Support productive struggle in learning mathematics. Provide opportunities for productive struggle, which is significant and essential to learning mathematics with understanding. Allow students to grapple with ideas and relationships. Give students ample time to work with and make sense of new ideas, which is critical to their learning and understanding.
Establish mathematics goals to focus learning. Set the stage to guide instructional decisions. Expect students to understand the purpose of a lesson beyond simply repeating the expectation.
Build procedural fluency from conceptual understanding. Provide experiences with concrete materials that allow students to make sense of important mathematics and flexibility choose from a variety of methods to solve problems.
Use and connect mathematical representations. Provide concrete representations that lead students to develop conceptual understanding and later connect that understanding to procedural skills.Provide a variety of representations that range from using physical models to using abstract models.
Elicit and use evidence of student thinking. Elicit and use evidence of student thinking which helps teachers access learning progress and can be used to make instructional decisions during the lessons as well as help to prepare what will occur in the next lesson. Assess student thinking and understanding by using formative assessment through student written and oral ideas.
Pose purposeful questions. Reveal students’ current understanding of a concept. Encourage students to explain, elaborate and clarify their thinking. Make the learning of mathematics more visible and accessible for students.
Facilitate meaningful mathematical discourse. Provide students with opportunities to share ideas, clarify their understanding, and develop convincing arguments. Advance the mathematical thinking of the whole class by talking and sharing aloud.
Implement tasks that promote reasoning and problem solving. Provide opportunities for students to engage in exploration and make sense of important mathematics. Encourage students to use procedures in ways that are connected to understanding.
Teacher asks questions that build on and extend student thinking while being intentional about the kinds of questions asked to make the mathematics more visible to students and using wait time to provide students with time to thinking and examine their ideas. Student thinks more deeply about the process of the mathematics rather than simply focusing on the answer, listening to and commenting on the explanations of others in the class.
Teacher supports student struggle without showing and telling a procedure but rather focusing on the important mathematical ideas and asks questions that scaffold and advance student thinking while recognizing the importance of effort as students work to make sense of new ideas. Student sticks to tasks and recognizes that struggle is part of making sense. Ask questions that will help them to better understand the task and support each other with ideas rather than telling others the answer or how to solve a problem.
Teacher considers broader goals, as well as the goals of the strand(s) and the actual lesson. Student makes sense of new concepts and skills, experiences connections, and deepens their understanding.
Teacher chooses tasks that are built on current student understandings, have various entry points with multiple ways for the problems to be solved, and are interesting to students. Student works to make sense out of the task and persevere in solving problems using a variety of models and materials to make sense of the mathematics in the task and convince themselves and others the answer is reasonable.
Teacher uses tasks that allow students to use a variety of representations and encourages the use of different representations, including concrete models, pictures, words, and numbers, that support students in explaining their thinking and reasoning. Student uses materials to make sense out of problem situations and connects representations to mathematical ideas and structures of big ideas, including operational sense with whole numbers, fractions and decimals.
Teacher engages students in explaining their mathematical reasoning in small groups and classroom situations and facilitates discussions among students that support making sense of a variety of strategies and approaches while scaffolding classroom discussions so that connections between representations and mathematical ideas take place. Student explains the ideas and reasoning in small groups and with the entire class, listening to the reasoning of other and asking questions of others to make sense of their ideas.
Teacher determines what to look for in gathering evidence of student learning, poses questions and answers students questions that provide information about student understanding, strategies and reasoning, then uses evidence to determine next steps of instruction. Student accepts reasoning and understanding are as important as the answer to a problem and uses mistakes and misconceptions to rethink their understanding. Asks questions of the teacher and peers to clarify confusion or misunderstanding and assesses progress toward developing mathematical understanding.
Teacher provides opportunities for students to reason about mathematical ideas, expects students to explain why their strategies work, and connects student methods to efficient procedures as appropriate. Student understands and explains the procedures they are using and why they work, uses a variety of strategies to solve problems and make sense of the mathematical tasks without relying on shortcuts or tricks to do mathematics.
