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Margaret Smith, Victoria Bill and Mary Lynn Raith

This article provides an overview of the eight effective mathematics teaching practices first described in NCTM's Principles to Actions: Ensuring Mathematical Success for All.

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Margaret S. Smith, Victoria Bill and Elizabeth K. Hughes

Mathematical tasks that give students the opportunity to use reasoning skills while thinking are the most difficult for teachers to implement well. Research by Stein and colleagues (Henningsen and Stein 1997; Stein and Lane 1996; Stein, Grover, and Henningsen 1996) makes the case resoundingly that cognitively challenging tasks that promote thinking, reasoning, and problem solving often decline during implementation as a result of various classroom factors. When this occurs, students must apply previously learned rules and procedures with no connection to meaning or understanding, and the opportunities for thinking and reasoning are lost. Why are such tasks so difficult to implement in ways that maintain the rigor of the activity? Stein and Kim (2006, p. 11) contend that lessons based on high-level (i.e., cognitively challenging) tasks “are less intellectually ‘controllable’ from the teacher's point of view.” They argue that since procedures for solving high-level tasks are often not specified in advance, students must draw on their relevant knowledge and experiences to find a solution path. Take, for example, the Bag of Marbles task shown in figure 1.