually away from team members. While mastery of a concept is not yet expected when using this strategy, students have worked with similar problems through collaboration in teams. One chair is placed in the front of the classroom for each team. Team Member (1) sits in the chair to work on the same problem as other team members. Team Member (1) remains silent while teams discuss and record strategies. Teacher assesses both Team Member (1)'s work and teamwork. Two points are assigned for Team Member (1) and one point for correct team responses. Team Member (2) rotates to the Hot Seat to continue the strategy.

  • One chair—Hot Seat—is placed in the front of the classroom for each team.

  • Using Numbered Heads, Team Member (1) sits in the Hot Seat.

  • Teacher gives everyone a problem to work on for a specified amount of time.

  • Hot Seat team member (1) remains silent, but teams discuss and record strategies.

  • Teacher assesses individual and team explanations.

  • Hot Seat team member (1) receives two points for correct answers.

  • Teams receive one point for correct answers.

  • Team Member (2) rotates to the Hot Seat to continue the strategy.



Huddle

Mode of Instruction: Teamwork           Purpose: Deliver instruction; collect information on team progress

Objective: To facilitate meaningful mathematical discourse, one team member is provided with information that will allow teams to analyze and compare student approaches to problem solving. The teacher monitors which teams need support to facilitate discourse.

Teacher monitors the learning progression of teams, disseminates information, or seeks consensus on topics or concepts. Teacher calls one team member to a Huddle in the classroom. Teacher shares a piece of information or checks for understanding. Team member (1) returns from the Huddle to share with other team members.

  • One team member is called to a location in the classroom.

  • Teacher shares a piece of information or checks for understanding.

  • Team member (1) returns to the team and shares information.



I

I Have...Who Has...

Mode of Instruction: Teacher-led            Purpose: Elicit final reflective comment

Objective: To facilitate mathematical discourse, students share understanding of a concept/topic and critique the sharing of others. Teacher makes instructional decisions about the concept/topic based on sharing.

To review, build vocabulary, connect mathematical representations, or connect mathematical threads, Teacher leads I Have...Who Has... Student receives a card with one problem and one answer. Student (1)—starter card—states, "Who has...[problem]." Student (2)—with the solution—says, "I have...[answer]." This continues throughout the set of cards.

This strategy may be modified for independent practice, partners, or teams. Consider time restrictions and multiple rounds.

  • Student receives a card with one problem and one answer to a different problem.

  • Student (1) asks, "Who has..." and states the problem.

  • Student (2)—with the solution—says, "I have..." and states the answer.

  • Process continues until all problems have answers.



I Spy

Mode of Instruction: Independent/Teamwork           Purpose: Share ideas

Objective: To support productive struggle in learning mathematics, students visit other teams to generate ideas about a concept/topic. Teacher makes instructional decisions after posing purposeful questions.

A team becomes stuck while problem solving. The Resource Manager becomes a spy for the team and silently circulates to listen for ideas from other teams. The Resource Manager refrains from any team interactions while spying on other teams. The Resource manager reports back to the team to share information about the problem.

  • Team becomes stuck at one point while problem solving.

  • The Resource Manager circulates around the classroom, silently listening to, or spying on the teams' work.

  • The Resource Manager reports back to the team to share information about the problem.



I Used to Think..., Now I Think... Protocol

reading strategy where group members read text and then reflect using “I used to think...” and “Now I think…”  Responses are shared with partners, groups and/or the whole class.


Implement Tasks that Promote Reasoning and Problem Solving

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 engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.


Implementation Support Visits

Also known as ISVs, these classroom visits are conducted by a trained and experienced CPM specialist using CPM's Implementation Progress Tool in order to provide individual support, non-evaluative feedback, and an opportunity for teacher reflection. Each new teacher is eligible to receive up to two visits per year during the first two years of implementation.


Instructional Strategies Outcome 1

Apply knowledge of NCTM’s Mathematics Teaching Practices and connect them to instructional strategies


Instructional Strategies Outcome 2

Implement instructional strategies that support the Standards for Mathematical Practice


Instructional Strategies Outcome 3

Understand how multiple modes of instruction ensure access for all students


Instructional Strategies Outcome 4

Establish and reinforce routines and roles that clearly define expectations for multiple modes of instruction


Instructional Strategies Outcome 5

Understand how intentional circulation and purposeful questioning provide feedback to students and teachers


Instructional Strategies Outcome 6

Experience and reflect on instructional strategies through model lessons and a typical day


Instructional Strategies Outcome 7

Know that lesson closure provides opportunities for students to make connections among key mathematical ideas and provides opportunities to reflect on the math goal


Instructional Strategies Outcome 8

Understand the importance of using a variety of instructional strategies and activities to engage students in chapter closure


Intentional Planning Outcome 1

Create, implement and reflect on purposefully planned CPM lessons


Intentional Planning Outcome 10

Integrate Desmos and other eTools into purposefully planned lessons that engage students with content


Intentional Planning Outcome 2

Create, implement, reflect on and revise an Implementation Action Plan that will guide classroom procedures and expectations


Intentional Planning Outcome 3

Incorporate suggested Universal Access strategies to support all students


Intentional Planning Outcome 4

Plan for intentional use of instructional strategies that support formative assessment


Intentional Planning Outcome 5

Use the course preparation resources to inform individual school decisions


Intentional Planning Outcome 6

Plan each chapter using the opening teacher notes


Intentional Planning Outcome 7

Plan for intentional use of instructional strategies that support status and equity


Intentional Planning Outcome 8

Incorporate multiple modes of instruction to support all learners


Intentional Planning Outcome 9

Utilize the Implementation Progress Tool to reflect on student learning and instructional strategies


IRE

The most common type of teacher to student discourse is IRE.  The teacher Initiates a question, the student Responds, and the teacher Evaluates the response. This type of discourse is the most widely used discourse technique but valuable only in a limited number of situations such as T: "Can someone tell me when the test is" S: "Tuesday" T: "Correct".  However, in most cases IRE is not very effective at getting students to think or for that matter motivating them to try. There is too much risk and it gives failure a bad rap. In addition, it positions the teacher as the only one in the room that can be correct.  Talk Moves are a better choice.


J

Jigsaw

Mode of Instruction: Teamwork           Purpose: Build understanding of information

Objective: To promote reasoning about a mathematical concept, team members share responsibility for learning different parts of a concept. Each member teaches other members about that part.

Each team member becomes an expert for one part of a topic or concept. Large reading assignments are broken into four parts, and each member of the team receives one part of the reading. Use Numbered Heads to assign a number to each team member. Team Member (1) learns about one part and prepares to share that learning. Team Members (2), (3), and (4) repeat this same process.

Jigsaw (Four Corners) - (Jigsaw variation)

Each team member becomes an expert for one part of the material presented. Each team member reports to the corner assigned, reads the assigned part, discusses with other students in that corner, decides on an explanation, and returns to the original team. Team members then take turns sharing with their team.

  • Each team member is assigned a different part of a topic or concept.

  • Team member (1) learns about the topic or concept.

  • Team member (1) presents the information to the team.

  • Team Members (2), (3), and (4) repeat this same process.



L

Lack of Mathematical Confidence

A student who lacks the ability to produce the desired result or perceives they lack the ability to do so, will be less likely to try when failure is certain.


Supporting Questions to Ask:

Does the student have a fixed mindset or in rare cases a significant learning disability? 

Are there a variety of opportunities and methods for students to demonstrate their mathematical understanding?


Lack of Motivation

Students may appear to be unmotivated when they have several root causes of unproductive struggle. Additionally, a student’s priorities may lead to lack of motivation in class.

Supporting Questions to Ask:

What matters to this student? 

Is there a way to relate the problem to something he or she cares about, or allow them to use their talent/interest in a way that benefits the team?


Learning Goals

Learning Goals specify the learning that is intended for a lesson.  Learning goals are usually restricted to a single lesson and may refer to understanding (i.e. a portion of the Lesson Objective), knowledge, skills, or applications.  They may also reference a process for doing math such as the Standards for Math Practice, or behaviors such as modeling quality collaboration.  These goals may use words such as know, develop, become fluent, apply, understand, use, or extend.  They are often accompanied by success criteria.  They can also be identified by their function:  concept goals, process goals, or product goals.


Learning Log

journal icon final  A writing tool used by students to reflect about understanding of mathematical concepts, consolidate ideas, develop new ways to describe mathematical ideas, and recognize gaps in understanding. It can also be used as a resource to refresh your students' memories.  Learning Logs are most powerful when they are revisited and edited with new understanding.


Learning Management System

Also referred to as the LMS, is a software application system that monitors professional development for CPM users; keeps track of progress; allows interaction with others and CPM specialists.


Listening Post

Mode of Instruction: Teamwork          Purpose: Focus attention

Objective: To establish mathematics goals to focus learning, specific roles situated actions to move through a learning progression. Teacher monitors each team role through circulation.

In teams, two team members are mathematicians and two team members are observers. Team Member (1) and Team Member (2) problem solve, sharing explanations aloud. Team Member (3) listens to Team Members (1) and (2) and asks clarifying questions, as needed. Team Member (4) records observations about explanations and attitudes of participants, but Team Member (4) remains silent throughout the activity. After the assigned time—15 minutes—Team Member (4) shares notes and observations. Team Members (1), (2), and (3) may share their perspectives, as well. Variations of this activity include multiple rounds with the roles rotated to other members.

  • Team Member (1) and Team Member (2) work on a math problem, explaining aloud.

  • Team Member (3) listens and may ask clarifying questions.

  • Team Member (4) only records what is discussed and verbalized—observes body language, team norms, or another focus area.

  • After 15 minutes, Team Member (4) shares notes and observations.


Literacy Tab

A section in the Teacher tab of the Navigation Bar in the eBook that gives teachers strategies to support students struggling with the English language, specifically reading and writing; includes Introduction, Literacy Guide, Student Strategies, Team Strategies, and Reading Strategies.


Look for and express regularity in repeated reasoning

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x2 + x + 1), and (x - 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.


Look for and make use of structure

Mathematically proficient students look closely to discern a pattern or structure. Students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.


M

Make sense of problems and persevere in solving them

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. 


Mastery Over Time

CPM's Mixed, Spaced Practice provides students an opportunity to achieve conceptual understanding over time.  Students must have the opportunity to engage meaningfully with and make sense of concepts before they are expected to have mastery.

Math Chat

Mode of Instruction: Independent/Teacher-led           Purpose: Silent reflection

Objective: To establish mathematics goals to focus learning, students silently contemplate topics that generate ideas within learning progressions. Teacher observes to make instructional decisions.

Students participate in the silent activity to reflect, summarize ideas, generate ideas, assess learning, or solve problems. Display posters with one topic or concept on each poster. Students use a writing utensil and circulate to each poster. Student (1) adds one brief note or explanation to the poster. Time for activity varies depending on the topic. You may want to consider using a timer to help pace the time at each poster.

  • Display posters with one topic or concept per poster.

  • Student has one writing utensil.

  • Silently, Student (1) circulates to each poster, writing a brief note or explanation on each one.

  • After rotation is complete, students return to seats.


Math Notes

Appearing routinely throughout the text, Math Notes consolidate core content ideas, provide definitions, explanations, examples, instructions about notation, formalizations of topics, and occasionally interesting extensions or applications of mathematical concepts. These boxes enable students to reference ideas that they missed or have forgotten.


Mathematical Content Outcome 1

Experience team-worthy math problems


Mathematical Content Outcome 2

Work through lessons to understand how the learning progressions support the coherence of the program


Mathematical Content Outcome 3

Identify and provide opportunities for students to make sense of the math goal throughout the lesson


Mathematical Content Outcome 4

Experience how a conceptual understanding of math leads to procedural fluency


Mathematical Content Outcome 5

Understand the use of mathematical strategies, structures and tools to develop conceptual understanding


Mathematical Content Outcome 6

Engage with the opening and closure activities through the chapter snapshot


Mathematical Content Outcome 7

Experience Desmos activities and eTools that support conceptual understanding


Mathematical Content Outcome 8

Understand the value and purpose of chapter 1


Mathematical Content Outcome 9

Deepen their own content knowledge


Mathematics Agency

Math Agency is Math Identity in action and the presentation of one’s identity to the world.  



Mathematics Identity

Mathematics identity includes beliefs about one’s self as a mathematics learner; one’s perceptions of how others perceive them as a mathematics learner, beliefs about the nature of mathematics, engagement in mathematics, and perception of self as a potential participant in mathematics.




Mathematics Teaching Practices


Metacognition Icon

Although a metacognitive approach is embedded throughout CPM's curriculum, many lessons in the Pre Calculus text contain specific prompts that encourage students to stop and reflect on becoming a better learner.


Micro Lab Protocol

A reading strategy where group members read a text and then share individually for two minutes while other members listen attentively without comment or interruption. Pause for 30 seconds of silence to take in what was said. This is repeated and then a group discussion occurs referencing the comments that have been made and making connections between the responses.


Mixed, Spaced Practice

Research says students learn ideas more permanently when they are required to engage and re-engage with those ideas for months or even years. 

Mixed, Spaced Practice is evident in a classroom when

  • both individual lessons and chapters are followed, using suggested pacing. 
  • Review & Preview problems are assigned and valued as an essential part of learning.

Model with mathematics

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.


Monitoring

  • Listen, observe, identify key strategies
  • Keep track of approaches
  • Ask questions of students to get them back on track or to think more deeply

Monitoring is Step 2 of the 5 Practices for Orchestrating Productive Math Discussions 


More Knowledgeable Other

Describes the role of the teacher as proactively supporting students' learning through co-participation. Stresses the importance of designing learning environments that support problematizing mathematical ideas, giving students mathematical authority, holding students accountable to others and to shared disciplinary norms, and providing students with relevant resources (Engle & Conant, 2002).




Multiple Modes of Instruction

Teachers use a variety of instructional strategies to engage students in teamwork, partner work, individual work, teacher-led discussions, presentations, and more.  Multiple modes of instruction provides differentiated learning opportunities for student engagement in a collaborative classroom.


MULTIPLE WAYS TO ACCESS INFORMATION AND KNOWLEDGE

This UDL principle connects to the Second Math Teaching Practice

Practice 2: Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.


N

Notice & Wonder

Mode of Instruction: Independent/Teamwork           Purpose: Introduce concepts

Objective: To establish mathematics goals to focus learning, students respond to two prompts as a topic is introduced. Teacher monitors notices and wonders to make instructional decisions.

Students view a picture, math problem, peer work, favorite mistake. Students critique a team poster, observe a team for their teamwork, etc. Students are prompted with the questions—What do you notice? and What do you wonder?

  • Student (1) receives a topic, picture, piece of work, math problem, sample student work, reading, etc.

  • Complete the prompt: I notice...

  • Complete the prompt: I wonder…



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.



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.



R

Reading protocols

Strategies useful for reading and processing longer passages in a team.


Reason abstractly and quantitatively

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize and the ability to contextualize. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.


Reciprocal Teaching

Mode of Instruction: Partner Work           Purpose: Articulate understanding

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

This is an activity that can be done at any time during the class period to check for understanding. At the beginning of the period, students could be asked to explain a concept or definition from the previous day or week. It would serve to remind them of what they have recently learned. Or this might be a way to introduce the topic of the day by remembering some of the parts from previous units which lead up to the new concept. If reciprocal teaching is used during the middle of class it can be used to check the comprehension of what is being learned in that day's lesson. The teacher listens to as many pairs as possible to formatively assess what they have learned and to identify questions and points of confusion.

  • Students work in pairs.

  • Partner (1) pretends that Partner (2) was absent and explains a concept.

  • Switch roles and Partner (2) pretends that Partner (1) was absent and explains a second concept.

  • Teacher circulates to listen to as many pairs as possible.



Recorder/Reporter

The Recorder/Reporter shares the team’s results with the class (as appropriate) and serves as a liaison with the teacher when he or she has additional information to share with the class and calls for a Huddle. In some activities, a they may make sure that each team member understands what information he or she needs to record personally. They may also take responsibility for organizing their team members’ contributions as they prepare presentations. 

Typically, a teacher could expect to hear a Recorder/Reporter asking:

“Does everyone understand what to write down?”

“How should we show our answer on this poster?”

“Can we show this in a different way?”

“What does each person want to explain in the presentation?”



Red Light - Green Light

Mode of Instruction: Teamwork/Teacher-led           Purpose: Sequencing of lesson

Objective: To establish mathematics goals to focus learning, teams check in regularly to move through the learning progress in a way that promotes successful completion of tasks. Teacher monitors and adjusts instruction throughout the check-ins.

Teacher ensures that teams verify the correct solution to each problem before proceeding to the next problem. Each team indicates their status—red indicates finished and green indicates in progress. The teacher circulates to red light teams to pose purposeful questions to check for understanding. Students continue to the next problem as instructed by the teacher.

  • Each team has a red cup and green cup, or laminated double-sided red/green card to indicate status.

  • While the team works together on a problem or problems, status is set to a green light.

  • When the team finishes a problem or gets stuck, status is set to red light.

  • Teacher goes over to a red light and poses questions to check for understanding.

  • After speaking with the teacher, teams go to the next problem or set of problems.


    Research Connections Outcome 1

    Gain knowledge of CPM’s Three Pillars of Research and how they are incorporated into the design of CPM curriculum


    Research Connections Outcome 2

    Understand and apply strategies and systems to establish CPM’s Three Pillars of Research in their classroom using the Implementation Progress Tool


    Research Connections Outcome 3

    Build an understanding of NCTM’s Mathematics Teaching Practices, connecting them to the design of CPM curriculum and to their own instructional practice


    Research Connections Outcome 4

    Incorporate knowledge of CPM’s Three Pillars of Research to intentionally build formative and summative assessments


    Research Connections Outcome 5

    Assimilate content from the CPM Newsletters, Position Papers, Principles, and the Teacher Redesign Corp’s (TRC) action research to reflect on and guide instructional practices


    Research Connections Outcome 6

    Use and connect research to instructional practices to formatively assess understanding


    Research Connections Outcome 7

    Examine beliefs about teaching and learning mathematics and its impact on all students based on research


    Research Connections Outcome 8

    Recognize the importance of the Triangle of Teacher Support and how all three components support effective implementation


    Resource Manager

    The Resource Manager gets necessary supplies and materials for the team and makes sure that the team has cleaned up its area at the end of the day. He or she also manages the non-material resources for the team, seeking input from each person and then calling the teacher over to ask a team question. 

    Typically, a teacher could expect to hear a Resource Manager asking:

    “Does anyone have an idea?”

    “Who can answer that question? Should I call the teacher?”

    “What supplies do we need for this activity?”



    Review & Preview

    The section after a lesson consisting of six to ten problems on a variety of topics and skills; a mixed spaced practice approach that leads to higher learning and better long–term retention.




    Review & Preview Feedback Strategies

    These are strategies for processing Review & Preview problems done independently.  


    Rick Rolled

    Rickrolling, alternatively rick-rolling, is a prank and an Internet meme involving an unexpected appearance of the music video for the 1987 Rick Astley song "Never Gonna Give You Up". The meme is a type of bait and switch using a disguised hyperlink that leads to the music video.


    S

    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.


    Selecting

    • 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.


    Sequencing

    • 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

    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.


    Swapmeet

    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.


    T

    Talk Moves

    Although IRE is the most common type of Teacher-Student discourse, talk moves are a better option.  The secret to talk moves is not to evaluate, but question for better discourse without acknowledging if the response was right or wrong.  Talk moves can work at the small team level or in a whole class discussion. Here are the four types of talk moves. (O'Connor and Chapin)

    Elicit Student Thinking (what are the students thinking and saying?)

    • Turn and Talk "Share your thinking with your partner"

    • Revoicing "Are you saying that.....?"

    • Say More "Could you give us an example?" or "Would someone else say that in their own words?" or "I'm not sure I understand what you are saying, could you say more?"

    Orient Students to the Thinking of Others (Are the students listening and understanding what others are saying?)

    • Repeating "Who can repeat what was said?"

    • Sharing Out "What did your partner think?"

    • Surveying Access "Can everyone hear what is being said?"

    • Focusing Attention on Student Thinking "As we listen to this response, think about how it is the same or different than what we talked about yesterday?"

    Deepen Student Understanding (How can I make this more meaningful?)

    • Press for Reasoning "Why do you think that?"

    • Having Reasoning Repeated in Multiple Ways "Who can put that in their own words?"

    • Find a Student Who is Unconvinced "Erin is not convinced, Who can explain why that is true?"

    • Turn and Talk, Prompting students to make sense of reasoning "Talk about Marla's idea"

    Students Response to the Reasoning of Others (How can students build on this idea?)

    • Press for reactions "Do you agree or disagree? Why?" or "What does Marla's statement make you think of?"

    • Compare or Contrast "Is what Erin said the same or different than what Marla said? How?"

    • Invite Challenges "Who sees it differently?" or "Can someone make a counter-argument?"

    • Turn and Talk "Talk to your partner about whether or not you agree with Erin?"



    Task Manager

    The Task Manager keeps the team focused on the assignment of the day. He or she works to keep the team discussing the math at hand and intervenes if anyone is talking outside of her/his team. Additionally, a Task Manager helps the team focus on articulating the reasoning behind the math statements they make as the well as the answers that are proposed.

    Typically, a teacher could expect to hear a Task Manager saying:

    “Ok, let’s get back to work!”

    “Let’s keep working.”

    “What does the next question say?”

    “Explain how you know that.”

    “Can you prove that?”

    “Tell me why!”



    Tasks that Promote Reasoning

    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 engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.


    Teacher Toolkit

    The Teacher Toolkit - Collaboration, Pacing, and Routines is a module within the Professional Learning Portal that provides teacher testimonials from experienced teachers regarding the routines and procedures they use to support student learning in their own classrooms.


    Teacher Transparency

    When implementing new instruction strategies it is not only important to make obvious the intellectual practices involved in completing and evaluating learning tasks, but to explain the intent of your practice to your students.  Each strategy and classroom expectation should be accompanied by an explanation of how that intentional act will positively impact the students' learning.

    Read the article Teacher Transparency , by John Hayes, in the May 2020 CPM Newsletter.  


    Team Roles

    CPM resources are designed around four Team Roles: Resource Manager, Facilitator, Recorder/Reporter, and Task Manager.  Click on the individual roles to see their descriptions.

    Teammates Consult

    Mode of Instruction: Teamwork           Purpose: Team discussion and decision making

    Objective: To establish mathematics goals to focus learning, teams utilize an established routine to begin problem solving by making sure all members know the goals and learning progression.

    Teammates Consult is an effective strategy to use for problem solving and concept development situations. It allows the students an opportunity to think and discuss the problem before actually writing anything down.

    • All pencils and calculators are set aside (no writing).

    • Students read the problem or question individually.

    • Students get approximately 1 minute of individual think time.

    • Students take turns sharing and discussing the problem for clarity.

    • Students share possible strategies or next steps.

    • Teacher gives okay for pencils to be picked up and written work to begin.



    They Say..., I Say..., So What...? Protocol

    A reading strategy where text is read silently and individuals are asked to describe what they say (the authors say) about the topic; interpret what the reader thinks about the topic (I say); and then the reader writes what the topic means to them (so what). This is shared with a partner, group and/or whole class. 




    THINK INK PAIR SHARE-ASYNCHRONOUS

    This could be done on your Learning Management System or on a document that all students have access to.

    You could assign each student a Team Role ahead of time and then pair up team roles (i.e. Facilitators are paired with Resource Manager)

    In your LMS, create a Forum with your prompt. Students write an entry and then read and comment on another a partners entry.  Partners are determined by their team role.

    On a Google Doc, put the prompt at the top.  Have each student write an entry and then read their partners and comment or question on what they read.


    Think Ink Pair Share-Socially Distanced

    This could be done with whiteboards so that the writing is large enough for distanced students to see each others work.

    It could also be done with a Google Doc.  The teacher posts a prompt in a document or sheet. Have each student write an entry and then read their partners and comment or question on what they read.  Teachers would monitor the time and give students verbal feedback about the amount of time left.



    Think Ink Pair Share-Synchronous

    This could be done on a document or a spreadsheet.  The advantage of a spreadsheet is that you could lock some of the columns or rows so that students could not change the data.  

    The teacher posts a prompt in a document or sheet. Have each student write an entry and then read their partners and comment or question on what they read.  Teachers would monitor the time and give students verbal feedback about the amount of time left.

    You could also do this with Private Chats or Breakout Rooms but be mindful that you may not always be able to monitor these chats.


    Think-Ink-Pair-Share

    Mode of Instruction: Independent/Partner/Teamwork           Purpose: Individual reflection prior to discussion

    Objective: To elicit and use evidence of student thinking, students utilize intentionally planned think time before responding to and sharing out understanding.

    To emphasize the importance of think-time, the teacher poses a question/problem for students to silently think about. After a short period of time, students write an explanation to share. When the teacher indicates, partners share explanations. Partners may share within the team or the whole class.

    Think-Pair-Share (Think-Ink-Pair-Share variation)

    Students receive a question—possibly about concepts covered in a unit, Diamond Problems, or mental math—and silently think for a short period of time. Without writing, partners discuss explanations of the question. Partners may then share out with the rest of the team or class.

    • Teacher poses a question/problem.

    • Students think for a period of time—one or two minutes.

    • Students silently prepare an explanation in writing to share.

    • Partners take turns sharing written explanations.

    • Partners may then share out with the rest of the team or class.


    Third CPM Principle of Assessment

    Students should be assessed only on content with which they have been meaningfully engaged, and with which they have had ample time to make sense of.


    Thread

    Related problems and/or lessons intentionally sequenced within and between courses to help students both deepen conceptual knowledge and build procedural fluency.


    Toolkit

    2kzpZqPIrDIYw5aoENW5md3NYg4ED8CeQXj75V0XO_huSaBi8oMjk68ikxT-FT_GdQEzVDRpzs4WnEhkM4WEfY8XeyuBDTDC0YzLmRrwkWk_6MOtVB7XjTnE91C3dekj_ET2jq5d Working documents in which students write Learning Logs, interact with Math Notes and create other personal reference tools.


    Tools and Technology Outcome 1

    Learn how to access CPM’s Synthesis of Research and course preparation resources


    Tools and Technology Outcome 2

    Experience how to engage students with content using Desmos and other eTools


    Tools and Technology Outcome 3

    Understand the structure and organization of the teacher and student ebooks


    Tools and Technology Outcome 4

    Locate Closure and Assessment resources


    Tools and Technology Outcome 5

    Know how to utilize Chapter Opening and Lesson Teacher Notes


    Tools and Technology Outcome 6

    Understand how to access Team Support and Strategies to establish and maintain classroom expectations


    Tools and Technology Outcome 7

    Examine Universal Access and Literacy resources to support the learning of all students


    Tools and Technology Outcome 8

    Learn how to access and navigate CPM’s Learning Management System


    Tools and Technology Outcome 9

    Locate resources to support parent and public relations


    Traveling Salesman

    Mode of Instruction: Teamwork/Teacher-led           Purpose: Communicate mathematical ideas

    Objective: To implement tasks that promote reasoning, students solve a problem and decide on an effective strategy within a team. One team member shares that strategy with other teams.

    Teams receive a topic or problem—same/different from other teams—and a presentation is prepared. Teams plan and practice within teams. Then one team member—Traveling Salesperson—rotates to another team to pitch the presentation. The team asks clarifying questions. Then the Traveling Salesperson returns to the team so that roles may rotate. The use of Numbered Heads allows all team members to rotate. The teacher circulates to make informed instructional decisions about which team member is assigned the role of Traveling Salesperson.

    • Teams receive a topic or problem from the teacher.

    • Teams complete the problem by planning a presentation.

    • Team Member (1) shares the presentation with another team.

    • The process continues with another topic or problem, and roles may rotate.



    Tuning Protocol

    Mode of Instruction: Teamwork           Purpose: Receive feedback

    Objective: To promote meaningful mathematical discourse, students share, analyze, and compare teammates' approaches and arguments.

    Team Member (1) presents an explanation to a problem to other team members in a short period of time—one or two minutes. The presentation may be a portfolio project, a report, a math problem, Learning Log, etc. When the time is up, team members ask Team Member (1) any questions or clarifications—about one minute. Then Team Member (1) turns away from the team while team members discuss the presentation and deepen understanding of the problem—five minutes—, while Team Member (1) listens and writes notes. Team Member (1) turns back and shares a reflection of the discussion. The role may rotate to Team Member (2) and the process continues. The teacher uses discretion for length of activity.

    • Team Member (1) presents an explanation to a problem in teams.

    • Team Members ask Team Member (1) questions or clarifications.

    • Team Member (1) turns away from the team while listening and writing notes.

    • Team Member (1) shares a reflection of the discussion.

    • The role is rotated to Team Member (2) and the process continues.


    Turn and Talk

    Mode of Instruction: Partner Work           Purpose: Share 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.

    To discuss a procedure or concept without writing, one team member explains while the other team member listens. If there is a disagreement, students continue to discuss the solution and agree on a single explanation. When partners have an explanation, they share with the rest of the team.

    • Students work in pairs.

    • Team Member (1) explains while Team Member (2) listens.

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

    • Partners agree on one explanation to share with other members of the team.

    • Roles are reversed for the next problem.



    Two Stars and a Wish

    Mode of Instruction: Independent/Teamwork           Purpose: Reflection and feedback

    Objective: To establish goals to focus learning, students identify strengths and areas for growth. Teacher monitors students’ understanding to guide instructional decisions.

    During any presentation, students record two things they really liked—Two Stars—and one suggestion that might improve the presentation—a Wish. At times, a team member may be present to explain or answer questions about the presentation.

    • Students record two things they like about a presentation.

    • Students record one thing that would improve the presentation.



    U

    Universal Access

    Support for teachers with students of varied abilities and backgrounds across a broad range of abilities. This section in the teacher’s edition ebook under the teacher tab includes an Introduction, Success for Students, Student Struggle, More Help, Special Needs, ELL, Advanced Learners, Unprepared Students and a Conclusion.


    Use and Connect Mathematical Representations

    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 engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.


    Use appropriate tools strategically

    Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations


    W

    Walk and Talk

    Mode of Instruction: Partner           Purpose: Movement and Communication

    Objective: To implement tasks that promote reasoning, students respond to questions or prompts about a topic while walking. The teacher monitors for multiple points of entry into a topic.

    To promote movement during learning, partners receive a topic or concept to discuss while walking—about the classroom, in the hallway, or outside. Partners summarize learning, clarify ideas, or ask questions. Partners report any remaining questions to the teacher after returning to desks.

    • Partners receive topics or concepts to discuss.

    • Partners summarize, clarify ideas, or ask questions while walking.

    • Partners share any remaining questions with the teacher after returning to their desks.



    Whiparound

    Mode of Instruction: Teamwork/Teacher-led           Purpose: Elicit final reflective comment

    Objective: To facilitate mathematical discourse, students share understanding of a concept/topic and critique the sharing of others. Teacher uses this time to make instructional decisions about the concept/topic.

    Students participate in a teacher-led discussion to share a final comment on a topic, concept, or lesson. Students turn in the direction of the speaker—possibly forming a circle around the classroom. Teacher states the topic or problem, and students take turns sharing brief comments. Students have the option to pass on their turn to share.

    Team Whiparound (Whiparound variation)

    Teams participate in a teacher-led discussion to share a final comment. Teams share out only one agreed upon response per team.

    • Teacher provides a prompt.

    • Students take turns sharing brief comments.

    • Students listen while others share.



    Word Wall

    A word wall is a collection of words which are displayed in large visible letters on a wall, bulletin board, or other display surface in a classroom.



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