publication of the Common Core State Standards for Mathematics (CCSSM; Common Core State Standards Initiative, 2010b). CCSSM presents grade-level-specific expectations that are intended to be the core expectations for mathematics learning in the United States. CCSSM diverges from CESSM and PSSM in certain ways, including how it names the strands of content to be taught and learned and how it distributes certain content across the grades, but it retains the same focus on the importance of teaching in ways that enable students to learn mathematics with understanding. The CCSSM states, “These Standards define what students should understand and be able to do in their study of mathematics” (Common Core State Standards Initiative, 2010b, p. 4). Not only is this a consistent theme across the reform documents, it is also a topic that has received considerable attention from the research community.

Research Perspectives on Teaching Mathematics for Understanding

Studies conducted over the past 60 years provide a solid body of evidence concerning the benefits of teaching mathematics for understanding. As summarized in Silver and Mesa (2011, p. 69), teaching mathematics for understanding is sometimes referred to as:

authentic instruction, ambitious instruction, higher order instruction, problem-solving instruction, and sense-making instruction (e.g., Brownell and Moser, 1949; Brownell and Sims, 1946; Carpenter, Fennema, and Franke, 1996; Carpenter et al., 1989; Cohen, 1990; Cohen, McLaughlin, and Talbert, 1993; Fuson and Briars, 1990; Hiebert and Wearne, 1993; Hiebert et al., 1996; Newmann and Associates, 1996). Although there are many unanswered questions about precisely how teaching practices are linked to students’ learning with understanding (see Hiebert and Grouws, 2007), the mathematics education community has begun to emphasize teaching that aims for this goal.

Among the hallmarks of this conceptually oriented version of instruction are (a) mathematical features, or tasks that are drawn from a broad array of content domains and are cognitively demanding, and (b) pedagogical features, or teaching practices that are suitable to support multiperson collaboration and mathematical discourse among students, as well as their engagement with mathematical reasoning and explanation, consideration of real-world applications, and use of technology or physical models (e.g., Hiebert and Carpenter, 1992; Fennema and Romberg, 1999).

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