The Mathematics Educator 2008, Vol. 18, No. 2, 11–22 Planning for Mathematics Instruction: A Model of Experienced Teachers’ Planning Processes in the Context of a Reform Mathematics Curriculum Alison Castro Superfine

Planning is an important phase of teaching, during which teachers make decisions about various aspects of instruction that ultimately shape students’ opportunities to learn. Prior research on teacher planning, however, fails to adequately describe experienced teachers’ planning decisions, and is unclear about the extent to which teachers use curriculum materials to inform their decisions. Using data from 6th grade mathematics teachers’ use of curriculum materials, this study presents a discipline-specific model of experienced mathematics teachers’ planning. The proposed model provides a lens for understanding the nature of teachers’ planning decisions, and the conditions under which such decisions change over time.

Planning is an important and often generally considered when planning their lessons underappreciated aspect of teaching practice, when (Popham & Baker, 1970; Taylor, 1970; Tyler, 1950). teachers make decisions that ultimately impact Under this model, teachers first consider the learning students’ opportunities to learn (Clark & Peterson, activities that take into account students’ interests and 1986; Floden, Porter, Schmidt, Freeman, & Schwille, abilities, then the learning goals and objectives of the 1980; Stigler & Hiebert, 1999). Planning commonly lesson, and finally the evaluation procedures to be used refers to the time teachers spend preparing and during the lesson. Some researchers later argued that designing activities for students. From tasks and linear models of teacher planning do not adequately activities to instructional practices employed during describe experienced teachers’ planning processes and lessons, teachers need to consider a variety of aspects do not account for the complexities inherent in of their instruction before students even enter the mathematics teaching. Rather, a variety of additional classroom. Teachers need to pay careful attention to factors, such as teachers’ experiences and conceptions designing their lessons; “effective teachers understand of mathematics teaching and learning, also influence that teaching requires a considerable effort at design. the ways in which teachers plan their lessons Such design is often termed planning, which many (Kilpatrick et al., 2001; Leinhardt & Greeno, 1986; teachers think of as a core routine of teaching.” Yinger, 1980). (Kilpatrick, Swafford, & Findell, 2001, p. 337). More recent research on teachers’ planning does Reviews of teacher planning and decision-making not clearly indicate the extent to which teachers draw further emphasize the centrality of planning processes from curricular resources when making planning in teachers’ practice (Clark & Peterson, 1986; Clark & decisions. Moreover, there is even less research that Yinger, 1977; Shavelson & Stern, 1981). Despite this focuses explicitly on teachers’ planning in the context general agreement about the importance of planning, of the reform mathematics curricula that provide much few researchers have explicitly examined the precise of the instructional design for teachers (Kilpatrick et ways in which teachers plan for mathematics al., 2001; Trafton et al., 2001). Such reform curricula instruction. are increasingly prevalent in classrooms in the United Prior research related to teacher planning presented States, embodying new modes of instruction (Reys, a “linear” or “rational” model of teacher planning by 2002). The challenges of planning lessons using such delineating the various lesson elements teachers curricula may be somewhat different from the challenges of planning lessons with more conventional Alison Castro Superfine is Assistant Professor of Mathematics mathematics curricula. Thus, exploring how teachers Education and Learning Sciences at the University of Illinois at plan in the particular context of reform curricula is Chicago. Her current work focuses on teacher-curriculum interactions and elementary preservice teacher education.. critical if mathematics educators want to understand this important phase of teaching. The research reported in this article is based upon the author’s In order to explore the theoretical considerations doctoral dissertation under the direction of Dr. Edward A. Silver at the University of Michigan. presented in this article, the author has selected

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examples of teachers’ planning routines taken from a process of teachers’ planning, which was not clearly larger study examining experienced 6th grade teachers’ addressed by proponents of the linear planning model. use of the Connected Mathematics Project (CMP) Some researchers have focused on the role of materials. CMP is a middle school reform curriculum curriculum materials as a resource for teachers to draw developed in response to the National Council of upon when making planning decisions. For example, Teachers of Mathematics (NCTM, 1989) Curriculum McCutcheon (1981) found that when planning for daily and Evaluation Standards. By emphasizing the lessons, teachers tend to rely heavily on suggestions in discovery of mathematical ideas through tasks, CMP the teacher’s guide. In a study of one teacher’s encourages students to make connections between planning throughout the school year, Clark and Elmore topics and important mathematical ideas in order to (1981) found that curriculum materials are primary help them apply their learning to real-world contexts. resources in the teacher’s planning. Similarly, Smith The larger study focused on how four teachers used the and Sendelbach (1979) studied this issue at the level of CMP teacher’s guide in both the planning and teachers’ unit planning. They found that teachers tend enactment of their lessons. The teacher’s guide to construct a mental image or plan of the unit and then includes summaries of the mathematical content, supplement their plan with notes based on the specific questions to ask students throughout a lesson, suggestions in the teacher’s guide. Additional research and examples of student errors. has highlighted various factors that influence how teachers use these curricular resources in the planning Teacher Planning and instructional processes (Ben-Peretz, 1990; Cohen Past research on teacher planning focused on the et al., 1990). broad features and order of teachers’ planning Research points to experience as a potential factor decisions and considerations, with minimal attention that influences teachers’ planning. Such research given to the particular ways that teachers considered suggests that experienced teachers have more extensive engaging students with the content. Adhering to a and well-organized knowledge of both pedagogy and linear model of teacher planning, Tyler (1950) and student learning, making them more flexible and Popham and Baker (1970) found that teachers specified attentive to the nature of the students’ learning ordered objectives, selected learning activities, opportunities that they create (Borko & Shavelson, organized learning activities, and specified evaluation 1990; Leinhardt & Greeno, 1986; Livingston & Borko, procedures. Similarly, Taylor (1970) found that 1989). For example, researchers found that when teachers sequentially considered four aspects of a given planning, experienced teachers make more extensive lesson when planning: materials and resources, mental plans than written plans and rely less on students’ interests, the aims and purposes of teaching, curriculum materials than their less experienced and evaluation. Implicit in these studies is the notion counterparts (Bush, 1986; Leinhardt, 1983; Livingston that teachers create their own objectives and activities & Borko, 1990). for students, which may reflect the design of the types As teachers’ experience with a particular of curriculum materials available to teachers at the curriculum program increases, they become more time in which these studies were conducted. familiar with the details, nuances, and presentation of In a later study on teacher planning, Brown (1988) the specific mathematics content in the curriculum. examined the extent to which 12 teachers adhered to a Thus, they may have developed daily routines in linear model of planning. Focusing on teachers’ planning and engaging with the curriculum in planning in different subject areas, Brown found that particular ways. For the purposes of this study, teachers tend to use curriculum materials and the experienced teachers are identified as having at least objectives expressly stated in these resources as a five years of teaching experience and at least three starting point for their planning. She noted, “teachers years experience using a certain curriculum program. operate as curriculum implementers and not curriculum Other possible factors influencing teachers’ planners as they consider objectives already written in planning decisions are the various conceptions teachers curriculum guides” (p. 79). Yackel and Cobb (1996) bring to bear on their practice. These conceptions noted that planning decisions about ways of facilitating contribute to what Remillard and Bryans (2004) refer students’ activity in a history or English classroom are to as teachers’ orientation toward the curriculum. considerably different from those in a mathematics Experienced teachers have refined their conceptions of classroom. Nevertheless, Brown’s (1988) study points mathematics teaching, learning, and curricula because to the integral role of curriculum materials in the they have spent considerable time formulating and

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applying these conceptions in the classroom. Teachers’ need to be able to resist the urge to tell students how to conceptions of mathematics content are also important work on the content so that they provide students with for understanding how teachers engage with the adequate time to think through what they are asked to curriculum (Lloyd, 1999; Remillard, 1999; Remillard do (Donovan, Bransford, & Pellegrino, 2000). & Bryans, 2004). Moreover, it is important to Anticipating student responses and having an understand the extent to which teachers’ conceptions of awareness of common errors can also help teachers mathematics teaching and learning align with the ideas effectively respond to and redirect students’ about teaching and learning underlying the curriculum discussions (Chazan & Ball, 1999; Fennema, Franke, (Manouchehri & Goodman, 1998; Remillard & Carpenter, & Carey, 1993). In addition, modifying Bryans, 2004). tasks based on students’ current knowledge and While linear models are useful for capturing abilities may help teachers to be mindful of the certain basic elements of teacher planning, these cognitive activity in which students should be engaged models fail to account for an array of factors that have (Stein et al., 2000). Employing such instructional been identified as influencing teachers’ planning practices to facilitate student learning in accordance processes, such as curriculum materials, teaching with the principles of reform mathematics, however, experience, and the various conceptions teachers have requires extensive and demanding work on the part of about teaching and learning. Although teachers have a teachers. Therefore, teachers face several challenges variety of conceptions, this article will focus on supporting students at such a high level of teachers’ conceptions of mathematics teaching, mathematical activity. learning, and curricula for the purposes of this study. These challenges to teachers’ work are considered the “problems” in mathematics teaching (Lampert, The Work of Reform-Oriented Mathematics 1992, 2001). “Problems” in mathematics teaching refer Teaching to the work teachers do to further students’ Although the various factors highlighted by understanding of mathematics. This includes researchers as influencing teachers’ planning are facilitating students’ discussion around the content, essential to consider when developing a new planning continuously pressing students to explain their ideas model, it is also important to consider the demands and and to communicate with each other, posing questions, characteristics of the particular discipline in which and selecting solution strategies to present to the class. such planning occurs. Most of the research discussed Teachers need to make important and often previously does not explicitly focus on planning or simultaneous decisions in ways that do not undermine instruction in the context of a specific discipline. students’ thinking or the mathematical opportunities Mathematics teaching, specifically in the context of afforded by the content in reform curricula. Hereafter, reform mathematics curricula, involves particular “problems” will be used to refer to the challenges and demands and challenges that may shape teachers’ decisions teachers face during mathematics teaching, planning processes. as described in Lampert (1992, 2001). The model developed in this study is grounded in a Teaching includes not only the physical act of specific conception of mathematics teaching, drawing teaching, during which teachers interact with students, from the works of Stein, Smith, Henningsen, and Silver but also includes the time teachers spend preparing for (2000), Lampert (1992, 2001), Clark and Elmore these interactions (Jackson, 1966, 1968). Planning for (1981), and Lampert and Ball (1998). These the demands and challenges of mathematics instruction researchers describe in detail the nuances and requires teachers to engage in a planning process that complexities of mathematics teaching in a way involves the development of skeletal frameworks typically embodied in reform curricula. Teachers need rather than detailed scripts for teaching lessons to consider the mathematical content and ways to (Rosebery, 2005). In particular, teachers must identify engage students in discussion about the content, while a particular mathematical topic to discuss and the simultaneously guiding students towards a particular means necessary to cover that topic, without goal. For example, during planning and instruction necessarily delineating the precise steps needed to teachers modify tasks and ask high-level questions in teach that topic. Therefore, planning for reform- order to promote students’ understanding of the oriented instruction requires teachers to select specific underlying ideas and concepts. topics or concepts and to identify particular activities, To support students’ understanding, teachers need instructional strategies, and suitable materials for a variety of pedagogical skills. For example, teachers discussing and engaging students with the topics or

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concepts. In addition, during planning, teachers must concept, in ways that both preserve the task’s anticipate potential problems that may arise during complexity and help students learn from working on instruction and then make decisions regarding how to the task (Stein et al., 2000). Though their intended manage these problems. Planning problems refer to the plans often differ from their enacted plans, teachers considerations and decisions teachers face when both need to carefully plan their lessons and anticipate how planning for and anticipating what will happen during students will interact with the content during a specific lesson. implementation in order to further students’ understanding of different mathematical ideas. In this A Model of Teacher Planning way, planning problems can be considered to be the There are several elements that a model of teacher anticipation of instructional problems. planning in the specific context of reform-oriented Planning problems are inherently different for each mathematics instruction must capture. These elements teacher depending on the teacher’s experiences, ideas, include the approaches to mathematics teaching and and conceptions, as well as the curriculum being used. learning embodied in reform curricula and teachers’ For instance, asking higher-order questions that press various experiences and conceptions they bring to their students to justify and explain their thinking is also planning decisions. Such a model must particularly only a planning problem for teachers who view the use capture how these elements interact with each other of such questions as contributing to student and ultimately influence teachers’ planning decisions. understanding. Planning problems also may be quite The concept of planning problems is well suited for different for a teacher who adheres to a more developing a conceptualization of planning because it conventional conception of mathematics teaching. incorporates the influence of these different elements. Such a teacher may need to determine how to The model developed in this study draws heavily on incorporate opportunities for students to practice the the notion of planning problems and highlights the application of certain skills and procedures within a various elements that drive the emergence of such curriculum the teacher perceives as deficient. planning problems in teachers’ practice. Furthermore, a teacher with experience implementing As discussed previously, the work of reform- multiple curriculum programs must consider how to oriented mathematics teaching includes facilitating and apply what they know of other mathematics curricula supporting students’ understanding in ways that will to their planning with a specific Standards-based neither constrain students’ opportunities to learn nor curriculum. Therefore, planning problems appear to be undermine students’ thinking. Consequently, teachers a useful lens for understanding the relationship need to plan for engaging in this sort of work during between teachers’ experiences, conceptions of instruction. For example, teachers must plan questions mathematics teaching and learning, and the curriculum they will ask students that will guide students’ thinking used in the planning process. about the content without giving them too much As the Planning for Mathematics Instruction (PMI) information, while also encouraging students to explain Model in Figure 1 illustrates, teachers’ various their ideas (Hiebert & Wearne, 1993; Maher & conceptions influence their engagement with Martino, 1992; Moyer & Milewicz, 2002). Teachers curriculum materials during planning. Additionally, the need to anticipate different solutions students may conceptions influence the type of planning problems offer, as well as alternative ways of thinking about a teachers encounter in the course of their work. These task, in order to facilitate students’ learning and planning decisions then influence teachers’ lesson discussion of these strategies in ways that foster a enactment and the types of learning opportunities they shared understanding of the ideas (Kilpatrick, 2003; create with students during instruction. This lesson NCTM, 1991). Teachers also need to anticipate enactment then informs experiences and shapes the potential errors in order to respond appropriately and information teachers have to draw upon when they help students learn from incorrect solutions. Finally, plan for and enact the lesson in subsequent classes or teachers should be prepared to change or modify a school years. task, in the case that students are struggling with a

14 Teacher Planning

Figure 1. Planning for Mathematics Instruction Model.

As teachers’ conceptions help to frame the years; Susan has been teaching 6 years, using CMP for planning problems they encounter, their various 3 years. These teachers were observed planning and conceptions also serve as a resource for managing enacting the same unit, Bits and Pieces III, which planning problems that arise in the course of their focuses on operations with rational numbers (Lappan et planning. When confronted with planning problems, al., 2006). The data collected includes interviews with teachers draw upon their previous experiences with the teachers prior to and immediately following classroom task and their ideas about what it means to learn and observations to understand teachers’ lesson plans and teach mathematics in order to make decisions about their reflections on their lesson enactments. Although ways of managing these different planning problems. post-hoc examinations of teachers’ planning were In some cases, teachers may also draw from the conducted, teachers were interviewed the same day of information and support provided within the actual the observation in order to increase the accuracy of mathematics curriculum materials. Notably, the CMP teachers’ responses. Field notes from classroom teacher guide provides the means for teachers to observations and artifacts from teachers’ lesson manage certain planning problems, such as anticipating planning are additional data sources. solution methods students may generate, questions to Prior to using CMP, Alicia used a more ask students, and errors and misconceptions students conventional mathematics curriculum for 13 years and may have in relation to a task. Though, the extent to claims to strongly believe that CMP does not provide which teachers use curriculum materials to inform their students with sufficient opportunities to practice skills planning decisions is largely dependent on the nature and procedures. She views her role as a teacher as that of their conception of the curriculum. of an intervener, providing direct guidance and explicit instructions for students, which is evident in her Teachers’ Planning Practices planning decisions. Alicia first reads through the entire The following teacher examples illustrate the lesson in the student book and solves the task: “And I various ways in which experienced teachers, with do the whole [lesson] myself, you know without distinct conceptions towards the CMP curriculum, can looking at the teacher’s guide or anything…And then I engage with reform curriculum materials in the course have an idea about what might be tricky and what of their planning and demonstrate how the PMI Model might not be.” Alicia says she then looks through the applies to actual teaching practices. Alicia, Richard, accompanying teacher’s guide and decides the most and Susan were selected for the present study because appropriate course of action, taking into consideration their planning decisions and considerations captured both who her students are as learners and the the range of variation in planning routines and constraints of class time. “And I kind of pick and problems encountered.1 All three teachers were choose what I think will work best with my students… teaching sixth grade at the same middle school at the And most times I won’t use all of it.” In general, Alicia time the study took place. As this study uses teaching claims to regularly modify the suggested content of the and curricular experience to define experience, Alicia lesson and the suggestions for how students should has been teaching 16 years, using CMP for 3 years; engage with the content during the lesson. Her view of Richard has been teaching 17 years, using CMP for 10 CMP, and ultimately her conception of what students

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should learn, is readily apparent in the nature of her materials. Notably, both of these teachers do not follow lesson additions and deletions: “I’m usually thinking the lesson suggestions in a prescriptive fashion, about what I need to add…Like word problems I including Richard who seems to only rely on the usually skip…and substitute other practice problems materials for content features of the lessons and not for that I feel like need to be emphasized more.” suggestions as to how to engage students with the Richard, on the other hand, claims to be a strong content. By engaging with the materials in such an proponent of the instructional approach embodied by adaptive, or even modified fashion, Richard and Alicia the program. Having taught CMP for over 10 years, he leave the lesson open to interpretation, making room believes that CMP is a desirable alternative to a for their conceptions of mathematics teaching and conventional mathematics curriculum because it allows learning to inform their lesson planning. him as a teacher to facilitate student discussion and to In contrast to the other two teachers, Susan, having play less of a central role in the classroom discussion: taught for 6 years and used CMP for 3 years, appears “I would characterize it less as teacher-driven…and to adhere closely to the lesson and corresponding more kid-driven…I think the focus becomes--to me suggestions in the curriculum. She believes the I’m giving up being the center of attention.” When instructional approach espoused in CMP is an ideal planning, to make sure that he understands the content way to support students’ mathematical development for himself, Richard says he first reads through the and their ability to communicate their understanding. lesson and solves the task that he will use with She views her role as that of a referee, mediating students. He then tries to ascertain, from reading the students’ discussions of their proposed solution student book, exactly what the lesson is about, the “big strategies during instruction: idea” students are to come away with, and also places And so sort of mediating that discussion is sort of where students may struggle or misconceptions the biggest part because…learning of the strategies students may have while working on the lesson: is supposed to take part amongst themselves. So “…[J]ust so I get a sense…of what problems they’re it’s like I have very little, this is how it works, you going to struggle with. Just looking at the answer know, it’s more, ok what are your ideas? Let’s put doesn’t help me out. If I actually work through the them together. problem, that gives me a sense of where they’re going When planning for a lesson, Susan follows the to struggle.” Richard says he uses the teacher’s guide suggestions in the teacher’s guide almost as a script for during planning only when an idea or concept or even the lesson. She will first read through and solve the the wording of a particular question in the student book task in order to think about how students will approach is unclear. The student book provides him with the and solve the task, as well as to consider potential necessary information he needs for enacting the task solution strategies that may arise. After working during instruction. through the content, Susan says she reads through the I think if I would read through the [teacher] manual longer lesson summary in the teacher’s guide to and not read through the kid edition, I’d just feel understand the overall direction and purpose of the like I’d be at a disadvantage to know what to lesson. In addition to reading the detailed teaching expect from the kids…It tells me as a teacher what notes, she reads both the suggested questions and to know conceptually…but it doesn’t help me answers to the task, and then includes these elements in understand quite how I think the kids are going to a slideshow presentation she uses during instruction. react to what I’m asking them to do. Though, she only includes suggested questions in her In general, Richard claims to plan for using most, if not lesson if she decides they are worthwhile and all, of the lesson elements as described in the appropriate for her students: “So…whenever there are curriculum. suggested questions, I see ok, is this a meaningful Although both Alicia and Richard use the question for my students?” Susan says she reads curriculum materials for the content features of the through the lesson suggestions to ensure she does not lessons (albeit to a limited degree in Alicia’s case), extend the discussion of a concept further than what is both teachers appear to have different views of the expected or stated in the materials, and also to gain an curriculum, and thus use the materials in very different overall sense of the lesson and the mathematical ideas ways when planning. While Alicia “picks and chooses” embedded in the lesson: “So, it’s good to read that from the lesson suggestions what she considers sometimes just to see ok, this is only where they need relevant and important for her students, Richard tends to get to at the end.” to plan for enacting the lesson as described in the

16 Teacher Planning

Table 1 Summary of planning problems teachers encountered during unit Planning Problem Alicia Richard Susan Anticipating students’ work on Recalled previous experience Read student book, solved task Planned for more teacher- task with lesson himself direction of task Treatment of content in Read teacher guide to clarify Read teacher guide to clarify Read teacher guide to clarify curriculum content, but focused only on content, but focused only on content, and planned to follow “important” aspects “important” aspects lesson suggestions

Susan also says the teacher’s guide provides her modifies the lesson suggestions as needed when with an image of how students will engage with the planning. In fact, all three teachers held varied task: “…it gets me ready for what they might say. conceptions of the curriculum – curriculum as a guide What’s the book going to be after? You know, what’s to varying extents (Alicia and Richard) and curriculum sort of the big idea that they want to come away with.” as a script (Susan). As these examples illustrate, She states that her purpose for using the student book teachers’ various conceptions influence the types of is to understand the task for herself. She uses the planning problems these teachers encounter and the teacher’s guide, on the other hand, to understand how ways in which teachers manage these problems as they students may approach and solve the task, including arise during planning for the unit. potential misconceptions students may have in relation Planning Problems to the task. In contrast to Alicia, Susan appears to agree with As discussed previously, planning problems the principles underlying CMP, and accordingly plans constitute a fundamental structural component of the for enacting lessons in the unit largely as described in PMI Model because they highlight the relationships the materials. Moreover, unlike Richard, Susan uses between teachers’ experience, conceptions of the curriculum not only for the content features of the mathematics teaching and learning, and the actual lesson, but also she uses the suggestions for how to curriculum program used by teachers. Applying this engage students with the content during the lesson. model to teachers’ practices requires close analysis of Thus, Susan, the teacher with less teaching experience the planning problems experienced by these teachers as compared to Alicia and Richard, draws heavily from and the factors underlying the emergence of these the suggestions in the curriculum when planning, and planning problems. The teachers in this study primarily plans to enact the lesson largely as described in the encountered two different planning problems – (1) materials. By adhering to the lesson suggestions in an anticipating potential errors and misconceptions almost prescriptive fashion, Susan leaves little room students may have in relation to a task and (2) for her own interpretation of the lesson. treatment of content in the curriculum. Although As these examples illustrate, the nature of teachers’ teachers encountered several planning problems engagement with curriculum materials during planning throughout the unit, these two problems were selected is determined by a variety of factors. Although Richard for analysis because they illustrate how the PMI Model agrees with CMP’s overall approach to teaching and depicts teachers’ planning practices. Though the two learning, he largely relies on the curriculum materials types of planning problems teachers primarily solely for its content. It appears that he does not require encountered during their planning were quite similar, much pedagogical support when planning; instead, he teachers varied considerably in the ways in which they typically limits himself to reading and working through managed these two planning problems. Table 1 the student book. Susan, on the other hand, seems to summarizes the planning problems these three teachers rely on the materials for both content and pedagogical encountered during their planning for the unit, and purposes during her planning for the unit, closely briefly describes the ways in which the teachers following the lesson suggestions. In contrast to Richard managed these different problems. Table 1 does not and Susan, Alicia does not seem to believe that CMP reflect the frequency in which participating teachers provides students with sufficient opportunities to encountered planning problems during the unit. practice basic skills and procedures, and therefore

Alison Castro Superfine 17

Anticipating Students’ Work on Tasks would struggle. This was also the case in Susan’s A specific planning problem all three teachers planning for a lesson involving computing discounts. encountered during their planning for the unit was In both situations, Susan’s previous experience, her anticipating how students would work on the content view of how struggling students should learn, and her of the lesson. Alicia, for example, relied on her proclivity to follow the curriculum suggestions closely, experience from previous classes to anticipate how influenced how she managed the problem of students would engage with the content of several anticipating how students would work on the lesson. lessons in the unit. Based on experiences with classes She often reduced the complexity of the tasks by in previous years, Alicia anticipated that students telling students how to solve them, taking away would struggle with lining up the decimal points when students’ opportunities to wrestle with the central adding and subtracting decimals in a certain lesson. ideas, but still enacted the lesson largely as written. She therefore planned to enact the entire lesson as a Treatment of Content in Curriculum whole-class activity to help students through the The treatment of the content within two lessons lesson, as opposed to providing opportunities for dealing with long division also emerged as a planning students to work collaboratively in groups during the problem for these teachers. All three teachers lesson, as was the suggested organization. Similarly, considered long division as a particularly important she anticipated students would struggle with another concept for students to know and to be able to do. task involving computing discounts by drawing upon However, the long division algorithm was not her previous experience with that lesson, and again explicitly presented in the unit; it was presented as a planned the lesson as a whole class activity. As she set of two interrelated lessons in order for students to described in her planning, Alicia believed that by understand the underlying rationale of the algorithm implementing lessons as either whole-class or and the role of place value when dividing decimals. All individual activities, she was better able to address three teachers had to consider how to enact these two student difficulties and “guide them in the right lessons in light of their conceptions towards the direction.” Richard regularly encountered this same content and the curriculum. Alicia and Richard planning problem but managed it quite differently than modified the lesson to focus on the procedural aspects Alicia. Rather than relying on his previous experiences, of decimal division in these two lessons. This Richard anticipated how students would work on the modification reflected their conceptions about what different tasks by working through the lessons himself they considered to be the most important aspects of the in the unit – he read the student book and solved the content. Moreover, this modification comported with task while thinking about how students would their conceptions of the curriculum as a guide rather approach the task and what potential aspects might than a script for their lesson planning. This particular confuse students. In doing so, Richard attempted to conception of the curriculum left room for the forecast the various ways in which students could teachers’ conceptions toward the content to dictate how engage with the content, which reflected his more non- teachers planned to enact the lessons. conventional conception of mathematics learning. While Susan also encountered this planning Susan also anticipated how students would work problem, she planned to enact the lesson largely as on the task, primarily drawing upon what she knew of written in the curriculum despite her reluctance to do her students’ previous work throughout the unit, but so. Her conceptions toward both the curriculum and the also the information included in the teacher’s guide. content influenced how she framed and managed this She became aware of this planning problem by not problem. Susan felt inclined to change the treatment of only reading the teacher’s guide, but also from her the content because her conception of the content previous experience with a particular lesson involving clashed with the treatment of the content in the decimal division in which students seemed to struggle curriculum. However, her desire to plan her lessons with long division. Although the materials alerted her largely in accordance with the lesson suggestions to this potential source of confusion for students, she provided a push in the opposite direction to teach the did not seem to use the suggestions in the teacher’s content as written: “I don’t know about this lesson guide to support students’ understanding of long because students have always struggled with division. Instead, she used her view of how students division….Though [the lesson] helps students should learn in order to address students’ difficulty understand, so I just have to be patient.” Susan had to with the content and planned to enact the lesson in consider what content to enact with students in light of particular ways to lessen the likelihood that students

18 Teacher Planning

these conflicting conceptions. Her conception to plan colleagues, experienced teachers have to consider how for lessons in accordance with the lesson suggestions to make use of their prior knowledge and experience. ultimately outweighed her conception of the content. With regard to teachers’ various conceptions, the The shape of Susan’s planning problem contrasts with “experience problem” consists of how to utilize that of Alicia’s and Richard’s in that they did not experienced teachers’ assumptions about and prior negotiate conflicting conceptions. In summary, the knowledge of mathematics curricula, and their ideas three teachers in this study encountered different about what it means to learn and teach mathematics. problems in the course of their planning for the unit. The planning routines of Alicia, Richard, and Despite the fact that the CMP materials provided the Susan reflect how the experience problem plays out in means to manage some potential planning problems, actual teaching practice. Experienced teachers do not teachers seemed to rely largely on their previous face significantly fewer or different planning problems experiences and particular conceptions to manage their as compared to less experienced teachers. On the planning problems. Therefore, in the case of all three contrary, all three teachers anticipated that a group of teachers, the PMI Model highlights how teachers with students would struggle with a particular aspect of a diverse conceptions and experiences frame and manage lesson, or even struggled themselves with certain particular planning problems. aspects of the mathematics content. Yet, these teachers encountered these planning problems differently. The Discussion and Conclusion differences among these teachers seem to be in their The previous discussion highlights the conceptions of the curriculum and content, the interrelationship among curriculum materials, teachers’ prevalence of their conceptions in their planning various conceptions, and the types of and ways in decisions, and ultimately their instructional decisions. which teachers frame and manage planning problems Regardless of their conceptions, teachers’ that arise in the course of their work. During planning, conceptions of curriculum and mathematics teaching teachers often use curriculum materials as a starting and learning can become calcified over time point for their lesson planning. The nature and extent (Leinhardt, 1983; Leinhardt & Greeno, 1986). As a of teachers’ engagement with the curriculum materials, result, teachers may become inattentive to how their however, is determined primarily by their various planning decisions influence students’ opportunities to conceptions. Teachers’ various conceptions then learn and they may become resistant to external influence the type of planning problems they influences such as new curriculum programs or encounter, and also how teachers manage these professional development experiences. Consider planning problems. Alicia, who seemed to adhere to a more conventional Teachers’ lesson enactment also contributes to the conception of mathematics teaching and learning pool of knowledge and information they have to draw during her planning. Because Alicia has quite from in subsequent years, thereby influencing their extensive teaching and curricular experience in using conceptions. As Figure 1 illustrates, the PMI Model more conventional mathematics curricula, she planned represents an iterative process that is continuously to focus students’ work on practicing computations and shaped by teachers’ experiences over the course of procedures and planned to modify lessons as whole- their careers. With every lesson, teachers potentially class discussions rather than collaborative work encounter unanticipated questions or new strategies groups. It appears that Alicia’s conceptions of teaching that contribute to the knowledge they can draw from and learning have become somewhat cemented when planning the same or related lessons in throughout her teaching career and seemed to have subsequent years. The ways in which teachers’ enact hindered her from planning for enacting CMP lessons lessons with students over time can also inform how in this unit in accordance with the curriculum’s teachers conceive of what it means to teach and learn underlying principles. The PMI Model also captures school mathematics. The proposed model provides a this common aspect of teachers’ practice – as this way to understand how teachers’ planning practices model represents an iterative process, allowing for change, or fail to change, over the course of their teachers’ conceptions to become reinforced as teachers careers. amass an increasing amount of knowledge and The PMI model suggests a possible cause for the experience. “experience problem,” perhaps one of the most The PMI Model has the potential for even broader significant problems teachers face as they advance utility because the planning problems and teacher through their careers. Unlike their less experienced conceptions discussed here constitute only a handful of

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the planning problems and conceptions that might teaching mathematics in reform oriented ways entails. influence teachers’ lesson planning. For example, At the very least, the PMI Model underscores teachers’ teachers’ perceptions of limited time during a lesson conceptions as a target for reform efforts because they may prove problematic for some teachers when structure and provide a major resource for managing deciding how much of the lesson to cover in the time planning problems that arise in the course of teachers’ allotted. Another planning problem may arise when work. teachers have to anticipate how to orchestrate the use In summary, the PMI Model highlights how of multiple solution strategies during a given lesson, teachers’ various conceptions frame and influence how thinking carefully about the order in which to present teachers’ manage planning problems that arise when certain strategies and the mathematical affordances of preparing for mathematics instruction. The model is discussing different strategies. In addition to structured around Lampert’s (2001) notion of teaching accounting for a broad range of planning problems, the problems, which can illuminate processes teachers proposed model also can account for a broad range of engage in during their daily planning, thus providing a teachers’ conceptions. For example, teachers’ useful lens to understand the nature of teachers’ conceptions of their role as teachers may influence planning routines and reasons underlying their how they engage with the curriculum materials during decisions during this phase of teaching. The concept of planning. This engagement, in turn, will give rise to teaching problems is useful for understanding teachers’ new planning problems and ways of managing these practice because it captures the interactions among problems. Regardless of the precise planning problems teachers’ various conceptions, their engagement with and conceptions that may influence teachers’ work, actual curriculum materials, and their previous these key elements of the PMI Model help to explain experiences. Although the discussion of the PMI teachers’ considerations and decisions made during the Model is specific to teacher planning in a reform planning process. mathematics context, such a model of teacher planning Given the PMI Model and the notion of planning is applicable to the planning that occurs in the context problems that provides its underlying structure, the of conventional curricula as well, though the nature of question for future research becomes how reformers teachers’ planning problems may be different. As the can work to improve teachers’ practice, and ultimately examples presented in this article illustrate, despite the student learning. Viewed by many as a driving force of principles of teaching and learning underlying a reform, mathematics curriculum materials have the curriculum, teachers’ various conceptions heavily potential to boost educational achievement while influence teachers’ engagement with the materials embodying new modes of instruction. However, during planning, thereby influencing the ways in which teachers can hold diverse conceptions that stand in teachers manage problems, and the types of planning contrast to the conceptions of teaching and learning problems teachers encounter. Applying the PMI Model underlying the curriculum, which can hinder teachers to understand planning problems, how these problems from planning in accordance with the curriculum. change over time, and under what conditions they Teachers who have more experience with conventional change highlights important elements in mathematics curricula and exhibit a more conventional conception teachers’ planning processes. of teaching are desensitized to the modes of instruction References entailed in implementing reform curricula. Researchers have found that for teachers with extensive experience Ben-Peretz, M. (1990). The teacher-curriculum encounter: Freeing teachers from the tyranny of texts. Albany, NY: State teaching with more conventional methods and University of New York Press. curricula questioned the value and relevance of reform Borko, H., & Shavelson, R. (1990). Teachers' decision making. In curricula (Manouchehri & Goodman, 1998; Preston & B. Jones & L. 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