Teacher EducaEtiroicna Q Nu.a Wrtaerlklye,r Summer 2007 Rethinking Professional Development for Elementary Mathematics Teachers By Erica N. Walker Introduction Researchers have found that despite reformers’ best efforts, teachers’ math- ematics classroom practice remains largely unchanged—in part because teachers hold fast to their own mathematics understandings, attitudes, and experiences (Ball, 1996; Raymond, 1997; Tzur, Martin, Heinz, & Kinzel, 2001). In particular, in the last decade, elementary mathematics teachers have found themselves balancing a number of sometimes competing requirements in their teaching: adhering to mathematics reform ini- Erica N. Walker is an tiatives in their school, district, and/or state; meeting assistant professor in the the expectations of principals and parents; and find- Program in Mathematics ing ways to ensure that their students are able to of the Department of perform adequately on standardized tests that have Mathematics, Science, significant ramifications for teachers and students if and Technology at students fail (Manouchehri, 1997; Raymond, 1997; Teachers College, Schoenfeld, 2002). In recent years, many teacher Columbia University, education programs have begun to address elemen- New York City, New tary mathematics instruction by helping prospective York. elementary teachers expand their knowledge of math- 113 Rethinking Professional Development ematics content. This has often occurred through mandating more mathematics courses (American Mathematical Society, 2001); but often these courses have not focused on the special needs of elementary teachers. Further, the support that these teachers receive once they leave teacher education programs is often sporadic and shallow (Borman & Associates, 2005). With the advent of new curricula, professional development for elementary teachers is often heavily focused on implementation of a particular curricular package, which may target organizational or logistical requirements of the curriculum rather than mathematics content or pedagogy aligned with content objectives. Many of these curricula, seeking alignment with National Council of Teachers of Mathemat- ics (NCTM) standards reform documents (1989; 2001), require substantially more teacher engagement with students than ‘traditional’ textbooks and their framers expect elementary teachers to deeply understand the underpinnings of elementary mathematics (D’Ambrosio, Boone, & Harkness, 2004). In order to spur student learning of mathematics, rather than just performance, teachers are expected to respond to student misconceptions, help students develop conceptual understand- ing, and provide multiple curricula and media to do it (American Mathematical Society, 2001; Frykholm, 1999). This can be difficult when teachers themselves may hold misconceptions, have limited rather than deep conceptual understanding of mathematical topics, and may not understand how working with different media and manipulatives can contribute to student thinking and learning in mathematics. I developed a professional development model designed to address these issues as part of a larger study of an intervention, Dynamic Pedagogy,1 targeting Grade 3 students and their teachers in an ethnically and socioeconomically diverse school district in upstate New York. My fellow researchers and I, through Dynamic Pedagogy, sought to improve student learning and performance in mathematics, as well as develop ‘habits of mind’ conducive to life long learning habits among these children. However, we soon realized that we first had to enhance teacher understand- ing of mathematics and help teachers to create mathematics classroom experiences that would foster student thinking, so that teachers would be able to effectively implement the Dynamic Pedagogy intervention. This paper discusses the profes- sional development model and describes how it was reflected in the classroom practice of participating teachers. Because much of the literature in teacher education is silent on the mechanisms by which teacher education and professional development affect actual classroom practice, I also report how this model influ- enced one teacher’s planning and instruction in mathematics. Background Teaching elementary mathematics requires both considerable mathematical knowl- edge and a wide range of pedagogical skills. For example, teachers must have the patience to listen for, as well as the ability to hear the sense. . . in children’s mathematical ideas. They need to see the topics they teach as embedded in rich networks of 114 Erica N. Walker interrelated concepts, know where, within those networks, to situate the tasks they set their students and the ideas these tasks elicit. In preparing a lesson, they must be able to appraise and select appropriate activities, and choose representations that will bring into focus the mathematics on the agenda. Then, in the flow of the lesson, they must instantly decide which among the alternative courses of action open to them will sustain productive discussion. (American Mathematical Society, 2001, p. 55) The American Mathematical Society and Mathematical Association of America’s (2001) emphasis on mathematical and pedagogical knowledge for elementary teachers has been underscored by the research literature documenting that elementary mathematics teaching and learning in the United States has too often been limited to rote, didactic experiences that do not propel mathematics thinking (Ball, 1996; National Research Council, 2001). Indeed, “[w]e must recognize that many current elementary teachers’ mathematical understanding is far from ideal (Ma, 1999)” (Farmer, 2003, p.333). Many elementary mathematics teachers “were not adequately prepared by the mathematics instruction they received” (AMS, 2001. p. 55). They may have limited content knowledge and in addition, may have an aversion to exploring mathematics content more deeply (D’Ambrosio, Boone, & Harkness, 2004; Frykholm, 1999; Thompson, 1992). This is a critical aspect, given that “subject matter knowl- edge significantly impacts classroom instruction as well as teachers’ decisions with respect to the selection and structure of teaching content, classroom activities, assignments, and choices in curriculum materials” (Shulman & Grossman, 1988, p. 3). Thus, it should be no surprise that professional development that focuses on the enhancement of content knowledge is linked to improvement in student mathematics achievement (Saxe, Gearhart, & Nasir, 2001). Good professional development for elementary teachers should effectively address this problem of enhancing teachers’ content knowledge. Content knowl- edge, however, is not enough: it is important that teachers develop effective teaching strategies and practice (Graham & Fennell, 2001). In doing so, they should relinquish their own reliance on procedural explanations for mathematics concepts2 (Frykholm, 1999). Echoing the recommendations of NCTM and AMS, Frykholm (1999) urges that “[t]eachers must understand mathematics deeply themselves if they are to facilitate the types of discussions and handle the various questions that emerge when learners are engaging in authentic mathematical experiences”(p. 3). However, it must be noted that elementary mathematics content development for teachers is hampered by the perception by many in the field that elementary mathematics is ‘simple’ and not worthy of extensive discussion, despite evidence of its rich conceptual underpinnings (Frykholm, 1999; AMS, 2001). Much of the understanding that both elementary and secondary teachers say they have is procedural and rule-based, rather than conceptually oriented. This understanding colors teachers’ views of instruction and limits their pedagogy to lecture and explication of procedures rather than expanding it to include students’ thinking and ideas and development of conceptual knowledge and understanding. 115 Rethinking Professional Development The idea that teachers’ content knowledge is linked to their pedagogical practices is not a new one. Shulman and Grossman (1988) have written extensively about the relationships between content knowledge, pedagogical knowledge, and pedagogical content knowledge. In particular, pedagogical knowledge can be discussed and shared, but is more powerfully “shaped through experiences with children” (Graham and Fennell, 2001). Pedagogical content knowledge (PCK) incorporates content knowl- edge of a discipline but also teachers’ “knowledge of the subject, supplemented with knowledge of students and of learning; knowledge of curriculum and school context; and knowledge of teaching” (Manouchehri, 1997, p. 201). Specifically, PCK requires teachers to understand “multiple representations of mathematical ideas, topics, and problems; the complexities of teaching and learning certain mathematical concepts; and the cognitive obstacles that learners face when they engage in certain topics in the mathematical curriculum” (Manouchehri, 1997, p. 201). This third category of knowledge reflects the importance of selecting and using appropriate activities and strategies in the classroom based on teachers’ understand- ing of what students know and need to learn. In short, teachers should know that it is not sufficient to just use pattern blocks3 as representations of fractions in a lesson and have students complete a worksheet about fractional parts; rather, teachers should know how and why these blocks could be used to develop students’ understanding of fractional concepts. Yet we see less developed research about this important but practical aspect of teaching—how teachers select activities appropriate for what their students need to learn (see Stein, Smith, Henningsen, and Silver (2001) for an exception). Further, what do teachers understand about their own knowledge and, as importantly, how do they implement this understanding in their classrooms? One way to discern teachers’ understanding of mathematics and their instructional practice is to analyze the kinds of tasks that teachers provide and the kinds of questions that teachers ask students during mathematics lessons (Tirosh, 2000). As Sullivan and Clarke (1991) state: Good questions have more than one correct mathematical answer; require more than recall of a fact or reproduction of a skill; are designed so that all students can make a start; assist students to learn in the process of solving the problem; and support teachers in learning about students’ understanding of mathematics from observing/ reading solutions (p. 337). But this level of questioning may be very different from the types of experiences that elementary teachers received themselves—in elementary school, secondary school and post secondary education institutions (AMS, 2001). Improving elemen- tary mathematics instruction, then, requires that we consider that the extent of mathematics preparation for teaching for many elementary teachers was attained in teacher preparation programs where they may have taken a single mathematics methods class. The role of Professional development becomes critical—and should be used as a site for content knowledge development. 116 Erica N. Walker In summary, effective professional development models should respect and address teachers’ existing beliefs about mathematics because these affect their instruction. Often this is a hidden agenda of providers. It should be noted, however, that teachers may not choose to participate in a professional development project to change their own beliefs about mathematics and its teaching, but this is often the “hoped-for outcome on the part of providers” (Farmer, 2003, p. 332). Professional development should also address the links between content knowledge, pedagogical knowledge, and pedagogical content knowledge (Shulman & Grossman, 1988; Ball & Bass, 2000). This underscores the need for professional development to be situated in school- and classroom-based contexts. Professional development should be sustained and ongoing. As Ma (1999) notes, “a single [PD] workshop, without periods of gestation and sustained support, does not afford sufficient time for teachers to develop a deep understanding of the mathematics they teach; such understandings develop, if at all, over longer periods” (p.333). Finally, but most importantly, I argue that it is necessary for teachers to be active participants in and constructors of the professional development experience so that they can share and analyze their own rich classroom experiences.4 In short, teachers should participate in the knowledge construction process that, hopefully, emerges from professional development. In designing the professional development expe- rience for teachers I asked, “How do we create opportunities for teachers to engage in collaboration that facilitates their own development—so that they can effec- tively model this important aspect of mathematics learning for students?” As we encourage opportunities for students of all ages to work together as communities of learners—a recognized effective teaching strategy—it is important that we create these communities for teachers as well (Lachance & Confrey, 2003). Method Designing the Professional Development Model Professional development for teachers was one major component of the Dynamic Pedagogy intervention. Our initial goals for professional development targeted three interrelated major problems affecting elementary school teaching: inflexible attitudes about mathematics and its teaching (Raymond, 1997), lack of deep understanding of basic mathematical concepts (Ma, 1999), and a teacher- centered approach to teaching that does not utilize students’ substantial knowledge of mathematics, gained from their in- and out-of school experiences (Walker, 2003). However, the primary goal was to address what the framers of Dynamic Pedagogy believed to be an underlying fundamental element: to enhance teachers’ mathemat- ics content knowledge meaningfully. In this section, I focus on sharing the process of developing and revamping the professional development model. The initial focus shifted to one that incorporated a framework to help teachers more effectively design and implement classroom 117 Rethinking Professional Development activities that demonstrate connections across mathematics topics; provide students with multiple representations of mathematics concepts to facilitate students’ math- ematical knowledge development; and address student misconceptions of mathemat- ics topics. These three components (connections, representations, and misconcep- tions) eventually formed the core of the professional development model, referred to as CRM (Figure 1). As the professional development seminars progressed, the research team expected that CRM professional development would also aid teachers in selecting appropriate curricular directions and providing students with rich problem- solving opportunities. In essence, I wanted to ensure that the CRM professional development framework, focusing on mathematics content, would help teachers better implement Dynamic Pedagogy in their classrooms. To address teachers’ mathematics knowledge and pedagogy, I designed and, along with co-facilitators, held monthly professional development seminars (Table 1) with nine grade 3 teachers participating in the Dynamic Pedagogy intervention. Seminars included four two and a half-hour after-school sessions, and two full day workshops. In addition, my co-facilitators and I held a three day Summer Institute the summer preceding the implementation of Dynamic Pedagogy. Each seminar targeted a particular elementary mathematics topic (e.g., fractions, number sense, geometry). Because the research team was interested in the learning that took place in professional development seminars and also evidence of teachers using what they had learned during class instruction, we collected data from multiple sources. During each seminar, held in September, November, February, and April, members of the research team took field notes, documenting problem solving and pedagogi- cal discussions. Members of the research team observed teachers’ lessons during Figure 1. CRM (Connections, Representations, and Misconceptions) Professional Development Model. 118 Erica N. Walker Table 1. Description of Professional Development Seminars Date Content Focus Duration June 2003 Dynamic Pedagogy Orientation; 3 days Number Sense; Fractions; Geometry; Measurement September 2003 Number Sense 2.5 hours October 2003 Dynamic Pedagogy Principles; 1 day Number Sense; Fractions November 2003 Fractions 2.5 hours January 2004 Dynamic Pedagogy Principles; 1 day Fractions and Geometry February 2004 Geometry 2.5 hours April 2004 Measurement 2.5 hours May 2004 Dynamic Pedagogy Principles; 2 hours Reflection each of the four Dynamic Pedagogy units (number sense, fractions, geometry, and measurement), and videotaped a teacher’s lesson at least once. Field notes (Denzin & Lincoln, 2000) were taken during observations. After observing lessons, research- ers conducted informal interviews with teachers and provided critical feedback. Teachers submitted portfolios for each unit, which included lesson preplans, plans, self-assessments, and samples of student work. Because the unit on fractions took place several months after the first unit, number sense, we were able to document evidence of the impact of professional development seminars. Later in this paper, I focus specifically on evidence of one teacher who incorporated elements of the CRM framework in her teaching as the year progressed. Developing the CRM Framework The CRM framework for professional development emerged, in part, because of the major focus of the professional development seminars, which was to develop teachers’ content and pedagogical content knowledge. Initially, we (the facilita- tors) began professional development sessions with teachers solving mathematics problems collaboratively. Although the problems were not at a level beyond high school algebra, we found that teachers were visibly anxious and reluctant to work together. Comments like “This is too hard,” “I forgot how to do this,” and “I was never good in math” abounded. Even though the problems teachers were asked by seminar leaders to solve were linked to the content of the curricular units that teachers were engaged in at those points in time with their students, the sessions could not progress in the ways we wanted them to without addressing teachers’ perspectives about mathematics (Raymond, 1997). 119 Rethinking Professional Development I realized that we needed to “work from what teachers do know-the mathematical ideas they hold, the skills they possess, and the contexts in which these are understood” (AMS, 2001, p. 57). I adjusted the organization of the sessions to address teachers’ strengths. While still maintaining a commitment to developing and expand- ing teachers’ mathematical knowledge, my co-facilitators and I did so by integrating discussions about problem solving, mathematics content, and pedagogical practice. We did this by organizing the professional development sessions so that teachers and seminar leaders talked about content as a part of discussions about teaching and learning episodes, student work, and lesson planning. I developed the CRM professional development framework based on the research literature and the recommendations of the NCTM and AMS pertaining to effective teaching and staff development for discussing teaching and learning episodes, student work, and lesson planning. The three major components of the framework— connections (between mathematical topics, students’ in- and out-of-school knowl- edge, and procedural and conceptual knowledge); multiple representations (of mathematical concepts, problems, and solutions), and misconceptions (about math- ematics concepts and procedures)—do not operate in isolation (Figure 2). Rather, I expect that these elements inform each other during the teaching-learning process. The teachers and facilitators thought and talked at length about students’ misconcep- tions, and how to help them develop multiple representations of concepts and see connections between mathematical topics. We worked to establish a space for teachers to engage in more rigorous mathematics, as well as develop their pedagogical knowledge. Because teachers worked collaboratively and heard from other teachers about their work, they were able to benefit from the experiences of others. Essentially, we attempted to model with the teachers during professional development sessions Figure 2. Hypothesized Relationships between CRM Professional Development, Classroom Instruction, and Student Mathematics Performance. 120 Erica N. Walker the way that we wanted them to help their students to engage in mathematics thinking and learning in the classroom. Like Farmer (2003), . . . [b]y using activities that were adaptable to the elementary level and sufficiently challenging to the teachers, we could address the desire of some teachers to add to their repertoire while still meeting a need to learn more mathematics. (p.333-334) My fellow researchers and I hypothesized that the work that teachers did during professional development seminars would inform their classroom teaching; simi- larly, their classroom teaching experiences would inform the content of the professional development seminars. This iterative and reciprocal process, we hypothesized, would enhance teachers’ teaching and could also help to spur student achievement in mathematics. We developed a coding scheme to analyze the presence of CRM indicators in teachers’ instructional practice. A team of two raters reviewed transcripts of videotaped lessons for the fractions units, and coded teachers’ questions posed to students, conversations with students, and responses to student questions (teacher- learner interactions) as encompassing connections, representations, misconcep- tions, or procedures. For example, raters coded the following: Teacher: We have been working with our fraction strips and our pattern blocks to study fractions [representations]. I want to start out by talking about a little bit of real life experience. When do you use fractions in your everyday life? [connection-real world] Very rarely did coders have different evaluations of the episodes; when this occurred it was usually because a particular exchange encompassed more than one CRM category. In these instances coders discussed the teacher-learner interactions and came to an agreement about how the episode should be coded. Some episodes, underscoring the interaction between the elements of CRM, belonged to more than one category and were coded accordingly. Following coding, raters counted the number of teacher-learner interactions within the lessons that revealed teachers’ emphasis on connections between mathematical topics, multiple representations, and/or student misconceptions. In addition, we found there were certain tasks that solely required students to recall a certain procedure, without making connections to conceptual understanding (Table 2). Findings Learning During Professional Development Seminars I first describe below how elements of the CRM model were enacted in a sample professional development seminar focusing on fractions (Table 1). I began by facilitating a discussion about participants’ prior classroom experiences teaching 121 Rethinking Professional Development Table 2. CRM Categorization of Representative Teacher Tasks and Questions. students about fractions. Teachers talked together about how they have introduced fractions to students and their rationale for introducing them in this way. The teachers’ ideas shared during this particular part of the session encompassed primarily activities (games or tasks to do with students) and questions they ask of students: Gloria: I ask my kids if they know what a fraction is. But they just came back with “numerator” and “denominator.” They don’t really understand what those words mean. Other teachers agreed with Gloria’s assessment. The discussion then turned to other common misconceptions that children have or mistakes their students made in working with fractions. Like Tirosh (2000), my co-facilitators and I found that teachers initially focused on procedural or algorithmic aspects of working with fractions. I then presented a problem from the 1992 National Assessment of Educational Progress mathematics assessment for 4th graders: Think carefully about the following question. Write a complete answer. You may use drawings, words, or numbers to explain your answer. Be sure to show all of your work. José ate ½ of a pizza. Ella ate ½ of another pizza. 122