Conceptual Understanding

Teachers focus on much more than “how to get the answer”; they support students’ ability to access concepts from a number of different perspectives. Students might demonstrate deep conceptual understanding of core mathematics concepts by solving short conceptual problems, applying mathematics in new situations, and speaking and writing about their understanding. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, such students may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, help other students understand a given method or find and correct an error, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

- CA Mathematics Frameworks

Five Practices for Orchestrating Productive Mathematics Discussions

Smith and Stein (2011) identify five practices that assist teachers in facilitating instruction that advances the mathematical understanding of the class:

      1. Anticipating students’ solutions to a mathematics task
      2. Monitoring students’ in-class, “real-time” work on the task
      3. Selecting approaches and students to share them
      4. Sequencing students’ presentations purposefully
      5. Connecting students’ approaches and the underlying mathematics