Patterns in curriculum use ABSTRACT The goal of this paper is to identify significant patterns of curriculum use among teachers using a reform-based elementary mathematics curriculum. Towards this end, we introduce the curriculum strategy framework as a way to characterize teachers' interactions with reform-based materials. The framework focuses on three key interpretive activities: reading, evaluating, and adapting curriculum materials. Describing an individual teacher's curriculum strategy involves identifying the manner in which a teacher engages with each of these activities before, during, and after instruction. Data for the study come from year-long observations and interviews with 10 elementary-school teachers who were using a reform-based mathematics curriculum for the first time.

IDENTIFYING PATTERNS IN TEACHERS' USE OF A REFORM-BASED ELEMENTARY CURRICULUM Prior research offers numerous images of teachers' attempts to implement mathematics education reform. There are reports of some teachers using new curricula and meeting the goals of reform (Davenport, 2000; M. S. Smith, 2000), and there are stories of other teachers consciously or unconsciously making changes in reform-based materials and bypassing the goals of reform (Peterson, 1990; Putnam, 1992). Yet, there is little in the way of empirically-based generalizations concerning the manner in which teachers use curriculum materials. And this is despite the fact that, to date, many new reform-based curricula have been developed and are being used in mathematics classrooms across the United States. We believe that in order to fully understand the teachers' role in the implementation of reform, more information is needed concerning the relationship between teachers and the reform-based curriculum materials they use. The goal of this paper is to identify significant patterns of curriculum use among teachers using a reform-based curriculum for the first time. Specifically, we want to identify the strategies that teachers use to work with these curriculum materials. For instance, what are they looking for as they read the teachers' guide? Do they focus on examples or on explanations? What are the main issues that teachers consider as they adapt a given lesson? And when do these decisions occur -- prior to, during, or after instruction? Certainly, the answers to these questions will be very complex. In any one instance, the particular manner in which a teacher makes use of a set of curriculum materials depends on many factors, including the nature of the materials, the teacher's own knowledge and beliefs, and how students react as the material is taught. Nonetheless, we have found that we can identify substantial regularities; there are recognizable patterns in how curriculum materials are used. In addition, these patterns are consistent for individual teachers, and similar patterns appear across multiple teachers. Furthermore, we have found that we can understand these regularities by focusing on just three categories of interpretive activities. First, we focus on how teachers read in order to digest

Patterns in curriculum use - 2 the information contained in the written materials themselves. Second, we look at how teachers evaluate the written materials, as well as the lesson-in-progress. Finally, we look at how teachers adapt the materials, based on their reading of the materials, and their evaluation of the materials and classroom events. To make our points, we draw on a study in which 10 elementary-school teachers enacted a reform-based mathematics curriculum called Children's Math Worlds. In the remainder of this paper, we begin by reviewing the existing literature on teachers' use of curriculum materials. This is followed by a detailed introduction to the curriculum strategy framework and a discussion of the methods used in this study. Next we provide examples of three teachers' curriculum strategies and then explore patterns across all 10 teachers in the study. Finally, we discuss the implications of this study for research on teacher cognition and for the design of reform-based curricula and professional development opportunities for teachers. TEACHERS' USE OF CURRICULUM MATERIALS Three main meanings of the term curriculum are common in the literature today. First, a curriculum can be thought of as the set of written materials provided to teachers -- the textbook, teachers' guide, assessment materials, etc. In addition, the term curriculum is used to refer to the lesson that is enacted in the classroom. Finally, for many teachers a curriculum also exists in the form of district or state-level learning objectives for students. In this research, we are working at the intersection of the first two meanings of the term curriculum. Specifically, we examine what happens when teachers use written curriculum materials that are based on the goals of mathematics education reform. For the sake of clarity, we will use the term curriculum to refer to the written materials, and the term enacted curriculum to refer to what happens when the lessons are implemented in the classroom.

Patterns in curriculum use - 3 Historical Perspective on Curriculum Use The perceived relationship between teachers and curricula has seen important changes during the last century. The twentieth century began with a strong emphasis on the use of textbooks. Rather than supplementing lessons designed by the teacher, the textbook had come to serve as the primary guide for instruction (Westbury, 1990). In the 1930s, however, as the Progressive Movement took hold, textbooks were viewed as less central to instruction and some teachers began to use a variety of materials and activities to support their teaching (Foshay, 1990). Another significant shift occurred in the 1950s with the advent of the Cold War and the Soviets' launching of Sputnik. There was a growing concern in the United States that we needed to provide a strong education for future scientists, and the creation of new textbooks was seen as a potential solution to this problem. The federal government provided funds for scientists and mathematicians to design school curriculum materials. Teachers were noticeably absent from this process. Not only were they excluded from the design process, the enactment of the materials by teachers was presumed to be a trivial step in the process. This New Math is widely considered to be a failure (Wilson, 2003). Although there is currently some debate and disagreement as to the reasons for this failure, there is one point on which many researchers and policymakers agree: Failure to consider teacher enactment as a critical link in the reform process was a significant difficulty (Ball & Cohen, 1996; Kilpatrick, 1997). More common today is the belief that no curriculum is "teacher-proof," and that there are by necessity changes that occur as a curriculum goes from written to enacted (Lappan, 1997). For example, Ben-Peretz (1990) discusses the notion of a "curriculum envelope" as a range of adaptations that teachers might make while still maintaining the intended goals of a lesson. Still, many people continue to assert that teachers and instruction can be changed as a result of using new materials. But the goal now is generally one of "mutual adaptation" -- that the materials will be changed as they are implemented by teachers, and that teachers will be changed as a result of using the materials (McLaughlin, 1976).

Patterns in curriculum use - 4 Curriculum Use in the Context of Reform The notion of mutual adaptation is particularly interesting in the context of current mathematics education reform efforts in the United States. First, new curricula are frequently used to introduce teachers to reform recommendations with the hope that using the materials will foster changes in instruction (Sykes, 1990). These curricula are generally quite different from what teachers have used in the past, often introducing new mathematical topics and new pedagogical approaches (M. S. Smith, 2000; Wilson & Lloyd, 2000). Thus, using such curricula calls for learning on the part of teachers. Some researchers have explicitly called on curriculum designers to address this need for teacher learning. They argue that new curricula should consider not only the needs of students, but also the needs of teachers in the context of reform (Ball & Cohen, 1996; Remillard, 1999; Russell, 1997). Second, teachers are expected to make changes in reform-based curricula as they use the materials. In particular, teachers are encouraged to respond to the ideas that students raise during instruction and to use these ideas to determine how to proceed with a lesson (J. P. Smith, 1996). Thus, despite detailed descriptions of activities and goals, teachers can no longer prepare a lesson completely in advance. Instead, using a reform-based curriculum involves using the materials with a degree of flexibility. An important goal of our research is to understand how teachers approach using a textbook or teachers' guide under these circumstances. Key Processes in Teachers' Use of Curriculum Materials Though the focus of this paper is teachers' use of a mathematics curriculum, we draw from related literature in a variety of disciplines including mathematics (Lloyd, 1999; Remillard, 1999, 2000), reading (Barr & Sadow, 1989; Grossman & Thompson, 2002), science (Ben-Peretz, 1990; Schneider, Krajcik, & Marx, 2000), and social studies (Kon, 1994; McCutcheon, 1981). Across all of these domains is an interest in understanding the decisions that teachers make as

Patterns in curriculum use - 5 they utilize curricula. In particular, researchers discuss three key processes in which teachers engage as they use curriculum materials. First, previous research most often studied curriculum use by exploring the ways in which teachers adapt curriculum materials. For instance, Freeman and Porter (1989) identified three styles of textbook use by focusing on the extent to which a teacher's instruction matched the topics and sequencing that were presented in a traditional mathematics textbook. In the context of reform, researchers have also identified a variety of ways in which teachers adapt curriculum materials. For example, some researchers describe cases in which teachers believe that they are implementing reform, but on close inspection, the researchers find that the teachers have incorporated only surface features of the reform (Cohen, 1990; Peterson, 1990). In other cases, researchers talk of "lethal mutations" in which teachers make substantial changes in written materials and no longer maintain the intended goals of a lesson (Brown & Campione, 1996). At the same time, however, many researchers provide examples of teachers who do meet the demands of reform as they use new materials. In some instances, the teachers appear to stick closely to the new curricula, particularly as they use the materials for the first time (Heaton, 2000; Lloyd, 1999). Yet in other cases, teachers adapt lessons in a variety of ways while still maintaining the goals of the lessons (Davenport, 2000; Sherin, 2002). Second, though not as extensive as the research on teachers' adaptation of curriculum materials, researchers also consider teachers' reading of curriculum materials. For example, in her year-long study of two teachers, Remillard (1999, 2000) found that the teachers chose to read different parts of the curriculum materials and that they drew on different resources to make sense of what they had read. While one teacher paid particular attention to the specific activities suggested in the curriculum, the other teacher focused on the mathematics that students were expected to learn. Morse (2000) also discusses the ways in which teachers read curriculum

Patterns in curriculum use - 6 materials. She explains that a powerful shift among teachers' reading occurred when the teachers began to "read the curriculum for intention," that is, to read the materials in order to "`get inside' the intentions of the curriculum" designers. Third, research also suggests that evaluating curriculum materials is part of the process through which teachers use a curriculum. For example, Sosniak and Stodolsky (1993) describe a teacher who thoughtfully judged the textbook in order to decide whether or not to follow the guidelines presented. Similarly, Ben-Peretz (1990) distinguishes between teachers' reading of curriculum materials and their interpretation of these materials. She claims that "`reading' means the attempt to understand what is in the text, without imposing on it one's own convictions," (p.66). In contrast, the goal of interpreting the curriculum is to assign some type of evaluation or critique to the materials. Ben-Peretz claims that teachers conduct subjective interpretations of curricula based on their personal experiences as teachers. Together, these three processes form the foundation for our own characterization of the ways that elementary-school teachers make use of reform-based curricula. In the next section, we describe this approach in detail by introducing what we call the curriculum strategy framework. CURRICULUM STRATEGY FRAMEWORK Ultimately, we would like to understand the chain of events whereby a set of curriculum materials leads to specific instances of classroom instruction, and to the learning of individual students. However, it is clear that the processes underlying this chain of events are complex. As the researchers discussed above have amply documented, a set of curriculum materials does not determine the nature of instruction in any straightforward manner. A teacher reads and interprets the materials, and this interpretation depends greatly on what the teacher knows, including his or her knowledge of the subject matter and of the students, and the teacher's beliefs about instruction. Furthermore, the manner in which the materials are enacted in a classroom depends

Patterns in curriculum use - 7 on many additional factors. For example, the enactment of the curriculum will depend on how students react as instruction proceeds. As researchers, our goal is to select a portion of this complex chain of events to understand. We must partition off a part of the scientific problem that is tractable, and that may help in the solving of problems that are of importance to educational practice. We believe we have found a way to focus our work that meets these criteria. Within the complex system of factors that determines how curriculum materials become instruction, we have evolved a focus on what we are calling teachers' curriculum strategies. The idea is that a teachers stands at the hub of a complex system that includes materials, students, and the rest of the instructional milieu. As they stand at this hub, they act as interpreters and evaluators of curriculum materials, and the classroom events that transpire. A teacher's curriculum strategy is the stance that they adopt as they play this role of interpreter. Our notion of curriculum strategy is intended to encompass a large range of the activities in which teachers engage. We intend, first, to include what teacher do prior to instruction, in order to prepare. Second, we include some aspects of the stances they adopt during instruction. Finally, our notion of curriculum strategy applies to what they do after instruction, as they reflect on what has transpired. Furthermore, during each of the time periods, we see a teacher's curriculum strategy as consisting of three interpretive activities: 1. Reading. A curriculum strategy includes a particular orientation toward the reading of the curriculum materials themselves. As documented by other researchers (Ben Peretz, 1990; Morse, 2000; Remillard, 1999, 2002), the ways in which teachers read curriculum materials can differ greatly. 2. Evaluating. A curriculum strategy also includes the types of evaluative stances that are adopted by teachers before, during, and after the enactment of a curriculum. These

Patterns in curriculum use - 8 stances can differ, both in what the teachers choose to attend to when evaluating, as well as the dimensions along which they conduct this evaluation. 3. Adapting. Finally, a teacher's curriculum strategy includes an orientation toward the adaptation of curriculum materials. In the simplest case, a teacher may tend to adapt simply by omitting portions of a lesson. But some teachers may be inclined to be innovative, and to occasionally make significant and creative changes to the curriculum.

During any of the three time periods that are our concern -- before, during, and after instruction -- the teacher stands at the hub of a system, reading, evaluating, and adapting according to their curriculum strategy. In principle, a teacher's curriculum strategy might lead the teacher to engage in all of these activities constantly. In practice, however, a teacher's actions are more circumscribed. For example, one teacher's orientation to reading might lead her to read curriculum materials before instruction, but not during instruction.

Before Instruction During Instruction After Instruction

Read

Evaluate Adapt

Figure 1. The Curriculum Strategy Matrix

It is the particular way that the three interpretive activities are manifested across the three

time periods that constitute a teacher's curriculum strategy. As shown in Figure 1, we can thus

think of a teacher's curriculum strategy as filling out a 3Ч3 matrix, with the three kinds of

interpretive activities across the top, and the three time periods down the side. In specifying a

curriculum strategy, we thus state how a teacher tends to act in each time period, with respect to

each of the activities.

Patterns in curriculum use - 9 It is not obvious, a priori, that it would be possible to specify a single curriculum strategy matrix for any given teacher. Empirically, it might be the case that a teacher's behavior is dependent on attributes of the context that are not easily captured in this way. However, we will attempt to show that, at least within the confines of our study, this is not the case. It is possible to specify a curriculum strategy matrix for each of the teachers we studied, and these matrices do a very good job of capturing what those teachers do, and the stances they adopt in enacting the curriculum. Furthermore, we hope to show that the characterizations encompassed in our curriculum strategies can help us to understand much that is important in how teachers enact a reform-based elementary mathematics curriculum for the first time, and much that is important in how the activities underlying teachers' enactments differ. RESEARCH DESIGN As part of this research, 10 teachers piloted the new reform-based elementary mathematics curriculum Children's Math Worlds (CMW) (Fuson et al., 2000). CMW is based on extensive theoretical and empirical research concerning how students learn number concepts, problem solving, and computation. The curriculum builds on the notion of an equity pedagogy in which all students are believed to be able to learn ambitious mathematics. Like other recent reform-based curricula, CMW is aligned with the standards of the National Council of Teachers of Mathematics (NCTM, 2000). CMW is designed to foster classroom environments in which students discuss mathematical ideas, and there is a strong emphasis on the use of students' drawings as a focus of this discourse. CMW also attempts to make the language of mathematics accessible to teachers and students by introducing a variety of new vocabulary. The CMW materials include a teachers' guide, a student activity book, an assessment packet, and a set of innovative classroom manipulatives. The teachers' guide provides several resources intended to support teachers' use of the accompanying materials. An initial section

Patterns in curriculum use - 10

presents the CMW philosophy and essays from teachers who have used CMW in the past. Next,

the teachers' guide presents a grade-level student learning trajectory in an effort to help teachers

get a sense of the "big picture" in terms of what students will learn across the year. Detailed

daily lesson plans are also included. These plans describe lesson objectives, activities, and

sample discourse that can help to guide implementation of the corresponding activities. The daily

plans also include occasional mathematical and pedagogical notes for the teacher such as

explanations of the different problem types given in a lesson or likely student solutions.

Participants

Ten teachers volunteered to use CMW as their mathematics curriculum for the school

year (Table 1). Six teachers participated in the study during the 1998-99 school year, and four

other teachers participated during the 1999-00 school year. The teachers represented a wide

range of years of teaching experience and were teaching grades one through three. The teachers

taught in four different schools, three of which were located in an urban public school district in

the Midwestern United States. The fourth school was a nearby suburban public school with an

ethnically-mixed student population. Administrators at all four schools were generally supportive

of the teachers' decision to use CMW, but provided no additional resources to aid this effort.

Teacher Anita Beth Fran Jan Kate Lauren Marta Paul Shelley Vera

Year in Study 99-00 98-99 99-00 98-99 99-00 98-99 98-99 99-00 98-99 98-99

Gender Female Female Female Female Female Female Female Male Female Female

Years of Teaching Experience

Grade

School Type

10

1

Urban School A

7

1

Urban School A

17

3

Urban School C

0

3

Suburban

19

2

Urban School C

6

1

Urban School B

12

3

Urban School A

7

1

Urban School C

0

3

Suburban

5

3

Urban School A

Table 1. Teacher Participants

Number of Observations 12 17 12 11 15 8 18 9 23 11

Patterns in curriculum use - 11 data collection All of the teachers were observed throughout one school year.1 During the 1998-99 school year, the number of observations per teacher ranged from 8 to 23. Three of the six teachers were observed approximately once a month and the three others were observed twice a month. During the 1999-00 school year, we observed four other teachers and each teacher was observed on average once a month.2 Across both years, we also conducted several "cluster" observations in which we visited a teacher's classroom for two to three consecutive days. The observation protocol remained the same across the two years of the study. During each observation, a researcher took field notes characterizing both teacher and student actions that occurred in the classroom. The majority of observations were also videotaped. In addition, the teachers were interviewed following each classroom observation. These interviews lasted approximately 15 minutes and were either conducted immediately following class, after school, or in the evening over the telephone.3 The post-observation interviews were audiotaped and transcribed. In these interviews, the researcher asked the teacher to discuss several related issues including: (a) the goals of the lesson, (b) how the teacher prepared for class and any questions that had come up for the teacher, (c) anything unexpected that occurred during instruction, and (d) what the teacher believed that he or she had learned from preparing for and carrying out the particular lesson. Finally, eight of the teachers participated in "mathematics story interviews" (Drake & Hufferd-Ackles, 1999) in which they were asked to relate their own personal history in terms of learning and teaching mathematics, pointing out high and low points, as well as their future plans in mathematics. The two other teachers (Shelley and Jan) wrote reflective essays in which they discussed what they were learning from the CMW materials. A team of four researchers shared responsibility for the data collection across the two years of the study.

Patterns in curriculum use - 12 data analysis Data analysis consisted of two stages. In the first stage, we examined whether it was possible to identify a predominant curriculum strategy for each of the 10 teachers in the study. In the second stage, we compared the curriculum strategies of all 10 teachers, looking for similarities and differences in the ways that the teachers interacted with the reform-based curriculum. Each of these stages is described in further detail below. Identifying a Teacher's Curriculum Strategy Characterizing an individual teacher's curriculum strategy involved synthesizing data from multiple sources (observations, interviews, and curriculum materials) and from across an entire school year. To do so, we used an approach in which one member of our research team reviewed all of the data from one teacher. In essence, this researcher became an expert on that teacher -- reviewing all videotapes, lesson transcripts, interviews, and relevant curriculum materials. This method allowed for fine-grained analysis of the data from each teacher and permitted triangulation on the different data sources. Across the two years, a team of seven researchers was involved in this process. Data analysis involved a constant comparative approach to develop grounded theory (Glaser & Strauss, 1967). In particular, the data sources for each teacher were examined for evidence of the teacher's interactions with the curriculum prior to, during, and after instruction. Within each of these time periods, analysis focused specifically on evidence concerning the manner in which the teacher read, evaluated, and adapted the curriculum materials. We were concerned not only with whether a teacher engaged in each of the interpretive activities at different points but also with how the teacher engaged in each activity, that is, what the teacher did when he or she read, evaluated, and adapted the curriculum.

Patterns in curriculum use - 13 To examine these issues, the researcher first reviewed the interviews for comments that the teacher had made concerning how he or she had prepared for instruction, the kinds of questions the teacher had about the curriculum, what changes he or she had made in the lesson, and what changes and concerns were likely to be addressed in the future if this lesson were re-taught. This was followed by a review of the observational data in which the researcher compared the written lesson with the enacted lesson in order to identify changes that had been made and to confirm or disconfirm changes that the teacher had described in the interviews. In addition, the researcher drew on the interview data to clarify whether these changes had been made before, during, or after instruction.4 This information was then organized into a table that listed, for each lesson observed, any evidence that the teacher had (or had not) read, evaluated, and adapted the curriculum and whether this had taken place before, during, or after instruction. These tables enabled us to examine the extent to which a teacher interacted with the curriculum in a consistent manner -- that is, whether the activities in which the teacher engaged in before, during, and after instruction changed from lesson to lesson or were, instead, consistent over time. Based on this, the researcher abstracted a curriculum strategy matrix intended to capture those consistencies detected in the table. Finally, the researcher compared each lesson in the table with the matrix in order to determine the number of observed lessons that were consistent with the identified curriculum strategy. Both confirming and disconfirming evidence were noted. This process revealed a curriculum strategy that was consistent for at least 78% of the observations for each teacher. We believe that our expert researcher approach was necessary in order in order to develop a detailed understanding of the manner in which each teacher used the curriculum across the school year. At the same time, however, we were aware of the need to check the reliability of the curriculum strategies that were developed. To do so, a subset of data from each teacher was reviewed by three researchers as a way of checking the reliability of the expert's coding. This

Patterns in curriculum use - 14 process helped to confirm that differences among the curriculum strategies were not due to differences among researchers, but rather, were due to differences among the teachers. Identifying Patterns Among Teachers' Curriculum Strategies After curriculum strategies were developed for each of the 10 teachers, two researchers then embarked on a second stage of analysis. The goal at this point was to identify significant similarities and differences among the teachers' curriculum strategies. Initially, a comparison was made of the interpretive strategies that different teachers engaged in before, during, and after instruction. Second, similarities and differences were explored in the manner in which teacher engaged in these activities over the course of preparing for, implementing, and reflecting on instruction. The results of this analysis are summarized in Appendix B and are discussed later in this paper. THREE TEACHERS' CURRICULUM STRATEGIES As described in the previous section, each of the 10 teachers was found to have a specific approach to interacting with the curriculum materials that was consistent over much of the school year. Here we illustrate three of these curriculum strategies. Our goal in doing so is to provide the reader with a flavor of the kind of information provided by and contained within a teacher's curriculum strategy. A curriculum strategy matrix for each of the remaining seven teachers is provided in Appendix A. Beth's Curriculum Strategy Beth's use of the CMW curriculum in her first grade classroom followed a consistent pattern across the year. Before instruction, Beth typically read the day's lesson, working to get a sense of the "big ideas" of the lesson in terms of the activities and language used. Beth provided the following explanation of her approach to reading for the vocabulary ideas: The book really has some great ways of introducing things...I know I don't have to say it verbatim, the way they say it, so I'm not trying to memorize it. But I'm trying to memorize how they said it...I'm trying to remember the lingo.

Patterns in curriculum use - 15 In other words, Beth was trying to learn and remember the flavor of the language used, but not the precise "verbatim" language. In a similar way, Beth read the curriculum for an overview of the activities in each lesson. She paid attention to the ordering of activities, the connections among activities, and the goals of the main activities, rather than to the precise details of each activity. Prior to class, Beth also evaluated specific aspects of the lesson and made any changes she felt were needed. Beth discussed this process with the researcher on many occasions and the researcher often observed her planning firsthand as well. For example, one day towards the middle of the school year, the researcher arrived early for an observation of Beth's mathematics lesson. Beth was preparing for class and began reading the description of the lesson in the teachers' guide. As she read the lesson, Beth discussed the sequence of activities with the researcher and noted the new terminology that the lesson introduced. Beth also raised questions about how the day's lesson related to prior and subsequent activities and lessons. Beth's concern on this day about the relationship between various lessons and activities in the curriculum was not unusual. She often questioned her understanding of how activities "flowed" together, and was concerned that parts of lessons were not sufficiently linked to prior activities or to the subsequent homework assignments. As a result, Beth frequently created new materials, similar to ones provided by the curriculum, that she felt could serve as transitions between one activity and the next or between schoolwork and homework. I ... [still had questions after examining] the resources that were in the book. I then had to create a page [myself]. So that was, "How am I going to do this? How am I going to match [the parts of the lessons]?"... So, that was how I prepared. During instruction, Beth continued to evaluate the lesson and made additional changes. Interestingly, before instruction, Beth was particularly concerned with whether she understood the mathematics involved in the lesson and the meaning and purpose of the activities provided. As she said during one post-observation interview, "So, I'm really having a hard time with the

Patterns in curriculum use - 16 book, like understanding what it's saying to me, what it's wanting me to do, the book." During instruction, however, the focus of Beth's evaluation shifted to her students -- she was now concerned with her students' understanding and their participation in the class activities. For example, she explained on one day that in class earlier she "was concerned about what their drawings looked like ... to [them] ...and whether they understood [it]." Based on her assessment of her students' understanding, Beth would invent new ways of explaining concepts during instruction and would create new terminology for students to use. After one lesson on word problems, she suggested that these changes in the midst of instruction were not accidents, but were instead purposeful aspects of her teaching. I was basically trying on hats during the lesson. What's working, what's not working, and trying to figure it out. Because sometimes, when you sit by yourself and do it, you don't really get clear...so it was kind of like, okay, I'm just gonna try this and see if it works or not. Because she recognized that she couldn't always correctly predict her students' responses to the lessons she created, Beth planned on making changes during instruction. Furthermore, both before and during instruction, Beth adapted lessons by creating new materials to supplement those provided by the curriculum. Finally, after instruction, Beth spent time, either in the context of post-observation interviews or on her own time, thinking about what was and was not successful in the enacted lesson. For example, she made the following comment about her use of counters to model word problems for her students: I was thinking, the counters that I used today I actually made last year for shapes and patterns. And they were ... definitely too big [for this lesson]. The use of these counters was itself a new element that Beth had introduced to the curricular lesson. This was a pattern that Beth followed many times over the course of the school year: she created new activities or materials before instruction and then evaluated their usefulness during and after instruction.

Patterns in curriculum use - 17

Beth's curriculum strategy is summarized in Figure 2. Each cell describes Beth' typical actions as she engaged in a particular interpretive activity at a given time. Those cells that are blank indicate that Beth did not engage in that activity during that time period. All 17 observations of Beth's instruction were consistent with the curriculum strategy presented in this matrix.

Before Instruction During Instruction After Instruction

Read Examines main activities in lesson; Examines new vocabulary introduced in lesson

Evaluate Considers own understanding of conceptual connections among activities in lesson

Adapt Creates transitional activities

Considers students' understanding of mathematics in lesson Considers whether students need more review; Considers whether she successfully managed activities in lesson

Creates new explanations and new terminology

Figure 2. Beth's curriculum strategy matrix

Fran's Curriculum Strategy

Fran taught third grade and, prior to instruction, Fran, like Beth, read, evaluated, and

adapted the lesson that she was preparing to teach. Furthermore, as was also the case with Beth,

Fran's reading of the curriculum was designed to give her a sense of the general outline, or big

ideas, of a lesson. Fran discussed this approach with the researcher, explaining that she paid most

attention to the objectives that were presented at the beginning of each lesson because "that gives

me my focus of what [the lesson] is going to be about." Moreover, on several occasions Fran

stated that she tended not to look closely at the specifics of a lesson prior to class.

Last night I [looked] at all the problems and I'm like, okay, we'll probably get through all these problems. But I didn't read the problems.

As she read the materials, Fran wondered about the extent to which her students would

understand the mathematics that was to be presented. When asked in interviews what questions

she had had prior to class, Fran repeatedly commented on whether the material was organized in

Patterns in curriculum use - 18 the best way for her students or for students at the level of mathematics understanding that she felt her class represented. "My big question...is where [is the lesson] going to have to go so that they understand it?" "They're going to need to go back over this so...[students] can get it." "The lesson [must be designed for] higher kids doing this math." In light of these concerns, Fran regularly made changes as she prepared for instruction, changes that typically involved substituting a new activity in place of one recommended in the curriculum. For example, rather than using a paper-folding activity described by the curriculum to help students explore fractions, Fran planned to ask students to shade in different portions of a square. Because when you hold up those little shapes, you can't see where they're folded. So how could they count the fourths if they couldn't see it? So I just thought [shading in] would help ...on the square, because then people could hold it up. Fran continued to evaluate lessons during instruction, though her approach to doing so was somewhat different than it had been prior to class. In the midst of instruction, Fran's evaluation of the lesson-in-progress focused more on her perception of her own ability to explain the mathematics effectively, rather than on her students' success with the mathematics as had been the case before instruction. Such evaluations were evident both during class ("Let me think about how to say this.") and in reflecting on lessons in interviews. The pacing wasn't right. I didn't have it right, first ask them for a problem, then a strategy, then an answer... it was [too] long and intricate. Unlike Beth, who often created new materials or activities, Fran's response to these concerns was to either replace one activity with another, as she had done before class, or to simply omit activities from a lesson. Interestingly, Fran was usually unaware of changes that she made during instruction. When asked by the researcher what changes she had made in the day's lesson, Fran typically responded "none." This contrasted with her tendency to be quite articulate about changes she made in a lesson before instruction.

Patterns in curriculum use - 19

Finally, Fran also read from the teacher's guide during instruction. While she had already reviewed the curriculum materials prior to class, she had done so quite generally. During class, she typically paid closer attention to specific portions of the teacher's guide, reading "for detail" rather than for "the big ideas." On different occasions, Fran referred to the teacher's guide for the specific directions to an activity, for the precise wording of a problem, and for the correct spelling and definition of new vocabulary. Fran followed the curriculum strategy described here in 10 out of 12 classroom observations for a total of 83% (Figure 3). Furthermore, the observed deviations from this pattern were minor. On one day, she physically carried the teacher's guide with her during instruction but did not refer to it directly, thus the lesson was not coded as including "reading during instruction." Second, on another occasion, Fran did not make changes to a lesson during instruction. The lesson, however, was cut short by other demands on the class' time that day. Had the lesson been carried out in full, it seems likely that additional adaptations would have taken place.

Before Instruction During Instruction After Instruction

Read Examines general outline of activities in lesson Examines precise wording of examples used in lesson

Evaluate Considers whether students understand mathematics in lesson Considers whether she is explaining mathematical ideas correctly

Figure 3. Fran's curriculum strategy

Adapt Substitutes activities Unintentionally omits or substitutes activities

Shelley's Curriculum Strategy

Shelley taught third grade and was observed 23 times over the year. All of these observations

reflected a single consistent curriculum strategy (Figure 4). Specifically, prior to instruction,

Shelley's goal was to carefully read the curriculum materials in order to familiarize herself with all of

the curriculum's details. This is in clear contrast to Beth and Fran, who read for the big ideas prior to

Patterns in curriculum use - 20

instruction. In talking about her preparation for class, Shelley more than once commented that she relied on the teacher's guide to "really clue her in" as to what a specific lesson involved. Furthermore, during instruction, she often recited directions or examples from the teacher's guide precisely as they were written, giving further evidence of her thorough reading of the materials before class. In interviews, Shelley indicated that she relied on the teacher's guide not only for information about the content of the lesson, but also for detailed guidelines about appropriate instructional strategies and about vocabulary and phrasing that she should use in talking with her students. [I look at] the terminology in the book, and the side... where it tells you the directions to the teacher, and here where... it talks about the problems, and the whole idea [of the lesson]. For example, in describing how she prepared for the day's lesson on adding and subtracting numbers up to 100, Shelley remarked, "[It's helpful] that [the teacher's guide] tells you to ask `Do I have enough tens here?' and `How can you subtract 20 cents?' and `You've got to trade in for two dimes.' The wording, the language...I'm seeing that." Interestingly, unlike Beth or Fran, Shelley did not evaluate or adapt the curriculum prior to instruction. Instead, her approach was simply to try to learn as much as possible about the written lesson prior to instruction.

Before Instruction During Instruction After Instruction

Read

Evaluate

Examines concepts of lesson in detail; Examines specific suggestions about how to implement lesson; Examines new terminology in lesson

Considers whether students understand mathematics in lesson; Considers whether students understand new terminology used; Considers whether manipulatives are distracting for students Considers whether lesson included sufficient review for students; Considers whether lesson included too much board work for students; Considers whether new terminology was confusing for her

Figure 3. Shelley's curriculum strategy

Adapt Adds new examples and new contexts for problems; Substitutes one activity for another Plans to use different activity in future

Patterns in curriculum use - 21 During instruction, Shelley engaged in a more flexible approach to using the curriculum, typically evaluating and adapting a lesson as it took place. She described this approach as "thinking on her feet," explaining that she wanted her students to get the most out of each lesson and this called for her to make changes during instruction. In particular, and similar to Beth, when she believed that her students were not fully understanding a concept, she would add new activities to offer them additional practice, or would substitute a new activity in place of one suggested by the curriculum. For instance, on one day in November, Shelley was leading a whole class activity on repeated groups in which she asked students to show different groupings of numbers with their hands (e.g., flashing five fingers six times for "six groups of five."). Shelley was concerned that her students weren't really seeing the "repeatedness in numbers" so in the midst of instruction she added a new activity that was not suggested in the curriculum. "I want to try something. Let's have three people come up [to the board]. Think of a situation where you have equal groups and see if you can come up with a drawing. A number sentence and a drawing to go with it." After class she explained that, "I wanted them to have a chance to come up with their own situations...to give them another way... to get the language of repeated ...groups and equal groups." On another occasion, she decided to add a grocery store context to an activity on addition and subtraction. "That just happened. I thought if they were thinking about it as a grocery store, they might understand it better." Thus, as she was teaching, Shelley evaluated the lesson, asking how her students were progressing. Moreover, she adapted by adding new activities and contexts to the existing lesson. After class, Shelley continued to evaluate and adapt the lessons. She made notes to herself in the margin of the curriculum concerning whether the organization of a lesson had met the needs of her students and whether she herself had understood the curriculum's terminology.

Patterns in curriculum use - 22 Such notes would also indicate places where, in the future, Shelley though it best to use an alternate activity. The language was [difficult] for me today. I want to try to put it into simpler language...and [next time] I definitely want to use that homework as part of a little review [in class]...that might take 10 or 15 minutes. PATTERNS IN TEACHERS' CURRICULUM STRATEGIES We now turn to a discussion of the similarities and differences among the curriculum strategies of all 10 teachers participating in the study. We begin by examining differences in when during the instructional process teachers read, evaluated, and adapted the curriculum. We then examine the ranges of ways in which the teachers engaged in these activities before, during, and after instruction. Some differences in teachers' curriculum strategies may already be familiar to the reader due to the prior characterizations of Beth, Fran, and Shelley's practice in this regard. Patterns in When Teachers Read, Evaluate, and Adapt the Curriculum Reading the Curriculum All of the teachers read portions of the curriculum materials. And as expected, most of the teachers read the curriculum while planning for instruction prior to class. Specifically, read occurred before implementation for 9 of the 10 teachers (Table 2), In addition, of these nine teachers, three teachers also read from the curriculum during instruction. Thus, despite having looked over the materials before teaching, these teachers referred directly to the text once again during instruction. One teacher, Vera, did not read the materials prior to class despite the fact that the curriculum was new for her (as it was for the other teachers). Instead, Vera read the curriculum materials only during instruction. She usually carried the teacher's guide with her during class, open to the lesson that was the focus for the day's instruction.5

Patterns in curriculum use - 23

Anita Beth Fran Jan Kate Lauren Marta Paul Shelley Vera

Read Before Instruction X X X X X X X X X

Read During Instruction X X X X

Table 2. Timing of teachers' reading of the curriculum

Read After Instruction

Evaluating and Adapting the Curriculum

Because most teachers tended to evaluate and adapt the curriculum during the same

instructional period, we discuss these together. Specifically, five teachers evaluated and adapted

the curriculum in preparing for class, and four of these continued to do so during instruction. One

additional teacher did not typically evaluate and adapt the materials before class, though she did

so consistently during instruction. Finally, there were two teachers who regularly evaluated and

adapted after instruction, considering what did and did not work in the enacted lesson and how

they planned to change the lesson in the future (Table 3).

Anita Beth Fran Jan Kate Lauren Marta Paul Shelley Vera

Evaluate and Adapt Before Instruction X X X X X (evaluate only) X

Evaluate and Adapt During Instruction X X X X X X (adapt only) X X (adapt only)

Evaluate and Adapt After Instruction X X (evaluate only) X (evaluate only) X (evaluate only) X

Table 3. Timing of teachers' evaluation and adaptation of the curriculum

Patterns in curriculum use - 24 There were also instances, both before and after instruction, in which teachers evaluated the curriculum but did not adapt the materials. For example, Lauren evaluated the curriculum in preparation for instruction, noting what aspects of the lessons she believed might not work. However, at this point, she chose not to make any changes in the lesson. In interviews, Lauren explained that while she often questioned how successful the curriculum materials might be, she felt that it was important for her, at least initially, to try out the materials as planned. For example, the curriculum suggests that teachers use an "equals board" to teach number sentences. While Lauren had some concerns about whether or not the equals board would work, she explained, "I'm not going to know to let go of the equals board until I've done the equals board." Thus she chose to evaluate but not adapt the lesson. Similarly, there were cases during instruction in which teachers adapted the curriculum materials without evaluating the ongoing lesson. For example, Vera often allowed students' comments to guide the direction of a lesson without considering whether or not these comments addressed her goals for the lesson. She explained that she had "not been sticking close to what's prescribed" and instead "kind of just goes along with the kids." Vera's primary goal was not to follow a particular lesson, thus she did not need to evaluate the lesson-in-progress in terms of specific goals that she had set. Instead, adaptations occurred without regard to how they might influence the evolving lesson. Patterns in How Teachers Read, Evaluate, and Adapt the Curriculum We next explore similarities and differences in the manner in which the teachers read, evaluated, and adapted, the curriculum. In what follows, we discuss each of these activities in turn. For the reader's reference, a summary by teacher across all three activities is provided in Appendix B. This is somewhat different than the curriculum strategies presented in Appendix A, because now we classify the ways in which the teachers read, evaluated, and adapted the curriculum as belonging to particular categories.

Patterns in curriculum use - 25 Ways of Reading the Curriculum In examining what the teachers looked for as they read the curriculum, we identified three general approaches (Table 4). One group of teachers consistently read the curriculum materials before instruction for the "big ideas" of a lesson, trying to get a general overview of what a lesson was about without paying as much attention to the details of the lesson. Furthermore, for most teachers, attending to the big ideas of a lesson involved focusing on the set of activities that were to be included in the day's lesson. Beth and Fran's curriculum strategies, described earlier, fit into this category. In reading the teachers' guide, both teachers focused on the general outline of the activities that comprised the lesson and on the order and pacing of these activities. In contrast, other teachers (such as Shelley) prepared for instruction by focusing on the curriculum at a more detailed level, carefully studying the materials provided. And as was the case with Shelley, examining the details of a given lesson involved paying close attention to the pedagogical strategies described in the curriculum. In other words, these teachers looked for very specific guidance about how to proceed with a lesson. In addition, three of the teachers, Fran, Lauren, and Paul, read the curriculum both for the big ideas and for details. As described earlier in discussing Fran's curriculum strategy, this took place as the teachers read for the big ideas of a lesson prior to instruction, but then referred back to the teacher's guide in a more detailed way during instruction. Furthermore, while the teachers' initial reading of the lesson was generally focused on the activities that were included in the lesson, their reading of the curriculum during instruction was more focused on the details of the language used in the curriculum -- selecting appropriate examples or reading an explanation word-for-word from the teachers' guide. For example, Lauren explained that she returned to the teachers' guide during instruction to refer to a particular explanation or example that she wanted

Patterns in curriculum use - 26

to use from the book. "I...tell my kids...`You know what? This part is really hard and I need to

use the teacher's book for this part...I'm going to read it straight out of the teacher's book.'"

Anita Beth Fran Jan Kate Lauren Marta Paul Shelley Vera

Read Before

Read During

Instruction

Instruction

Big Idea

Big Idea

Big Idea

Detail

Detail

Detail

Big Idea

Detail

Big Idea

Big Idea

Detail

Detail

Detail

Table 4. Teachers' ways of reading the curriculum

Read After Instruction

These three approaches illustrate for us that when teachers "read the curriculum," they do

not all read the same information. Similar results have been reported by Remillard (1999, 2000).

However, while Remillard focused on differences in the topics that teachers considered as they

read the curriculum (mathematical concepts versus student activities), here we argue that

differences in reading also occur in terms of the extent to which teachers attend to the details of a

given lesson and when in the instructional process this reading takes place.

In addition, identifying these different approaches to reading the curriculum raised questions

for us about teachers' ability to understand the intent of a given lesson. First, those teachers who

read for the big ideas in a lesson were not always aware of the specifics of how a lesson was

designed to meet those big ideas. In contrast, teachers who read for detailed information about what

they were expected to do during instruction at times missed out on the bigger picture and did not

understand how and why the pieces of a lesson were intended to fit together. Yet one group of

teachers seemed to find a way to deal with this tension, reading first for the big ideas, and later

during instruction, referring to the details of a lesson. What we find particularly interesting is that

this group of teachers all read for the big ideas before instruction. Apparently, what was critical to

Patterns in curriculum use - 27 them in preparing to teach was that they had a sense of what the goals of the lesson were and what the activities of the lesson would be. These teachers would concern themselves with the details of the lesson during instruction, once they figured out precisely what information they needed during the moment of instruction. In doing so, they seemed to balance the need for an understanding of both the big ideas and the details of a given lesson. Ways of Evaluating the Curriculum We were also able to characterize distinct approaches to evaluating the curriculum. In particular, each teacher typically had a specific audience in mind as the focus of their evaluation. Three such audiences were identified: Evaluation occurred either in terms of the teacher, the students, or others such as parents or administrators. For example, Marta frequently evaluated in terms of the teacher asking, "Do I understand the mathematics in the lesson? Am I sure what this concept means?" Alternatively, Jan evaluated the curriculum in terms of the students in the class. "Did my students understand how to find the partner numbers? Were they able to follow the procedure introduced in the lesson?" In another variation, Lauren asked, "What will the parents think about the ideas introduced in this lesson? What will the parents think of the invented algorithms that students are using?" Looking at how the teachers evaluated the curriculum before, during, and after instruction, an interesting pattern appeared. One set of teachers (Anita, Beth, Lauren, Shelley) tended to evaluate the lesson in terms of the teacher prior to instruction and in terms of the students during instruction (Table 5). Thus, before class, these teachers examined the curriculum in terms of their own expertise, trying to determine whether or not they were prepared to teach the lesson. During instruction, however, this perspective changed, and these four teachers examined the lesson in terms of the students' progress. For example, recall that Beth, prior to instruction, was concerned with what the book was "wanting her to do" while during instruction, she was concerned with

Patterns in curriculum use - 28 whether her students understood what she was "wanting from them." We include Shelley here, because, although she did no evaluation prior to instruction, she evaluated exclusively in terms of the students during instruction. Furthermore, Shelley had several different themes along which she tended to evaluate during instruction, and for each of these she focused on the students. Thus her focus on the students during instruction appears to be quite robust. We have also grouped Anita with these teachers. Like the other teachers in this group, Anita evaluated exclusively in terms of her students during instruction. However, prior to instruction, while she evaluated some aspects of the lesson in terms of the teacher, when it came to issues related to language, Anita evaluated in terms of the teacher, students, and parents (e.g,, others). We attribute this to the fact that Anita was teaching a bilingual class and considered issues of language more broadly than she did other aspects of teaching. Thus it was not a focus on students per se in contrast to the teacher. Instead, she had a general concern with language, whether it was the language of the teacher, the students, or others such as parents. Her tendency overall, nevertheless, was to evaluate first in terms of the teacher, and later during instruction, in terms of the students. In contrast to these four teachers, two other teachers, Fran and Kate, evaluated first in terms of the students and then in terms of the teacher. For example, as described earlier, in preparing for class, Fran was generally concerned about her students' understandings of mathematics and whether or not they would be able to grasp the material to be covered. During instruction, however, she focused on her own knowledge of the subject matter and her ability to explain the ideas effectively to students. "What am I missing here in terms of the math? How can I make sense of this?" It seemed that because Fran had evaluated only in terms of students beforehand, she was not always sure of her own role during instruction and thus, evaluated in terms of the teacher in the midst of instruction. We note that Kate also maintained her focus on

Patterns in curriculum use - 29 the students during instruction. Still, her initial focus on the students and then on the teacher during instruction provides an important contrast with the group described above that focused first on the teacher. Of the four remaining teachers, only Marta and Jan consistently evaluated lessons and they do not fit neatly into either trend. Marta evaluated in terms of both her students and the teacher prior to instruction. But because she did not evaluate during or after instruction, we cannot say whether she might tend towards one category or another. Jan did no evaluation before or during instruction. However, she did evaluate after instruction, and at this point her evaluation was focused exclusively on the students. Interestingly, the four teachers who focused initially on the teacher and later on the students also evaluated regularly after instruction, while Kate and Fran did not. Prior research (Ben-Peretz & Tamir, 1981) found that teachers reported little interest in using the curriculum as a resource for their own learning of content and pedagogy. In contrast, 7 of the 10 teachers we studied were concerned at some point with their own understanding and use of the CMW materials. This difference may be due in part to the fact that the CMW curriculum was new for the teachers. In addition, teachers today may be more aware that they are expected to play a central role in shaping how curricula are implemented. We also want to mention that two of the teachers consistently evaluated the curriculum in terms of people other than the teacher or the students. As mentioned above, prior to instruction Anita evaluated the language of the lesson in terms of not only the teacher and the students, but also the parents asking, for example, "Will the parents understand the homework instructions?" Lauren considered both parents and administrators in her evaluation of CMW lessons. In particular, she asked how the mathematical ideas of a lesson were related to district standards and how parents would respond to the mathematics that their children were learning. Though not widespread among the teachers, we find this interest in evaluating a lesson beyond the

Patterns in curriculum use - 30

boundaries of the classroom particularly noteworthy in today's context of reform. Mathematics

education reform in the U.S. receives widespread attention in the media and may in fact

influence the way that teachers approach the use of new curricula.

Teacher Audience for Evaluation Audience for Evaluation Audience for Evaluation

Prior to Instruction

During Instruction

After Instruction

Evaluate before instruction in terms of teacher and during instruction in terms of students

Anita Teacher & Students & Others Students

Teacher & Students

Beth Teacher

Students

Teacher & Students

Lauren Teacher

Students

Others

Shelley

Students

Teacher & Students

Evaluate before instruction in terms of students and during instruction in terms of teacher

Fran Students

Teacher

Kate Students

Teacher & Students

Pattern not apparent

Jan

Students

Marta Teacher & Students

Do not evaluate

Paul

Vera

Table 5. Teachers' ways of evaluating the curriculum

Ways of Adapting the Curriculum

The final interpretive activity we discuss is adaptation. Again, to be clear, no lesson can

be implemented exactly as written: changes inevitably occur. Here however, by adaptation, we

refer to significant changes that teachers make in the intended curriculum such as changes in the

structure of a lesson, in the activities that comprise the lesson, or in the purpose of the lesson.

In examining the ways in which the teachers adapted lessons, we found three general

approaches that can be thought of as lying along a continuum. At one end, making changes to a

lesson involved creating new components for the lesson. For example, as mentioned earlier, Beth

developed transitional activities to help students move from one activity in a lesson to the next

activity. Similarly, Shelley consistently created new activities in which her students could

explore the mathematics of a given lesson.

Patterns in curriculum use - 31 In the middle of the continuum is the idea of replacing, substituting one component of a lesson with something different. For example, Kate regularly replaced the materials suggested in the curriculum; she would use CMW "penny strips" instead of "secret-code cards."6 Similarly, Kate would change the participant structure described in the lesson, having students at the board each solve a different problem, rather than, as described in the curriculum, having them solve the same problem in order to illustrate multiple solution strategies. In another case, Vera often replaced the mathematical topic of a lesson, returning to single-digit addition, for example, on a lesson intended to work on multi-digit addition. Finally, at the other end of the continuum, adapting can be thought of as omitting components of a lesson, that is, deleting one part of a lesson without adding something in its place. For example, before teaching, Marta typically evaluated a lesson in terms of her own understanding of mathematics. When she felt that the lesson addressed a topic that she did not understand, she simply omitted that part of the lesson. Similarly, Paul omitted what he thought of as games from the lessons that he taught. Table 6 shows four teachers at one end of the continuum either creating, or creating and replacing. Two teachers exclusively replaced as their form of adaptation and represent the middle of the spectrum. In contrast, three teachers were at the other end, omitting, or omitting and replacing sections of the lesson7. The fact that no teachers both omitted and created as they adapted the curriculum supports our claim that the teachers' approaches to adaptation form a continuum. Other researchers also discuss teachers' selective use of curriculum materials. For example, Remillard (2000) contrasts one teacher's tendency to invent new material with that of a second teacher who would pick and choose from among the activities suggested. In presenting these different approaches as a continuum, we provide a new avenue for comparing the ways in which teachers make changes in curriculum materials.

Patterns in curriculum use - 32

Adapt by creating Anita Beth Lauren Shelley Adapt by replacing Kate Vera Adapt by omitting Fran Marta Paul Did not adapt Jan

Adapt Before Instruction Replace Create Replace Replace Omit

Adapt During Instruction Replace Create Create Create/Replace Replace Replace Replace/Omit Replace/Omit

Adapt After Instruction Create Replace

Table 6. Teachers ways of adapting the curriculum

Before concluding this section, there are two final points about the teachers' approaches

to adaptation that we wish to make. First, the evidence for the teachers' approaches to adapt (as

well as read and evaluate) comes from both videotapes and observations of instruction as well as

interviews with the teachers. However, in the case of adapt, we found three teachers who seemed

to be unaware of the changes they were making in a lesson during instruction -- unaware, at

least to the extent that when asked about changes they had made in the curriculum, these teachers

consistently did not mention changes that we noticed in our analysis. In particular, Fran and Vera

seemed genuinely unaware that they were in fact making substantial changes to the curriculum.

In contrast, Anita discussed changes she made in the lesson segments and choice of language,

thus she was aware that she purposefully made changes to some aspects of the curriculum. Yet at

the same time Anita seemed unaware of her tendency to replace the mathematical focus of a

given lesson with other mathematical ideas. Furthermore, for all three teachers, these

unintentional adaptations of the curriculum occurred during instruction. Thus teachers may be

more aware of changes they make in lessons before and after instruction than changes made

Patterns in curriculum use - 33 during instruction. This is not too surprising given the intense rate at which teachers make decisions during instruction (Clark & Peterson, 1986). Second, as mentioned above, we found that the ways in which the teachers adapted the curriculum generally remained consistent throughout the instructional process. Thus, for example, no teacher was found to "create" before instruction, and then "omit" during instruction. However, which feature of the lesson was changed did depend on when in the instructional process the adaptation took place. Specifically, before, during, and after instruction teachers tended to make changes in the activities that comprised a given lesson. Similar findings have been reported by other researchers in which teachers made changes to the activities within a given lesson but not to the mathematical focus of the lesson (Ben-Peretz & Silberstein, 1982; Freeman et al., 1983; Stodolsky, 1989). In contrast, we found that mathematics was a focus of adaptation, but that his occurred almost exclusively during instruction. Thus, even though the curriculum was new for the teachers, they felt comfortable making changes before instruction at the level of activity -- how a class would be organized and managed, what materials would be used, who would be doing what. In contrast, teachers were less able to envision before instruction what kind of changes in the mathematics of the lesson they wanted to make. Instead, they had to first try to teach the mathematical concepts described in the curriculum. Then during class, once having seen the mathematics in action to some extent, they were willing to make changes in the concepts addressed in the lesson. Connections between Approaches to Evaluate and Adapt As mentioned previously, in most cases, teachers evaluated and adapted a lesson at the same point in the instructional process. Thus, we wondered if there might be connections between the teachers' approaches to these two processes. What we found was that the four teachers who evaluated the curriculum in terms of the teacher prior to instruction and in terms of

Patterns in curriculum use - 34

the students during instruction shared a distinct approach to adaptation: they created new

materials as part of their adaptation process. Moreover, none of the other teachers we studied

engaged in creating materials as they made changes to a lesson (Table 7).

Teacher Anita Beth Lauren Shelley Fran Kate Marta Jan Paul Vera

Approaches to Evaluate

Before

During

After

Instruction

Instruction

Instruction

Teacher, Ss & Other Students

Teacher

Students

Teacher

Students

Others

Students

Teacher & Ss

Students

Teacher

Students

Teacher & Ss

Teacher & Ss

Students

Approaches to Adapt

Before

During

After

Instruction Instruction

Instruction

Replace

Replace

Create

Create

Create

Create

Create/Replace Replace

Replace

Replace/Omit

Replace

Replace

Omit

Replace/Omit Replace

Table 7. Comparing ways of evaluating and adapting the curriculum

We believe that this connection between the teachers' approaches to evaluation and

adaptation illustrates a proactive sense of collaborating with the curriculum. Specifically, before

instruction, these teachers focused on their own understandings. The curriculum was new to them

and they could not yet fully anticipate what students would do with the materials. Nonetheless,

they tried to understand for themselves what was involved in each lesson. Thus, when a question

or concern about a lesson arose, before, during, or after instruction, these teachers could turn to

their own understandings of the materials to decide how to proceed. And because they had a sense

of why a particular part of a lesson had been included in the first place, they were more likely to

try to work through the issue rather than simply omit that part of the lesson.

This approach was very different from teachers who, prior to instruction, asked "What

will my students understand?" In preparing for instruction, this group of teachers focused on

students' understandings and did not use the curriculum as a resource for their own learning.

Therefore, when they had the sense that part of a lesson would not be effective for students, they

Patterns in curriculum use - 35 chose to either omit that part or replace it with more familiar activities. Furthermore, when a concern arose during instruction, they did, at this point, consider their own understanding of the lesson, yet there was not always enough time for teachers to work through their questions. Instead, they often responded by simply moving on to the next portion of the lesson. Interestingly, the four teachers who focused on teachers prior to instruction and adapted by creating new materials did not share a single approach to reading the curriculum. Beth and Anita read the curriculum before class in terms of the big idea while Shelley read for detailed instructions. Lauren read beforehand for the big idea but during instruction for detail. Thus, teachers' approaches to read do not appear to have constrained their approaches to evaluating and adapting the curriculum. Sharing a common approach to evaluating and adapting the curriculum is not explained by the number of years of teaching experience of these four teachers (Table 1). These teachers were not the most or the least experienced of the group. Their years of teaching experience ranged from 0 to 10 and other teachers in the study had similar years of teaching experience. Furthermore, the similarities among the teachers' approaches are not explained by collaboration among the teachers. Three of the teachers were observed during the first year of the study, while the fourth was observed during the second year. Of the three who were observed during the first year, each taught at a separate school and they did not meet together to prepare or discuss instruction. Moreover, two of these teachers did have colleagues at their schools who were also using the CMW materials, yet their colleagues illustrated a different approach to evaluating and adapting the curriculum. Thus, the particular context in which a teacher works does not predict one's curriculum strategy. Arguing for the Validity of Curriculum Strategy Framework A central goal of this paper is to argue that the curriculum strategy framework captures key features of the ways in which elementary-school teachers interact with a reform-based

Patterns in curriculum use - 36

curriculum. To further support this argument, here we raise four points that speak to the validity

of this framework as an analytic tool.

First, individual teachers maintained a particular curriculum strategy over time. Thus it

was not the case that one minute a teacher had one strategy, and the next minute, the teacher

switched to a different strategy. Instead, applying the framework we have outlined in this paper

led to stable characterizations of individuals with respect to curriculum use. Specifically, we

noted whether each lesson observation confirmed or disconfirmed the curriculum strategy matrix

for a teacher. We found that across all 10 teachers, the percentage of confirming observations

was 78% or above, with an average of 86% of the observations confirming a stable Curriculum

Strategy for each teacher. And in fact, 4 of the 10 teachers exhibited a stable Curriculum Strategy

for 100% of the lessons observed (Table 8).

Teacher Shelley Jan Vera Lauren Beth Marta Paul Anita Fran Kate

Total # observations 23 11 11 8 17 18 9 12 12 15

# observations consistent with curriculum strategy matrix 23 9 9 7 17 14 9 12 10 12

# observations inconsistent with curriculum strategy matrix 0 2 2 1 0 4 0 0 2 3

% observations consistent with curriculum strategy matrix 100% 82% 81% 88% 100% 78% 100% 100% 83% 80%

Table 8. Percentage of observations confirming a teacher's curriculum strategy

Second, the models were consistent not only across time, but also across data types. Thus,

it was not the case that we identified one curriculum strategy when observing instruction and a

second when talking with a teacher about his or her instruction. Instead, both data sources

supported the same curriculum strategy. For example, in interviews, Beth explained that she

changes lessons by adding transitional activities to help provide a bridge from one activity to the

Patterns in curriculum use - 37 next. And when we examined her instruction, we saw these transitional activities in practice -- an explicit explanation of how two parts of a lesson fit together or a new homework sheet to prepare students for the next day's lessons. Third, through our presentation of examples, we believe that we have established the plausibility of the framework. For example, we discussed that we actually saw Vera carrying the teacher's guide with her as she taught. We could see that she "read" during instruction. Similarly, Lauren stated explicitly that her intention was to try out the curriculum as written. Thus, although she had questions and concerns (i.e., she evaluated), she was willing to go ahead and try a lesson before she made any changes. Thus, it makes sense that Lauren tended to "evaluate but not adapt" a lesson prior to instruction. These kinds of examples taken together constitute a prima facie case for the validity of the framework. Fourth, the framework enabled us to describe not just the way that one teacher interacted with the curriculum, but the ways that all 10 teachers in our study interacted with the materials. The teachers represented a range of backgrounds, instructional types, and teaching experience. Our ability to develop stable curriculum strategies for all 10 teachers suggests that our framework has scope and can capture the practices of a variety of teachers with different teaching practices. IMPLICATIONS AND FUTURE RESEARCH In this paper, we set out to accomplish two related goals. First, we wanted to introduce a new framework for exploring elementary-school teachers' use of reform-based curricula. Second, we wanted to use the framework to search for patterns among 10 teachers who were in their first year of using a new curriculum. The results support the validity of the framework and point to a variety of ways in which teachers use curriculum materials. The phenomena we were studying are complex and, to a certain extent, our data and analyses reflect this complexity. For example, there was some variety in the curriculum strategies

Patterns in curriculum use - 38 we observed. However, our analyses were able to uncover substantial order in this complexity. Individual teachers could be characterized in terms of a single, curriculum strategy. And though the strategies employed by different teachers certainly varied, we were able to characterize this variety with a moderately compact framework, that makes only a small number of distinctions; reading involved a focus on big ideas or on details; evaluate involved a focus on students, teachers, or others; and adapt involved creating, replacing, or omitting activities or materials. This research adds to our understanding of teacher cognition by delineating the range of strategies that teachers use in a key area of their practice. Research on student learning has sought for many years to identify student strategies in specific content areas (e.g., Carpenter & Moser, 1983). The idea is that understanding the range of student approaches to a problem or concept can help us to better understand what it means for students to learn mathematics and how we can better scaffold and support that learning. We claim that the same is true of teachers. Thus, while instruction may look very different among a set of teachers, the curriculum strategy framework gives us a common lens that we can use to characterize teachers' practices. Furthermore, we find that the use of reform-based materials by teachers, even in their first year of use, is not haphazard. Reform-based curricula are used with consistency and with attention to particular issues and questions. Identifying these common themes in teachers' use of curriculum provides insights into what teachers see as the key issues to consider in preparing for and carrying out instruction. This research also has important implications for curriculum design. Ball and Cohen (1996) and others have called for the creation of educative curricula, curricula that are designed to support the learning of both teachers and students. We argue that such development must go hand in hand with research on how teachers use curriculum materials. Thus, it is not enough to say that curriculum materials should provide explicit information for teachers on likely student errors. We must also consider whether and how teachers will read and evaluate this information, and how,

Patterns in curriculum use - 39 given what we know about curriculum strategies, we could makes such information usefully accessible to teachers. We claim that new curricula will be most effective when their designs support the multiple approaches to curriculum use that we have identified. For example, now that we know that some teachers read curricula only for big ideas, materials can be designed so that the big ideas (i.e., goals, objectives, lesson overviews) provide enough information for teachers to be able to teach, or at least to begin teaching, a lesson successfully within the "curriculum envelope." In terms of professional development, we think that it would be instructive for teachers to learn about these different approaches to using curricula. For example, a teacher who usually evaluates in terms of the order of the activities in a lesson might find it useful to learn to think about mathematical issues also. Similarly, a teacher who tends to read for the big ideas might find it useful to have an alternate approach available (i.e., reading for details) at times when it seems like the big ideas do not provide enough information. This is similar to Palinscar & Brown's (1984) work on introducing students to meta-Cognitive Strategies in the area of reading. Here our goal would be to introduce teachers to meta-cognitive strategies for interacting with curriculum materials -- strategies that they could actively choose among given their goals for a particular lesson and their own understanding of the lesson under consideration. Many researchers and teacher educators assert that professional development should be directly related to teachers' classroom experiences; that teachers should have something "to bring back" from professional development directly into their classroom. At the same time, others argue that professional development should help teachers to reflect and learn from their teaching practices, even if this means they do not have a new activity to try out on the following day. We suggest that professional development around curriculum strategies might be able to achieve both of these goals simultaneously. On the one hand, studying new curriculum materials could help teachers to better understand the activities and ideas that they are expected to implement and

Patterns in curriculum use - 40 could help the teachers assess whether the adaptations they intend to make are appropriate given the goals of the materials. At the same time, examining with colleagues the ways that they use the curriculum materials can be a productive avenue for self-reflection and growth as teachers consider their typical approach as well as what other strategies might offer them. This paper introduces a new analytic framework for examining the ways in which teachers interact with curriculum materials in the context of reform. While we believe that this is an important contribution to research in mathematics education, we realize that the work described here is only the beginning of a program of research in this area. In the future, we plan to conduct similar studies that investigate the following three issues. The relationship between teachers' curriculum strategies and the implementation of mathematics education reform. An important goal of research on teachers' use of reform-based mathematics curricula is to examine how using such materials can help teachers manage the complex demands of reform. To date, researchers have explored this issue by studying how new curricula foster teacher learning and help teachers navigate key challenges of implementing reform (Remillard, 2000; M. S. Smith, 2000; M. R. Wilson & Lloyd, 2000). We believe that the curriculum strategy framework can also contribute productively to this conversation. Specifically, we want to understand whether teachers with different curriculum strategies have more or less success with the goals of mathematics education reform. Of particular interest are two patterns of curriculum use that we believe may help to support teachers' efforts to implement reform. First, three teachers read the curriculum both for the big ideas of a lesson and for detailed information about a lesson. We suspect that a focus on both of these is needed if teachers are to understand the goals and intentions of new materials as well as the specifics of how they might carry out those goals and assess the ongoing lesson. This is particularly critical in the context of reform, when curricula often include new content and new pedagogical approaches with which

Patterns in curriculum use - 41 teachers may not be familiar. Second, four teachers shared a specific approach to evaluating and adapting the curriculum. This approach involved a focus on the teachers' understanding of a lesson before instruction and a focus on students' responses to the lesson during instruction. It seems to us that this perspective supports teachers in one of the main goals of reform, namely the ability to make changes during instruction in response to students. Furthermore changes made in a lesson, either before, during, or after instruction, consisted of the creation of new materials. This is also an important tenet of reform -- that lessons cannot be completely planned out in advance and that teachers need to be able to make changes that maintain the goals of a lesson. The relationship between teachers' curriculum strategies and teachers' beliefs and knowledge. As stated earlier, in examining how teachers read, evaluate, and adapt curriculum materials at different points in the instructional process, we have tackled only a small portion of the factors that influence teachers' use of curriculum materials. In future work, we hope to broaden this perspective by examining how teachers' beliefs and knowledge influence their interactions with curriculum materials. Towards this end, in related work (Sherin, Drake & Wrobbel, under review) we examine teachers' own experiences as mathematics learners and explore how this may have influenced the approach to curriculum adaptation that they take as teachers. We also plan return to the observation and interview data used in this study for evidence of the teachers' subject matter knowledge and pedagogical content knowledge. This information will allow us to examine whether differences in content knowledge reflect differences in the teachers' curriculum strategies. Furthermore, we believe that it would be useful to investigate the curriculum strategies of secondary mathematics teachers. The teachers in this study were all elementary-school teachers who teach a variety of subjects. Studying the curriculum strategies of high school teachers who specialize in teaching mathematics might reveal particular ways in which subject matter knowledge influences patterns of curriculum use. Prior research has shown

Patterns in curriculum use - 42 that both at the elementary and second levels, teachers' existing content knowledge can constrain and enable the implementation of reform (Sherin, 2002; Lloyd & Wilson, 1998; M. S. Smith, 2000). Comparing curriculum strategies for teachers at these different grade levels use could help to explain more precisely how one's knowledge of the domain affects one's approach to a new curriculum. The role of the curriculum. It is also important to consider how much of what we found was specific to the CMW curriculum that the teachers were using. That is, what differences might we find if the teachers were using a more traditional mathematics curriculum? Moreover, how do particular aspects of reform-based materials influence teachers' curriculum strategies, e.g., organization of the materials, mathematical content, information about student learning. Similarly, to what extent are the curriculum strategies reported here a result of teachers' using a reform-based curriculum for the first time? We are currently studying a subset of teachers who continued to use CMW for more than one year and plan to examine whether and how their curriculum strategies change over time. New curricula play an increasingly important role in the implementation of mathematics education reform. They not only communicate the central goals of reform to teachers, they also guide teachers' daily activities. Furthermore, both of these aspects of reform-based curricula are very often new for teachers. What do teachers do in these situations? How do they proceed? When and why do they make changes when using new curricula? Our introduction of the curriculum strategy framework provides a context for beginning to answer these questions.

Patterns in curriculum use - 43 NOTES 1 Three of the teachers, Fran, Kate, and Beth, continued with the project for a second year. However, only the data from their first year in the project is included in this study. 2 Paul was observed only nine times during the year because he had a student teacher during the months of September and October. 3 One teacher, Paul, was not available for post-observation interviews on a regular basis. Instead, he was interviewed more extensively periodically throughout the year. 4 A variety of data sources contributed to our analysis of teachers' activities prior to instruction. First, on occasion we observed teachers preparing for class and monitored which CMW materials they paid attention to during that process. Yet even when we did not observe this process directly, we were able to collect evidence concerning what took place. For example, in many cases, teachers prepared for a lesson by making notes in the teachers' guide, by designing new materials, or by accessing special materials that they expected to use during instruction. In addition, evidence for the teachers' prior reading of the curriculum was often apparent during instruction. For instance, if a teacher used an example verbatim from the teachers' guide, but did not refer to the teachers' guide during instruction, that provided evidence of a close reading of the example prior to instruction. Furthermore, as noted earlier, in the post-observation interview, teachers were asked to describe how they prepared for the day's lesson. 5No teachers read from the curriculum after instruction. Thus reading the teachers' guide or student pages was not the basis for evaluating or adapting the lesson following teaching of the lesson. 6 Secret code cards are manipulatives designed to help students build and decompose multi-digit numbers with conceptual understanding. Students have cards with "10", "20", "30", etc. through "90" and single-digit cards sized to fit over the zero's in the tens cards. 7 To be clear, in those instances where two approaches to adaptation are listed (e.g., create/replace or replace/omit), the teacher consistently engaged in both approaches at the corresponding point in the instructional process.

REFERENCES

Patterns of curriculum use - 44

Ball, D. L. (1991). Research on teaching mathematics: Making subject-matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching: Vol. 2 (pp. 1-48). Greenwich, CT: JAI Press. Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is - or might be - the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6-8,14. Barr, R., & Sadow, M. W. (1989). Influence of basal programs on four-grade reading instruction. Reading Research Quarterly, 24, 44-71. Ben-Peretz, M. (1990). The teacher-curriculum encounter: Freeing teachers from the tyranny of texts. Albany: SUNY Press. Ben-Peretz, M. & Silberstein, M. (1982). A curriculum development case study in biology: Two levels of interpretation. European Journal of Science Education, 4(4), 377-389. Ben-Peretz, M. & Tamir, P. (1981). What teachers want to know about curriculum materials. Journal of Curriculum Studies 13(1), 45-54. Brown, A.L. & Campione, J.C. (1996). Psychological theory and the design of innovative learning environments: On procedures, principles, and systems. In L. Schauble & R. Glaser (Eds.), Innovations in learning: New environments for education (pp. 289-325). Mahwah NJ: Erlbaum. Carpenter, T. P., & Moser, J. M. (1983). The acquisition of addition and subtraction concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp.7-44). NY: Academic Press. Clark, C., & Peterson, P. (1986). Teachers'thought processes. In M. C. Wittrock (Ed.), Handbook of research on teaching (Third edition, pp. 255-314). New York: Macmillan. Cohen, D. K., & Ball, D. L. (1990). Policy and Practice: An Overview. Educational Evaluation and Policy Analysis, 12, 233-240. Davenport, L. R. (2000). Elementary mathematics curricula as a tool for mathematics education reform: Challenges of implementation and implications for professional development. Newton, MA: Center for the Development of Teaching, Education Development Center, Inc. Drake, C. & Hufferd-Ackles, K. (1999). Living Math Histories: The Influence of Teachers' Prior Math Experiences on Their Implementation of a Reform Math Curriculum. In F. Hitt and M. Santos (Eds.). Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education: 709-715. Fennema, E., & Nelson, B. S. (1997). Mathematics teachers in transition. Mahwah, NJ: Erlbaum. Foshay, A. W. (1990). Textbooks and curriculum during the Progressive Era: 1930-1950). In D. Elliot & A. Woodward (Eds.), Textbooks and schooling in the United States: Part 1: Eightyninth yearbook of the National Society for the Study of Education (Vol. 1). Chicago: University of Chicago Press.

Patterns in curriculum use - 45 Franke, M. L. Fennema, E., & Carpenter, T. (1997). Teachers creating change: Examining evolving beliefs and classroom practices. In E. Fennema, & B. S. Nelson (Eds.), Mathematics teachers in transition (pp. 255 - 282). Mahwah, NJ: Erlbaum. Freeman, D.J., Belli, G. M., Porter, A. C., Floden, R. E., Schmidt, W. H., & Schwille, J. R. (1983). The influence of different styles of textbook use on instructional validity of standardized tests. Journal of Educational Measurement, 20(1), 259-70. Freeman, D.J. & A. C. Porter (1989). Do textbooks dictate the content of mathematics instruction in elementary schools? American Educational Research Journal, 26(3), 403-21. Fuson, K. C., De La Cruz, Y., Lo Cicero, A. M., Smith, S. T., Hudson, K., Ron, P., & Steeby, R. (2000). Blending the best of the 20th century to achieve a mathematics equity pedagogy in the 21st century. In M. Burke (Ed.), Learning mathematics for a new century, 2000 Yearbook of the NCTM. Reston, VA. Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Chicago: Aldine. Grossman, P. L., & Thompson, C. (2002, April). Visions of language arts: Curriculum materials as opportunities for secondary teacher learning. Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA. Heaton, R. M. (2000). Teaching mathematics to the new standards: Relearning the dance. Teachers College Press. Kilpatrick, J. (1997, October). Five lessons from the new math era. Paper presented at a National Academy of Sciences symposium, Washington, D.C. http://www.nas.edu/sputnik/kilpatin.htm. Kon, J. H. (1994). The thud at the classroom door: Teachers' curriculum decision-making in response to a new textbook. Dissertation Abstracts International, 54(12), 4345. (UMI No. 9414599) Lappan, G. (1997, October). Lessons from the Sputnik era in mathematics education. Paper presented at a National Academy of Sciences symposium, Washington, D.C. http://www.nas.edu/sputnik/lappan1.htm. Lloyd, G. M. (1999). Two teachers' conceptions of a reform-oriented curriculum: Implications for mathematics teacher development. Journal of Mathematics Teacher Education, 2(3), 227-252. Lloyd, G. M.,, & Wilson, M. (1998). Supporting innovation: The impact of a teacher's conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education, 29, 248-274. McLaughlin, M. W. (1976). Implementation as mutual adaptation: Change in classroom organization. Teachers College Record, 77(3), 339-351. McCutcheon, G. (1981). elementary school teachers' planning for social studies and other subjects. Theory and research in social studies education, 9, 45-66. Morse, A. (2000). Forging a partnership: Intent, decision making, and curricula. Newton, MA: Center for the Development of Teaching, Education Development Center, Inc.

Patterns in curriculum use - 46 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA. Palinscar, A. & Brown, A. (1984). Reciprocal teaching of comprehension-fostering and comprehension monitoring activities. Cognition and Instruction, I (2), p. 117-175 Peterson, P. L. (1990). Doing more in the same amount of time: The case of Cathy Swift. Educational Evaluation and Policy Analysis, 12, 261-280. Putnam, R. T. (1992). Teaching the "hows" of mathematics for everyday life: A case study of a fifth-grade teacher. Elementary School Journal, 93, 163-177. Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers' curriculum development. Curriculum Inquiry, 29(3), 315-341. Remillard, J. T. (2000). Can curriculum materials support teachers' learning? Two fourth-grade teachers' use of a new mathematics text. The Elementary School Journal, 100(4), 331-350. Russell, S. J. (1997). The role of curriculum in teacher development. In S. Friel & G. Bright (Eds.), Reflecting on our work: NCTM enhancement in K-6 mathematics (pp.247-254). New York: University Press of America. Schneider, R. M., Krajcik, J., & Marx, R. (2000). The role of educative curriculum materials in reforming science education. In B. Fishman & S. O'Connor-Divelbiss (Eds.), Fourth International Conference of the learning sciences (pp.54-61). Mahwah, NJ: Erlbaum. Sherin, M. G. (2002). When teaching becomes learning. Cognition and Instruction 20(2), 119-150. Sherin, M. G., Drake, C., & Wrobbel, R. M. (under review). Teacher identity: An examination of teachers' views of themselves as mathematics teachers and learner. Smith, J. P. (1996). Efficacy and teaching mathematics by telling: A challenge for reform. Journal for Research in Mathematics Education, 27(4), 458-477. Smith, M. S. (2000). Balancing old and new: An experienced Middle School teacher's learning in the context of mathematics instructional reform. Elementary School Journal, 100(4), 351-375. Sosniak, L. A., & Stodolsky, S. S. (1993). Teachers and textbooks: Materials use in four fourthgrade classrooms. Elementary School Journal, 93(3), 249-75. Stodolsky, S. (1989). Is teaching really by the book? In P. W. Jackson & S. Haroutunian-Gordon (Eds.), From Socrates to software (88th Yearbook of the National Society for the Study of Education, Pt. 1) (pp. 159-184). Chicago: National Society for the Study of Education. Sykes, G. (1990). Organizing policy into practice: Reactions to the cases. Educational Evaluation and Policy Analysis, 12, 349-353. Westbury, I. (1990). Textbooks, textbook publishers, and the quality of schooling. In D. Elliot & A. Woodward (Eds.), Textbooks and schooling in the United States: Part 1: Eighty-ninth yearbook of the National Society for the Study of Education (Vol. 1). Chicago: University of Chicago Press.

Patterns in curriculum use - 47 Wilson, M. R., & Lloyd, G. M. (2000). Sharing mathematical authority with students: The challenge for high school teachers. Journal of Curriculum and Supervision, 15, 146-169. Wilson, S. (2003). California dreaming: Reforming mathematics education. New Haven, CT: Yale University Press.

Patterns in curriculum use - 48

APPENDIX A

Before Instruction During Instruction After Instruction

Read Examines main concepts in lesson; Examines general outline of activities in lesson

Evaluate Considers whether she has proper materials for lesson; Considers whether she understands concepts in lesson; Considers whether translations into Spanish make sense to her; Considers whether translations will make sense to her students and their parents Considers students' understanding of mathematics in lesson Considers whether students need more review of lesson concepts; Considers whether she understands mathematics of lesson

Adapt Translates materials into Spanish Unintentionally changes purpose of worksheets; Translates materials Creates new activities to use to continue lesson

Figure A1. Anita's curriculum strategy matrix

Before Instruction During Instruction After Instruction

Read Examines precise sequence of activities in lesson; Examines specifically what is involved in each activity; Examines how new vocabulary is used in each part of lesson

Evaluate

Considers whether students understood lesson; Considers whether students explained and discussed mathematics; Considers whether reviewing homework was a good use of students' time

Figure A2. Jan's curriculum strategy matrix

Adapt

Patterns in curriculum use - 49

Before Instruction During Instruction After Instruction

Read Examines specific concepts introduced in lesson; Examines terms introduced and specifically how they are used in lesson

Evaluate Considers whether recommended ways of grouping students will work for her students Considers whether she is maintaining control of class; Considers whether students are following directions

Adapt Replaces group work with individual work and wholeclass discussion Changes the way that students are grouped for selected activities

Figure A3. Kate's curriculum strategy matrix

Before Instruction During Instruction After Instruction

Read Examines main activities in lesson; Examines overall teaching method suggested for lesson Examines specific explanations of concepts provided in text

Evaluate Considers whether she can use recommended discovery methods in a class as large as hers; Considers whether activities are worth energy she will need to expend on class management Considers students' understanding of main ideas in lesson Considers how focus of lesson aligns with school district standards

Adapt Adds direct instruction after using recommended CMW pedagogical approach

Figure A4. Lauren's curriculum strategy matrix

Before Instruction

Read Examines main activities in lesson; Examines new vocabulary introduced in lesson

During Instruction After Instruction

Evaluate Considers own understanding of mathematics in lesson; Considers own understanding of CMW philosophy illustrated in lesson; Considers whether students will be able to engage in lesson appropriately

Adapt Omits activities; Omits mathematical concepts from lesson

Figure A5. Marta's curriculum strategy matrix

Patterns in curriculum use - 50

Before Instruction During Instruction After Instruction

Read Examines main concepts in lesson; Examines general CMW approach to teaching mathematics Examines specific examples from curriculum to use in class

Evaluate

Adapt Omits and substitutes activities and representations

Figure A6. Paul's curriculum strategy matrix

Before Instruction During Instruction

Read Examines specific explanations of concepts provided in text

Evaluate

After Instruction

Adapt Unintentionally changes mathematical focus of lesson; Unintentionally changes meaning of terminology introduced in lesson

Figure A7. Vera's curriculum strategy matrix

Before Instruction Anita Beth Fran Jan Kate Lauren Marta Paul Shelley Vera During Instruction Anita Beth Fran Jan Kate Lauren Marta Paul Shelley Vera After Instruction Anita Beth Fran Jan Kate Lauren Marta Paul Shelley Vera

Read Big Idea Big Idea Big Idea Detail Detail Big Idea Big Idea Big Idea Detail Read Detail Detail Detail Detail Read

APPENDIX B

Patterns in curriculum use - 51

Evaluate Teacher/Students/Other Teacher Students Students Teacher Teacher/Students

Adapt Replace Create Replace Replace Omit

Evaluate Students Students Teacher Teacher/Students Students Students Evaluate Teacher/Students Teacher/Students Students Other Teacher/Students

Adapt Replace Create Replace/Omit Replace Create Replace/Omit Create/Replace Replace Adapt Create Replace

MG Sherin, C Drake

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