Background/Context Which ideas should be included in the K–12 curriculum, how they are learned, and how they should be taught have been debated for decades in multiple subjects. In this article, we offer mathematics as a case in point of how new standards-related policies may offer an opportunity for reassessment and clarification of such debates. Purpose/Objective Our goal was to specify instructional models associated with terms such as “reform” and “traditional”—which, in this article, we refer to as “dialogic” and “direct”—in terms of perspectives on what it means to know mathematics, how students learn mathematics, and how mathematics should be taught. Research Design In the spirit of “adversarial collaboration,” we hosted a series of semi-structured discussions among nationally recognized experts who hold opposing points of view on mathematics teaching and/or learning. During those discussions, the recent consensus regarding what students should learn—as represented by the Common Core State Standards for Mathematics (CCSSM)—was taken as a common goal, and additional areas of agreement and disagreement were identified and discussed. The goal was not to reach consensus but to invite representatives of different perspectives to clarify and come to agreement on how they disagree. Findings/Results We present two instructional models that were specified and refined over the course of those discussions and describe nine key areas that distinguish the two models: (a) the importance and role of talk; (b) the importance and role of group work; (c) the sequencing of mathematical topics; (d) the nature and ordering of mathematical instructional tasks; (e) the nature, timing, source, and purpose of feedback; (f) the emphasis on creativity (i.e., authoring one's own learning; mathematizing subject matter from reality); (g) the purpose of diagnosing student thinking; (h) the introduction and role of definitions; and (i) the nature and role of representations. Additionally, we elaborate a more nuanced description of the ongoing debate, as it pertains to particular sources of difference in perspective. Conclusions/Recommendations With this article, we hope to advance ongoing debates in two ways: (a) discrediting false assumptions and oversimplified conceptions of the “other side's” arguments (which can obscure both the real differences and real similarities between different models of instruction), and (b) framing the debates in a manner that allows for more thoughtful empirical investigation oriented to understanding learning in the discipline.
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