EXAMINING INSTRUCTIONAL PRACTICES IN CORE ... - Springer Link

Journal of Mathematics Teacher Education (2006) 9:517–550 DOI 10.1007/s10857-006-9019-3

Springer 2006

FRAN ARBAUGH, JOHN LANNIN, DUSTIN L. JONES and MEREDITH PARK-ROGERS

EXAMINING INSTRUCTIONAL PRACTICES IN CORE-PLUS LESSONS: IMPLICATIONS FOR PROFESSIONAL DEVELOPMENT

ABSTRACT. In the research reported in this article, we sought to understand the instructional practices of 26 secondary teachers from one district who use a problemsbased mathematics textbook series (Core-Plus). Further, we wanted to examine beliefs that may be associated with their instructional practices. After analyzing data from classroom observations, our findings indicated that the teachers’ instructional practices fell along a wide continuum of lesson implementation. Analysis of interview data suggested that teachers’ beliefs with regard to students’ ability to do mathematics were associated with their level of lesson implementation. Teachers also differed, by level of instructional practices, in their beliefs about appropriateness of the textbook series for all students. Results strongly support the need for professional development for teachers implementing a problems-based, reform mathematics curriculum. Further, findings indicate that the professional development be designed to meet the diverse nature of teacher needs. KEY WORDS: Core-Plus, curriculum implementation, instructional practices, professional development, secondary mathematics teachers, teaching, teacher education

Educational change depends on what teachers do and think – it’s as simple and complex as that. Fullan (2001, p. 115)

In the United States, creating mathematics classrooms where students actively engage in learning important mathematics is a challenging endeavor, even for the most ‘‘reform-minded’’ teachers. Often teachers teach as they have been taught (Ball, 1987), and the transition from instructional ‘‘show and tell’’ practices to encouraging students to engage in learning through mathematical inquiry is a difficult shift in philosophy and practice for many teachers (Fennema & Nelson, 1997). As a mathematics teacher education community, we have amassed a number of empirical studies from around the world (e.g., Boaler, 2002; Clarke, 1997; Lloyd, 2002; Watson & DeGeest, 2005) that help us to

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understand better the mathematics classrooms where teachers are committed to creating the types of environments described in reform documents (Australian Education Council, 1994; Department for Education and Skills, 2001; National Council of Teachers of Mathematics [NCTM], 1991, 2000). However, in the U.S., more work needs to be done to understand the specifics of mathematics teachers’ instruction, particularly in response to the calls for reforming mathematics teaching and learning (National Research Council, 2001, 2005). In the 1990s, the National Science Foundation (NSF) funded 12 projects to develop mathematics textbook series that embodied the vision of the NCTM Standards documents (NCTM, 1989, 1991, 1995). These 12 projects created comprehensive textbook series at either the elementary, middle, or high school levels. In addition, all 12 projects partnered with commercial textbook publishers to make their textbook series available for use in U.S. schools. Each NSF-supported textbook series was designed to constitute a coherent set of materials that develop mathematical ideas through a problems-based approach. Consequently, the instructional sequence (commonly characterized by words such as ‘‘Launch, Explore, and Summarize’’) depends heavily on active student involvement in exploration and sense-making and less on the ‘‘Demonstrate and Practice’’ instructional model typically found in U.S. mathematics (and described in Stigler & Hiebert, 1999). The increased use of these textbook series in the U.S. has created a challenge for mathematics teacher educators: how to support teachers in the use of mathematics textbooks that are designed to engage students in actively constructing mathematical knowledge. Addressing this challenge has, thus far, proven to be a complex task (Stein, Smith, & Silver, 1999). One of the NSF-funded high school textbook series is Contemporary Mathematics in Context [commonly referred to as ‘‘Core-Plus’’] (Coxford et al., 1997, 1998a, 1999, 2001). Core-Plus is a ‘‘complete three-year high school mathematics [textbook series] for all students, plus a fourth-year course continuing the preparation of students for college mathematics’’ that ‘‘builds upon the theme of mathematics as sense-making’’ (http://www.ithaca.edu/compass/). By engaging students in the study of algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics in each of the four courses, the Core-Plus authors sought to create a high school mathematics textbook series that made more mathematics accessible to more students, while at the same time challenging the most able students.

EXAMINING INSTRUCTIONAL PRACTICES IN CORE-PLUS LESSONS

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In the research reported in this article, we sought to understand the instructional practices of 26 teachers from one district who used CorePlus in mathematics classes in grades 8–12. Further, we wanted to examine beliefs that may have influenced the teachers’ instructional practices. In this study we did not seek to tie instructional practices to student learning outcomes, as did Schoen, Cebulla, Finn, and Fi (2003). These researchers found that a number of ‘‘teacher practice and concern variables [were] significantly and positively associated with growth in student achievement’’ (p. 255) for students using Core-Plus. A subset of these variables included: • use of more group and pair work and less teacher presentation and whole-group discussion when completing the investigations; • minimal class time spent on nonacademic activities; • use of the instructional and assessment materials as provided by the curriculum with minimal replacing or revising to make them more skill-oriented and less open-ended; and • a high observer rating of teaching based on the criteria for effective reform teaching. (p. 255) Schoen and his colleagues (2003) argued that teachers implement Core-Plus in different ways; further, they argued that those differences affect student learning. However, an awareness of general differences in teacher implementation of a mathematics textbook provides little guidance for professional development. What remains to be documented are specific examples of how teaching from a problems-based textbook like Core-Plus differs for a number of teachers. Thus, in this study we focused on the instructional practices of teachers who used the Core-Plus textbook series; we seek to provide the mathematics education community with a ‘‘thick description’’ (Geertz, 1973) of such divergent teaching practices. Overarching this investigation is our contention that we need to understand better the instructional practices of teachers who use Core-Plus (and textbook series like Core-Plus), and what beliefs may influence those practices, in order to provide authentic, worthwhile professional development experiences. We view our work through a lens similar to the one that we encourage teachers to use in the mathematics classroom. In the classroom, we ask the question: How can instruction be organized to meet the needs of all students? It is essential that we ask a similar question about the teachers engaged in professional development activities. One way to understand teachers’

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needs is to understand better their instructional practices. Specifically, our study addresses these questions: 1. How do 26 teachers across one district, using the Core-Plus textbook series, differ in their instructional practices? 2. What teacher beliefs may have influenced these differences? We believe that this research is important for a number of reasons. First, it provides the mathematics education community with an indepth look into high school classrooms where textbook series like Core-Plus are being used, as called for by Clarke (1997) who argued, ‘‘there is a need for clear descriptions of the role of teacher in such Ôreformed’ classrooms’’ (p. 278). Second, this research systematically documents and supports anecdotal accounts of instructional practices in Core-Plus classrooms. Third, this research has implications for the professional development of teachers who use the NSF-funded textbook series that are available in the U.S., as well as for mathematics teachers in transition around the world.

THEORETICAL CONSIDERATIONS AND LITERATURE REVIEW Our initial theoretical lens for examining teachers’ instructional practices with Core-Plus included three interacting components: (a) the impact of Core-Plus, (b) teachers’ beliefs about instruction, and (c) their instructional practices. As shown in Figure 1, we view Core-Plus and teachers’ instructional practices as influencing each other, through the filter of teachers’ beliefs about learning and instruction. Teacher beliefs, instructional practices, and use of the textbook occur within classroom and school contexts that often serve to further constrain teacher decision-making. We believe that using Core-Plus impacts

Figure 1. The Interplay amongst the textbook, beliefs, and instructional practices.


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teachers’ instructional practices and that the teachers’ instruction practices, in turn, influence their use of the textbook (Remillard, 2005). We consider their beliefs as a mediating factor between the textbook series and their instructional practices. As such, we recognize that teachers interpret Core-Plus through their beliefs about teaching and learning, and that their interpretations guide their instructional decision-making with the textbook. The influences of context on this relationship, while important to consider, is outside of the scope of this particular study.

MATHEMATICS TEXTBOOKS AND TEACHER INSTRUCTIONAL PRACTICES Unlike the broader literature on mathematics teaching and teachers (as represented in Koehler & Grouws, 1992), research describing teaching while implementing NSF-funded textbook series like Core-Plus is less comprehensive and cohesive. Nonetheless, this growing literature provides some insight into the initial challenges that teachers face. For example, Lambdin and Preston (1995) created three caricatures to describe teachers who were in the initial stages of piloting the NSFfunded middle school mathematics textbook series Connected Mathematics Project [CMP]: The Frustrated Methodologist, The Teacher on the Grow, and The Standards Bearer. Based on data from 16 teacher interviews, these researchers took a broad look across the teachers in their study to describe three teacher types. These authors provide caricatures of teachers that can help professional developers consider teacher needs; however, more specific descriptions of teacher instructional practices are warranted. Other researchers have focused on specific aspects of teachers who use Core-Plus and their instructional practices. Wilson and Lloyd (2000) studied three teachers’ instruction through the lens of shared mathematical authority and found that the teachers’ efforts to change classroom culture is a complex undertaking, and that the three teachers experienced different degrees of ‘‘success’’ at sharing mathematical authority. In a related study, Lloyd (1999) studied two teachers’ conceptions of cooperation and exploration components of Core-Plus. She found that ‘‘although the teachers appreciated the themes of cooperation and exploration in theory, their conceptions of these themes with respect to their implementations of the curriculum [textbook] differed’’ (p. 227). Lloyd (2002) also reported an in-depth case study of one

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teacher’s use of Core-Plus over a two-year period. She documented the impact of the teacher’s use of Core-Plus on his conceptions about mathematics and teaching mathematics. Lloyd found that the teacher ‘‘developed new pedagogical content conceptions based on his classroom experiences with innovation’’ (p. 51). In addition, Lloyd and Wilson (1998) specifically studied this teacher’s content understanding of function and how that understanding impacted the teacher’s instructional practice during a six-week Core-Plus unit focused on functions. They found that the teacher’s deep understanding of function, ‘‘dominated by graphical representations and covariation notions...played crucial roles in the teacher’s practice when he emphasized the use of multiple representations to understand dependence patterns in data’’ (p. 248). While beginning to give us a glimpse of teachers’ instructional practices in classrooms where Core-Plus is used, the studies described above leave much to be investigated about these instructional practices. The research reported in this paper contributes to the community’s understanding and extends the existing literature in a number of ways. First, we investigated the instructional practices of a relatively large number of teachers (26) who worked in a district that has been using the Core-Plus textbook for an extended period of time, unlike those studies cited above that reported on instructional practices of a small number of teachers who were in initial stages of Core-Plus implementation. In addition, we propose connections between the teachers’ instructional practices and underlying beliefs that may inform their instructional decision-making.

TEACHER BELIEFS RELATED TO INSTRUCTIONAL PRACTICE Thompson (1992) contrasts beliefs and knowledge, stating: A characteristic of knowledge is general agreement about procedures for evaluating and judging its validity; knowledge must meet criteria involving canons of evidence. Beliefs, on the other hand, are often held or justified for reasons that do not meet those criteria, and thus, are characterized by a lack of agreement over how they are to be evaluated or judged. (p. 130)

Thus, teacher beliefs are statements that are accepted to be true without the rigorous analysis that characterize evidence for knowledge. As such, beliefs differ markedly from individual to individual.


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As discussed in Thompson (1992), evidence of teacher beliefs is difficult to capture and the link between teacher practice and beliefs is not a simple, cause–effect relationship. Some researchers have found teacher beliefs to be closely aligned with instructional practices (e.g., G. B. Shirk, unpublished dissertation), though other studies report teacher beliefs that are inconsistent with instructional practice (e.g., Cooney, 1985; K. Shaw, unpublished dissertation 1989). More specifically, studies have shown that teacher mathematical beliefs and mathematical knowledge can impact how teachers design lessons (Thompson, 1984), interact with students (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989), and react to new school policies and practices (Putnam, Heaton, Prawat, & Remillard, 1992). Teacher beliefs are more likely to influence how teachers act in the classroom than their knowledge of content (Brown & Cooney, 1982; Ernest, 1989). Pajares (1992) noted that teacher beliefs provide a ‘‘filtering effect’’ through which classroom interactions are processed. However, Thompson (1992) noted that other factors, such as the social context of the school environment, also influence teacher practice. We sought to examine teacher beliefs in light of the other potential factors that may impact teacher instructional decision-making. Thus, we viewed teacher beliefs through the lens of the idiosyncratic judgments that they made in their classrooms in combination with the statements that they made about their instruction. We sought to understand how teachers viewed the textbook series as they enacted it within their classrooms and how their beliefs about the use of CorePlus with their students could impact their instructional practices.

THE STUDY In this section, we present the frameworks that guided our research, the study setting and participants, and the data sources and analyses. Guiding Frameworks Studying teachers and their practices is a complex endeavor. Several frameworks have been developed to help researchers analyze instructional practices (e.g., Weiss, Pasley, Smith, Banilower, & Heck, 2003). Three existing frameworks guided our data collection, data analysis, and the reporting of our results (Hiebert et al., 1997; Stein, Smith, Henningsen, & Silver, 2000; Weiss et al., 2003). We begin by

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explaining how the three frameworks were identified and the rationale for adapting the three frameworks into one. We then present our framework, which we call the Classroom Implementation Framework. Our framework begins with the five interrelated classroom dimensions that promote mathematical understanding (Hiebert et al., 1997). The five dimensions include examination of: (a) the nature of classroom tasks, (b) the role of the teacher, (c) the social culture of the classroom, (d) mathematical tools as learning supports, and (e) equity and accessibility. Each dimension is further delineated by core features. For example, within the nature of classroom tasks dimension, attention is paid to whether the tasks in which students engage make the mathematics problematic, connect with where students are, and leave behind something of mathematical value. Core features of this and other dimensions are contained in Table I. We extended the Heibert et al. framework to include portions of the Mathematical Tasks Framework (Stein et al., 2000) in the following manner. Within the ‘‘nature of classroom tasks’’ dimension of the Hiebert et al. framework, we included classifying the tasks used during lessons by their Level of Cognitive Demand (LCD), a set of criteria, designed by the QUASAR project, for use in analyzing the ‘‘kind of thinking (the task, as written) demands of students’’ (Stein et al., 2000, p. 15). Using this framework, tasks, as they are written in textbooks, can be analyzed and characterized as requiring one of four levels of cognitive demand: (1) memorization; (2) procedures without connections to concepts or meaning; (3) procedures with connections to concepts or meaning; and (4) doing mathematics (which, to the QUASAR researchers, means conjecturing, collecting data, and making arguments supported by mathematical evidence). See Stein et al. (2000) for an extended discussion of these criteria. We further adapted the Heibert et al. (1997) framework, in the role of the teacher dimension, by including the factors associated with the maintenance and decline of high-level demands (Stein & Smith, 1998) from the Mathematical Tasks Framework. These factors provide descriptions of ‘‘the classroom activity that surrounds the way in which...problems are set up and actually carried out by teachers and students’’ (Stein et al., 2000, pp. 24–25). For example, these researchers found that one of the classroom factors associated with a decline of high-level demand is: Problematic aspects of the task become routinized (e.g., students press the teacher to reduce the complexity of the task by specifying explicit procedures or steps to perform; the teacher ‘‘takes over’’ the thinking and reasoning and tells students how to do the problem) (p. 27)


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TABLE I The Classroom Implementation Framework Nature of classroom tasks • Tasks make mathematics problematic • Tasks connect with where students are • Tasks leave behind something of mathematical value • Tasks require a high level of cognitive demand Role of the teacher • Teacher selects tasks with goal in mind • Teacher shares essential information • Teacher establishes classroom culture • Teacher exhibits factors associated with the maintenance of high-level demands Social culture of the classroom • Teacher and students value ideas and methods • Students choose and share their methods • Teacher uses mistakes as learning sites for everyone • Argument determines correctness of mathematical processes and answers (not the teacher or textbook) Mathematical tools as learning supports • Students construct meaning for tools • Teachers and students use tools with purpose – to solve problems • Teachers and students use tools for recording, communicating, and thinking Equity and accessibility • Tasks are accessible to all students • Every student is heard • Every student contributes Capsule description of the quality of the lesson • Level 1: Ineffective instruction • Level 2: Elements of effective instruction • Level 3: Beginning stages of effective instruction • Level 4: Accomplished, effective instruction • Level 5: Exemplary instruction

See Stein et al. (2000) for a further discussion of these factors. Finally, we utilized the ‘‘Capsule Description of the Quality of the Lesson’’ portion of the classroom observation protocol developed by Horizon Research, Inc. (HRI) (see Appendix A). This capsule description allowed us to characterize each lesson concisely and then assign the lessons to one of three groups, as we discuss further in the methodology section. In summary, Table I contains our framework, which we call the Classroom Implementation Framework. We believe that combining

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three existing frameworks provided a powerful lens for capturing many different aspects of lessons and teachers’ instructional practices in classrooms where Core-Plus was used. Setting Dickinson Public Schools (DPS) is a mid-western school district of over 16,000 students, with over 5800 students and approximately 50 mathematics teachers in Grades 8–12. Teachers in DPS high schools (grades 10–12) began implementing Core-Plus during the 1997–1998 school year with approximately 10–15% of the ‘‘lower track’’ students; the following year DPS junior high schools (grades 8–9) offered courses in which Core-Plus was used as the mathematics text. After that time, all four years of the textbook series were phased into courses that constituted approximately 50% of the school population by the 2002–2003 school year, when data collection for this research occurred. During that year, the other approximately 50% of students self-selected a mathematics course pathway traditional to U.S. high school mathematics (Algebra 1, Geometry, Algebra II) to complete their high school mathematics requirements. By 2008, the school district expects that approximately 75% of their students in grades 8–12 will take mathematics classes in which they use the Core-Plus textbook series. Study Participants Twenty-six teachers participated in this study. At the time of data collection, all teachers in the study taught at least one mathematics course in which Core-Plus was used as the textbook. All study participants had, previous to data collection, agreed to engage in two years of professional development that would focus on implementing CorePlus. This group of 26 teachers comprised approximately 70% of DPS teachers who used Core-Plus, and 87% of the teachers who would be in the professional development project. Participation in both the professional development project and in this study was voluntary. Participants included two beginning teachers as well as four teachers with over 25 years of experience. The mean number of years of teaching experience, at the time of data collection, for the group was 13. The group also had varied years of teaching using Core-Plus, ranging from zero to four years. The group contained 20 female teachers and six male teachers. Eighteen teachers had earned advanced degrees; the other eight teachers held bachelors’ degrees.


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Data Collection We collected data for this study during Fall 2002. The timing of data collection was an important consideration because we wanted to collect baseline data before the teachers began participating in a two-year professional development project. We gathered two types of data that are pertinent to this particular research study: classroom observation data and teacher interviews. Classroom Observation Data. One of five members of the data collection group (which included the first three authors and two other researchers) observed each teacher for one class period (between 50 and 110 minutes, depending on the school-dictated length of classes). During the observation, we took detailed field notes, which included the classroom activities and the sequencing of those activities. These field notes also included scripting of classroom discussions. Soon after the actual observation, the observer filled out an observation protocol that was developed to align with the Classroom Implementation Framework described earlier in this paper. After the observer completed the protocol for a particular lesson, (s)he considered the overall impact of the lesson and assigned a rating based on the Capsule Description of the Quality of the Lesson rating sheet (see Appendix A). Prior to data collection we established reliability among the observers by completing the entire process twice, for two different teachers’ lessons. After each observation, we completed protocols individually and then met to compare these completed protocols. After two rounds of the process, we were satisfied that the observers were gathering data consistently across lessons. Teacher Interview Data. We conducted interviews to examine the teachers’ beliefs and attitudes regarding teaching with Core-Plus. We asked every teacher questions about their background with Core-Plus, what they thought about the textbook (e.g., its strengths, weaknesses, what students think about it), and how they approached teaching from Core-Plus. We asked them to describe their ‘‘best’’ and ‘‘worst’’ days using this textbook in their mathematics classes. Finally, we asked them what they wanted to learn from the professional development project they were about to begin. Each interview lasted approximately one hour and was audio-taped for subsequent transcribing.

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Data Analysis In their report Looking Inside the Classroom: A Study of K-12 Mathematics and Science Education in the United States, Weiss and her colleagues (2003) used the Capsule Description of the Lesson criteria (Appendix A) to describe 364 mathematics and science classrooms across the United States. To facilitate their analysis, the HRI researchers grouped the lessons into three categories and assigned lessons to those categories based on their ratings for the Capsule Description of the Lesson (see Table II). Following their lead, we began our analysis by separating the teachers’ lessons, based on the Capsule Description of the Lesson rating, into the same three groups (see Table II). It is important to note that we assigned lessons to the three groups (LLQ, MLQ, and HLQ), not teachers. Horizon’s Capsule Description Ratings are meant to characterize lessons, not the teachers of those lessons. While this may be ‘‘splitting hairs’’ to some, we believe that the distinction is important. We are researching teachers’ instructional practices; we argue that by analyzing teachers’ lessons, we can investigate their instructional practices. It is also important to report that after we grouped the lessons into LLQ, MLQ, and HLQ categories, we conducted analyses to see if certain demographic characteristics of the teachers could help us understand the instructional practices of the teachers in the three groups. The Mann–Whitney test (Hinkle, Wiersma, & Jurs, 1988) is a nonparametric test appropriate for ordinal data. It is sensitive to both the central tendency and the distribution of scores, and indicates whether the distributions of two sets of scores are significantly different. Analyses using this test indicated that the distributions of number of years teaching mathematics within the three categories were not significantly different at the / = 0.05 level. Likewise, the distribution of the assigned quality of the lesson rating for teachers with a bachelor’s degree was not significantly different from the distribution for teachers with more advanced degrees. Regarding the number of years teaching TABLE II Capsule Description Categories as Defined by Weiss et al. (2003) Weiss et al. (2003) Capsule description categories

Associated capsule description ratings

Number of Core-Plus lessons assigned to each category

Low-lesson quality (LLQ) Medium-lesson quality (MLQ) High-lesson quality (HLQ)

1 and 2 Low 3 and Medium 3 High 3, 4, and 5

11 8 7


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Core-Plus, the distributions of this variable were not significantly different for the MLQ and HLQ categories, but the LLQ distribution of number of years teaching Core-Plus was significantly different from the distributions of the MLQ (p = 0.042, 1-tailed) and HLQ (p = 0.012, 1-tailed) categories. It is not surprising that teachers with more experience with a curriculum may have a higher quality lesson; however, the number of years of teaching Core-Plus does not completely explain the diversity of quality of the lesson ratings. Classroom Observations From the 26 completed lesson protocols, we produced a summary of the five dimensions for each lesson, and a preliminary analysis of the classroom observations was completed. Given the assignment of lessons to the three groups as described above (LLQ, MLQ, and HLQ), we separated the summaries into corresponding groups. Up to this point, our unit of analysis had been individual lessons. We then switched our unit of analysis to the group level, analyzing the LLQ, MLQ, and HLQ groups’ observation summaries, searching for commonalities and differences among the lessons in each group. Teacher Interviews We developed an original list of codes by simultaneously coding one teacher’s interview and we refined the list over the course of coding the remaining transcripts. Ultimately we used the following six broad codes to aid in reducing the interview data: teacher’s attitudes toward students; the textbook and basic skills; the role of the teacher; description of best and worst days of teaching Core-Plus; perceived strengths and weaknesses; and what the teacher hoped to gain from the professional development project. We then wrote a summary of each teacher’s interview responses using these codes as headings. At this point in our analysis we used the LLQ, MLQ, and HLQ groups as our unit of analysis. Once the individual teacher summaries were completed, we developed descriptions for each code across teachers in the same group. This process allowed us to search for similarities and differences among the teachers in a particular group.

FINDINGS We concentrate on two groups of lessons in this section, the LLQ lessons and the HLQ lessons. We present an extended discussion regarding these two groups, as the instructional practices and related

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beliefs are in stark contrast between the two groups. We found that the MLQ groups’ characteristics were sometimes similar to the LLQ group and sometimes to the HLQ group. While we do not report on the MLQ group in as much detail as the other groups, we do comment on the salient differences between groups in the discussion section. For the LLQ and HLQ lesson sections, we first present a vignette of a classroom episode that exemplifies the instruction that we observed for that particular lesson level. The vignettes were constructed from detailed field notes, and the classroom discussions are verbatim. We then report on the instructional practices of the teachers in the group through the lens of the Classroom Implementation Framework. Finally, we present teacher beliefs that may influence their instructional practices.

LOW LESSON QUALITY LESSONS In this section we present data regarding the LLQ lessons. As a reminder to the reader, we identified 11 of 26 lessons as falling into the LLQ group. We provide interpretive comments, through the lens of the Mathematical Tasks Framework, in italics throughout the vignette. Classroom Vignette: Ms. Taylor In this Core-Plus: Course 1 class, 15 10th- and 11th-graders sat in desks, placed in groups of three, facing the front of the room. During the previous class period, the students had begun working on modeling human population growth in various countries using iterative calculations that they had not yet formalized symbolically. At the beginning of this class, Ms. Taylor provided population data that she had found on U.S. Census Website. The task required students to model population growth with these data. [This task was rated as requiring a high level of cognitive demand, as it was presented to the students] The following exchange occurred just after she presented the data: Ms. Taylor: The population in the U.S. in 2001 was 288 million. In 1990 it was what? Student: 240. Ms. Taylor: So it increased by 48 million. The birth rate is 1.3% of the population. The death rate is 0.2%. So what is the net (population change)? Student: 1.1% Ms. Taylor: Yes. The birth rate makes (the population) go up and the death rate makes it go down.


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[Ms. Taylor lowered the level of cognitive demand during this exchange by focusing the students’ responses on correct answers and not pressing for justification.] Mrs. Taylor then encouraged the students to follow a procedure similar to the one used during the previous class to make predictions for the U.S. population for 2002 through 2005. The students worked individually on this task for a few minutes, using their graphing calculators to complete various arithmetic computations. Ms. Taylor asked a few students to put their tables on the board after which Ms. Taylor uncovered the correct population for 2002 on the overhead. Ms. Taylor: I have the first row (for 2002) up on the overhead, so you know if you did it right. I want you to be able to check it because after you make a mistake, everything after it is wrong.

[Here, and throughout the vignette, Ms. Taylor shifts the emphasis from meaning and understanding to the correctness of the answer.] After a short time, Ms. Taylor displayed the correct answers for 2003 through 2005 on the overhead. Some students, including those who had written on the board and those who had not, changed their answers to match those displayed on the overhead. Ms. Taylor did not reference the work that students put on the board before moving on. The lesson continued with Ms. Taylor using a ‘‘Check Point’’ with the students. According to the Core-Plus teacher materials, a Check Point is intended for a full-class discussion of the content and methods developed by different small groups (that) provides an opportunity to share progress and thinking...Varying points of view and differing conclusions that can be justified should be encouraged. Teacher is moderator. (Coxford et al., 1998b, p. 12, authors’ emphasis) Ms. Taylor: Okay, let’s look at the [problem] on p. 112. What calculations are needed to estimate the population growth from one year to the next in the two different countries? Student: The year before. Ms. Taylor: What do you mean by ‘‘the year before’’? [Here, Ms. Taylor asks for an explanation.] Student: You need to find the previous. Ms. Taylor: Because the population in 2002 depends on the population in 2001. What else? Student: The increase. Ms. Taylor: The birth rate, the death rate, and the immigrants. Let’s write out a step-by-step process. What do you do first? Student: Find the rate of change. Ms. Taylor: Birth rate minus death rate. It may be negative. What was the percent (for our previous problem)?

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Student: 1.1% Ms. Taylor: 1.1%. Did we use that in our calculations? Student: No. Convert it to a decimal. Ms. Taylor: How do we do that? Student: Move the decimal place two to the left. Ms. Taylor: So we used 0.011. Now what do we do? Student: Multiply by the population. Ms. Taylor: Which population? Student: The previous year’s. Ms. Taylor: You need to know the previous population to find the next year’s population. Now what do we do? Student: Add the immigrants. Ms. Taylor: Is it possible we could have more people leaving than incoming? Student: Yes Ms. Taylor: These are your steps. You may want to put a circle around this.

As she was saying this, Ms. Taylor wrote the following step-by-step process on an overhead transparency: (1) Find the net change (birth rate)death rate); (2) Convert to a decimal (two places to the left); (3) Multiply by the previous year’s populations; (4) Add the immigrants. [Above, Ms. Taylor reduced the level of cognitive demand by specifying explicit steps for the students to perform, thus Ôtaking over’ the thinking for the students.] Ms. Taylor then moved to the next problem in the book. The problem read: ‘‘Using the word NOW to stand for the population of the United States in any year, write an expression that shows how to calculate the population in the NEXT year’’ (p. 112). The population growth rate was provided as 1.9% and the death rate was 1.2%. Ms. Taylor: What is the net change? Students: (No response.) Ms. Taylor: 1.9 minus 1.2 is 0.7%. We still need to change it to a decimal. It already looks like a decimal. So we’re going to multiply it by NOW and add the immigrants. Does this give you the population for next year? (During this statement, Ms. Taylor writes NEXT = 0.007 NOW + 0.9 on the board at the front of the room).

[Above, Ms. Taylor Ôtakes over’ the thinking for the students and provides a step-by-step procedure for them to follow.] Student: Yes. Ms. Taylor: I’m saying it doesn’t. Why not? Student: You need to add the previous population

Ms. Taylor wrote ‘‘+ NOW’’ at the end of the previous equation. She then asked students to create a similar equation for the population change in Brazil and write their equations on the board.


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LLQ Instructional Practices In this section we look more broadly at the instructional practices of the teachers whose lessons were rated as Low Lesson Quality. Using Ms. Taylor’s vignette, as well as data from other classrooms, we present findings on: (a) the nature of classroom tasks; (b) the role of the teacher and social culture of the classroom; (c) mathematical tools; and (d) equity and accessibility. Nature of Classroom Tasks Ms. Taylor primarily used tasks from the Core-Plus textbook during this observation; she supplemented with non-text data at the beginning of the class so that her students could have more practice with the previous day’s concepts. The other ten LLQ lessons also involved instructional tasks from the Core-Plus textbook. Initially these tasks involved problematic situations—tasks for which students had developed no prior solution strategy. Approximately half of the tasks utilized in the LLQ lessons were coded as requiring a high-level of cognitive demand; the remainder of the classrooms tasks were categorized as requiring a low-level of cognitive demand. In a few of the LLQ lessons, teachers made modifications to the tasks provided in the Core-Plus textbook; these modifications often involved simplifying the language in the text and/or limiting the reference to the context in the textbook. For example, one task, as written in the textbook, required students to ‘‘Write a formula for the perimeter of an equilateral triangle that is different from the formula for a general triangle’’ (Coxford et al., 1997, p. 360). Mr. Cooke modified this task so that it read, ‘‘An equilateral triangle has three sides that are all the same length. Write a formula for its perimeter that does not use addition.’’ In his revision to this task, Mr. Cooke defined what an equilateral triangle was for the students and placed constraints on the type of response that was acceptable for this situation. Role of the Teacher and Social Culture of the Classroom Although Ms. Taylor used tasks that require a high-level of cognitive demand, she consistently reduced the level of cognitive demand required for the tasks. Like Ms. Taylor, the teachers of LLQ lessons often provided students with procedures to solve instructional tasks, did not follow-up on unclear or incorrect student responses, or shifted the focus away from meaning toward a procedure to follow. The teachers whose lessons were assigned to the LLQ group tended to dispense information to students, focusing on the correctness of the

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answer without discussing the underlying mathematical concepts. If students made mistakes, they were quickly corrected without reference to understanding the reason why such mistakes were made, as was the case in Ms. Taylor’s lesson. In addition, in every LLQ lesson the teacher served as the mathematical authority for the correctness of answers, providing quick positive and negative feedback. Just as Ms. Taylor did in her lesson when students asked questions, teachers in LLQ lessons often provided students with direct answers without investigating possible sources of student misunderstanding. A similar situation occurred when Mr. Dunbar reviewed the formulas for the areas of various geometric shapes. He asked the question, ‘‘Why does it [the formula for the area of a triangle] have 1/2 in it?’’ A student responded, ‘‘because it [a triangle] has only three sides.’’ Neither Mr. Dunbar nor other students questioned the student’s explanation. Although not evident in Ms. Taylor’s lesson, behavior management issues arose in half of the LLQ lessons. These issues ranged from students exhibiting off-task behavior that was not redirected by the teacher to student–teacher interactions that resulted in administrative disciplinary actions being taken. Behavior management problems are one of the classroom factors associated with the decline of cognitive demand of a mathematical task (Stein et al., 2000). In the lessons we observed, students’ off-task behavior was a contributing factor to a decline in the cognitive demand of the task during implementation. Mathematical Tools Various mathematical tools were available for use during the LLQ lessons. Calculators, graphs, drawings, measurement instruments, and mathematical symbols were used in these classrooms. However, these tools were often employed to record information or perform computations rather than to analyze information or communicate mathematical ideas. Little emphasis was placed on developing meaning for the tool or connecting the meaning of various tools to one another. No discussions occurred regarding the advantages and disadvantages of various tools. Equity and Accessibility All lessons in the LLQ group contained tasks that were accessible to students, though we observed little effort to connect student prior experiences to the tasks. In addition, limited opportunities existed for students to participate in discussion. The LLQ lessons varied considerably in structure; lessons were either predominately conducted in small groups or whole-group format – little mixed structure occurred.


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Although opportunity for discussion existed in both types of settings, the discussion in these lessons often focused on following a procedure rather than developing understanding, as occurred in Ms. Taylor’s lesson. Teacher Beliefs Two distinct categories of beliefs about learning and teaching appeared in the interviews conducted with the teachers whose lessons were assigned to the LLQ group. The first is a set of beliefs about their students’ ability to learn. Throughout their interviews, these teachers expressed low expectations with regard to their students’ ability to succeed in learning mathematics using Core-Plus. They were concerned with their students’ lack of basic skills, which the teachers attributed to a poor mathematical background, and felt that the students needed more practice than that provided in the Core-Plus textbook. The following statement made by Ms. Martin is representative of the statements made by all of the teachers whose lessons were assigned to the LLQ group: I think it is material they are expected to do that they have no background on. They are supposed to discover things, but they are clueless. They plug everything into a calculator but they don’t know what anything means or how to find it on their own...I’m bogged down by kids who know nothing about what they should know.

The second set of beliefs dealt with the teachers’ abilities to orchestrate the type of learning environment necessary for the use of Core-Plus. These teachers perceived themselves as poor in moderating discourse and organizing/managing small groups. As Mr. Tanner expressed, ‘‘A weakness I feel I have in Core-Plus is the use of group work. Sometimes the kids work together very well and other times some work in isolation in a group. I need ways to get them to work well in groups.’’ This theme of pedagogical difficulty was prevalent in the interviews of teachers in this group. LLQ Lessons Summary To summarize, teachers used mathematical tasks that required a highlevel of cognitive demand about 50% of the time in the LLQ lessons. However, during implementation of those tasks, the teachers took over the thinking for the students, gave step-by-step instructions to the students on how to solve the demanding problems, and served as the mathematical authority. While the lessons incorporated the tools of mathematics, those tools were most often used to record information

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or perform calculations. Teachers of LLQ lessons consistently expressed beliefs that indicated they had serious doubts that students could learn worthwhile mathematics from a problems-based textbook like Core-Plus.

HIGH LESSON QUALITY LESSONS In this section we present data regarding the HLQ lessons. As a reminder to the reader, we placed 7 of 26 lessons in the HLQ group. We provide interpretive comments, through the lens of the Mathematical Tasks Framework, in italics throughout the vignette. Vignette: Ms. Andrews The 27 students in this class were mostly 11th graders; they were seated in groups of three, facing the front of the classroom. Ms. Andrews began her Course 3 class by directing the students toward problem #4 from their mathematics textbook [this task was rated as requiring a high level of cognitive demand as it was written in the textbook]: When John and Carla were 14 years old, they had summer jobs. John saved $600 from his earnings, and Carla saved $550 from her earnings. They both decided to put their money into savings accounts. John found an account that would pay interest at an annual rate of 8%, but at the end of each year he took out the interest. Each year his original deposit stayed in the account. Carla’s account only paid 7% interest, but she decided not to withdraw any money. The interest was added to her account at the end of each year, and she began earning interest on the interest added. (a) How much money will each person receive from the banks if they closed their accounts at the end of 12 years? (Coxford et al., 1999, p. 58)

The students worked individually on this problem for a few minutes until Ms. Andrews engaged the class in a whole-class discussion; a number of different students engaged in this exchange with Ms. Andrews: Ms. Andrews: At the end of one year, how much interest has John earned? Student: $48. Ms. Andrews: So how would you find the amount of money John has after one year? Student: Take 600 plus 48. Ms. Andrews: How much would John have after 12 years? Student: $600 + 48*12, which is $1176. Ms. Andrews: How much after 7 years? Student: 600 + 48 * 7. Ms. Andrews: How much after y years?


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Student: 600 + 48y. Ms. Andrews: How would you use your (graphing) calculator to figure this out? Student: Enter ‘‘y = 600 + 48x’’ and look at the table. Ms. Andrews: [continuing the problem] Carla invested her money at a rate of 7%, starting with $550 in her account. How would we determine the amount of money she has?

During the following exchange, Ms. Andrews writes 550*.07 = 38.50. She then adds *1.07 to the equation each time the students direct her to. [Above, Ms. Andrews scaffolds student thinking.] Student: Take 550 + 38.50. Ms. Andrews: What do I do to find the amount of money she has after 2 years? Student: That amount stays in, so you multiply that amount by 1.07. Ms. Andrews: How much does Carla have after 3 years? Student: Multiply the total amount by 1.07. Ms. Andrews: What am I doing repeatedly? Student: Multiply by 1.07. Student: Raise it to the number of years. Ms. Andrews: For any number of years? Student: To the x power. [Ms. Andrews writes on the board: 550 1.07x] Ms. Andrews: What kind of model is this? Student: Exponential Ms. Andrews: Invest now, leave it in, and it grows. [Ms. Andrews writes both models on the board: y = 550 1.07x and y = 600 + 48x] Ms. Andrews: What kind of model is y = 600 + 48x? Student: Linear Ms. Andrews: Why doesn’t it curve? Student: It increases by the same amount each time.

[Above, Ms. Andrews uses scaffolding to support students to make conceptual connections] Later in the class, the students examined data for the projected income from ticket sales for a fictitious summer theater (see Table III). Students took a few minutes to answer various questions from the textbook about the data before Ms. Andrews guided them in discussion about these items. Ms. Andrews: (Reading from the text) According to this table, how will ticket income be affected by choice of ticket price? Does the pattern in the table seem reasonable? Why or why not?

(At this time many different students are involved in the conversation) Student: Student:

Doesn’t ticket income ‘‘peak’’? It goes up to a certain point, then drops. It’s like a bell curve.

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TABLE III Projected Income from Ticket Sales Ticket price (dollars) Number of tickets Ticket income (dollars)

0 2500 0

5 2250 11,250

10 2000 20,000

20 1500 30,000

30 1000 30,000

40 500 20,000

50 0 0

Ms. Andrews: Does that make sense?

[Above, Ms. Andrews presses for justification and encourages student to draw conceptual connections.] Student:

Yes, when the price of the tickets is small, you would sell more tickets, but not make as much. When the price of a ticket is high, you would sell fewer tickets. Somewhere in the middle you would sell a lot of tickets and make more money. Ms. Andrews: (Reading from the text) Someone proposed the equation I = P(2500)50P) for the relation between the ticket price and ticket income; someone else proposed I = 2500P)50P2. Which of these two rules is correct? Why? Student: The first one? Student: They are the same. If you factor the first equation you get the second equation. Ms. Andrews: [Referring to the first equation] What do we do with P? Student: Distribute. (The teacher draws arrows demonstrating that the P is multiplied by both terms of (2500)50P).) Student: What is P times 50 times P? Ms. Andrews: Is that the same as 50 times P times P? Student: Yes Ms. Andrews: What else can we do? Student: P times P is P squared. So when we distribute the answer is 2500P–50P2.

[Above, Ms. Andrews asks questions to support students in making conceptual connections, thus modeling high-level performance.] The discussion continues about what the shape of the graph of I = 2500P)50P2 looks like when graphed on the coordinate plane. The students ‘‘trace’’ the graph using their graphing calculators to examine what the ticket price should be to maximize the profit. HLQ Instructional Practices In this section we look more broadly at the instructional practices and beliefs of the teachers whose lessons were rated as High Lesson Quality. Using Ms. Andrew’s vignette, as well as data from other lessons, we present findings on: (a) the nature of classroom tasks;


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(b) the role of the teacher and social culture of the classroom; (c) mathematical tools; and (d) equity and accessibility. Nature of Classroom Tasks Just as occurred in Ms. Andrews’ classroom, each of the HLQ lessons involved tasks from the Core-Plus textbook. These tasks were classified as requiring a high level of cognitive demand and were problematic in nature. Role of the Teacher and Social Culture of the Classroom The HLQ lessons contained many factors associated with the maintenance of high-level cognitive demands. As in Ms. Andrews’ class, the students in HLQ lessons were allowed to find solutions to mathematical tasks using strategies they devised, and the teacher encouraged the students to reflect on their strategies and responses. Also evident in these lessons were a few times when the teacher lowered the level of cognitive demand; however, these occurrences were far less prevalent than instances of teachers maintaining a high level of cognitive demand. The residence of authority for mathematical reasoning varied in the HLQ lessons. When working individually or in small groups, students were generally encouraged to use whatever strategy they desired and the teacher did not provide feedback on the correctness of student responses. However, in large group discussions in the HLQ classrooms, the teacher often exhibited mathematical authority tacitly by moving on to the next question or not requiring further student input. Mathematical Tools In the HLQ lessons, typical mathematical tools were available for students’ use. These tools included calculators, measuring instruments, various graphical representations (e.g., coordinate graphs, box plots), and symbolic representations. In Ms. Andrews’ classroom, the students used their graphing calculators to examine the graphs of various equations and to ‘‘trace’’ a graph to determine the ticket price that would be likely to lead to the maximum profit. Teachers of the HLQ lessons generally encouraged students to refer to tools such as charts or tables for recording information to communicate to others as well as to consider the advantages and limitations of the tools that they used. Equity and Accessibility All mathematical tasks were accessible to the students in the HLQ lessons. Further, the teachers of the HLQ lessons supported access to mathematical ideas by allowing students to investigate their own strategies for the mathematical tasks and to consider mathematical

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relationships. For example, Ms. Andrews encouraged students to describe their strategies for determining whether two expressions, P(2500)50P) and 2500P)50P2 are equivalent. The resulting discussion benefited many students who may have been confused about how to simplify this expression. Teacher Beliefs In general, teachers in the HLQ group expressed a positive attitude towards the overall structure of the Core-Plus textbook. They advocated the use of the textbook because they felt that it encouraged students to develop patience, perseverance, and critical thinking skills. One of the teachers whose lesson was included in the HLQ group told us, ‘‘In general I think the textbook is excellent. It gives the students a chance to think on their own – I think it gives them a chance to work through problems on their own. I think a student that...goes through the training with this and is successful is going to be very well off mathematically.’’ Other teachers whose lessons were included in this group echoed this statement in their interviews. However, the teachers of the HLQ group expressed concern with the prerequisite basic skill and reading levels required in the textbook. For example, Ms. Nivens said, ‘‘I am concerned about the manipulation skills and again...right or wrong...I think kids should be exposed to that just because of what could be on college entrance exams, what could be expected of the in college.’’ Despite these concerns, teachers in the HLQ group collectively agreed that their students regularly were able to engage successfully in the mathematical tasks in the textbook, a belief that Ms. Byer implied when she stated, ‘‘[When problems are put] in context, I haven’t seen anything come their way that they couldn’t do.’’ Though the teachers of HLQ lessons valued small-group discussion, they expressed concern with their own abilities to direct classroom discourse and manage small groups. For example, Ms. Nelson stated, I think maybe I need to improve on some of the class activities. I sometimes let the groups diverge a little bit, and that’s tough because half of a group will already have something, and I’ll have half of the group who is behind and sometimes instead of pulling them all back together as foursome or whatever I’ll let them go two and two...I love it when I look at a group of four people or 3 people and they are all working on it together, and so I need to work on that.

Despite these concerns, the teachers whose lessons were assigned to the HLQ group valued the conceptual approach to the textbook and application problems that connected to situations which were familiar to the students.


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HLQ Lesson Summary To summarize, teachers used mathematical tasks that required a high-level of cognitive demand in all observed HLQ lessons. During implementation of those tasks, teachers allowed students to find solutions to problems using multiple student-generated strategies. Teachers of HLQ lessons rarely acted explicitly as the mathematical authority; they encouraged students to make sense and justify their mathematical processes. Students and teachers in HLQ lessons used tools to investigate mathematical relationships as well as for recording information and to perform calculations. Teachers of HLQ lessons consistently expressed beliefs that indicated they had confidence that students could learn worthwhile mathematics from a problems-based textbook like Core-Plus. DISCUSSION In the U.S., many districts are adopting problems-based textbook series across all grades. Evidence is building that students who use these types of textbooks to learn mathematics benefit both in the ways that they understand mathematics and in their performances on standardized assessments (see, for example, Riordin & Noyce, 2001). Evidence also exists that the ways in which teachers use these textbooks influences student achievement (Tarr, Reys, Chavez, & Shih, 2006). As is supported by the study reported here, it is not the case that putting these textbooks into the hands of teachers will result in them using the textbooks as intended by the authors. Consider, for example, the way in which Ms. Taylor reduced the development of a mathematical model to a step-by-step procedure for students to follow. Based on the intended focus of the Checkpoint (as explicated in Ms. Taylor’s vignette earlier in this paper), Ms. Taylor was not implementing the textbook as intended by the authors. She was also not delving into her students’ understandings of the mathematics they were learning. We suggest that Ms. Taylor’s beliefs about students’ abilities to learn mathematics through a problems-based, inquiry approach may motivate her to decrease the cognitive demand required by the mathematical tasks. However, this offering is tentative as our research was not designed to investigate deeply teachers’ motivations for their instructional decision-making; further research is warranted. As stated earlier in this paper, it was also not our intention in this study to tie instructional practices to student learning outcomes. Thus, we do not suggest that the DPS students learned something only in

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mathematics classrooms where we observed HLQ lessons. However, as Boaler (2002) and Henningsen and Stein (1997) argue, the opportunities for students to think mathematically in classrooms where lessons would be classified as HLQ are qualitatively different from the opportunities for students to think mathematically in classrooms where lessons would be classified as LLQ. Although we anticipated many of the results in the HLQ and LLQ cases, based on other research documenting instructional practices (e.g., Weiss et al., 2003), we believe that documenting instructional practices that occurred in the HLQ and LLQ lessons through the lens of the Classroom Implementation Framework helps us, as a community, better to understand mathematics lessons at the secondary level, particularly when a textbook like Core-Plus is used. Two striking comparisons between these two groups of teachers are noteworthy. First, teachers whose lessons were rated as HLQ believed that they could learn to use the Core-Plus textbook series to its potential; they believed that the textbook series was appropriate for all students and that all students would benefit from using Core-Plus to learn mathematics. In contrast, teachers whose lessons were rated as LLQ tended to believe that the success of students using Core-Plus depended upon the Ôtype’ of student in the class. This is not a new phenomenon; Thompson, Warren, and Carter (2004) found that certain teacher characteristics correlated positively with teachers’ propensity to place blame on students and parents for students’ underachievement. Because of our use of the Classroom Implementation Framework, we did not collect data for this study that would allow us to determine if the teachers whose lessons were rated as LLQ had characteristics similar to those found by Thompson and her colleagues. Nor did our use of the Classroom Implementation Framework support collecting data with regard to examining any constraints that the teachers might experience that stem from the contexts in which they teach. Sullivan and Mousley (2001) propose that a three-way influential relationship exists between teacher beliefs, classroom- and pupil-based constraints, and teacher actions. In the case of our study, perhaps earlier in the year Ms. Taylor experienced little success in having her students engage in mathematical discourse. Perhaps her observation occurred on an ‘‘off’’ day for her students or herself. A study designed to examine the types of constraints that teachers feel they face when using problems-based mathematics textbook would enhance our understandings of teacher instructional practices.


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A second striking comparison is that the teachers whose lessons were rated as HLQ believed that any deficiencies in their students’ basic skills could be addressed through solving problems provided in the Core-Plus textbook series. As a result, these teachers did not limit student access to Core-Plus tasks. In contrast, the teachers whose lessons were rated as LLQ believed that basic skills were a prerequisite for using the textbook, often leading to the conclusion that Core-Plus was not the Ôright textbook’ for lower-level students and limiting students’ access to the textbook as it was intended to be used. Also interesting to us as researchers and professional developers are the lessons rated as MLQ. These lessons had characteristics that were common to the HLQ lessons and at the same time they also had characteristics that were common to the LLQ lessons. While we saw evidence of the use of high-level tasks that came from Core-Plus during MLQ lessons, the teachers’ actions during class appeared to have a great effect on whether students were provided the opportunities to wrestle with the mathematics they were learning. Further, the teachers of the lessons assigned to the MLQ group really did ‘‘sit the fence’’ when it came to their beliefs about using this textbook in their mathematics classrooms. Many of these teachers spoke of the lack of basic skills (i.e., ability to do arithmetic) that students brought with them into the use of this textbook. For most of these teachers, this perceived student deficiency kept the teachers from allowing students real access to the mathematics in Core-Plus textbooks. They felt that the students could not succeed with a textbook like Core-Plus without being led, step-by-step, through the problems. However, some of the teachers whose lessons were assigned to this group were beginning to see glimpses of the students’ ‘‘aha’’ moments – moments that have the capacity for influencing teachers’ beliefs about students’ ability to learn. In the end, this study allowed us to understand better the instructional practices and beliefs of a number of teachers across one district who use Core-Plus in their mathematics classrooms. Further, conducting this study has provided us with findings that inform the design and implementation of professional development for such teachers, as explicated in the next section. IMPLICATIONS FOR PROFESSIONAL DEVELOPMENT Many school districts in the U.S. have attempted to implement problems-based mathematics textbook series like Core-Plus in an attempt to accelerate changes in instructional practices (Corcoran, 2003),

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despite the lack of evidence that a direct link exists between textbook implementation and teacher change (Remillard & Bryans, 2004). Such adoption decisions are often made under the pressure to demonstrate quick improvements in student performance and have sometimes lead to districts abandoning the use of problems-based textbook series after only a few years of implementation. Perhaps contributing to the abandonment of the use of these types of mathematics textbooks is a lack of attention to teacher professional development that focuses on implementing the textbook. Our research strongly supports the need for professional development for teachers who use these types of textbooks in their mathematics classes. What we do not know from this research is the extent to which professional development needs to be textbook-specific in order to impact teachers’ instructional practices. Future research studies need to be designed that investigate questions that inform the community about this issue. The research reported here does suggest that professional developers who are designing professional development for teachers using a textbook series like Core-Plus consider the wide range of instructional practices and beliefs that teachers will bring with them into the learning activities. Specifically, this research suggests that professional developers consider designing learning activities for mathematics teachers who use problem-based textbooks like Core-Plus that: • allow for varied points of entry for teachers whose lessons fall along the instructional practices continuum (low lesson quality to high lesson quality); • provide opportunities for teachers to learn to manage day-to-day classroom-based organizational issues (e.g., checking homework, small-group work, collecting/grading papers); • incorporate opportunities for teachers to be challenged about their beliefs on student learning (e.g., how students learn mathematics, expectations for what students can learn); • include discussions of mathematical tools and their uses for recording, communicating, and thinking. STUDY LIMITATIONS Our data collection for this study was determined by the framework we chose to use. While the use of a framework is a strength of the study (Eisenhart, 1991), we also consider our use of the framework as a limiting factor with regard to data collection. In particular, since we designed our observation protocol to capture data in relation to the


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Classroom Implementation Framework, we ‘‘missed’’ data that could have informed us about such things as constraints (as discussed above). We also acknowledge that only observing one lesson per teacher limits our study. However, consider the following argument: The teachers chose the day and class period that we observed. We assume that the teachers selected these lessons because they considered them to be of high quality. We also assume that teachers and students cannot shift classroom norms for just one observable class period. Thus, while some of the specifics of the lessons may have changed based on the day and class we observed, we argue that the types of mathematical tasks chosen by the teacher, the role of the teacher, the classroom culture, the ways mathematical tools are used, and the access students have to the mathematics is more consistent from day to day than not. Secondly, a tension existed for us with regard to breadth vs. depth. We chose to observe one lesson from each of 26 teachers as opposed to more lessons from fewer teachers. Although our decision to focus on breadth may be considered a limitation of this study, we believe that this breadth can also be considered a strength of the study. CONCLUSION If we, as a community, believe that developing deeper student understanding of mathematical ideas through reform textbook series can increase student achievement at the high school level, then we need to know much more about what is occurring in the classrooms where these textbooks are being used. Further, the more we understand about the teachers and their instructional decisions, the better able we will be to support the necessary and often difficult transitions they need to make in order to enact the textbook series in the manner intended by the their authors, and consistent with the recommendations of the National Council of Teachers of Mathematics (2000). As a community, we are also in the beginning stages of understanding what types of professional development will support teachers to move to implementing high quality lessons as described here (Stein, Smith & Silver, 1999). The research reported in this article directly contradicts the belief that adopting a problems-based textbook series and using it in a classroom is enough, in itself, to have an effect on teacher instructional practices – to get them to teach in a more reform-oriented manner. Professional development is a critical part of the adoption process. Reform in mathematics classrooms is an ambitious and worthy cause. Understanding teachers’ current instructional practices and the

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implications of those practices for professional development is essential for attending to the diverse needs of classroom teachers. In the end, the more we know about and support teachers, the more K-12 students benefit. APPENDIX A Capsule Description of the Lesson Level

Description

Level 1: Ineffective instruction

There is no evidence of student thinking or engagement with important ideas of mathematics. Instruction is unlikely to enhance students’ understanding of the discipline or to develop their capacity to successfully ‘‘do’’ mathematics. Lesson was characterized by either: • Passive ‘‘Learning’’: Instruction is pedantic and uninspiring. Students are passive recipients of information from the teacher or textbook; material is presented in a way that is inaccessible to many of the students • Activity for Activity’s Sake: Students are involved in hands-on activities or other individual or group work, but it appears to be activity for activity’s sake. Lesson lacks a clear sense of purpose and/or a clear link to conceptual development Instruction contains some elements of effective practices, but there are substantial problems in the design, implementation, content, and/or appropriateness; instruction may not successfully address the difficulties that many students are experiencing, etc. Overall, the lesson is quite limited in its likelihood to enhance students’ understanding of the discipline or to develop their capacity to successfully ‘‘do’’ mathematics. Instruction is purposeful and characterized by quite a few elements of effective practice. Students are, at times, engaged in meaningful work, but there are some weaknesses in the design, implementation, or content of instruction. For example, teacher may short-circuit a planned exploration by telling students what they ‘‘should have found’’; instruction may not adequately address the needs of a number of students; or the classroom culture may limit the accessibility or effectiveness of the lesson. Overall, the lesson is somewhat limited in its likelihood to enhance students’ understanding of the discipline or to develop their capacity to successfully ‘‘do’’ mathematics.

Level 2: Elements of effective instruction

Level 3: Beginning stages of effective instruction


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APPENDIX A (Continued) Level

Description

Level 4: Accomplished, effective instruction

Instruction is purposeful and engaging for most students. Students actively participate in meaningful work (e.g., investigations, teacher presentations, discussions with each other of the teacher, reading). The lesson is well designed and the teacher implements it well, but adaptation of content or pedagogy in response to student needs and interests is limited. Instruction is quite likely to enhance most students’ understanding of the discipline and to develop their capacity to successfully ‘‘do’’ mathematics. Instruction is purposeful and all students are highly engaged most or all of the time in meaningful work (e.g., investigation, teacher presentations, discussions with each other or the teacher, reading). The lesson is well designed and artfully implemented, with flexibility and responsiveness to students’ needs and interests. Instruction is highly likely to enhance most students’ understanding of the discipline and to develop their capacity to successfully ‘‘do’’ mathematics.

Level 5: Exemplary instruction

ACKNOWLEDGEMENTS Support for this article was funded in part by the National Science Foundation under two grants: 1) ESI-0138556 and 2) ESI-0333879 – The Center for the Study of Mathematics Curriculum. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

REFERENCES Australian Education Council (1994). Mathematics: A curriculum profile for Australian schools. Carlton, VIC: Curriculum Corporation. Ball, D. (1987). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1), 40–48. Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning. Mahwah, NJ: Lawrence Erlbaum Associates. Brown, C. A. & Cooney, T. J. (1982). Research on teacher education: A philosophical orientation. Journal of Research and Development in Education, 15(4), 13–18.

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Sam Houston State University Box 2206, Huntsville, TX 77341 USA E-mail: [email protected]

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Meredith Park-Rogers