Constructivist theory suggests that math is learned when an individual constructs a new mental model to solve a math problem. This requires active engagement as the learner works to fit new information into what he already knows. In a constructivist math classroom, students are not taught a set of rules or methods for solving problems. Instead, they are challenged to find their own solutions to problems. Various ways of arriving at solutions are accepted as long as they are rational. Problems are presented strategically, each one designed to push a learner toward the discovery of more complex math ideas.
Because constructivist theory states that learners construct meaning from sensory input, the constructivist math classroom is rich in manipulatives. Students are given time to explore these materials before using them to solve problems so they can discover the material's unique characteristics and gain an understanding of how to use it. However, simply playing with manipulatives does not produce learning. Materials must be coupled with challenges that engage the mind. In a constructivist math classroom, the specific tool used in any lesson is carefully chosen for its usefulness in leading students to the construction of a math idea.
The constructivist approach believes that social interaction is vital to learning. Learners must verbalize their ideas to clarify their thinking and reach deeper understandings. Therefore, the constructivist math classroom is noisy at times. Students are expected to work through problems in pairs or small groups, explaining their thinking to classmates. At times, students explain their solutions to a large group. The use of language to support their methods is as important to the speakers' mental constructions as it is to the listeners. Discussion regarding the logic of a student's work is encouraged, since this can solidify a learner's thinking.
The constructivist approach incorporates the idea that learners are not just learning, they are learning how to learn. Therefore, as students find ways to solve math problems, they are teaching themselves how to approach other math problems. This requires treating learners as individuals who are at different stages in the construction of various math ideas. It also means that problems are differentiated to allow students to build on structures they already possess. Finally, the constructivist approach requires that concepts be revisited again and again. This "spiraling" allows students the time to ponder concepts and construct deeper understandings.