Math Out of the Box™ uses a learning cycle to foster inquiry-based learning. The learning cycle used in the lessons gives teachers the structure that is needed to reach their students, whether they are traditional or inquiry-based in practice. The learning cycle provides teachers with a template that promotes the development of active inquiry and critical thinking. The learning cycle allows students to make connections between past and present learning experiences and is based in the “cognitive principle of assimilation,” which implies that understanding cannot be imposed on the learner, but instead is developed progressively by the learner, beginning with concrete and progressing to abstract opportunities. The learning cycle provides the opportunity for students to share ideas with others and to more formerly connect what they have learned with what they already know. The learning cycle has four parts.

Engage. The engage phase of the learning cycle allows students with varying prior experiences and abilities to make connections between past and present learning experiences. Such connections provide a natural pre-assessment opportunity for both the teacher and student.

Investigate. In the investigate phase of the learning cycle students are given common, concrete experiences that challenge them to solve problems and investigate mathematical ideas. Information is gathered, patterns are observed and analyzed, connections are made and applied, and conclusions are drawn and defended

Reflect. The reflect phase of the learning cycle is where students think about how what they have learned fits into what they already know. The teachers‘ role is especially crucial because it is their knowledge of mathematics that enables them to assist students in summarizing and structuring their thinking into meaningful models of the mathematical ideas they have explored.

Apply. In the application phase of the cycle, students and teachers can assess the depth of understanding of the newly formed ideas. The new knowledge then becomes old knowledge on which to connect new learning—and the cycle of learning continues.