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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, 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.


  • 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.


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.

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