In the descriptions of classrooms where students are learning with
understanding, instruction is not portrayed as the presentation of clear,
precise explanations of procedures to be practiced by students. Instead,
understanding is constructed by each student as she or he engages in the various mental activities we have found to be critical: actively constructing relationships between and among mathematical ideas by reflecting on problem solutions, extending knowledge by relating the new solutions to what has been known previously, and articulating thinking about the
mathematics they have explored. By engaging in these activities, each learner assumes ownership of the mathematical knowledge that she or he has constructed.