CPM Professional Learning Portal

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.  

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

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.

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.

• 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.

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

Reflect on and make connections between formative assessment and instructional strategies

Utilize CPM’s assessment tools and resources

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

Understand the purpose and value of team tests

Develop feedback and expectations for all forms of assessments

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.

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.

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.

 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.

 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.

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?

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 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.

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.

• 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. 

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.

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."

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.

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?

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

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

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.

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 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

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.

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?

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

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 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.

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.

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.

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 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.

A summary of the research that supports the Three Pillars of CPM - collaborative learning, problem based learning and mastery over time.

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.

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?”

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.

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.

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? 

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.  

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

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.

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.

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?

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.

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.

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. 

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.

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.

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.

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.

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.

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.

A 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.

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.

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.

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

Implement instructional strategies that support the Standards for Mathematical Practice

Understand how multiple modes of instruction ensure access for all students

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

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

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

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

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

Create, implement and reflect on purposefully planned CPM lessons

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

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

Incorporate suggested Universal Access strategies to support all students

Plan for intentional use of instructional strategies that support formative assessment

Use the course preparation resources to inform individual school decisions

Plan each chapter using the opening teacher notes

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

Incorporate multiple modes of instruction to support all learners

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

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.

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.

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?

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 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.

  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.

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.

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.

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.

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.

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. 

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.

Experience team-worthy math problems

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

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

Experience how a conceptual understanding of math leads to procedural fluency

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

Engage with the opening and closure activities through the chapter snapshot

Experience Desmos activities and eTools that support conceptual understanding

Understand the value and purpose of chapter 1

Deepen their own content knowledge

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

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

Research indicates that these eight practices need to be consistent components of every mathematics lesson

• Establish Mathematics Goals to Focus Learning
• Implement Tasks that Promote Reasoning and Problem Solving
• Use and Connect Mathematical Representations
• Facilitate Meaningful Mathematical Discourse
• Pose Purposeful Questions
• Build Procedural Fluency from Conceptual Understanding
• Support Productive Struggle in Learning Mathematics
• Elicit and Use Evidence of Student Thinking

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.

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.

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.

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.

• 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 

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).

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.

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.

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…

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.

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.

Mode of Instruction: Independent/Teamwork           Purpose: Collaboration

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

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

• Teacher posts a list of strengths.

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

• Students take turns sharing their strengths with their team.

• Students use strengths as they work on the lesson.

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

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

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

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

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

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

• Team Member explains a topic or concept.

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

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.

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

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.

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.

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 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 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.

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 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.

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

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

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

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

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

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

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

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.

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?”

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

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

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

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

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

Use and connect research to instructional practices to formatively assess understanding

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

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

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?”

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.

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

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.

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.

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

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

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

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

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

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

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.

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.

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.

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.

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.

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.

• Make sense of problems and persevere in solving them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated reasoning.

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.

 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.

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

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 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.

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.

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.

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?"

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!”

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.

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.

When implementing new instruction strategies, it is not only important to make obvious the intellectual practices involved in completing and evaluating learning tasks, but also 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.  

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.

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. 

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.

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.

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.

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.

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.

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

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

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

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

Understand the structure and organization of the teacher and student ebooks

Locate Closure and Assessment resources

Know how to utilize Chapter Opening and Lesson Teacher Notes

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

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

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

Locate resources to support parent and public relations

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.

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.

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.

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

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.

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.