The role of instructional materials in the relationship between the official curriculum and the enacted curriculum

ABSTRACT

We studied how the distal policy mechanisms of curricular aims and objectives articulated in official curriculum documents influenced classroom instruction, and the factors that were associated with the enactment of those curricular aims and objectives. The study was set in the U.S. context, where there is an ambitious effort to transform curriculum and instruction via the Common Core State Standards for Mathematics (CCSSM). The CCSSM represented the curricular aims and objectives in most of the U.S. at the time of the study. We analyzed enactments of this official curriculum in terms of the rigor of mathematical activity in 47 middle school mathematics lessons from multiple state and curriculum contexts. The enactment of the CCSSM was not uniform across contexts, and the lack of uniformity was associated in part with the type of instructional materials used by teachers. The use of instructional materials classified as delivery mechanism was associated with activity we characterized as routine procedural rigor. In lessons involving instructional materials classified as thinking device, we found greater variation and more occurrences of non-routine forms of rigor. These differences between types of instructional materials occurred despite the finding that teachers across the sample held similar views of the CCSSM. We conclude that the teachers responded more to features in the instructional materials than to the curriculum aims and objectives articulated in the CCSSM while planning and enacting lessons, which has implications for policy makers who aim to influence instruction through national standards and for school districts as they select materials.

KEYWORDS:

• Curriculum enactments
• standards
• instructional materials
• teacher practices

Policy makers at national and regional levels establish and adopt curriculum policy with the intention of influencing instruction and ultimately student learning; however, research has shown in multiple national contexts that educators do not always follow the official curriculum in classrooms (Thrupp, 2013; Wu, 2012; Yemini & Bronshtein, 2016). Furthermore, intermediate-level policy choices that reflect localized interpretations of the official curriculum influence what content is taught, how it is taught, and ultimately students’ opportunities to learn (Remillard & Heck, 2014). Other potential factors influencing how curriculum policy is enacted include features of the local context (e.g., demographics, socio-economic status), as well as resources associated with curriculum and instruction (Cohen et al., 2003; Remillard & Heck, 2014). Cohen, Raudenbush, and Ball state that resources closest to classroom practice, such as investments in curriculum and curriculum-based professional development, are more likely to transform mathematics instruction than expenditures for resources that are more distal or nebulous. This has implications for policy: policy decisions addressing local factors, those things closest to the classroom, may influence instruction more than policy made at the level of official curriculum, creating a dilemma for policy makers in terms of how best to effect changes they desire at scale. Consequently, there is a need to understand how distal policy mechanisms, such as the national and regional articulations of curriculum aims and objectives, influence classroom instruction, and what factors are associated with the ways distal policy mechanisms are interpreted and enacted.

Conceptions of the official curriculum and the operational curriculum

To conceptualize aspects of curriculum, we turn to Remillard and Heck (2014). They explain that the official curriculum is comprised of curriculum goals and objectives, the content of consequential assessments, and the designated curriculum. The goals and objectives include “expectations for student learning or performance and, in some cases, the instructional or curricular resources, and pathways for learning to be employed” (p. 708). Remillard and Heck characterize national or regional standards as part of the official curriculum. Consequential assessments do not formally stipulate a scope and sequence of content; however, due to the stakes involved in these assessments, these assessments influence the content to which teachers attend. The designated curriculum is the “set of instructional plans specified by an authorized, governing body, be it a ministry of education or local province” (p. 710), and can include instructional materials adopted by local educational officials. Instructional materials can be part of the designated curriculum when their adoption is closely linked at the policy level to the official curriculum, as is the case in highly centralized education systems such as Korea and Japan, and in other systems that stipulate the alignment between textbooks and standards.

The operational curriculum includes the teacher-intended curriculum (the lessons teachers design from the designated curriculum) and the enacted curriculum. These two phases (from designated curriculum to planned lessons and from planned lessons to enacted lessons) involve transformations related to a variety of factors, including teacher characteristics and experiences, nature of curriculum resources, and student characteristics and experiences (Remillard, 2005). The enacted curriculum involves what happens in classrooms when students and teachers interact around content and tasks.

Developments related to the official curriculum in the United States

Over the last decade, the United States embarked on an ambitious experiment related to national standards for curriculum aims and objectives, a part of the official curriculum. During that time, the Common Core State Standards for Mathematics (CCSSM) (Common Core State Standards Initiative, 2010) represented the curricular aims and objectives within the official curriculum, constituting a broad and rare consensus among diverse stakeholders about the mathematical content in the official curriculum in the U.S. The CCSSM were intended to represent a common desired curriculum (Munter et al., 2015) to prepare students for college and careers (Common Core State Standards Initiative, 2010), with the intent of pushing teachers to take up ambitious forms of curriculum and instruction. At one time, the CCSSM were the official standards for 45 states plus the District of Columbia, and, despite the rollback in some states, the CCSSM or CCSSM-based standards were the official curriculum in roughly 40 states when this study was conducted. Analyses of the CCSSM found that the standards were more rigorous, focused, and coherent than prior state standards (Porter et al., 2011; Schmidt & Houang, 2012; Woolard, 2013), and thus more in line with the standards of high-achieving countries as measured on the Trends in International Mathematics and Science Study (TIMSS), suggesting an ambitious effort to overhaul mathematics curriculum and instruction in the U.S.

In addition to specifying content objectives, the CCSSM documents articulated a vision of mathematical learning (Common Core State Standards Initiative, 2010; http://www.corestandards.org/other-resources/key-shifts-in-mathematics/). These documents critiqued the conceptually weak presentation of mathematical topics in most U.S. textbooks, which they contrasted with their view of rigorous mathematical learning. The CCSSM documents stated that learners need to pursue equally three aspects of rigor: conceptual understanding, procedural skills and fluency, and application. The authors described these aspects of rigor in the following ways:

• Conceptual understanding is viewed as learners understanding concepts from a number of perspectives and seeing math as “more than a set of mnemonics or discrete procedures;”

• Procedural skills and fluency are viewed in terms of practicing “core functions” and providing “access to more complex concepts and procedures;” and

• Applications are viewed as using mathematics “in situations that require mathematical knowledge,” which necessitates that students use both conceptual understanding and procedural fluency (http://www.corestandards.org/other-resources/key-shifts-in-mathematics/).

In short, these conceptualizations of what it means to learn and understand mathematics emphasized procedural competency and making sense of mathematics, including knowing when to apply procedures. Indeed, the CCSSM documents articulated the view that rigorous learning of mathematics involves rich communication and interactions around problems that provide opportunities for students to make connections and to think flexibly.

Thus, the CCSSM represented a well-defined set of curricular aims and objectives and an ambitious attempt at the national and state level to influence local curriculum and instruction. However, the impact of the official curriculum is not straightforward, as noted earlier, and it was not clear what impact this ambitious effort would have on classroom instruction. Prior efforts to reform mathematics instruction in the U.S. show how difficult it is to make substantive changes to classroom practice (e.g., Spillane & Zeuli, 1999; Weiss et al., 2003). Although the CCSSM represented a scope and sequence of content at each grade level as well as a perspective on the mathematical learning that should occur with respect to that content, few explicit connections were evident between the two in the CCSSM documents (cf. Cobb & Jackson, 2011). Thus, it was possible to view the CCSSM only in terms of a scope and sequence of mathematical content, with little regard to an articulation of pedagogy or theory of learning. Furthermore, the authors did not specify characteristics of curriculum and instruction that should accompany the CCSSM, stating that local stakeholders should make such decisions (McCallum, 2012), leaving parts of the official curriculum unarticulated. This unusual combination of centralized articulation of curriculum aims and objectives and localized choices regarding how to implement those aims and objectives provided us with an opportunity to explore factors that mediated the enactment of national standards. The primary factor on which we focused were the instructional materials. We explored how these materials were associated with the different ways teachers enacted the rigorous learning envisioned in the CCSSM documents.

Research questions

Our research questions were:

1. How do middle grades teachers’ lesson enactments using different types of instructional materials – delivery mechanism or thinking device – compare in terms of rigor of mathematical activity?

2. What is the role of instructional materials in the relationship between the official curriculum and the operational curriculum?

Framework

We drew from frameworks that explore how curriculum policy impacts classroom practice. These frameworks focus on how curriculum policies such as national standards are operationalized in classrooms. We conceived of curriculum policy in terms of components in Remillard and Heck (2014) “curriculum policy, design, and enactment system” (p. 709). In particular, we explored the relationships between instructional materials and the official and operational curriculum. Scholars argue that the alignment of standards with other parts of instructional systems, such as accountability mechanisms, instructional materials, and professional development, is critical if the ambitious goals of the official curriculum are to be realized (Coburn et al., 2016). However, local educational leaders often interpret curriculum policy in ways not intended by policy makers (Hill, 2001), including their decisions about which instructional materials align with curriculum policy. The instructional materials sanctioned by local policy makers and educational leaders influence teachers’ interpretations of the goals, aims, and expected student learning in the national standards; in fact, they may serve as a proxy for those standards (Remillard & Heck, 2014). Another component of the system we drew from in the Remillard and Heck model is the operational curriculum, representing teachers’ work as they sequence, plan, and enact lessons. In doing so, teachers draw on their interpretations of the official curriculum, as stipulated in state or local policy frameworks and consequential assessments, the local education authority’s decision to adopt a particular curriculum program, and the characteristics of that program. The enacted curriculum, what actually happens in classrooms, is a key component of the operational curriculum. As noted by Stein et al. (1996), changes can occur between the planned and enacted lesson, particularly with tasks that are challenging or complex. Thus, the framework does not assume a perfect match between the planned and enacted lessons, and stipulates the important role teachers play in enacting curriculum.

The framework is shown in Figure 1. The figure shows that we conjectured that there could be two potential effects from the CCSSM, one direct and one indirect. Teachers might directly respond to national standards as they design and enact lessons, and/or they might respond indirectly to national standards via the instructional materials adopted by the district. One of the topics we explored in this study is the extent to which the direct effect is evident; that is, given that all the teachers in this study were responding to the same national standards, and had similar perceptions of the national standards, would they enact the standards in similar ways? For teachers, a potential role for instructional materials is that they serve as a clarifying mechanism for how to interpret and enact national standards, as instructional materials typically provide greater specificity than standards with respect to mathematical activity and instructional practices. So, instructional materials have the potential to exert considerable influence on the operational curriculum, especially if local educational leaders assume or determine that the materials align with the national standards. Below, we conceptualize two distinct approaches to the design of instructional materials found in the U.S. and elsewhere; this conceptualization framed the analysis of materials for this study, described in more detail in the methods section.

Figure 1. Components of Curriculum Policy Implementation.

Characterizing instructional materials

In our characterization of instructional materials, we avoided the labels traditional and reform, broad terms that have little meaning outside of the U.S. context. We sought instead to characterize materials based on the prevalence and absence of particular design features. Despite differences in approaches in the design of the materials, we assumed that the characteristics of instructional materials reflected the goals and purposes of the designers (i.e., we assumed characteristics were designed intentionally), and that those goals and purposes were to support students in the rigorous learning of mathematics. We chose to use the lens of communication function to characterize curriculum materials, recognizing that other lenses, such as anti-racism or ideology (cf. Gutierrez, 2013; Iseke-Barnes, 2000), may lead to different ways to characterize curriculum materials (and subsequent findings.

We describe two contrasting sets of design features that serve as endpoints of a conjectured continuum of design perspectives; each set of features is associated with a curriculum metaphor described in more detail below. The continuum of design features was evident in the materials used by the U.S. teachers in our broader study conducted in the years 2013–2016, a period that coincides with the widespread adoption of the CCSSM. The design features mirror curriculum movements in the U.S. and elsewhere over the last 100 or so years, with some important distinctions. The first set of features is associated with the delivery mechanism (DM) metaphor, which derives largely from the scientific approach to curriculum development (Choppin et al., 2015). This approach focuses on efficiency, in which activities are designed with respect to disciplinary logic or expert performance so that they can be learned in the most efficient way. In the U.S., this perspective stems from the technical-rational approach to curriculum that has been dominant over the last century (Kliebard, 1975; Larson, 2014). Early adherents to this approach describe curriculum development as entailing a highly detailed analysis of disciplinary experts’ knowledge and performance (Bobbitt, 1918; Charters, 1923), dividing the content into discrete bits of knowledge and skills that can be efficiently delivered and, when mastered, constitute competence in a discipline (e.g., the mastery perspective). Curriculum perspectives and development processes derived from the scientific tradition are still the basis of the design of most instructional materials in the U.S. (Larson, 2014).

The second set of features is associated with the thinking device (TD) metaphor, which stems from progressive curriculum movements that focus on the development of the child, though with some important distinctions (Choppin et al., 2015). Prior progressive movements, such as Dewey’s Laboratory School, focused on creating curriculum from the perspective of how children think and develop, rather than from a disciplinary perspective (Kliebard, 1975). There are key differences between those efforts and the curriculum programs developed in the U.S. in the 1990s and 2000s. Some curriculum programs in mathematics education in the U.S. in the 1990s and 2000s were based on cognitive and sociocultural research conducted over the last 50 years, research that builds on understanding of how children think in particular mathematical domains (A. Schoenfeld, 2006). These mathematics curriculum materials are quite bounded in terms of the selection of content and open-endedness of learning activities, perhaps more so than approaches from the earlier progressive movement (Tanner, 1991).

According to our framework, the basic distinguishing trait of DM versus TD metaphors is the orientation to communicating what people should learn and how they should learn it (Choppin et al., 2015). Drawing on Wertsch and Toma (1995), we focus on two primary perspectives on the function of communication: the monologic and the dialogic. The monologic perspective aligns with the delivery or conduit metaphor (Reddy, 1979), in which the main processes related to text are encoding, transmission, and decoding. The dialogic perspective views text as thinking devices (Lotman, 1988, as cited in Wertsch & Toma, 1995). The monologic perspective conveys the message that learning occurs through unambiguous transmission of meaning, which occurs through explicit and formal communication. This view reflects a positivist perspective that knowledge is objective and neutral and can unambiguously reflect meaning. These assumptions have been strongly critiqued (cf. Halliday & Matthiessen, 2014; Jewitt et al., 2016), but nevertheless are widely evident in the design of instructional materials in the U.S. and elsewhere (Kliebard, 1975; Larson, 2014; Valverde et al., 2002).

In the dialogic view, communication involves sense making in which parties coordinate semiotic resources according to their interpretations of the context and of their goals (Halliday, 1978; Kress, 2010). The dialogic perspective stresses that interactions between interlocutors are the mechanism for learning: the meanings of texts are ambiguous and contested and require explicit attempts to negotiate a shared understanding, which results in new meanings being generated, both individually and collectively. In this perspective, knowledge is fluid, contested, and situated within particular contexts (cf. Gee, 1999), rather than fixed and universal.

Though we recognize that all texts have both monologic and dialogic functions, in the design of instructional materials typically one of these functions is dominant. Below, we describe dimensions of instructional materials that provide the analytic means to categorize programs according to their communicative functions.

Features of instructional materials

The features of instructional materials on which we focused emerged from the assumptions rooted in the metaphors above about what mathematics is and what it means to learn mathematics. From the DM perspective, concepts are formal ideas that are represented by precise definitions and well-defined algorithms that can be unambiguously communicated. Rigorous learning involves algorithmic fluency, including applying those algorithms in real-world contexts (Stein et al., 2007; Valverde et al., 2002). Conceptual understanding involves mastery of formalized bits of knowledge and skills that represent the knowledge and skills of experts.

From the TD perspective, concepts represent big ideas that coherently connect multiple topics and representations of those concepts (Carpenter & Lehrer, 1999; Hiebert & Carpenter, 1992), and, more generally, conceptual understanding involves the inter-connectedness of ideas and concepts (Vygotsky, 1934/2012). Given the interconnected nature of conceptual ideas, understanding is developed through experiences that elicit students’ intuitive and idiosyncratic ideas, which are then refined in subsequent experiences to more conventional disciplinary understandings, in a process of progressive formalization (e.g., Gravemeijer, 2004; National Research Council, 2001). Developing deep understanding of disciplinary concepts involves an iterative and dynamic interplay between students’ intuitive and idiosyncratic ideas and conventional formulations of disciplinary concepts (Vygotsky, 1934/2012). Rigorous understanding in this perspective involves making connections between topics and between multiple representations, developing effective if idiosyncratic algorithms, and flexibly applying knowledge in new situations (National Research Council, 2001).

We present four features of instructional materials and then describe each of those features with respect to the two metaphors. Together, these features form a collection associated with each metaphor that represents distinct views of mathematics and the learning of mathematics.

Feature 1: development of terminology

In DM materials, following a transmission model of learning, formal terminology is presented at the beginning of lessons so that students can use precise terms during the lesson. Formalization occurs at the beginning of lessons with the intention of reducing ambiguity and representing expert forms of knowledge and skills (Stein et al., 2007).

In TD materials, the formalization of terminology occurs after students experience problem-solving in contexts that provide opportunities for the subsequent discussion of terms. Key assumptions are that these experiences will recruit students’ intuitive understanding of big ideas, that these understandings will initially be idiosyncratic and ambiguous, and that there will be opportunities to refine these understandings into more conventional, precise, and formal mathematics concepts. The timing of formalization of concepts aligns with notions of progressive formalization in which students’ initial informal and intuitive ideas are used as the basis of subsequent generalization and abstraction (Gravemeijer, 2004; National Research Council, 2001).

Feature 2: development of procedures

Similar to terminology, in DM materials algorithms are modeled at the beginning of lessons so that students can accurately apply procedures during the lesson. Formalization occurs at the beginning of lessons to represent expert performance and reduce the use of inefficient or inaccurate procedures. Procedures are modeled before students engage in independent work or problem solving and primarily focus on practicing the algorithms modeled earlier in the lesson (Stein et al., 2007).

In contrast, TD materials develop procedures through iterative processes, providing opportunities for collective negotiation of what constitutes appropriate and efficient strategies. Students initially solve problems without having a prescribed procedure, an approach that entails flexibility and ambiguity. Subsequent problems provide opportunities for continued refinement of procedures, with a focus on identifying efficient procedures and an understanding of why procedures work. Key assumptions are that initial problem-solving experiences will recruit students’ intuitive understanding of big ideas, that students’ approaches will initially be idiosyncratic, and that there will be opportunities to revise procedures into more conventional or efficient versions (Van den Heuvel-panhuizen & Drijvers, 2014).

Feature 3: nature of tasks

The DM perspective emphasizes well- bounded tasks that represent a discrete bit of expert knowledge or skill. The tasks usually have a well-defined approach that is connected to conventionally- or historically-developed approaches that has been previously presented by the teacher. Tasks may vary with respect to complexity, in terms of the number of steps necessary to complete the task; furthermore, the variety of tasks emphasize opportunities for students to select from among multiple known approaches to develop flexible methods (Munter et al., 2015, p. 10).

Tasks from the TD perspective emphasize internal heterogeneity by providing multiple entry points and varied solution paths and outcomes (Jackson et al., 2013; Stein et al., 2000). These tasks are relatively easy to access at a basic level but can be approached using more complex and difficult approaches; the diversity of approaches provides opportunities for collective reflection and discussion that emphasizes connections between approaches and establishes criteria for what constitutes appropriate and efficient strategies. Following Munter et al.’s (2015) description of dialogic instruction, these tasks afford opportunities for students to initially attempt a problem before encountering challenges that require additional personal and collective resources.

Feature 4: collections of problems or tasks within a lesson

From the DM perspective, sequences form incremental structures (like bricks in a wall, such that the whole wall is viewed as the composite of separate, distinct bricks) in which mathematics is presented in discrete bits that represent formalized knowledge or skill that incrementally build toward a greater (coherent) whole. The ordering of task sequences is dictated by the structure of mathematics (Munter et al., 2015). Problem contexts and skills vary across lessons, but within lessons there is often a singular skill or approach emphasized. In terms of how these characteristics are manifest in instructional materials, problem sets for one lesson are designed for mastery of the skill or procedure that is the basis of the lesson, with repetitive tasks designed to build fluency. Tasks are typically sequenced from lower to higher level tasks, with repetitive skill practice provided before tasks that require an application of those skills in context (Stein et al., 2007). In this view “basic knowledge is a prerequisite for applications” (A. Schoenfeld, 2006, p. 16). The initial focus on fluency is aimed at providing a basis for more complex problems in which students progressively adapt their emerging procedural fluency to new problem types (Munter et al., 2015), a key feature of rigorous learning described in the CCSSM.

From the TD perspective, tasks and lessons in a sequence are connected by a coherent thread (overarching mathematical concept and/or common context) that links the mathematics and tasks. These sequences communicate a view of that learning occurs as students build meaning from a variety of experiences, from which conceptual understanding is developed. Initial problems in a sequence are intended to invoke intuitive understandings of the big idea and prior mathematical knowledge they have developed: in dialogic approaches, students are initially presented tasks that “initiate students to new ideas and deepen their understanding of concepts,” whose solution is not readily available, and which provide opportunities for students to “wrestle … for a while without the teacher’s interference” (Munter et al., 2015, p. 10). Subsequent problems vary with respect to representation, given information, or task demand, as opposed to sets of uniform problems. Instructional sequences develop an essential idea over time, with topics varying in ways that convey a connected and comprehensive view of the big idea. This variation and repeated considerations of the big idea over multiple lessons provides opportunities to refine initial approaches and understandings and connect them to more formal disciplinary understandings, a process of progressive formalization (Gravemeijer, 2004; Van den Heuvel-panhuizen, 2003), with a lesser emphasis on procedural fluency. The notion of progressive formalization suggests that tasks early in an instructional sequence should recruit students’ pre-formal experiences and relate them to a mathematical context (Van den Heuvel-panhuizen, 2003). Thus, lessons later in instructional sequences are designed to formalize student thinking and relate that thinking to conventional mathematical terminology and notation.

Summary of features

In summary, the DM perspective communicates a view of mathematics as precise, discrete, unambiguous, logical, and deductive. DM emphasizes direct explanations as well as modeling of problems, with the goal of developing procedural fluency. Curriculum as TD communicates a view of the learning of mathematics as idiosyncratic, connected, ambiguous, intuitive, and inductive. This perspective focuses on the dialogic affordances of tasks that offer the potential for interactions and that promote understanding through their multiple entry points, ambiguity, and connections to big mathematical ideas. See Table 1 for a summary of the features of each type.

Table 1. Manifestations of Dimensions of Curriculum Messages and Structures

Features	Delivery Mechanism	Thinking Device
Development of terminology	Formal definitions for key terms are provided at or near the beginning of the lesson; emphasis is on precise use of vocabulary	Terminology is formalized after students have had experiences that ground the development of terminology; initial emphasis is on using informal language
Development of procedures	Examples or models for procedures are provided at or near the beginning of the lesson; emphasis is on accurate use of established procedures	Procedures or algorithms are developed as a result of problem solving experiences; there is initial ambiguity with respect to approaches, allowing for multiple and idiosyncratic approaches
Nature of tasks	The tasks are narrowly bounded, belonging to a well-defined class of problems. More complex problems allow students to select from multiple known approaches.	Many problems can be approached from a variety of methods, and the selection of method is initially ambiguous. Problems emphasize connections, including connections between representations and to context.
Collections of problems or tasks within a lesson	The ordering of tasks is dictated by a disciplinary progression in which each new set of problems involves a slightly different topic or skill from preceding problems. Sets of problems are designed for repetitive practice, with increasing complexity to develop strategic flexibility.	Sequences of tasks are connected by an overarching concept and/or common context that connects the mathematics and tasks. Initial problems provide opportunities for students to wrestle with a concept and an approach. Subsequent problems vary with respect to representation, given information, or task demand.

Methods

For each curriculum program in the study, we explored the role of instructional materials in how teachers enacted the official curriculum, and sought to understand how instructional materials were associated with the rigor of mathematical activity in observed lessons. We analyzed features of a set of lessons from each of the nine curriculum programs used by participating teachers, and we also analyzed teachers’ lessons enactments with those materials. Previous research that examined the interaction between curriculum type and instructional practices has shown that, when teachers are newly introduced to high demand tasks, they often reduce the complexity and demand of tasks (Stein et al., 1996). Other research has shown that instructional practices and enactments often do not align with the designers’ intentions (Brown et al., 2009; Tarr et al., 2008). In the U.S., this lack of alignment is especially prevalent for instructional materials based on the standards documents from the National Council of Teachers of Mathematics [NCTM] (National Council of Teachers of Mathematics, 1989, 2000). Reasons cited for the difficulties in taking up resources in standards-based programs include a range of teacher and contextual characteristics (Arbaugh, Lannin, Jones & Park-Rogers, 2006; Lloyd, 1999; Remillard & Bryans, 2004).

We video-recorded 47 lessons, one from each of the participants. These lessons were recorded during the 2013–2014 school year, when all of the states in which the participants taught used the CCSSM (or a very near version in the case of Indiana) as the state curriculum framework. This study was part of a larger NSF-funded project that explored teachers’ perceptions of the CCSSM, the ways teachers were prepared to teach the CCSSM, how teachers drew upon instructional materials to plan lessons they viewed as CCSSM-aligned, and how they enacted those lessons (Choppin et al., 2012).

Participants

The 47 participating teachers were middle grade mathematics teachers from six states and used nine different sets of instructional materials. The demographic contexts included urban, rural, and suburban districts. Teachers reported a range of experience with the instructional materials they used in the lesson observations, though teachers using CMP were the only teachers who reported using versions of their current instructional materials for five or more years. See Table 2 for a summary of the participants’ district pseudonyms, states, instructional materials, and years of experience with their respective materials.

Table 2. Participants’ State, Curriculum, and Experience with Curriculum

State/District Pseudonyms	Teachers	Instructional Materials	Years using curriculum
Colorado/Pablo	3	Math in Focus	2
Indiana/Anna	5	digits (digital)	1
Michigan/Diamond	1	Holt On Core	>3
Michigan/Jackson	1	Holt Algebra 1	1
Michigan/Karry	4	Prentice Hall	3
Michigan/Trenton	2	Glencoe	2 or more
New Hampshire/George	6	Glencoe	2
New York/Denton	5	Glencoe	2
Michigan/Adam	1	CMP2	>5
Michigan/Brad	2	CMP3	2 (CMP3); > 5 for CMP2
Michigan/Ellen	2	CMP3	2
New York/Chester	4	CMP3	2 (CMP3); > 5 for CMP2
New York/Harriett	7	CMP3	1
Wisconsin/Sanders	4	CPM	3

At the start of data collection, 32 of the teachers worked in districts that had recently adopted programs specifically to address the CCSSM. Specifically, nine teachers used versions of CMP (Ellen, Harriett), four teachers used CPM (Sanders), and 19 teachers used a range of commercially-developed programs (Anna, Denton, George, Jackson, Pablo). The districts of the other 14 teachers had adopted their programs before the advent of the CCSSM, including seven teachers who used versions of CMP (Adams, Brad, Chester) and seven teachers who used commercially-developed programs (Diamond, Karry, Trenton). These teachers reported that their districts had determined that the materials they were already using were sufficiently aligned with the CCSSM.

The observed lessons were intended by the teachers to be aligned with the CCSSM. All of the teachers identified specific content and practice CCSSM standards addressed in the lessons; moreover, in the recruitment scripts and in multiple interviews, the research team explained to the teachers that we were interested in observing lessons that represented their typical efforts to address the CCSSM. Furthermore, all of the local curriculum frameworks were based on the CCSSM and the teachers reported aligning their lessons to meet these frameworks, further evidencing teachers’ intentions.

Participants’ initial perceptions of the CCSSM

In order to understand how our participants interpreted the CCSSM, we administered a survey in the same year of the lesson observations reported here. We found participants held a common perception that the CCSSM emphasized more active and interactive forms of mathematical activity than previous standards. For example, 80% of them agreed with the statement that the CCSSM “will require you to emphasize communication more with your students.” At the same time, 88% of the teachers agreed with the statement that the CCSSM will provide “more opportunities for students to struggle while solving problems.” Thus, the participants viewed the CCSSM as more than a set of content objectives; they viewed it is as incorporating a more active and problem-based instructional style.

Participant interviews also showed evidence of these perceptions of the CCSSM. For example, one teacher emphasized the conceptual focus of the CCSSM, stating “it’s not acceptable anymore to know just the answer to something, but understand the meaning behind it and why they’re doing it and being able to explain it” (Pressley, Birch School District). Another teacher commented on the problem-solving focus, stating that the CCSSM:

really pushes students to be able to think mathematically and be able to given a problem that they’ve never seen before apply the concepts that they have learned in new and different ways in order to tackle any kind of problem and build a problem solving in a critical thinking mind in our students as opposed to like this is a percent problem do the percent formula. (Lester, Harriett School District).

A third teacher reported an emphasis in the CCSSM on applying mathematics to real-world problems:

There’s more of a focus on real world application and kind of real life situations, as opposed to just here’s a formula here’s a problem but any situation where you might actually apply this or use that skill or that knowledge. (Bernard, Pablo School District)

Coding process for instructional materials

We selected cases in which teachers used an identifiable set of instructional materials to plan the lesson, specifically those materials adopted by their school district. To characterize each set of instructional materials, we focused on lessons involving proportional reasoning and linearity, using at least three lessons from each set of materials. We purposefully selected lessons on these mathematical topics to maintain consistency among the lesson topics. Consistent with Student Achievement Partners (achievethecore.org/category/774/mathematics-focus-by-grade-level), we viewed these topics as central topics in the CCSSM middle grade standards. We coded the materials along the characteristics listed in Table 1. We focused on the primary student resources associated within the district-adopted materials, recognizing that most instructional materials have supplemental materials that may serve different purposes (e.g., a TD program may have supplemental skill worksheets, or a DM program may have a set of tasks intended for occasional extended problem-solving activities).

We coded the materials along a seven-point continuum, with a 1 representing the most extreme case of a DM program, a 7 representing the most extreme case of a TD program, and a 4 representing a program whose attributes were equally represented by the two sets of features. We conceived of the 1s and 7s as extremes that were unlikely to be present in our data but were conceptually possible. Thus a 2 indicated material with strong and consistent evidence of DM features. These materials predominantly exhibited tendencies associated with the DM metaphor. Similarly, a 6 indicated material with strong and consistent evidence of TD features. These materials predominantly exhibited tendencies associated with the TD metaphor. A 3 or a 5 indicated that the materials showed evidence toward one end of the continuum but did not exhibit features as strongly as a 2 or a 7, respectively. See Table 3 to see the continuum.

Table 3. Continuum of Features

Category	Descriptor
1	Delivery Mechanism Features Exclusively
2	Substantial Presence of DM Features
3	Moderate Presence of DM Features
4	Balance of DM and TD Features
5	Moderate Presence of TD Features
6	Substantial Presence of TD Features
7	Thinking Device Features Exclusively

We found differences within programs that had similar ratings. These differences were related to the representations of mathematics and mathematical tasks in the materials. For example, we found that Math in Focus was rated similarly to digits, both of which we rated a 2. However, we found the representations of mathematics and mathematical tasks in the Math in Focus materials to be more coherent and powerful than in the digits materials. This difference was not captured in our analysis because our analytic framework focused on the communicative aspects of the materials, not on whether the representations of the mathematics or tasks were coherent or powerful.

To code the materials, the four authors wrote memos for each of the four categories from Table 1 and then made a single holistic evaluation of the materials using the 7-point scale. The initial coding process began with the four authors independently coding at least one lesson from each of the programs, using the characteristics from Table 1. We conducted a group reconciling process to establish consensus about how to rate the programs on our scale. Once we reached shared agreement on the ratings, the remaining lessons were pair-coded, with differences resolved via consensus discussions. For the instructional materials we characterized as having features predominantly in the DM category, we found little ambiguity or disagreement amongst the coders. The ambiguities we did identify represented minor deviations from the descriptions of features in our framework. For example, in some DM materials, the lessons began with a launch problem before any procedure was explained. These launch problems, however, had a narrow range of intended responses (as noted in the teacher resources), and were followed immediately by formal explanations of terminology or procedures before the main portion of the lesson. Consequently, we still considered that terminology and procedures were formalized before students had opportunities to engage with mathematical problems, consistent with the characteristics of DM materials. For the TD materials, the authors coalesced quickly around designating CMP as a 6. We deliberated about CPM longer to determine its placement on the scale. We identified some features as strongly connected to the TD perspective (e.g., students worked on problems before a procedure was modeled or explained) while other features limited this work (e.g., the problem space was more constrained to the point where we felt the materials were emphasizing a single approach). Given the mix of TD and DM features, we rated CPM a 5.

Characterizing the enacted curriculum

Our research questions focused on how instructional materials played a role in the relationship between the official curriculum and the enacted curriculum. Consequently, we studied the lesson enactments of 47 teachers using instructional materials analyzed with the framework outlined above. Though we collected data on teachers’ lesson planning, we focused on enactments in this study because we were interested in how the plans manifested themselves in terms of mathematical activity. In prior analysis, we did not find major discrepancies between the lesson plans and how the teachers presented the lessons, so we did not see a need to expand the focus to planning to address our research questions. Below, we describe the process for analyzing the enacted curriculum.

Video-recorded lessons

The 47 lessons were videotaped with the purpose of analyzing instructional practices related to the CCSSM and documenting teachers’ enacted curricula. We viewed and transcribed the videos in Transana (Woods & Fassnacht, 2014). The 47 lessons represented the maximum number of participants for whom we had adequate data and who used an identifiable curriculum program, specifically the one adopted by their district.

Analysis of video-recorded lessons

In this section, we describe the analysis of the video-recorded lessons, which we connect to the enacted curriculum. In our analysis of the video-recorded lessons, we focused on the rigor of mathematical activity. Because we contend that instruction can vary based on content, we observed lessons primarily related to linearity and proportionality, with a few geometry lessons, to keep the content area common across observations. We note here that a limitation in our study is that we only analyzed one lesson per teacher. We did so because we aimed to study instruction across a larger number of teachers rather than several lessons from fewer teachers.

Observation instrument

We decided to analyze at the level of lesson segments rather than lessons as a whole, given that lesson segments serve different instructional purposes, a decision similar to designs of other instruments (cf. A.H. Schoenfeld & the Teaching for Robust Understanding Project, 2016). We established codes for mathematical rigor to capture the aspects of rigor described in the CCSSM documents, including whether the activity emphasized procedural skills, procedural fluency, application, or conceptual understanding.

Time sampling approach

We utilized a modified time-sampling approach (A.H. Schoenfeld & the Teaching for Robust Understanding Project, 2016; Hill et al., 2012). We transcribed the whole class portions of the lessons and most of the group work, as the audio quality permitted. The first author parsed the transcripts into roughly four-to six-minute chunks delineated by participation structure and topical focus, similar to what Mehan (1979) termed a topically related set. To determine each chunk, the first author bounded the segments first by participation structures (e.g., seat work, whole class discussion, group work), then by a combination of duration and topical focus. Within participation structures, the first author chunked segments into intervals, with variation due to efforts to maintain continuity of a line of questioning within a given segment. We found that using a lesson segment as a unit of analysis was a consistent and sensible way to capture enactments regardless of curriculum type while still preserving the natural breaks and shifts in classroom instruction. Coding by segments allowed variability within lessons to be captured, while still identifying the predominant forms of mathematical activity, defined below. The number of segments per lesson ranged from four to 17, with a median of nine segments for each lesson. There were on average two more segments per lesson for the TD lessons (roughly 10 per lesson for the TD and eight per lesson for the DM), a difference we attributed to the prevalence of block scheduling in the TD classes. We then grouped the lessons according to the predominant type of rigor evident in the lesson. Description of the rigor categories is given below.

Analytic categories

We coded each segment with one of four rigor categories: procedural skills, procedural fluency, application, and conceptual understanding. Within each of these categories, we developed definitions for three different types of mathematical activity or topic focus that we observed in our study: (1) facts, definitions, and formulas; (2) procedures and algorithms; and (3) representations. We note that other types of mathematical activity are possible (e.g., proof), but the activities we listed comprehensively covered the activities we observed in our lessons, most likely because the lessons were focused on proportionality, linearity, and, to a lesser extent, geometry.

We developed the definitions within each category using language from CCSSM and the documents on which the CCSSM were based, mainly Adding it Up: Helping Children Learn Mathematics (National Research Council, 2001). Distinguishing among different types of mathematical activity allowed us to elaborate in detail the four categories of rigor. This is turn proved helpful when coding the segments, as we had highly specific language and descriptions to reference. See Table 4 for our definitions of categories of rigor (shown in the second through fifth columns) with descriptors for each type of mathematical activity/topic (shown in the first column).

Table 4. Definitions of categories of rigor

Mathematical Activity/Topic	Categories of Rigor in the Common Core
	Procedural skills (Focus on routine activity aimed at accuracy and speed.)	Procedural Fluency (Includes flexibility with and understanding of procedures. Focus on non-routine activity)	Application (Applying math in situations; Non-routine)	Conceptual Understanding (Accessing concepts from a number of perspectives and making connections that show mathematics as more than a set of mnemonics or discrete procedures; Non-routine)
Facts, definitions, and formulas (formulas is an established equation [conventional rule] relating to a specific mathematical object/relationship)	Memorization or recall of basic facts, definitions, or formulas in established or familiar contexts (routine)	Recalling facts, definitions, and/or formulas appropriately and using flexibly when solving a problem	Applying basic facts, definitions, and/or formulas in new contexts, including mathematical contexts, and the context matters for understanding and/or using the mathematics, not just “window dressing”	Developing or making sense of facts, definitions and/or formulas, including conventional forms. This might include drawing on students’ informal/experiential knowledge to make sense of a mathematical term and develop a mathematical definition/formula.
Procedures and algorithms (algebraic or numeric manipulations	Describing and/or accurately and efficiently carrying out computations or well-established procedure in a routine fashion.	Students strategically and flexibly select and carry out procedures	Applying procedures, and including explanations with connections to new contexts (e.g., mathematical, the real world), and the context matters for understanding and/or using the mathematics, not just “window dressing”	Developing or making sense of procedures and/or algorithms that have not been previously established.
Representations	Interpreting or generating representations with a well-established method, without strategic choices or justification for those choices	Generating information or interpreting mathematical claims and/or relationships from a table, graph, equation, or other standard or nonstandard representation, involving strategic choices regarding representations, translating information across types of representations or within a representation, and/or extracting and describing a pattern evident in a representation. Includes making sense of components of an equation.	Interpreting information in a table, graph, equation, or other representation in relation to the disciplinary or real world context, and the context matters for understanding and/or using the mathematics, not just “window dressing”	Students create novel representations and/or use representations in novel ways, and provide justification for their choices and translation between choices

We separated procedural skills from procedural fluency. The language in the CCSSM distinguishes between use of procedures (procedural skills) and understanding when procedures should be used (procedural fluency). In one document, they state:

The Standards for Mathematical Content are a balanced combination of procedure and understanding … Students who lack understanding of a topic may rely on procedures too heavily. (CCSSM, p. 8).

In another document, the authors state:

The standards call for speed and accuracy in calculation. Students must practice core functions, such as single-digit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others. (http://www.corestandards.org/other-resources/key-shifts-in-mathematics/)

Both passages equate procedure with skills, and fluency is associated with automaticity developed through repetition. This departs from the National Research Council (2001)’s definition of procedural fluency, which incorporates strategic understanding of why a procedure works and when it is appropriate to use a procedure. Given this dichotomy, and attempting to incorporate both views of procedural rigor, we separated procedural skill from procedural fluency as distinct forms of rigor. As will be seen below, both forms of procedural rigor were evident in our data.

Our first category of rigor is thus procedural skills that are developed through repetitive tasks. We felt that this category represented routine forms of activity, in comparison with non-routine forms of activity associated with the other forms of rigor. We defined routine activity as activity whose primary purpose is to apply previously encountered terms or accurately and efficiently carry out highly-defined and unambiguous procedures. Conversely, our next category, procedural fluency, reflects non-routine activity. This category emphasizes flexibility with and understanding of procedures.

The next two categories of rigor discussed in the CCSSM also focus on non-routine activities. Application refers to activity in which students apply mathematics in situations in ways that are non-routine and for which the context is not superfluous. For example, for our definition of application in the procedures and algorithms mathematics activity, we stated that this included applying procedures to new contexts in ways that the context mattered for understanding and using the mathematics.

The fourth category of rigor, conceptual understanding, refers to when students have the opportunity to access concepts from a number of perspectives and make connections (between mathematical topics or between mathematics and students’ lived experiences) and that show mathematics as a non-routine activity. We also coded segments in this category if activities set up opportunities for engaging in conceptual understanding, even if conceptual understanding was not yet evident.

We make no evaluative claims about which of these types of rigor is desirable; instead, our focus remained on characterizing activities within a lesson. However, the CCSSM argue for a balance between routine and non-routine forms of rigor; the CCSSM document states that “mathematical understanding and procedural skill are equally important” (p. 4). Too much emphasis on routine rigor neglects non-routine rigor and too much emphasis on non-routine rigor neglects opportunities for students to practice. In our discussion we return to this notion of balance.

We coded segments as non-mathematical when the teacher was conducting managerial tasks such as providing directions, taking attendance, attending to classroom rules, or the like, which occurred regularly in all classrooms (about 12% of all lesson segments) and equally across curriculum programs.

Coding process for observed lessons

After generating our categories, we (the four authors) team-coded six lessons. In this process we refined our category definitions and came to consensus on each of the coded lesson segments. Subsequently, we pair coded the remaining 41 lessons, alternating pairs to diminish rater drift. We used a process of consensus coding (Saldaña 2013, 2013), which means we resolved our disagreements via discussion until each member of the pair was satisfied with the rating. We also met four times as a whole group between rounds of coding to refine our interpretations of the codes and to ensure our interpretations were shared across the four authors.

As we coded, we noticed that many lessons had a substantial number of segments coded as procedural skills. We also noticed that the three forms of non-routine activity (fluency, applications, and connections) were individually less evident and often happened together within the same lesson. Thus, to group the observed lessons, we collapsed the three non-routine categories into one, which we labeled FAC, an acronym for the three categories (fluency, applications, and connections).

Grouping the lessons

We then grouped the lessons according to the extent to which segments were coded as FAC or procedural skills, to examine balance between routine and non-routine rigor. The first grouping of lessons was labeled as highly FAC. The 11 lessons in this grouping had more than 80% of the segments in the lesson coded as FAC. The second grouping we labeled mostly FAC; these seven lessons had between 50% and 75% of the segments rated as FAC. The third grouping was labeled mostly skills-based. The ten lessons in this grouping had over 60% of the segments rated as procedural skills but also had segments rated as having other types of rigor. The fourth and final grouping was labeled exclusively skills-based. The 19 lessons in this group had all of their segments rated as procedural skills. We note that these percentages were calculated after removing all of the segments that were coded as having non-mathematical activity.

Teachers’ views on rigor in the CCSSM

As noted above and discussed in further detail below, the teachers across all curriculum programs held consistent views of the CCSSM, including about whether the CCSSM required them to teach more conceptually, solve complex problems, struggle while solving problems, explore topics, and emphasized communication. Notably, in another survey (Choppin et al., 2013), we found that US middle school teachers felt the CCSSM emphasized both conceptual understanding and procedural fluency, so it is likely that the teachers in this study were aware of the focus on both of these aspects of rigor. What we investigated in this study was whether the district’s choice of curriculum program was associated with the distribution of routine and non-routine rigor in the enacted curriculum. Our purpose was to determine whether the use of curriculum materials with different characteristics was associated with forms of rigor.

We realize that there is potential overlap between our conceptualization of the DM curriculum type and the procedural skills category of mathematical rigor. It should be noted that the DM categories represent a view of rigor that involves precision, accuracy, and repetition to develop procedural fluency and appropriation of mathematical terminology. A focus on precision, accuracy, and repetition does not preclude non-routine forms of rigorous activity from occurring; however, the messages and structures in the materials may influence the activity that occurs in classrooms in ways anticipated (eliminating ambiguity and imprecision in the development and use of terminology and procedures) and unanticipated (activity remains highly scaffolded and students do not have opportunities to apply these procedures and terminology strategically and flexibly).

Results

We aimed to study how instructional materials influenced the relationship between the official curriculum and the operational curriculum. In the results, we explore how the use of instructional materials with different characteristics was associated with distinct forms of rigor. First, we report on our analysis of the curriculum materials and then describe the results from our analysis of the enacted lessons.

Analysis of curriculum materials

We characterized six of the curriculum programs as predominantly having features associated with the DM metaphor, as displayed in Table 5. These programs were commercially produced materials from major U.S. publishers. As noted earlier, though we rated these programs the same in our continuum, some differences existed among the programs. In particular, one program, Math in Focus, had integrated, interconnected, and consistent use of models and representations (e.g., cube bars, double number lines) and more rigorous explanations of the mathematics than other programs in the same category. Nevertheless, we still rated Math in Focus as a 2 because it provided established methods and procedures before students had opportunities to solve problems, provided explicit definitions of terminology in the lesson, modeled tasks (including word problems) after detailed examples, included tasks with a narrow range of viable approaches, and sequenced tasks in a progression of increasing mathematical complexity. The grouping of Math in Focus with the other programs reflects our focus on the communicative emphasis in curriculum materials; as we noted earlier, other perspectives might have resulted in a stronger contrast between Math in Focus and the other programs we rated as having a substantial presence of DM features.

Table 5. Analysis of Instructional Materials

Curriculum Program	Characterization of Program	Rating on Continuum
Math in Focus	Delivery Mechanism	2
digits (digital)	Delivery Mechanism	2
Holt On Core	Delivery Mechanism	2
Holt Algebra 1	Delivery Mechanism	2
Prentice Hall	Delivery Mechanism	2
Glencoe	Delivery Mechanism	2
CMP2/CMP3	Thinking Device	6
CPM	Thinking Device	5

We also found differences between the two programs characterized as having a TD perspective and these differences are evident in our ratings of the two programs. We rated CMP2/CMP3 as showing strong and consistent evidence of the TD perspective because the materials did not formalize terminology in the text, did not provide worked-out examples or models before they presented problems to students, incorporated a common context for the problems that allowed students to reason about the mathematics, and emphasized a big mathematical idea that connected the tasks across the lesson. At the same time, aspects of the programs reflected some DM features, such as incorporating well-bounded tasks that offered constrained solution spaces and sequencing mathematical topics using disciplinary logic. The CPM materials had many of the same features of the CMP2/CMP3 materials, but we found the problems were more constrained, often affording only one approach, so we rated the materials a 5, indicating moderate evidence of the TD perspective. We also found that the tasks in CPM, in addition to being narrowly bounded, were sequenced to funnel students to a particular mathematical conclusion. These constrained problems and funneled sequencing distinguished the CPM materials from the CMP2/CMP3 materials.

The interaction between enacted lessons and curriculum perspective

We analyzed lesson enactments to explore how the use of instructional materials with particular characteristics was associated with particular forms of rigor. We analyzed 47 lessons across a range of curriculum and geographic contexts. Among the 47 lessons we analyzed, we found that 11 lessons were highly (80% or higher) composed of segments rated as involving non-routine forms of rigor (FAC) (see Table 6), seven lessons were mostly (50% – 75%) composed of FAC segments, 10 lessons were highly procedural (60% or higher, but not 100%), and 19 lessons that were exclusively procedural. We want to remark on the distribution of the non-routine forms of rigor – fluency, application, and conceptual. Out of 438 segments, we coded 55 as fluency, 72 as applications, and only 21 as conceptual. The 21 segments coded as conceptual involved making sense of terminology (e.g., slope inequality) or creating definitions or formulas related to geometric figures. We do not see the forms of rigor as a continuum leading to a highest form, but rather consider that each of the forms of rigor are important, and the three non-routine forms of rigor all involve sense making and opportunities to make connections. See Table 6 for a summary of the results. We describe results from each of the four groups of lessons.

Table 6. Enactment Grouping vs. Curriculum Perspective of the Lessons

Enactment Grouping by Forms of Rigor	Curriculum Perspective of Lessons (N = 47)
	Delivery Mechanism (rating of 2)	Thinking Device (rating of 5)		Thinking Device(rating of 6)
Highly FAC Lessons	0 lessons	1 lesson	10 lessons
Mostly FAC Lessons	5 lessons	1 lesson	1 lessons
Highly Procedural Lessons	7 lessons	1 lesson	2 lessons
Exclusively Procedural Lessons	15 lessons	0 lessons	4 lessons

Highly FAC lessons

The 11 lessons rated as highly FAC were from classrooms using instructional materials from the TD perspective (i.e., those curricula rated 5 or 6). Five of these lessons had 50% or higher of the segments rated as application, four had 50% or higher of the segments rated as fluency, and the remaining two had 50% or more of the segments rated as conceptual understanding. Of these lessons, seven had 70% or higher of the segments rated as a singular form of rigor, while the other four had a mix of rigor types (e.g., fluency and application).

Mostly FAC lessons

Five of the seven lessons rated as mostly FAC involved materials rated as DM, including all three of the lessons that use Math in Focus. These three lessons focus primarily on applications or fluency, with only 25% to 33% of the segments rated as procedural skills.

Highly procedural lessons

The ten lessons we rated as Highly Procedural had between 50% and 86% of their segments rated as procedural skills. Eight of these lessons were from classrooms using curriculum materials from a DM perspective. In terms of the presence of non-routine forms of rigor, five lessons had segments rated as involving procedural fluency, three had segments rated as involving application, and three had segments rated as involving conceptual forms of rigor.

Exclusively procedural lessons

The 19 lessons rated as exclusively procedural comprised a mix of lessons in terms of curriculum perspective, with 15 lessons involving the use of DM materials and four involving TD materials.

We note three trends in the results. First, all 11 of the highly FAC lessons involved materials from the TD perspective. Second, 22 of the 29 lessons rated as highly or exclusively procedural involved lessons using materials from a DM perspective. Of the seven lessons with teachers using materials from a TD perspective that also rated as highly or exclusively procedural, four were from a large district in which there was minimal preparation and support for using the materials, and the materials had been adopted the year in which the lessons were observed. Third, in terms of balance, we noted that 21 lessons had a mix of routine and non-routine forms of rigor. These lessons included the ten lessons rated as Highly Procedural, the seven lessons rated as Mostly FAC, and four of the lessons rated as Highly FAC. The other seven lessons rated as Highly FAC had a mix of non-routine forms of rigor but no routine procedural rigor, and the 19 lessons rated as exclusively procedural had only routine forms of rigor. We return to these trends in the discussion section.

Discussion

A primary goal of the study was to explore the role of instructional materials in enacting the official curriculum: We conceptualized various forms of rigor found in the official curriculum, how routine and non-routine forms of rigor were distributed in the enacted lessons, and the role of instructional materials in the forms of rigor we observed. In short, we wanted to ascertain the impact of localized interpretations of policy to help understand how teachers responded to the CCSSM as an articulation of the official curriculum. Specifically, we investigated the role of the district-adopted instructional materials on teachers’ enactments of the official curriculum.

We begin by describing three trends from the results. First, lessons involving materials characterized as TD emphasized non-routine forms of rigor more than DM materials. Second, lessons involving materials characterized as DM emphasized routine procedural rigor more than TD materials. Third, we noted that less than half of the lessons exhibited a balance of both routine and non-routine forms of rigor. After we describe these trends, we discuss the factors that may have contributed to them, the role of instructional materials, the adoption of curriculum programs as a proxy for the official curriculum, and the role of the official curriculum.

Discussion of trends

The first trend is that lessons involving materials characterized as TD consistently included non-routine forms of rigor. Notably, all of the lessons we rated as highly FAC were in classrooms using curriculum materials from the TD perspective. However, this trend was not uniform; seven of the 20 lessons involving materials from the TD perspective were rated as Highly or Exclusively Procedural.

The second trend is that lessons involving materials characterized as DM more often involved routine procedural rigor; 22 of the 29 lessons rated as highly or exclusively procedural involved lessons using materials from a DM perspective. The association between DM materials and routine forms of rigor was not uniform either; five of the seven lessons in the Mostly FAC (non-routine forms of rigor) category were in classrooms using curriculum materials characterized as DM.

The third trend is that roughly half of the lessons had mixed forms of rigor, with 20 of the 47 lessons rated as having a mix of routine and non-routine forms of rigor. By contrast, we rated the segments of 19 lessons as exclusively routine procedural rigor and three lessons as having a single form of non-routine rigor.

What do these trends tell us? First, the communicative perspective of the curriculum materials does not determine forms of rigor in enacted lessons. Teachers using materials from either perspective engaged students in both routine and non-routine forms of rigor. Nevertheless, these trends suggest an association between the features of the instructional materials and the forms of rigor that occur. These patterns emerged despite the consistent views across the sample of teachers of the kinds of curriculum and instruction emphasized in the CCSSM. Consequently, we conclude that the choice of curriculum program played a role in how the official curriculum was taken up in classroom practice.

Second, while the CCSSM argued for a balance of rigor between routine and non-routine forms, less than half of the lessons we observed had a mix of routine and non-routine forms. The lessons using TD materials had greater variation with respect to routine and non-routine forms of rigor, with 12 of the 20 lessons having multiple forms of rigor, than we found in lessons involving DM materials. Most lessons involving DM materials involved only routine rigor (15 of the 27 lessons), and seven of the remaining 12 lessons were predominantly comprised of routine forms of rigor. Thus, the balance of rigor in DM materials was heavily skewed toward routine rigor while there was considerable variation across routine and non-routine forms of rigor in lessons involving TD materials. Thus, while the dominant features related to the communicative perspectives of the curriculum materials did not determine the forms of rigor, they appeared to influence them.

Other factors that may account for trends

We describe two factors that may partly account for the trends we noted above, including teacher experience with materials and contextual factors such as support to use materials. Teacher experience with particular sets of materials may lead to more skilled and productive use of those materials, especially with materials from the TD perspective (Choppin, 2011). In our data, we found that seven of the 11 teachers whose lessons were rated as Highly FAC were very experienced with using CMP (current and/or prior editions of CMP), while an eighth teacher had used CPM for three years prior to the study. The other three teachers whose lessons were rated as Highly FAC were relative novices to CMP. By contrast, there was one very experienced CMP teacher whose lesson was rated as Highly or Exclusively Procedural. Thus, teachers’ experience with CMP seemed to associate with High FAC for most teachers, but this was not evident for all teachers’ lessons.

In contrast, we found that teacher experience with specific DM materials was not related to the forms of rigor we observed in their lessons. Two teachers using DM materials whose lessons were rated as Mostly FAC were long-time mathematics teachers (10 or more years) and had taught at the grade level in which they were observed for five or more years, while the three other teachers using DM materials whose lessons were rated as Mostly FAC had used Math in Focus for only two years. However, the lessons of other teachers with similar experience using DM programs were not rated as Highly or Mostly FAC. Overall, the teachers using DM materials had a wide range of teaching experience and a range of experience with their particular programs. While these teachers typically had only a few years of experience with a specific DM program, most had significant prior experience with other DM programs.

In terms of contextual challenges, we also noted that four of the teachers using CMP whose lessons were rated as Highly or Exclusively Procedural taught in a large district that had recently adopted the CMP curriculum program and had only provided minimal curriculum-specific professional development. This pattern suggests that teachers using TD materials with less experience default to enacting lessons with an emphasis on routine procedures in the absence of support for using TD.

The data show that teachers play an important role in how instructional materials are enacted, and while the instructional materials can exert a considerable influence on the kinds of activity enacted in classrooms, instructional materials alone do not determine the enacted activity and discourse, as evidenced in our participants and in prior research (Brown et al., 2009; Tarr et al., 2008). The variation in both DM and TD groups speaks to the role of the teachers and to factors related to the local instructional context, as noted above.

Connections between the official and operational curriculum

In Figure 1 we conjectured that the operational curriculum was directly and indirectly influenced by the official curriculum. If the CCSSM (the official curriculum) directly influenced the operational curriculum, then the teachers’ perceptions of the CCSSM would be associated with the enacted lessons (which is how we represented the operational curriculum) and there would be little variation across curriculum programs. If instead the adoption of instructional materials signaled to teachers how the CCSSM should be interpreted, then we would expect to see variation across curriculum program types. We discuss these two influences below.

Influence of CCSSM on the operational curriculum

A key question in the study is the extent to which teachers attended to the national standards in their daily lesson enactment. While the teachers cited CCSSM content standards and, though less fluently and frequently, the Standards for Mathematical Practice, in their lesson planning interviews and usually during the lessons, it was hard to say whether these standards served as more than simple validation of the selection of the content in the lessons. That is, we found little evidence that participants focused on the CCSSM as they enacted lessons. While the participants were clearly aware of the standards, it was not clear how the CCSSM influenced their daily planning and enactment beyond stipulating topics in their lesson plans and articulating content objectives in their lessons.

In addition, as we noted earlier, there was considerable consensus on teachers’ views of the CCSSM (80% – 90% responded in similar ways) and that this consensus appeared to indicate that non-routine forms of rigor were emphasized in the standards. For example, 88% agreed that the CCSSM emphasized more solving of complex problems and 83% agreed that the CCSSM would provide more opportunities for students to struggle while solving problems.

Our results did not show such a clear consensus in terms of the operational curriculum; our results showed considerable variation in types of rigor and that the majority of lessons involved mostly or exclusively routine forms of rigor. Thus, we turn our discussion to the role of instructional materials.

Instructional materials as proxy for the official curriculum

The choice of curriculum programs by local schools and districts has two possible mechanisms for influencing the operational curriculum. First, the choice serves as a validation of forms of curriculum and instruction, which would imply that the materials communicate messages about the nature of mathematics and the learning of mathematics. Second, the materials themselves directly influence the nature of the enacted curriculum via the features we articulated in Table 1. We discuss these possibilities below.

A basic premise of this study is that curriculum programs convey messages about the nature of mathematics and the learning of mathematics. Given the longstanding empirical research on what has typically constituted mathematics instruction in US middle schools (Hiebert et al., 2003; Jacobs et al., 2006; Stigler & Hiebert, 1999) and the association of these forms of instruction with conventional US middle school mathematics textbooks (Common Core State Standards Initiative, 2010; Schmidt et al., 1997; Valverde et al., 2002), the adoption of a conventional textbook would signal that there is little need to alter the typical focus on routine forms of rigor. In our analysis of materials, we characterized all of the conventional, commercially-published mathematics textbooks as adhering to the DM perspective. Consequently, it is possible that teachers ignored the CCSSM in favor of the messages indicated in the adoption of conventional textbooks.

A second premise is that the dominant communicative features of the materials have a direct influence on the forms of curriculum and instruction in classrooms. In our Results section, we noted associations between the types of instructional materials and the forms of rigor in the enacted lessons. While we noted variation across curriculum types, we also noticed some clear trends, discussed above. One of the patterns we noticed in an earlier analysis of the same data (Choppin et al., 2016), was that all of the lessons involving DM materials were teacher-focused in terms of the classroom discourse, while roughly half of the lessons involving TD materials were teacher-focused. This suggests another dimension of the communicative perspective of the materials, which is the distribution of mathematical authority in the classroom.

We also want to note that if indeed instructional materials involve messages about the nature of mathematics and the learning of mathematics, then skilled uses of materials from different perspectives would vary. Teachers using materials from a DM perspective who focus on routine procedural rigor may be following their interpretation of what the materials are intended to do, and they may become skilled at reinforcing that form of rigor in their classrooms. Similarly, teachers using materials from a TD perspective might emphasize varied forms of rigor, particularly non-routine forms, and as they become more skilled are able to accomplish this.

The role of instructional materials in the operational curriculum

Our results show that instructional materials influence the nature of mathematical activity in classrooms. The data raise doubts about whether the use of DM programs can support non-routine forms of rigor, as recommended in the CCSSM and associated documents (Common Core State Standards Initiative, 2010; http://www.corestandards.org/other-resources/key-shifts-in-mathematics/). We trace this claim to the characteristics of DM materials we articulated in Table 3. Characteristics of DM materials – such as explaining terminology and procedures before students have an opportunity to grapple with problems – conveyed that students need to be told how to talk about and do mathematics so that their work can be precise, accurate, and efficient, which are crucial disciplinary traits of mathematics. The messages were interpreted in a way that resulted in teachers spending most of the lessons explaining the mathematics, with typically very few opportunities for students to engage in non-routine activity or talk about mathematics on their own terms. The structure of the materials emphasized applying and practicing terminology and procedures modeled at the start of the lesson, to emphasize precision, accuracy, and efficiency. The result was that students engaged predominantly in routine forms of rigor, with little evidence of the non-routine forms emphasized in the CCSSM. This is not a critique of the materials rated as DM or of the teachers using them; in fact, we could say that the teachers used the materials as they were designed to be used. Simply put, the materials were not designed for the non-routine forms of rigor called for in the CCSSM documents.

For TD programs, the emphasis on exploration before formalization, making connections among multiple representations, and drawing on big ideas that build over time similarly conveyed other messages about the nature of mathematical activity. Students in those classes generally had more opportunities to make sense of mathematics and to explore mathematics without explicitly being told what procedures to follow. So, while we argue above for an important role for the teacher in enacting instructional materials, it is also apparent that characteristics of instructional materials adopted by districts influenced the nature of mathematical activity and discourse in classrooms. The differences across program types occurred despite the participating teachers holding similar views of the national standards.

Implications

Our results provide a deeper understanding of how the participatory relationship unfolds between teachers and instructional materials (i.e., teachers and tools) (Choppin et al., 2018; Remillard & Bryans, 2004; Roth McDuffie et al., 2018a, 2018b). While a considerable body of literature has documented how the characteristics of teachers influence how they draw from instructional materials (e.g., Lloyd, 1999), little research that has explored how the dominant features of instructional materials influence the forms of rigor in mathematics classrooms.

An implication of our curriculum analysis is that other curriculum types are possible. We envision other possibilities besides those that were evident in this study. For example, materials could be designed to emphasize mathematical structure (e.g., the field properties) as a means of connecting mathematical topics across lessons. This focus may influence, even constrain, exploration by requiring students to engage in specified forms of proof and algebra, but may offer a means of clearly articulating connections across mathematical topics in ways not done in either of the types articulated in this study. Alternatively, the profusion of digital curriculum resources may fragment the coherence of any set of curriculum features so that lessons may vary considerably from day to day in terms of the balance among the characteristics in Table 2.

A second implication from our findings is that the choice of instructional materials – whether by district, school, or teacher – mediates between the official and enacted curriculum. It may be the case, as Remillard and Heck (2014) note, that local interpretations of the official curriculum may not fully reflect the goals and objectives in the official curriculum. An underlying question, then, is what districts and schools were responding to in their choice of instructional materials, especially if the ostensible goal – to align instruction with the CCSSM – was the same across the contexts we studied. What messages or information were the schools and districts responding to, and what messages did they want to send with their choice of programs? What does their choice of curriculum program say about the depth with which or the evidence on which they based their decisions about curriculum adoption? Suggestions for districts adopting instructional materials could be to consider: (1) the extent to which the Standards for Mathematical Practice are fully integrated in the curriculum, as these potentially support a greater balance across forms of rigorous activity as we characterized it; (2) adoption processes that challenge current forms of curriculum (i.e., open up the problem space with respect to discussions of curriculum characteristics); and (3) the validity of publishers’ claims with respect to alignment.

A third implication is that educators and stakeholders should be skeptical of claims that DM materials can support a full range of rigor, especially non-routine forms of rigor. Many schools, including some in our data, turned to DM materials while still communicating expectations to teachers that they should incorporate more problem solving, communication, and exploration (i.e., focus on the Standards for Mathematical Practice in the CCSSM). Our findings suggest that the structure of the DM materials in our study did not support or promote these forms of activity. As Wertsch and Toma (1995) noted, materials that emphasize monologic forms of communication cannot be simply revised to be thinking devices. Furthermore, years of teaching and teacher experience with a particular curriculum seemed to play a more important role with TD materials than with delivery mechanism materials. While we noted substantive differences based on teacher experience among teachers using TD materials, there was not such an obvious relationship with the participants using DM materials. We note that years of experience with a TD program does not guarantee that teachers will enact lessons involving non-routine forms of rigor.

A fourth implication is that interpretations of the official curriculum (i.e., the CCSSM in this study) are heavily mediated by decisions and curricular choices made at the local level by teachers, school administrators, and, more recently, even chief technology officers, who now play a large role in curriculum adoption processes (Yettick, 2015). This suggests challenges for policy makers who hope to change classroom instruction without providing a stronger articulation of what classroom practices should look like or without articulating qualities of materials and investigating those that have been developed with these practices in mind. Future iterations of the official curriculum should articulate desired forms of curriculum and instruction, including listing existing programs that, while not a perfect match to content specifications, may specify forms of mathematical activity deemed rigorous in the official curriculum.

A fifth implication of our study relates to methodological contributions, particularly around the focus on curriculum features and on characterizing the multiple forms of rigor. With respect to characterizing rigor, we described how forms of rigor appeared across different kinds of mathematical activity (e.g., facts, definitions, and formulas; procedures and algorithms; and representations; and we could add proof). This will be helpful for the field to analyze rigor not in context-free terms, but tie their analysis to the mathematical focus of lessons. In terms of curriculum features, we attempt to move the curriculum conversation beyond labels such as reform, traditional, or standards-based. These terms are both contested and loaded, conveying little actual insight into the features and how they afford or constrain pedagogical opportunities.

Limitations

One of the limitations of this paper is that we did not analyze how the demographics of our research sites may have contributed to our findings. We note that demographics were not a focus of our data collection or analysis, but recognize that this is an area that needs further attention, especially for researchers exploring the kinds of curriculum and instruction typically present in high-need contexts.

A second limitation refers to the notion of balance and our focus on a single lesson for each teacher. We recognize that balance is more productively considered across a range of lessons; there are times when it is necessary to focus strictly on either routine or non-routine forms of rigor. Given our narrow time scale, we may have missed more relevant indications of balance.

Additional information

Funding

This work was supported by the US National Science Foundation under Grant DRL # 1222359.

Notes on contributors

Jeffrey Choppin

Jeffrey Choppin is a professor of mathematics education at the University of Rochester. He has worked as a mathematics teacher, mathematics teacher educator, and mathematics education researcher for over 30 years, primarily in Washington, DC and Rochester, NY. His research focuses on teachers’ use and understanding of curriculum resources, focusing on the participatory relationship between teachers and curriculum materials. Recent work has explored the concept of curriculum ergonomics, which conceptualizations the relationship between curriculum features and teachers’ characteristics and goals. He also researched digital curriculum materials, focusing on trends and forces related to the development and uptake of digital curriculum resources. His current research projects focuses on online professional development for rural teachers and coaches to support ambitious instrurctional strategies. His work has appeared in Mathematical Thinking and Learning, Journal of Mathematics Teacher Education, ZDM Mathematics Education, Action in Teacher Education, Educational Policy, Mathematics Education Research Journal, and Curriculum Inquiry

Amy Roth McDuffie

Amy Roth McDuffie is a professor of mathematics education for the College of Education at Washington State University. She teaches prospective and practicing teachers and conducts research on the professional development of teachers in mathematics education. Specifically, she focuses on supporting teachers’ learning in and from practice in the with attention to teachers’ use of curriculum resources and equitable pedagogies. She has served as a PI/co-PI on National Science Funded projects including: Developing Principles for Curriculum Design and Use in the Common Core Era (the project that funded the work reported here), Mathematical Modeling with Cultural and Community Contexts, Teachers Empowered to Advance Change in Mathematics, and Applicant Information, Selection, & STEM Teacher Retention and Effectiveness. She was the series editor for the National Council of Teachers of Mathematics’ Annual Perspectives in Mathematics Education (2014-2016) and has published in venues including: Journal for Mathematic Teacher Education, Journal of Teacher Education, Journal of Curriculum Studies, and Teaching Children Mathematics.

Corey Drake

Corey Drake is a Professor of Teacher Education and Mathematics Education at Michigan State University. For nine years, she was also the Director of Teacher Preparation at Michigan State. She received her PhD in Human Development and Social Policy from Northwestern University in 2000. Her research and teaching interests center on the roles of curriculum materials and teacher preparation in supporting teachers’ capacity to teach elementary mathematics to diverse groups of students in transformative and justice-oriented ways. Her work has been published in Educational Researcher, Journal of Mathematics Teacher Education, Journal of Teacher Education, and Teaching Children Mathematics, among other venues, and funded by the National Science Foundation and the Spencer Foundation.

Jon Davis

Jon Davis is a Professor of Mathematics Education in the Department of Mathematics at Western Michigan University in Kalamazoo, Michigan. He teaches mathematics content courses to undergraduate students, mathematics content courses to preservice elementary teachers, mathematics methods courses to preservice secondary mathematics education students, and mathematics education courses to graduate students. His research interests include: teachers’ use of instructional materials; how students and teachers use mathematical action technologies to develop students’ conceptual understanding; and how classroom communities of practice operate to develop technology-rich curricular resources. He has served as a co-PI on the National Science Funded project Developing Principles for Curriculum Design and Use in the Common Core Era. He has served on the editorial panel of Mathematics Teacher and his research has appeared in the Journal for Research in Mathematics Education, Journal of Mathematics Teacher Education, Educational Studies in Mathematics, and Mathematics Education Research Journal.