

Dialogic and Direct Instruction: Two Distinct Models of Mathematics Instruction and the Debate(s) Surrounding Themby Charles Munter, Mary Kay Stein & Margaret S. Smith  2015 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 standardsrelated 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 semistructured 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. Which ideas should be included in the K12 curriculum, how they are learned, and how they should be taught have been debated for decades in reading (Pearson, 2004), social studies (Evans, 2006), science (Barrow, 2006), and mathematics (Klein, 2003; Schoenfeld, 2004). Many labels have been used to characterize the sides of these debates, with the umbrella terms reform and traditional often applied. We refer to two such instructional models as dialogic and direct,^{1} respectively. Although the reading wars and math wars are not being waged with such fervor as in the 1990s, debates about teaching and learning have continued into the 21st centuryas evidenced by declarations of both dialogic (Applebee, Langer, Nystrand, & Gamoran, 2003; Nystrand & Gamoran, 1997; Schoenfeld, 2002; Senk & Thompson, 2003) and direct (Clark, Kirschner, & Sweller, 2012; Izumi & Coburn, 2001; Kirschner, Sweller, & Clark, 2006; Klahr & Nigam, 2004; Mayer, 2004) models of instruction as the clear winner in journals and books in education and psychology. And yet, in mathematics, the National Mathematics Advisory Panel (2008) located only eight comparative studies, which presented a mixed and inconclusive picture of the relative effect of these two approaches to instruction (p. 45), and from which they could draw no generalizations. Recent developments in state policy have the potential to reshape these debates. The recently released common standards for K12 education, including the Common Core State Standards Initiative for English language arts (2010) and mathematics (CCSSM, 2010), and the Next Generation Science Standards (2013), purportedly reflect a newly achieved consensus regarding what should be taught in our nations classrooms. The list of organizations and individuals who have signed on to these new standards appears to represent a heretofore unattainable agreement among those who have typically been on opposite sides of the debates. A closer examination of the new standards, however, reveals that they do not cover the entire territory of past disputes. Although they may represent consensus in some areas, they leave open to interpretationand perhaps bring into sharper focusother facets of the debate. For example, in mathematics, the CCSSM represent an unprecedented agreementacross previously divided partiesregarding which ideas should be included in the K12 mathematics curriculum. However, as with previous standards documents, instructional decisions remain open to interpretation. In Venezkys (1992) terms of a curricular chain, the CCSSM represents a new desired curriculumone that has succeeded in satisfying different conceptions of the needed curriculum, but will be reinterpreted in each subsequent link of the chain, including the prescribed curriculum (e.g., textbooks representation and structuring of content) as well as the delivered (i.e., taught) and received (i.e., learned) curriculum. In short, the CCSSM specify what but not how mathematics should be taught in schools. So, even with a seemingly common goal, the debate continues, albeit in an altered form. Across the subject areas, we argue, the new standards identification of agreement in some facet(s) of the debate provides the opportunity to reassess the debate in general and to identify and bring into focus additional facets. In this article, we offer mathematics as a case in point of how these new policies may offer an opportunity for reassessment and clarification of the ongoing debates in reading, science, mathematics, and any other subject areas in which a contemporary consensus around standards or policies is reached. Our approach involved hosting a series of semistructured discussions among nationally recognized experts who hold opposing points of view. During those discussions, the object of newfound consensuswhat students should learnwas held constant; that is, although the participants may interpret particular standards differently, they agreed not to debate the inclusion or exclusion of any idea or the grade level by which an understanding or skill should be developed. Instead, additional areas of agreement and disagreement were identified and discussed. The goal was not to come to consensus or resolve disagreements but rather, in the spirit of adversarial collaboration (Gilovich, Medvec, & Kahneman, 1998; Mellers, Hertwig, & Kahneman, 2001), to invite representatives of different perspectives to clarify and come to agreement on how they disagree. In so doing, our aim was to surface and highlight the underlying sources (i.e., rationales, perspectives, theories, priorities) that give rise to disagreements, thereby increasing the fields depth of understanding regarding how and why each model of instruction is meant to work. The structure of the article is as follows. First, we explain ways in which we both built upon and departed from similar past efforts in education. Then we describe our approach to convening and summarizing the results of the series of meetings that we hosted. Following that, we present a systematic account of the similarities and differences between the direct and dialogic instructional models that emerged from our workan account that we argue is more highly specified than has been available in the past. Following that, we elaborate a more nuanced description of the debate, one that goes beyond the popularized surface version (e.g., discovery learning vs. direct instruction; curriculum X vs. curriculum Y) by drawing on each sides underlying perspectives regarding what it means to know mathematics, what is involved in learning mathematics, and how mathematics is best taught. In so doing, we hope to advance the debate in two ways: (1) discrediting false assumptions and oversimplified conceptions of the other sides arguments (which can obscure both the real differences and real similarities between different models of instruction), and (2) framing the debate in a manner that allows for more thoughtful empirical investigation oriented to understanding learning in the discipline. We view this frame of learning in the discipline and the implications that it has for education reform as complementary tobut distinctly different fromthe broader school and teacherchange literature that typically uses movement from traditional to reform instruction to frame discussions (e.g., Fullan, 2007; Tyack & Cuban, 1995). In the concluding section, we argue that, insomuch as the problems that hinder informed debate in mathematics are endemic to other fields, our approach can provide the broad outlines of a method that will be useful to untangling issues commonly argued about in other subject areas as well. RELATIONSHIP TO PREVIOUS EFFORTS Convening players on different sides of such a debate is not a novel effort. The National Reading Panel produced Preventing Reading Failure in Young Children (Snow, Burns, & Griffin, 1998). Resnick convened a panel in early literacy to produce K3 literacy standards under the auspices of the New Standards Project (New Standards Primary Literacy Committee, 1999). In mathematics, both Adding it Up: Helping Children Learn Mathematics (NRC, Kilpatrick, Swafford, & Findell, 2001), which provided a fivestranded definition of mathematical proficiency, and the CCSSM (2010) involved diverse groups of educators, mathematicians, and professionals in writing, reviewing, and/or validating the documents. The Curriculum Focal Points (NCTM, 2006) and National Mathematics Advisory Panel report (2008) similarly provided foundational work toward building consensus. In addition, in 2005, a small group of mathematicians and mathematics educators were invited to attempt to identify common ground between their respective communities (Ball et al., 2005). This group articulated seven areas of agreement with respect to previously contentious topics, including basic fact recall; calculators; algorithms; fractions; real world contexts; teacher knowledge; and instructional methods. For example, regarding the last of these, the panel stated, Students can learn effectively via a mixture of direct instruction, structured investigation, and open exploration. Decisions about what is better taught through direct instruction and what might be better taught by structuring explorations for students should be made on the basis of the particular mathematics, the goals for learning, and the students present skills and knowledge. (p. 4) By convening experts on different sides of the debate, our approach followed the playbook from previous efforts, but with two key exceptions. The first was an affordance of recent policy developments: As noted earlier, we were able to single out the CCSSM as a reasonable representation of what students should learn in K12 mathematics and obtain agreement from both sides not to reargue the appropriateness of the mathematical ideas included therein, but rather to focus on additional facets of the debate.^{2} This was not because we consider the CCSSM flawless; like those who participated in our discussions, we hope that, over time, the standards will indeed continue to be refined. Rather, we were attempting to capitalize on a moment that, in the history of U.S. mathematics education, we perceived as different and significantthe identification of a set of ideas and practices that previously opposed parties viewed as an adequate representation of the goal for K12 students. Although the CCSSM have, recently, come under greater scrutiny in political arenas, even leading to a handful of states withdrawal of support, mathematicsrelated professional organizations maintain their support for these standards (Briars, 2014; CBMS, 2013; NCTM, 2013). Our discussions also differed from previous efforts in that we invited representatives of different perspectives to first outline their sides stance with respect to knowing, learning, and teaching mathematics, and then to meet with the opposing side to clarify and come to agreement on exactly how they disagree on issues of mathematics knowledge, learning, and teaching. From our perspective, focusing strictly on areas of consensus renders real differences (and the sources of those differences) invisibledifferences that, if left unresolved (or at least unspecified), can undermine the potential influence of the consensus documents. Thus, our goal was not to characterize contentious issues at a level of breadth sufficient for individuals with opposing perspectives to agree (as we interpret Ball et al.s, 2005, statement on instruction quoted previously), but to zoom in in order to identify genuine distinctions. In general, our intent was to convert disagreement into something productive; by asking opposing groups to collaboratively explicate how their perspectives differed, we hoped to add clarity and depth to the debate. CONVENING THE EXPERTS Over the course of a year (September 2011August 2012), we convened five meetings that brought together mathematicians, educators, psychologists, and learning scientists, each time separated into two groups representing different sides of the debate (see the Appendix for a list of participants). Based on what we knew of their work, previous involvement in addressing such issues, and, in some cases, suggestions of other scholars, we identified participants to ensure a range of backgrounds, expertise, and perspectivesbetween and within groups. Framing the conversations as preparatory work for an eventual comparative study, we invited participants to particular meetings to contribute to the specification of a direct or dialogic instructional model (in the initial invitations, we used the placeholder terms traditional and reform). The meetings focused on: (a) defining what it means to know and learn mathematics and specifying two distinct instructional models; (b) identifying and describing curricular materials and student assessments that align with the models; (c) describing teacher professional development necessary to support the enactment of each model; and (d) assessing fidelity of implementation through classroom observation. Two meetings addressed topic (a) and each of the other three topics was addressed in one meeting. By focusing the initial meetings on the articulation of the theories of learning and teaching on which the two instructional models are built, discussions of curriculum, assessment, professional development, and implementation at subsequent meetings could then be framed in terms of the models underlying theories. Each meeting consisted of a combination of simultaneous smallgroup discussions among proponents of the same model and wholegroup discussions in which each of the two groups shared and contrasted their ideas with members of the other group. A focus of the combined discussions was identifying and articulating areas of distinction (the results of which led to the points included in Table 1 in the following section). The discussions were guided by a set of questions related to the aspect of the model that was being addressed at the meeting, which we developed and disseminated to participants ahead of time (see the Appendix for a complete list of guiding questions we used to structure the discussions). The authors served as facilitators for the discussions and intervened only as needed to clarify the task, refocus the conversation, or, in the combined discussions, nominate a point of distinction to be explicated. All of the meetings were audio recorded and all of the artifacts created for and during the meetings were archived. Following each meeting, we reviewed audio recordings, artifacts, and notes to create a summary of what each group produced and similarities and differences between groups. We then shared the summary with the meetings participants in order to solicit their feedback regarding whether we got it right. Based on their suggestions, we revised the summary and then shared it with all past and present participants. Thus, the instructional model descriptions in the next section are based on multiple iterations of our summary reports, repeated requests for participant feedback, and, ultimately, final versions that all participants approved. Before presenting those descriptions, however, we wish to acknowledge the biases that we, the authors, brought to (and incorporated into) this work. Although our many conversations may have revealed minor ways in which our perspectives differ, fundamentally we align in our advocacy for a dialogic approach to mathematics instruction. It is this very alignment that led us to solicit the input of those whose perspectives differ from our own: We were interested in the prospect of a comparative study, but not in setting up a straw man. With participants in the meetings, we were always careful to clarify our stances and to use them for generating questions rather than arguments. While we did contribute some to the specification of the dialogic model, our primary commitment was to facilitating discussions in which two instructional models, their underlying theories, and the ways in which they differ were made explicit. As a result, we found that we developed a better appreciation for arguments in favor of a direct approach and a clearer picture of what highquality enactments of such a model would look like. TWO INSTRUCTIONAL MODELS Based on the input of the experts at the meetings that we convened, we have labeled and specified two mathematics instructional models that differ in a number of important ways.^{3} Abbreviated accounts of teaching in each of those models are as follows. SUMMARIES OF THE TWO PEDAGOGICAL MODELS In the direct instruction model, pedagogy consists of describing an objective, articulating motivating reasons for achieving the objective and connections to previous topics; presenting requisite concepts (if they have not been presented previously); demonstrating how to complete the target problem type; and providing scaffolded phases of guided and independent practice, accompanied by corrective feedback. Across these phases, lessons should be made engaging, which can be accomplished through keeping a brisk instructional pace, inviting group unison responses to questions, encouraging student motivation by supporting them in experiencing success, and providing focused praise. In the dialogic model, across a series of lessons, students must have opportunities to (a) wrestle with big ideas, without teachers interfering prematurely, (b) put forth claims and justify them as well as listening to and critiquing claims of others, and (c) engage in carefully designed, deliberate practice. This requires teachers, first, to engage students in two main types of taskstasks that introduce students to new ideas and deepen their understanding of concepts, and tasks that help them become more competent with what they already know; second, to orchestrate discussions that make mathematical ideas available to all students and steer collective understandings toward the mathematical goal of the lesson; third, to introduce tools and representations that have longevity (i.e., can be used repeatedly over time for different, but likely related, purposes, as students understanding grows); and, finally, to sequence classroom activities in a way that consistently positions students as autonomous learners and users of mathematics. SIMILARITIES BETWEEN MODELS That there are differences between these models of teaching will hardly be surprising to anyone familiar with debates about mathematics education. However, in addition to consensus on the content of the CCSSM (which we have already noted), we found a number of similarities. For example, contrary to the popularized bifurcation of two key strands of mathematical proficiency, in both models, both conceptual understanding and procedural fluency are not only valued as important forms of knowledge, but are viewed as being developed together. Additionally, we found that both models emphasize using carefully designed, purposefully sequenced, mathematically rigorous tasks (despite task design typically being associated more with dialogic instruction and rigor with direct instruction); closely monitoring students reasoning (typically associated more with dialogic instruction); and providing regular opportunities for practice (typically associated more with direct instruction). DIFFERENCES BETWEEN MODELS However, although both models share a number of features, their respective rationales often differ. In Table 1 we provide a summary of differences between the two models with respect to nine key areas of distinction. The table is necessarily detailed; many of the sidebyside comparisons explicate differences discussed in this section, but othersincluding group work, definitions, and representationswe present only in table form. Table 1. Distinctions Between Dialogic and Direct Mathematics Instruction
Knowing Our discussions suggest that pedagogical differences likely stem from differing perspectives on what it means to know and how children learn mathematics. For example, although advocates of both models take the National Research Council's (NRC, 2001) five strands of mathematical proficiency and the CCSSM (standards for both content and mathematical practice) to be reasonable representations of knowing mathematics, they emphasize different components of those strands and practices. For example, in the direct instruction model, the communication aspect of the third Standard for Mathematical Practice is not emphasized. While a good student may have an internal dialogue concerning all of the other aspects of the third standard, communicating effectively with others is not a necessary capability. In the dialogic model, communicating effectively with others is fundamental to knowing (and learning). Similarly, in the direct instruction model, to make conjectures and build a logical progression of statements to explore the truth of their conjectures (CCSSM, p. 6) is limited to trying solution strategies for solving a problem posed to them; student questions that drive instruction or lead to new mathematical investigations are not emphasized as they are in the dialogic model. Learning Between these two models, perspectives on learning are even more distinct than those on knowing. The perspective underlying the direct instruction model is that, when students have the required prerequisite conceptual and procedural knowledge, they will learn from (a) watching clear, complete demonstrationswith accompanying explanations and accurate definitionsof how to solve problems; (b) practicing on similar problems sequenced according to difficulty; and (c) receiving immediate, corrective feedback. The perspective underlying the dialogic model, on the other hand, is that students must (a) actively engage in new mathematics, persevering through challenges as they attempt to solve novel problems; (b) participate in a discourse of conjecture, explanation, and argumentation; (c) engage in generalization and abstraction, developing efficient problemsolving strategies and relating their ideas to conventional procedures; and, to achieve fluency with these skills, (d) engage in some amount of practice. As noted in Table 1, perspectives on learning across the two models also vary with respect to the sequence in which mathematical topics are best learned. In the direct model, the sequence is dictated primarily by a disciplinary progression (i.e., prerequisites determined by the structure of mathematics). In the dialogic model, the sequence of learning experiences reflects both the progression of ideas that the structure of the discipline would suggest, and the developmental pathways students current understandings and capabilities take. Such a perspective places an importance on building on prior knowledge, which, in this case, refers to the skills and concepts required for students to meaningfully engage in learning experiences and struggle for understanding, rather than knowing exactly how to solve the problem due to prior exposure to very similar examples. SUMMARY As described in this section, our approach not only produced descriptions of what both direct and dialogic instruction can be expected to look like, but also served to clarify nine important areas of distinction and the differing perspectives on knowing and learning mathematics that underlie them. In addition to developing a more thorough understanding of two models of mathematics instruction and how they compare, our efforts have shed light on the nature of the debate itself, which we discuss in the next section. THE NATURE OF THE DEBATE(S) The popularized versions of any subject areas debatessuch as those reported in the evening news (cf. Costello & Williams, 2008)are likely inadequate (or even inaccurate) characterizations of the complexities and histories of what is actually being debated among researchers, disciplinary representatives, and other stakeholders. For example, the math wars have been largely about curriculum and particular textbook series (Klein, 2007a). But more fundamental issuesincluding the nature of the mathematics that gets taughthave likely underlain the arguments (Schoenfeld, 2004). Through our approach in mathematics, we have found that preparing for and participating in a series of conversations with individuals of diverse backgrounds and areas of expertise has made clear to us that the debate has been and continues to be multifaceted, and that its participants likely enter the debate to take a position on one but not necessarily all of those facets. Specifically, our conversations have further illuminated four possible sources of difference in perspectives, accounting for at least four distinct facets of the debate: (a) content: different opinions on which mathematical ideas should be included and how those ideas should be represented in K12 mathematics curricula; (b) epistemology: commitments to different definitions of mathematical knowledge or proficiency, in particular whether constructs such as student identity or mathematical authority should be included in such definitions (and goals of school mathematics); (c) learning: alignment with different theories of learning; and (d) pedagogy: commitments to different kinds of instruction as providing superior learning experiences. Below, we continue our examination of mathematics as a case of ongoing debates in education in which opportunities for reassessment and clarification of those debates may now exist, thanks to new standards or state policies. We draw on relevant literature to discuss each of the above listed sources of difference in turn, indicating how over the course of the conversations we have hosted, we have come to understand that the popularized versions of the debate are not always reflective of (and sometimes even obscure) the true differences. A number of the points we raise are, necessarily, specific to mathematics education, but many others originate in arguments about learning and teaching in general and are therefore applicable to other subject areas. Additionally, we view the general approach we have taken to understanding and elaborating the facets of a subjectspecific debateand even the facets (content, epistemology, learning, and pedagogy) themselvesas potentially useful and relevant to other subject areas. CONTENT AS A SOURCE OF DIFFERING PERSPECTIVES Over the last two decades, issues of content have attracted more attention than any other facet of the math wars debates (Klein, 2007a; Schoenfeld, 2004). Indicatively, in the third review of states mathematics standards by the Thomas B. Fordham Foundation, published in 2005 (the first two having been released in 1998 and 2000), the evaluators considered the same four criteria as in past reviewsclarity, content, sound mathematical reasoning, and the absence of negative featuresbut increased the weighting of content from 25% to 40% of a states total score because [t]he consensus of the evaluating panel of mathematicians is that this revised weighting properly reflects what matters most in K12 standards today (Klein et al., 2005, p. 9). Contentrelated disagreements concern which concepts and skills should be learned in grades K12, by when those concepts and skills should be learned, and whether and how they should be represented in textbooks. As noted previously, the adoption of the CCSSM has quieted many of these arguments. The CCSSM authors selection and sequencing of topics have been deemed satisfactory by multiple stakeholders who were previously at odds. In a joint statement from the Conference Board of the Mathematical Sciences (CBMS, 2013), presidents from 15 professional societies of both mathematics educators and mathematicians expressed their strong support for the CCSSM. Additionally, authors of the fourth Fordham Foundation review of states mathematics standards (for which content and rigor comprised 70% of a states grade) gave the CCSSM a grade of Aminus (Carmichael, Martino, PorterMagee, & Wilson, 2010). Though each community has its minority dissenters, this firsttime consensus is exactly why the efforts described in this report are timely. The usual complaints about the over or underrepresentation of some topics (e.g., to what extent data analysis should be emphasized in the elementary grades), the omission or inclusion of particular skills (e.g., whether students should become fluent with standard^{4} algorithms), or poor sequencing (e.g., whether the Standards pacing matches that of higherperforming countries curricula) will surely be lodged, but these can now be treated as ideas for future refinements, rather than fatal flaws. With content goals as a steady, mutually agreedupon backdrop, we can more carefully consider what might be other facets of the debate. However, some of these additional facets are still in the content realm. In particular, the CCSSM do not settle debates about whether and how mathematical ideas should be represented in textbooks, including those concerning the presence and role of definitions, worked examples, and derivations (Garelick, 2005; Klein, 2007b). Of course, mathematically inaccurate statements in textbooksthe target of much criticism over the yearswill never be acceptable to either side. But opinions differ with respect to what is not in textbooks. What some perceive as omissionsinformation that the author failed to give to studentsothers see as opportunities afforded because the author successfully refrained from giving away the conclusion. For example, it is unlikely that anyone who values any of the NRCs (Kilpatrick et al., 2001) strands of mathematical proficiency other than procedural fluency would argue that the quadratic formula should be simply provided to and memorized and practiced by students. But should its derivation, which is critical for understanding the formulas applicability and utility, be read from a textbook, or performed in response to an emergent need for a way to identify the roots of any quadratic equation? In general, are definitions, explanations, derivations, and formulas more often things to be given (e.g., at the start of a lesson), or things to be achieved (e.g., in the course of a lesson)? Throughout our discussions, this remained an unresolved issue, with some arguing the former and others the latter. Thus, we found difference in opinion not on which mathematical ideas and procedures should be learned, but on whether and how that content should be provided and represented in textbooks and instruction. This difference in opinion is likely due to different, perhaps implicit, assumptions about pedagogy or some other facet(s) of the debate, but is manifest in discussions about content. EPISTEMOLOGY AS A SOURCE OF DIFFERING PERSPECTIVES Contrary to how the debate in mathematics is often framed, no one whose opinion we have solicited has suggested that conceptual understanding and procedural fluency are not both fundamental aspects of mathematical knowledge (or that one should be learned in isolation from the other); they agree that any dichotomy drawn between the two is false (Wu, 1999). More specifically, nearly all participants agreed that the two consensus documents noted previouslythe definition of mathematical proficiency in NRCs Adding it Up and the standards for mathematical practice listed in the CCSSMare reasonable articulations of what it means to know and do mathematics. What we did find was a difference between the two sides with respect to the relative value they placed on the various strands and standards. For example, some are more likely to emphasize the importance of communication (as something that both affords learning and gets learned). Others are more likely to emphasize precision (e.g., in definitions) and fluency with symbolic representations. These differences in relative emphases on specific components (when there is agreement on the broad descriptions of mathematical proficiency and practice) may appear subtle, but likely influence very observable (and consequential) decisions, including textbook selection, curriculum development, the nature of classroom activity, and assessments. For example, an emphasis on communication will likely lead to lessons with substantial time devoted to student presentations and wholeclass discussions in which students language and argumentation skills are shaped, whereas less regard for the importance of communication can be expected to result in more teacher demonstration and individual seatwork with teacherprovided corrective feedback. Conversely, clearer distinctions can be drawn with respect to constructs that likely are manifested in classroom learning in very subtle ways. Specifically, we have encountered forthright differences in perspective with respect to whether student identity and mathematical authority should be considered when defining mathematical knowledge and practice. For example, in discussing mathematical knowledge and capability, we have found that some are more likely to view the construction of identities as central to knowing and practicing mathematics (Boaler, 1999; Gutstein, 2006; Martin, 2007, Moses & Cobb, 2001), that coming to know mathematics involves becoming a different personmathematically or otherwiseas one becomes able to participate in new activities with others (Lave & Wenger, 1991, p. 53). Those same individuals typically express similar epistemic commitments to the development of intellectual authority in mathematics (Carpenter & Lehrer, 1999; Gravemeijer, 2004; Lampert, 1990; Nathan & Knuth, 2003; Stein, Engle, Smith, & Hughes, 2008). Generally, their view is that students who have developed such authority are able to participate in a zigzag between conjectures and arguments for their validity (Lampert, 1990, p. 32). They are also are able to only accept new mathematical knowledge of which they can judge the validity themselves (Gravemeijer, 2004, p. 109), the achievement of which likely requires that the authority for correctness lies in logic and the structure of the subject, not in the teacher or textbook (Kilpatrick et al., 2001, p. 425; Simon, 1994). These very real differences in perspective on what it means to know and do mathematics can be inferred based on observations of classroom lessons, but requires an ability and willingness to view classroom interactions through that lens. LEARNING AS A SOURCE OF DIFFERING PERSPECTIVES A cursory reading of public arguments about how to improve mathematics instruction might lead one to expect that any differences on the subject of how children learn mathematics are between those who espouse constructivism as a valid theory of learning and those who do not. In our view, this accounts for a significant portionthough not allof the disagreement on learning, but not for the reasons that are typically articulated. Often, the issue emerges in disagreements about instruction, where theories of learning are mistaken for (or are overextended to describe) approaches to instruction, producing a popular understanding of constructivism, as Thompson (2000) put it, as being about discovery learning, cooperation, and a ban on lecturing (p. 415). A number of authors whose works are cited in arguments about mathematics learning and teaching have addressed this conflation. For example, after reviewing decades of research on different models of instruction, all loosely categorized as either discovery learning or direct instruction, Lee and Anderson (in press) concluded, In some way students must construct the knowledge by understanding how it applies to their problem solving. . . . The key question is how students can be guided to construct this knowledge efficiently in a form that will transfer across the desired range of situations. (p. 463) Mayer (2004) drew a similar line between learning and instruction when he described a constructivist teaching fallacy, pointing out that being cognitively active (in knowledge construction) does not require being active behaviorally (e.g., working in small groups, using manipulatives, participating in discussions). Likewise, Anderson, Reder, and Simon (1998) argued, denying that information is recorded passively does not imply that students must discover their knowledge by themselves, without explicit instruction (p. 232). All of these arguments echoed Simons (1994) debunking of one of two myths of constructivism, namely that the teaching that some teachers do is constructivist teaching (p. 75), when, in actuality, [t]here is no simple function that maps teaching methodology onto constructivist principles (Simon, 1995, p. 117).^{5} A slightly closer read of the abovecited workspast the misleading constructivist teaching layermight suggest that constructivism is not actually a bone of contention in the debates. In some of the most often cited works in antireform arguments, authors seem to have endorsed constructivist perspectives. For example, Kirschner et al. (2006) concurred with Steffe and Gale (1995): The constructivist description of learning is accurate (p. 78). Mayer (2004) agreed, there is merit in the constructivist vision of learning as knowledge construction (p. 14). But such instances of apparent alignment are possibly attributable to differences in interpretations of constructivism. What Mayer (2004) conceded might be a naïve interpretation von Glasersfeld (1989) termed a trivial rather than radical view of knowledge construction. One may hold the view that knowledge is actively constructed and not passively received, but not the radical view that each individual constructs knowledgeor, realityuniquely (Richards & von Glasersfeld, 1980). This distinction helps to explain Mayers (2004) suggestion that often some appropriate amount of guidance is required to help students mentally construct the desired learning outcome (p. 15). Cobb, Yackel, and Wood (1992) suggested this perspective is typical of a representational view of mind (though Anderson, Reder, & Simon, 1998, contended that modern cognitive theories actually assume that the mind transforms external representations into internal structures that are not at all isomorphic to the stimuli, p. 234). But differences in perspectives on learning extend beyond the cognitivist realm. Steffe and Kieren (1994) argued that, as the 1960s modern mathematics movement (or, the new math) faded, behaviorist interpretations of mathematics became prevalent in the practice (but not the research) of mathematics educationexemplified in the development of various backtobasics programs, which are still widely used in schools today (Slavin, Lake, & Groff, 2009). More recent than a behaviorismcognitivism divide, however, are theories of learning that do not emphasize acquisition at all (Sfard, 1998), but rather participation in communities of practice (Lave & Wenger, 1991). As with all perspectives on learning, taking a participationist stance potentially has implications for the nature of classroom activity, and the roles that teachers and students play. For example, in elaborating on his assertion that constructivist views of learning have too often led to assumptions about how active a learning environment should be, Mayer (2004) argued, [i]nstead of depending solely on learning by doing or learning by discussion, the most genuine approach to constructivist learning is learning by thinking. Methods that rely on doing or discussing should be judged not on how much doing or discussing is involved but rather on the degree to which they promote appropriate cognitive processing. (p. 17) But those who hold a view of learning that emphasizes participation would likely argue that the doing and discussing are goals in themselvesthe communitys forms of practice in which students participate more fully over time. Thus, the evidence of learning is not a change in longterm memory (Kirschner et al., 2006), but rather a transformation of participation in shared endeavors (Rogoff, 1994). Clearly, individuals bring a variety of theories of learning to discussions of curriculum and instruction. Broadly speaking, from our view, those who would espouse the dialogic instruction model are more likely to employ participationist, or radical or social constructivist explanations of learning, while those who would espouse direct instruction are more likely to employ explanations from a cognitive constructivist, information processing, or behaviorist perspective. More importantly, however, we have found that those whose work is often cited in arguments against constructivist teaching or rote learning are seldom guilty of propagating the conflation noted by numerous researchers and described above (cf. Anderson, Reder, & Simon, 2000; Kirschner et al., 2006; Mayer, 2004). But we have also found that other participants in the debate, particularly those whose participation is motivated by content concerns, often do not attend to how children learn mathematics. One potential implication of this concerns the sequencing of mathematical topics. As discussed previously and included in Table 1, for some, sequences should be dictated primarily by a disciplinary progression (i.e., prerequisites determined by the structure of mathematics). For others, developmental progressions (i.e., supporting students in building new knowledge from prior knowledge) should be considered as well. Building sequences of topics from these different perspectives likely leads to very different experiences for students. PEDAGOGY AS A SOURCE OF DIFFERING PERSPECTIVES After the question of what should be learned (content), the question of how it should be taught has likely been the facet of the debate most associated with the math wars. As suggested in the previous section, pedagogical approaches often follow from perspectives on knowledge and learning (indeed, attempts have been made to build instructional models from each of the perspectives described earlier). Some have even argued that pedagogy is dictated by the mathematics that is to be learned (cf. Wu, 2004). Thus, disagreements about mathematics pedagogy are often rooted in different theories of learning or different epistemologies, and are therefore more complicated than the popularized pitting of traditional or direct instruction against reform or discovery learning would suggest. Much of this article has already been devoted to this facet of the debate. Previously, we summarized two distinct models of mathematics instruction and identified particular points of contrast between them so that we might understand what each model is and is not, in relation to the other. One aim in doing so is to push conversations about either into a level of depth commensurate with the models themselvesparticularly among each models detractors. As was shown, however, ideas from one perspective often do not map neatly onto alternatives from another, which brings up one last point we wish to make concerning the pedagogy facet of the debate. Often, arguments for particular pedagogies are based on unique sets of assumptions, frame problems of instruction differently, or look for teaching in different places (Greeno, 1998). Consequently, different and incompatible questions get asked about instruction (Greeno, 1997). For example, Lee and Anderson (2013) suggested that Koedinger and Alevens (2007) assistance dilemmawhether to provide or withhold instructional assistance to studentsshould be considered as a continuum, with discovery learning at one end and direct instruction at the other. But How much assistance? is not a question that those approaching instruction from a transformation of participation perspective would likely ask. Rogoff (1994), for example, suggested that what Lee and Anderson cast as extremes of a continuumadultrun and childrenrun, as she called themare actually closely related because they correspond to a view of learning as a onesided acquisition (whether transmitted or discovered). She instead proposed a community of learners instructional model, in which students learn as they collaborate with other children and with adults in carrying out activities with purposes connected explicitly with the history and current practices of the community (p. 211). Rogoff was clear that she did not view this model as a compromise or balance of the extremes identified by Lee and Anderson; her argument was that what is learned differs . . . not that learning occurs best in one model (p. 210). All of this suggests that comparisons of different pedagogies are not simply a matter of asking how much or how little one model employs the strategies of another. We must determine the ways in which the instructional models approaches differ and, to the extent that they employ similar strategies, we must determine how and why those strategies are used in each model. SUMMARY In this section we identified four^{6} facets of the mathematics debate on which individuals may or may not take a stance (whether explicit or implicit), and we have attempted to articulate insights we have developed concerning the subtleties of those facets, as we reflected on arguments in the literature in light of the conversations we have hosted. In general, it is our perception that misconceptions of others arguments, incompatibilities among perspectives, and ways that participants in the debate frame problems and solutions have masked the true differences that underlie the multiple facets of the debate, allowing more superficial versions of the arguments to define the debate in the public arena and impede empirical progress. Sfard (1998) has argued that perspectives on facets of the debate, such as those discussed in this section, are not independent. While we suspect that this is true, space constraints prevent us from articulating specific conjectures about all of the potential relationships. We do, however, wish to discuss what we perceive as a conflation of the two most popular, public arguments, namely those concerning content and pedagogy. This conflation is an aspect of the debate that our pulling apart clarified for us. For some, matters of content (or teachers knowledge of content) transcend those of pedagogy (cf. Ma, 1999; Wu, 2004); others emphasize elements of highquality instruction regardless of what content is being taught (cf. Rosenshine, 2012); and, for some, the two are inextricably linked (cf. Lampert, 1990). The mathematics reform movement over the last 25 years (including NCTM standards, NSFfunded textbook development, and much of mathematics education research on learning and teaching) has advanced ideas about both. Stances on content have been contentious, particularly once reform textbooks were released in the mid1990s, as have been the ideas put forth about how best to support students in learning mathematics in the classroom. But more than being contentious, these ideas have, unfortunately (for everyone), been combined into a single target of criticism. Consequently, terms such as reform math, fuzzy math, discovery learning, and constructivist teaching have, for many, become synonymous, as have what are typically named as their alternatives, traditional math, backtobasics, direct instruction, and drill and kill. But whether students should learn particular algorithms, or whether complex, inquiryoriented tasks support learning, for example, are distinct questions, which can be answered separately. And, regardless of which kind of content or pedagogy one espouses, such questions should be answered based only on the relevant criteria, which sometimes originate in structures of the discipline and other times in research on learning and teaching. To the extent that debates in other subject areas are also multifaceted, we suspect that those debates complexities have been masked by similar conflations, but also that those conflations can be similarly teased apart. In Figure 1 we represent our assertion that the two most contentious facets of the mathematics debatewhat content should be taught and howcan be asked simultaneously, but separately, of any instance of mathematics classroom instruction. One could determine both whether the mathematics (e.g., represented in textbooks, on shared spaces such as a whiteboard, or in classroom talk) is sound and rigorous, and what the nature of the instruction iswhich, in Figure 1, we exemplify with the two models detailed in the previous section. Figure 1. Separating questions of content and pedagogy.
DISCUSSION Recent developments in state policyand the consensus on new standards in reading, science, and mathematics in particularoffer opportunities for reassessment and clarification of ongoing debates in education. In this article, we have framed our efforts in mathematics as a case of capitalizing on such opportunities. To the extent that informed debate has been similarly hindered in other fields, our approach to untangling issues commonly argued about in mathematics may provide some guidance in other subject areas as well. In addition to specifying two distinct models of mathematics instruction (dialogic and direct), we have proposed a more nuanced description of the disciplines ongoing, multifaceted debate. It is our perception that equally thoughtful rationales have been provided for very different perspectives on mathematics instruction and that, in general, many who have participated in the debate have not adequately understood or characterized the views of those with whom they disagree. Those more aligned with direct instruction do not argue that teachers should simply provide facts and procedures to students and ask students to memorize and practice them over and over until proficient. Likewise, those more aligned with dialogic instruction do not advocate pure discovery learning, in which students somehow manage to reinvent the mathematics curriculum. But such misunderstandings are likely attributable, in part, to the fact that neither of the models described in this article is typical, and therefore opportunities to observe either model are limited. Consequently, arguments between proponents and critics of any given model are often not about the same pedagogy. In each case, participants in the debate are likely drawing comparisons between an ideal version of the model that they support and a diluted version of a model that they do not support. Therefore, from our view, it behooves advocates of any instructional model in any subject area to increase public access to highquality instantiations of that instructional model. Though they may not agree on how, everyone agrees that widespread instructional improvement is needed. The models described herein represent two distinct perspectives on what the goals of those efforts should be in mathematics. To be clear, our intention is not to dichotomize all imagined pedagogies for mathematics. We have merely specified twonot the two. Furthermore, we do not view the two instructional models as consisting of mutually exclusive sets of teaching strategies. Teachers in dialogic classrooms may very well demonstrate some procedures, just as the students in a direct instruction classroom may very well engage in smallgroup, projectbased activities. In fact, instruction likely involves elements of both direct and dialogic within every lesson. Our suspicion is that it is not a matter of the strategies per se, but rather how they are used and experienced in relation to all other forms of learning opportunities throughout a school year. That said, we contend that, in any subject, careful, comparative studies of faithful implementations of wellspecified, distinct modelssuch as those specified in this articlewill help us better understand what students are more or less likely to learn within a wellexecuted version of a particular model. Too often, one carefully designed model is examined in relation to some unspecified, business as usual form of instruction. In the worst cases, comparison groups are cast as legitimate versions of what critics of the treatment support. For instance, one might report on the effects of reformoriented teaching or textbooks compared to those of traditional, direct instruction, when in fact the comparison model is not nearly as coherent or well specified as what was described in this article. Additionally, we encourage all participants in conversations and debates about curriculum and instructionand certainly anyone who engages in the type of effort described in this articleto assume that those who propose alternative arguments and models are rational, concerned individuals; to press each other to identify exactly which of the facets of a debate have motivated ones concern, and how perspectives on multiple facets are related; and to hold each other accountable for articulating rationales and, perhaps most importantly, citing evidence. The consensus on recent standards represents significant progress in decadeslong debates, and affords all interested communities a common foundation on which to make progress with respect to better understanding learning and teaching in their disciplines of interest. To capitalize on this opportunity requires understanding true differences between viable options across all of a debates multiple facets. Notes 1. The labels we use were suggested by participants in the meetings we hosted. The direct instruction model we specify in this article is distinct from the programs Direct Instruction and Explicit Direct Instructionthough all three share many elements. To emphasize the distinction, we use a lowercase direct. 2. We also relied on agreement that the five strands of mathematical proficiency from NRCs Adding it Up (Kilpatrick et al., 2001) were reasonable representations of the learning goals for K12 mathematics, representations acceptable to both sides. 3. More elaborate descriptions of the two models can be found on the Teachers College Record website at http://www.tcrecord.org/content.asp?contentid=18115. 4. Our use of quotations around the word standard is intended to imply only that, from our perspective, the words connotation has been a source of contention. 5. Simons (1994) other myth: A constructivist perspective results in teachers who have no agenda for what mathematics students will learn (p. 74). 6. We do not suggest that the four sources of difference we have discussed comprise an exhaustive list. At least three other potential sources include: different perceptions of evidence (e.g., claims that evidence of the effectiveness of either approach to instruction is lacking); different views on the purpose of mathematics education (e.g., social mobility vs. social efficiency); and different perceptions of equity (e.g., many on both sides think that only the most capable students will succeed in the other sides model). We do not elaborate on these potential sources here because we did not address them with sufficient explicitness and thoroughness with our participants. Acknowledgment The work reported here was supported by The Learning Research and Development Center. We are deeply grateful to the 26 individuals who agreed to participate in the meetings we hosted, and also appreciate the comments on an earlier draft provided by Janine Remillard and by the anonymous reviewers. Correspondence concerning this article should be addressed to Chuck Munter, University of Pittsburgh, Department of Instruction and Learning, 230 S. Bouquet St., 5517 Wesley W. Posvar Hall, Pittsburgh, PA 15260. Email: cmunter@pitt.edu, Phone: (412)6481079. References Anderson, J. R., Reder, L. M., & Simon, H. A. (1998). Radical constructivism and cognitive psychology. In D. Ravitch (Ed.), Brookings papers on education policy 1998. Washington, DC: Brookings Institute. Anderson, J. R., Reder, L. M., & Simon, H. A. (2000). Applications and misapplications of cognitive psychology to mathematics instruction. Texas Education Review, 1(2), 2949. Applebee, A. N., Langer, J. A., Nystrand, M., & Gamoran, A. (2003). Discussionbased approaches to developing understanding: Classroom instruction and student performance in middle and high school English. American Educational Research Journal, 40(3), 685730. Ball, D. L., FerriniMundy, J., Kilpatrick, J., Milgram, R. J., Schmid, W., & Schaar, R. (2005). Reaching for common ground in K12 mathematics education. Notices of the American Mathematical Society, 52(9), 10551058. Barrow, L. H. (2006). A brief history of inquiry: From Dewey to standards. Journal of Science Teacher Education, 17, 265278. Boaler, J. (1999). Participation, knowledge, and beliefs: A community perspective on mathematics learning. Educational Studies in Mathematics, 40, 258281. Briars, D. J. (2014). Core truths. Retrieved from http://www.nctm.org/ Carmichael, S. B., Martino, G., PorterMagee, K., & Wilson, W. S. (2010). The State of State Standardsand the Common Corein 2010. Washington, DC: Thomas B. Fordham Institute. Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennema & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 1932). Mahwah, NJ: Lawrence Erlbaum. Clark, R. E., Kirschner, P. A., & Sweller, J. (2012). Putting students on the path to learning: The case for fully guided instruction. American Educator, 36(1), 611. Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 233. Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/assets/ CCSSI_Math%20Standards.pdf Common Core State Standards Initiative. (2010). Common Core State Standards for English Language Arts & Literacy in History/Social Studies, Science, and Technical Subjects. Retrieved from http://www.corestandards.org/assets/ CCSSI_ELA%20Standards.pdf Conference Board of the Mathematical Sciences. (2013). Common Core State Standards for Mathematics Statement by Presidents of CBMS Member Professional Societies. Retrieved August 5, 2013, from http://amte.net/sites/all/themes/amte/resources/ccssm/CCSSM_SupportStatement.pdf Confrey, J. (1994). Splitting, similarity, and the rate of change: New approaches to multiplication and exponential functions. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 293332). Albany, NY: State University of New York Press. Costello, T. (Reporter), & Williams, B. (Anchor). (2008, February 18). Math Wars: Traditional Math vs. Math Lite [Television series episode]. NBC Nightly News. Retrieved from https://archives.nbclearn.com/portal/site/k12/browse/?cuecard=5978 Conference Board of the Mathematical Sciences. (2013). Common Core State Standards for Mathematics Statement by Presidents of CBMS Member Professional Societies. Retrieved from http://commoncoretools.me/wpcontent/uploads/2013/07/SupportStatementforCCSSMath.pdf Evans, R. W. (2006). The social studies wars, now and then. Social Education, 70(5), 317321. Fullan, M. (2007). The new meaning of educational change (4th ed.). New York, NY: Teachers College Press. Garelick, B. (2005). The new, amazeing approach to math: A mathematician with a child learns some politics. Education Next, 5(2), 2936. Gilovich, T., Medvec, V. H., & Kahneman, D. (1998). Varieties of regret: A debate and partial resolution. Psychological Review, 105, 602605. Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6, 105128. Greeno, J. G. (1997). On claims that answer the wrong questions. Educational Researcher, 26, 517. Greeno, J. G. (1998). Where is teaching? Issues in Education, 4(1), 111119. Gutstein, E. (2006). Reading and writing the world with mathematics: Towards a pedagogy for social justice. New York, NY: Routledge. Izumi, L., & Coburn, K. G. (2001). Facing the classroom challenge: Teacher quality and teacher training in Californias schools of education. San Francisco, CA: Pacific Research Institute for Public Policy. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press. Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discover, problembased, experiential, and inquirybased teaching. Educational Psychologist, 41(2), 7586. Klahr, D., & Nigam, M. (2004). The equivalence of learning paths in early science instruction: Effects of direct instruction and discovery learning. Psychological Science, 15(10), 661667. Klein, D. (2003). A brief history of American K12 mathematics education in the 20th century. In J. Royer (Ed.), Mathematical cognition: A volume in current perspectives on cognition, learning, and instruction (pp. 175225). Greenwich, CT: Information Age Publishing. Klein, D. (2007a). A quarter century of US math wars and political partisanship. Journal of the British Society for the History of Mathematics, 22, 2233. Klein, D. (2007b). School math books, nonsense, and the national science foundation. American Journal of Physics, 75(2), 101102. Klein, D., Braams, B. J., Parker, T., Quirk, W., Schmid, W., & Wilson, W. S. (2005). The state of state math standards. The Thomas B. Fordham Foundation. Retrieved June 2011 from www.edexcellence.net Koedinger, K. R., & Aleven, V. (2007). Exploring the assistance dilemma in experiments with cognitive tutors. Educational Psychology Review, 19, 239264. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 2963. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, UK: Cambridge University Press. Lee, H. S., & Anderson, J. R. (2013). Student learning: What has instruction got to do with it? Annual Review of Psychology, 64, 445469. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum. Martin, D. B. (2007). Beyond missionaries or cannibals. The High School Journal, 91, 628. Mayer, R. (2004). Should there be a threestrikes rule against pure discovery learning? The case for guided methods of instruction. American Psychologist, 59, 1419. Mellers, B., Hertwig, R., & Kahneman, D. (2001). Do frequency representations eliminate conjunction effects? An exercise in adversarial collaboration. Psychological Science, 12, 269275. Moses, R. P., & Cobb, C. E. (2001). Radical equations: Civil rights from Mississippi to the algebra project. Boston, MA: Beacon Press. Nathan, M. J., & Knuth, E. J. (2003). A study of whole classroom mathematical discourse and teacher change. Cognition and Instruction, 21, 175207. National Council of Teachers of Mathematics. (2006). Curriculum Focal Points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author. National Council of Teachers of Mathematics. (2013). Supporting the Common Core State Standards for Mathematics. Retrieved from http://www.nctm.org/ccssmposition/ National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. New Standards Primary Literacy Committee. (1999). Reading & writing grade by grade: Primary literacy standards for kindergarten through third grade (Illustrated by Garin Baker). Washington, DC: National Center on Education and the Economy. Next Generation Science Standards. (2013). Next generation science standards. Retrieved from http://www.nextgenscience.org/nextgenerationsciencestandards Nystrand, M., & Gamoran, A. (1997). The big picture: Language and learning in hundreds of English lessons. In M. Nystrand (Ed.), Opening dialogue. Understanding the dynamics of language and learning in the English classroom (pp. 3074). New York, NY: Teachers College Press. Pearson, P. D. (2004). The reading wars. Educational Policy, 18(1), 216252. Richards, J., & von Glasersfeld, E. (1980). Jean Piaget, Psychologist of epistemology: A discussion of Rotmans Jean Piaget: Psychologist of the real. Journal for Research in Mathematics Education, 11, 2936. Rogoff, B. (1994). Developing understanding of the idea of communities. Mind Culture and Society, 1(4), 209229. Rosenshine, B. (2012). Principles of instruction: Researchbased strategies that all teachers should know. American Educator, 36(1), 1219, 39. Schoenfeld, A. H. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational Researcher, 31, 1325. Schoenfeld, A. H. (2004). The math wars. Education Policy, 18, 253286. Senk, S., & Thompson, D. (2003). School mathematics curricula: Recommendations and issues. In S. Senk & D. Thompson (Eds.), Standardsoriented school mathematics curricula: What does the research say about student outcomes? Mahwah, NJ: Lawrence Erlbaum. Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27, 413. Simon, M. A. (1994). Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational Studies in Mathematics, 26, 7194. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114145. Slavin, R. E., Lake, C., & Groff, C. (2009). Effective programs in middle and high school mathematics: A bestevidence synthesis. Review of Educational Research, 79, 839911. Snow, C., Burns, S., & Griffin, P. (1998). Preventing reading failure in young children. Washington, DC: National Academies Press. Steffe, L., & Gale, J. (Eds.). (1995). Constructivism in education. Hillsdale, NJ: Lawrence Erlbaum. Steffe, L. P., & Kieren, T. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25(6), 711733. Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10, 313340. Thompson, P. W. (2000). Radical constructivism: Reflections and directions. In L. P. Steffe & P. W. Thompson (Eds.), Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld (pp. 412448). London, England: Falmer Press. Tyack, D., & Cuban, L. (1995). Tinkering toward utopia: A century of public school reform. Cambridge, MA: Harvard University Press. Venezky, R. L. (1992). Textbooks in school and society. In P. Q. Jackson (Ed.), Handbook of research on curriculum (pp. 436462). New York, NY: Macmillan. von Glasersfeld, E. (1989). Constructivism in education. In T. Husen & N. Postlethwaite (Eds.), International Encyclopedia of Education (Supplementary Volume; pp. 162163). Oxford, England: Pergamon. Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23(3), 1420. Wu, H. (2004). Must content dictate pedagogy in mathematics education? Unpublished manuscript, Department of Mathematics, University of California, Berkeley. Appendix Guiding Questions Used to Structure Discussion at Each Meeting ^{ } Participants in the Meetings Hosted at the University of Pittsburgh^{*}
Facilitators: Charles Munter, Mary Kay Stein, and Margaret Smith, University of Pittsburgh *Although all participants contributed to and reviewed the specifications of the two instructional models, inclusion of an individuals name on the above list is not to imply that the individual necessarily agrees with the assertions made in this article regarding the debates surrounding those models.





