Students develop conceptual understanding, procedural skills and fluency, and application through the Discover, Practice, and Apply cycle of learning within each Math Techbook concept.
Discover: Conceptual Understanding
The first phase of the instructional cycle, Discover, includes scenarios relevant to the concept being learned. Students are exposed to a rich set of investigations that enable them to:
- actively explore each scenario,
- develop mathematical models,
- make and test conjectures, and
- solidify their own understanding.
Students also develop solutions for problems that appear within the Discover section using what they’ve previously learned or discovered. Within each concept, students encounter multiple problems of increasing complexity based on the conceptual learning target. The problems include a blend of online and hands-on investigations.
Practice: Procedural Fluency
The Practice phase of the instructional cycle develops procedural fluency and reinforces concepts. After demonstrating understanding within Discover, students move to Practice, which offers two types of support: Coach and Play.
With Coach, students get direct feedback on their constructed responses. In this guided pathway, students practice solving exercises similar to those that were solved in Discover. Exercises in Coach are purposefully varied so that students must apply their understanding of the concept to solve them, not just the algorithm or problem-solution method they developed.
In Coach, students’ responses are monitored, and the system provides feedback based on those responses. The feedback is also aligned with the most common student errors and misconceptions.
Play provides independent practice. It consists of exercises through which students practice at their own pace based on the current concept about which they are learning. Unlike in the Coach exercises, students in the Play segment are on their own to complete assignments, and they receive no support on how to answer questions correctly.
Play exercises provide encouragement in a gamified environment where students earn badges and awards. Students make progress toward a badge each time they correctly answer a self-paced, interactive question, and five correct answers within a concept earns a badge.
Apply: Application
In Apply, the final phase of the learning cycle, students are given several contextual problems in which they must independently apply what they have learned during Discover and practiced through Practice. Students are often required to research a topic, and then use their research to construct a mathematical model, pose a solution, and justify their work. Some problems are machine-scored, and students receive feedback on their responses from the computer; others use a digital rubric to allow for easy online scoring by the teacher, and teachers can enter personalized feedback when scoring student responses.
The variety of problems in each concept range from relatively straightforward application of the material to “messy” problems that add variables, require additional thought, expect students to conduct research and make assumptions, necessitate the creation of a mathematical model, and insist that students provide a well-articulated solution with justification.
Apply problems involve students in modeling with mathematics. Teachers are encouraged to use them not only as assessment activities, but also as opportunities for classroom dialogue and discussion about effective mathematical modeling and problem solving.
