7: Case Stories: Supporting Teacher Reflection and Collaboration on the Implementation of Cognitively Challenging Mathematical Tasksⁱ

Research shows that mathematical tasks that provide the greatest opportunities for students to think and reason are the most difficult for teachers to implement well during instruction (Stein, Grover, & Henningsen, 1996; Stigler & Hiebert, 2004). All too frequently, students’ opportunities for thinking and reasoning are lost as cognitively-challenging tasks decline into procedural exercises that require the application of rules and memorized facts. Factors that contribute to task decline include: routinizing problematic aspects of the task; not holding students accountable for high-level products; shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer; and providing students with insufficient time to wrestle with the demanding aspects of the task (Henningsen & Stein, 1997).