Examining novice teacher leaders’ facilitation of mathematics professional development

Journal of Mathematical Behavior 33 (2014) 149–167

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The Journal of Mathematical Behavior journal homepage: www.elsevier.com/locate/jmathb

Examining novice teacher leaders’ facilitation of mathematics professional development Hilda Borko a,1 , Karen Koellner b,∗ , Jennifer Jacobs c a b c

Stanford University, School of Education, 485 Lausen Mall, Stanford, CA 94305-3096, United States Hunter College, CUNY, School of Education, United States University of Colorado Boulder, Institute of Cognitive Science, United States

a r t i c l e

i n f o

Article history: Available online 11 December 2013 Keywords: Mathematics professional development Professional development leaders Teacher leaders Facilitating professional development Knowledge for professional development

a b s t r a c t This paper reports on novice teacher leaders’ efforts to enact mathematics PD through an analysis of their facilitation in workshops conducted at their schools. We consider the extent to which teacher leaders facilitated the Problem-Solving Cycle model of PD with integrity to its key characteristics. We examine the characteristics they enacted particularly well and those that were the most problematic to enact. Facilitators were generally successful with respect to workshop culture and selecting video clips for use in the PD workshops. They had more difficulty supporting discussions to foster aspects of mathematics teachers’ specialized content knowledge and pedagogical content knowledge. We suggest a number of activities that may help to better prepare novice PD leaders to hold effective workshops. Furthermore, we conjecture that leaders of mathematics PD draw from a construct we have labeled Mathematical Knowledge for Professional Development (MKPD), and we posit some domains that may comprise this construct. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Professional development (PD) opportunities for mathematics teachers are widely recognized as a critical component of efforts to support increased student achievement. A growing body of empirical research on the structure, content, and outcomes of effective PD offers insights into the characteristics of programs that provide high-quality, high-impact learning opportunities (Borko, Jacobs, & Koellner, 2010; Desimone, 2009; Wei, Darling-Hammond, Andree, Richardson, & Orphanos, 2009). However, having a deep understanding of what PD should look like is only part of the equation. An often overlooked variable is having well-prepared facilitators to ensure the PD’s effectiveness (Katz, Earl, & Ben Jaafar, 2009; Stein, Smith, & Silver, 1999). Presently in the United States, PD facilitators represent a new cadre of prominent players on the educational scene, and in the field of mathematics most PD facilitators are only in the beginning stages of honing their leadership skills (Zaslavsky & Leikin, 2004). In order to promote powerful and lasting change in the teaching profession there is an urgent need to prepare novice PD facilitators to successfully facilitate newly developed PD models that offer high-quality learning opportunities for teachers. Practicing mathematics teachers appear to be an obvious personnel source to assume leadership positions, especially for the delivery of local, site-based PD. However, having inservice teachers work with adult learners—typically teachers in their schools—on issues involving mathematics learning and instruction is very different compared to their usual work

∗ Corresponding author. Tel.: +1 6463307144. E-mail addresses: [email protected] (H. Borko), [email protected] (K. Koellner). 1 Tel.: +1 650 723 7640. 0732-3123/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmathb.2013.11.003

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of teaching mathematics to K–12 students. How can we foster the leadership capacity of a large base of mathematics PD facilitators to keep up with the demand for widespread, high-quality professional learning opportunities? What sort of guidance and support do novice facilitators need in order to be successful? Researchers are just beginning to investigate these questions, with an eye toward characterizing the mathematical knowledge and skills that leaders require (e.g., Borko, Koellner, & Jacobs, 2011; Elliott et al., 2009; Koellner, Jacobs, & Borko, 2011; Schifter & Lester, 2005). Especially critical is research on the knowledge and practices necessary to be an effective leader of mathematics PD. We must understand what PD leaders should do to cultivate improvements in mathematics classroom instruction that will result in increased student learning of complex subject matter (Even, 2008; LeFevre, 2004). Articulating the practice of leading high-quality PD is a key component in defining the process through which these programs contribute to advances in mathematics teacher knowledge, instructional practice, and student learning. Without closer attention to facilitation PD programs, although designed in accordance with general criteria of effectiveness, may, when enacted, fail to produce increases in student achievement (e.g., Garet et al., 2011). To meet an increasing demand for teacher learning opportunities, PD programs must be both sustainable and scalable (Marrongelle, Sztajn, & Smith, 2013; Wilson, 2013). They must adjust to local contexts so that the work can be carried out by schools and districts on a long-term basis, using internal resources (Loucks-Horsley, Love, Stiles, Mundry, & Hewson, 2003). Borko (2004) suggested a three-phase research agenda for designing, implementing, and investigating scalable PD. In Phase 1, researchers design a PD program and provide initial evidence that it can have a positive impact on teacher learning. In Phase 2 researchers “determine whether the professional development program can be enacted with integrity in different settings and by different professional development providers” (p. 9). “Major design activities in Phase 2 include refining a professional development program’s tasks and materials for teachers. . ., specifying the role of the facilitator, and developing resources and training for facilitators” (p. 10). Finally, in Phase 3, researchers compare multiple PD programs, investigating their impact on teacher and student learning, and their resource requirements for successful enactment across sites. To date, very little research has been conducted on efforts to scale up professional development. The vast majority of research on PD programs is Phase 1 research. Phase 2 and Phase 3 investigations are rare; as Borko (2004) noted; her literature review “did not yield any professional development programs for which there is adequate evidence that they can be enacted with integrity by multiple facilitators or in multiple settings” (p. 10). Since that time, a small number of Phase 2 (e.g., Bell, Wilson, Higgins, & McCoach, 2010) and Phase 3 (e.g., Heller, Daehler, Wong, Shinohara, & Maritrix, 2012; Penuel, Gallagher, & Moorthy, 2011) investigations have been conducted, providing important information about the effectiveness of established PD programs. The study discussed in this paper can be considered a Phase 2 investigation, where researchers provided initial preparation and ongoing support to local providers of PD who, in turn, implemented the PD model with teachers in their schools while researchers documented the process and the impacts. This paper reports on these novice teacher leaders’ efforts to enact PD, through an analysis of their facilitation skills as they conducted PD workshops in their schools. 2. Conceptual framework: Scalable high-quality mathematics professional development 2.1. Structure and content of high-quality mathematics professional development High-quality PD refers to both the process and structure of the PD program and the PD content. With respect to process and structure, high-quality PD programs provide opportunities for teachers to participate actively and collaboratively in a professional learning community, situated in the practice of teaching (Brodie & Shalem, 2011; Hawley & Valli, 2000; Knapp, 2003; Wilson & Berne, 1999). Skillfully selecting and using artifacts, such as video clips or student work, is one way for facilitators to situate PD in practice (Jacobs, Borko, & Koellner, 2009; Sherin, 2007; Taylor, 2011; van Es & Sherin, 2010;). Trust and respect are also important aspects of PD. Leaders in effective PD programs pay explicit attention to creating a safe and supportive professional community where teachers are willing to share information often considered “private,” such as video from their own classrooms, and to engage in constructively critical conversations about their practice (Borko, Jacobs, Eiteljorg, & Pittman, 2008; Brodie & Shalem, 2011; Gerard, Varma, Corliss, & Linn, 2011; Little, 2002). Impactful leaders also model instructional strategies—for example, practices for engaging students in mathematics problem solving and for facilitating productive discussions (Clark, Jacobs, Pittman, & Borko, 2005; Stein, Engle, Smith, & Hughes, 2008). These characteristics ensure that participants experience effective instructional strategies as learners and then reflect collaboratively on their learning (Borko et al., 2010; Desimone, 2009; Taylor, 2011; Wei et al., 2009). With respect to content, high-quality PD helps teachers to develop the knowledge and skills they need to support their students’ learning (Borko et al., 2010; Desimone, 2009; Wei et al., 2009). Shulman (1986) introduced a framework for articulating the relevant domains of professional knowledge, a structure expanded upon in the field of mathematics education by Ball et al. in the Learning Mathematics for Teaching (LMT, 2006) project, through which they identified and elucidated the construct “mathematics knowledge for teaching” (MKT)—the mathematical knowledge that teachers must have in order to teach mathematics effectively. Within the broader construct of MKT, they explored four categories that are central to performing the recurrent tasks of teaching mathematics to students: (1) common content knowledge, (2) specialized content knowledge, (3) knowledge of content and students, and (4) knowledge of content and teaching (Ball & Bass, 2000; Ball, Thames, & Phelps, 2008). Bringing this research into the design of PD, educators have highlighted the importance of strengthening skills focused on teaching with rich mathematics problems (Lampert, 2001) and eliciting and building on

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student thinking (Kazemi & Franke, 2004; Roth et al., 2011). Studies have shown a positive impact on student achievement when PD leaders forefront the development of MKT (Hill & Ball, 2004; Hill, Rowan, & Ball, 2005). 2.2. Preparation of mathematics professional development leaders A central component of a sustainable, scalable PD model is the ability to prepare PD leaders who can adapt the model to a variety of local contexts and advocate for school and district support of the PD while maintaining integrity to its goals and design (Cobb & Smith, 2008; Goos, Dole, & Makar, 2007). For site-based PD programs, building the leadership capabilities of local professional developers is critical. PD leaders provide the human resources for building a district’s internal capacity to promote ongoing learning among its teachers. They can foster professional learning communities and collaborative cultures, increasing the human and social capital of schools and districts (Little, 1990). Without strong leadership, professional learning communities are likely to be ineffective and fail to promote either teacher learning or increased student achievement (McLaughlin & Talbert, 2006). Despite its importance, developing the knowledge base, experience, and leadership skills of local leaders is often a missing step in educational reform efforts (Loucks-Horsley et al., 2003; McLaughlin & Talbert, 2006). In the field of mathematics education, a small number of researchers are focusing on what PD leaders should know and be able to do, and on how to prepare and support them (Elliott et al., 2009; Even, 2008; Koellner et al., 2011). In the project Research on Mathematics Leaders’ Learning (RMLL, Elliott et al., 2009) researchers conducted a series of seminars for mathematics PD facilitators, aimed at developing knowledge and skills for cultivating mathematically rich learning opportunities for teachers. They are studying the specialized knowledge that leaders need to facilitate teachers’ learning of MKT. Their assertion, based on initial analyses, is that leaders must be able to identify mathematics problems and discussion prompts that promote in-depth conversations focused on the mathematics content, support productive social interactions, and orchestrate discussions that help teachers unpack their often highly symbolic or incomplete reasoning (Elliott et al., 2009). Findings from RMLL suggest that to successfully facilitate mathematics PD, leaders require knowledge of mathematics that includes MKT along with knowledge about how to facilitate the development of MKT. 3. Project design: preparing teacher leaders to facilitate the Problem-Solving Cycle We report on the results from a multi-year design research project, Toward a Scalable Model of Mathematics Professional Development: A Field Study of Preparing Facilitators to Implement the Problem-Solving Cycle (iPSC). The iPSC project investigated the scalability and sustainability of the Problem-Solving Cycle (PSC) model of mathematics PD (described in the next section), and in particular the degree to which the PSC could be implemented with integrity by novice local facilitators. The study included 2½ years of preparation and support for teacher leaders (TLs) from the participating school district, teachers who volunteered (or were selected by school or district administrators) to facilitate the PSC with the mathematics teachers in their schools. Our research incorporated video, classroom artifacts, and interviews to document the preparation and support provided to the TLs; the range and quality of their implementation of the PSC; and the impact of the intervention on TLs, teachers, and students. In this paper, we focus on the range and quality of the TLs’ implementation of the PSC. Elsewhere we have written about the preparation and support provided to the TLs and the impact of the intervention (Jacobs, Koellner, & Funderburk, 2012; Koellner et al., 2011). As depicted in Fig. 1, our ultimate goal in the iPSC project was improved student learning. Our theory of action focused on teaching quality as a key factor in impacting student learning. Teaching quality was addressed by preparing TLs to provide PD for mathematics teachers, using the Problem-Solving Cycle. The PD of both TLs and teachers was informed by a process of mutual adaptation between the model and the organizational and cultural/linguistic contexts of the district and schools (Berman & McLaughlin, 1978; Snyder, Bolin, & Zumwalt, 1992). This article highlights one central component of our theory of action: the impact of PD for the TLs. In particular, we focus on novice teacher leaders’ efforts to enact the PSC model of PD effectively. We address the following research questions: 1. To what extent did the teacher leaders enact the Problem-Solving Cycle with integrity to its key characteristics? 2. Which characteristics of the PSC did the teacher leaders enact particularly well? Which characteristics were the most problematic to enact? 3.1. The Problem-Solving Cycle The Problem-Solving Cycle (PSC) is an iterative, long-term approach to mathematics PD (Borko et al., 2005; Jacobs et al., 2007; Koellner et al., 2007; Koellner, Schneider, Roberts, Jacobs, & Borko, 2008). The key characteristics of the PSC are derived from the research on the nature of high quality, effective PD (Borko, 2004; Borko et al., 2010; Desimone, 2009): • The PD program is ongoing, long-term, and adaptive to participants’ needs and priorities; • Communities of practice play a central role in determining what and how people learn;

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Fig. 1. Implementing the Problem-Solving Cycle: theory of action.

• PD activities are situated in teachers’ classroom instruction through tangible artifacts of practice, particularly video; • The PD aims to improve content knowledge in a specific domain; • The PD focuses on student thinking and provides teachers with opportunities to make connections to their own instructional practice. As implemented in the iPSC project, the PSC entails multiple cycles of three interconnected PD workshops, all organized around a rich mathematics task (see Fig. 2). For iPSC we selected tasks within the mathematics domain of ratio and proportion. Each cycle uses a different mathematical task and highlights specific topics related to student learning and instructional practices. During Workshop 1 of a given cycle, teachers collaboratively solve the selected mathematics task and develop plans for teaching it, taking into consideration the needs of their students. The goals of this workshop are to help teachers develop a deeper knowledge of the subject matter and strong planning skills. After the first workshop, teachers implement the problem with their own students and their lessons are videotaped. The facilitators then select video clips that highlight key moments in the instruction and in students’ thinking about the problem. Workshops 2 and 3 of the cycle focus on the teachers’ classroom experiences and rely heavily on the selected video clips. The goals of these two workshops are to help

Fig. 2. The Problem Solving Cycle.

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Fig. 3. Implementing the Problem-Solving Cycle: structure of support for TLs.

teachers learn how to elicit and build on student thinking, and to explore a variety of instructional strategies for teaching with rich problems based on targeted learning goals. 3.2. Teacher leader preparation A central goal of the iPSC project was to prepare novice teacher leaders to implement the PSC with integrity to its key characteristics. By integrity we mean that the teacher leaders adhere to the intended goals and design of the PD; maintaining integrity does not imply rigidly implementing a specific set of activities and procedures (Borko, 2004; LeFevre, 2004). Implementing the PSC with integrity entails using a rich mathematics problem as a shared experience; facilitating productive discussions about the mathematical content, student thinking and instructional practices; focusing attention on multiple representations and solution strategies; and using video from the teachers’ own classrooms. Based on prior research in which the developers of the PSC model also served as the PD facilitators, we have documented preliminary evidence of the effectiveness of the PSC as well as an emerging understanding of the characteristics of successful facilitation (Borko et al., 2008; Borko, Jacobs, Seago, & Mangram, in press; Clark et al., 2005; Jacobs et al., 2007; Jacobs, Koellner, John, & King, in press; Koellner et al., 2008). This research identified basic supports that novice facilitators of the PSC would be likely to need in order to (1) create a professional learning community, (2) facilitate mathematics discussions with teachers, and (3) facilitate video-based discussions to help teachers examine student thinking and classroom instruction. The iPSC project prepared full-time mathematics teachers to take on the role of PSC facilitator and learn how to lead PSC workshops in their schools. The project provided 2½ years of ongoing, yet gradually decreasing, support for the TLs. This support involved two major components (see Fig. 3): a summer leadership academy and multiple cycles of structured guidance for facilitating the PSC. During Year 1, for the first wave of TLs, the project incorporated one academic semester in which the TLs participated in a cycle of the PSC facilitated by the research team, as shown in Table 1. (For future waves of TLs, this type of introduction to the PSC “as teacher participants” was not feasible.) All TLs attended a summer leadership academy focused on explicating the key characteristics of the PSC and preparing to facilitate PSC workshops. Summer academies were held during each year of the project, and included both new and returning TLs. During these academies the TLs reflected on their experiences as PSC “teacher participants” (in Year 1 only), viewed and discussed selected video clips from prior PSC workshops with respect to key PSC characteristics, developed a general plan for implementing the PSC with mathematics teachers in their schools, and participated in PSC simulations (mini-cycles) using the mathematics problems selected for the upcoming academic year PSC cycles. In initial simulations, members of the research team modeled practices central to the successful enactment of PSC workshops. Subsequently TLs planned and then took turns leading simulations of the various activities that compose the three workshops. These simulations are examples of the approximations of practice—“opportunities to rehearse and develop discrete components of complex practices in settings of reduced complexity” that Grossman and McDonald (2008, p. 190) recommended incorporating into teacher education programs. In addition, members of the research team provided ongoing structured guidance as the TLs facilitated the PSC. Prior to conducting each PSC workshop, TLs attended a full-day Instructional Support Meeting led by the research team. These Table 1 iPSC project implementation timeline. Year 1 7 TLs (4 schools) Experience PSC as teachers January–May 2008

Summer Academy 1 June

Year 2 5 TLs (3 schools) Facilitate PSC Cycle 1 August–December

Year 3 8 TLs (6 schools) Facilitate PSC Cycle 2 January–May 2009

Summer Academy 2 June

Facilitate PSC Cycle 3 August–December

Facilitate PSC Cycle 4 January–May 2010

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Table 2 Number of schools, leaders, and teachers participating in iPSC project by date. Participation dates

Teacher leaders

Year 1 (spring–summer 2008) Year 2 (fall 2008–summer 2009) Year 3 (fall 2009–spring 2010) Totalsa

Teachers

Middle schools

7 5 8

0 13 45

4 3 6

12

54

8

a

Owing to a variety of factors, schools, leaders, and teachers participated for either 1, 2, or 3 years. Numbers in this row indicate total number of schools, TLs, and teachers participating in project for 1 or more years. In Year 1, TLs engaged in a series of PSC workshops in their role as classroom teachers. Facilitation of workshops began in Year 2.

meetings were designed to assist TLs in planning and conducting all aspects of their upcoming PSC workshops. The first Instructional Support Meeting (prior to PSC workshop 1) supported TLs to lead discussions with teachers that would help them identify and unpack the mathematics content embedded in the selected PSC problem. After the first PSC workshop, the teachers (as well as the TLs) taught the PSC problem in one of their classes. Digital copies of the videotaped lessons were provided to the TLs. The second and third Instructional Support Meetings (held prior to PSC workshops 2 and 3) supported TLs to choose appropriate video clips from the lessons taught by their teachers using the PSC problem, write guiding questions, and lead discussions based on the video clips. The meetings also addressed ways the TLs might tailor their workshops to each school’s context (e.g., cultural/linguistic diversity in the student population, specific workplace norms, constraints on time and scheduling). The research team remained available to TLs on an as-needed basis throughout the school year to address questions and concerns that arose outside of the workshop preparation meetings. All TLs received a copy of the Facilitator’s Guide to the Problem-Solving Cycle (available on our website, psc.stanford.edu) during the first summer academy they attended. The Facilitator’s Guide is designed to help facilitators learn about the key characteristics of the PSC and provide ongoing support for its successful implementation. It provides a description of and rationale for the types of activities that constitute each PSC workshop, describes the various decisions facilitators need to make as they prepare for and conduct each workshop, and includes examples from the research team’s experiences in developing the PSC model and conducting PSC workshops. The research team continually made use of the Guide during the summer academies and Instructional Support Meetings, drawing TLs’ attention to relevant sections as they learned about and prepared to conduct each PSC workshop. 4. Research methods 4.1. Participants The iPSC project involved a partnership with a large urban school district in the Western United States. The district mathematics coordinator, with support from other district administrators, agreed to help recruit mathematics teachers from their middle schools to serve as Teacher Leaders and learn to implement the PSC in their own schools. As shown in Table 2, the project began in Spring 2008 with seven TLs from four schools. One school dropped out of the project after the first summer academy owing to a change in their local administration. The five TLs from three schools who continued with the project facilitated two cycles of the PSC during the 2008–2009 academic year (1 cycle per semester). In the following year, another school dropped out of the project, but four new schools joined. Thus, eight TLs from six schools participated in the second summer academy and facilitated two cycles of the PSC during the 2009–2010 academic year. Table 3 provides additional information about the TLs: the year they joined the project, their school name (pseudonym), years of teaching experience, whether they were the mathematics department chair in the school, and the number of teachers with whom they worked. Six of the TLs individually facilitated PSC workshops, and two of the TLs co-facilitated. Table 3 Teacher leader demographic information. Participant (TL) namea c

Jordan (J) Mandy (M)c Robert (R) Candace (Cn) Carla (Cr) Kaitlyn (Kt) Kyla (K)d Jason (J)d a b c d

Date joined project

School name1

Years’ experienceb

Department chair

January 2008 January 2008 January 2008 June 2009 June 2009 June 2009 June 2009 June 2009

Pride Pride Champion Woodlawn Torrence Road Four Reed Fire Crest Fire Crest

10 1 32 13 15 19 13 20

No No Yes No Yes Yes No Yes

All names of teacher leaders and schools in this paper are pseudonyms. Years’ experience refers to number of years of teaching experience at time individual first joined iPSC project. Jordan and Mandy taught at the same school, but led separate PSC workshops with different groups of teachers within their school. Kyla and Jason co-facilitated all of the PSC workshops within their school.

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Fig. 4. Lemonade problem. Adapted from Van de Walle (2007).

4.2. Data sources and analyses This paper focuses on two cycles of PSC workshops conducted by the TLs. We concentrate on the first cycle of workshops facilitated by the three TLs who participated in the entire 3-year intervention, and the last cycle of workshops facilitated by the eight TLs who participated in the final year of the intervention. The first PSC cycle used the Lemonade problem, a mixture problem adapted from Van de Walle (2007) (Fig. 4). The last PSC cycle used the Fuel Gauge problem, a ratio-and-rate problem adapted from Jacob and Fosnot (2008) (Fig. 5). In all cases, the TLs led three workshops per PSC cycle. With two exceptions (due to last-minute scheduling changes), all of these workshops were videotaped by members of the research team. We engaged in a detailed examination of this subset of 28 videotaped workshops to address our research questions about adherence of TL workshops to key PSC characteristics, and about characteristics of the PSC model that were particularly easy or difficult to enact with integrity over time. Interviews with TLs—conducted at the beginning of their participation in the project and after each PSC cycle—along with field notes from our observations of the workshops, served as secondary data sources. 4.3. Rating and analysis of PSC workshops In rating the TLs’ videotaped Lemonade and Fuel Gauge workshops, we used an observation protocol adapted from the Professional Development Observation Protocol (PDOP, Banilower & Shimkus, 2004). The PDOP was designed by Horizon Research, Inc. to evaluate PD sessions using standards for exemplary practice derived from NCTM’s Principles and Standards for School Mathematics (2000) and NRC’s National Science Education Standards (1996). The PDOP involves holistic ratings of an entire PD workshop within six major categories: design, implementation, mathematics/science content, exploring pedagogy/instructional materials, leadership content, and culture of the session). Each category is composed of a set of more specific indicators, ranging in number from six to eleven. In addition, each category includes space for the observer to add one or more indicators for attributes of the workshop not captured by the other indicators. Furthermore, the coding manual specifies that there may be categories that are not applicable to a particular PD session and provides guidelines for observers to use in choosing appropriate categories to rate. Trained observers watch the video of an entire workshop session and then rate the workshop holistically on each indicator using a 5-point Likert scale with possible ratings of 1 (not at all) to 5 (to a great extent), 6 (do not know), and 7 (not applicable). Thus one number on the Likert scale is used to indicate the extent to which an attribute is characteristic of an entire workshop session. Our adaptations to the PDOP included combining the design and implementation sections, adding indicators to the mathematics content category to address TLs’ knowledge of mathematics for teaching, adding indicators related to use of video in the pedagogy/instructional materials category, eliminating the leadership content section, and eliminating indicators within remaining categories that the researchers agreed were not relevant to our project. The additional content indicators were derived from indicators of knowledge of the mathematical terrain in the Mathematical Quality of Instruction instrument developed by the Learning Mathematics for Teaching project (LMT, 2006) and the five practices for orchestrating productive

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Fig. 5. Fuel Gauge problem. Adapted from Jacob and Fosnot (2008).

mathematical discussions identified by Stein et al. (2008). We also renamed the categories to better reflect the focus of our project. The results reported in this paper are from three categories, each with several indicators. The three categories are (1) workshop culture, (2) specialized content knowledge, and (3) pedagogical content knowledge. Indicators of specialized content knowledge include, for example, teachers’ discussions of: mathematical skills, procedures, and concepts; a variety of solution strategies; relationships among solution strategies; and affordances and constraints of solution strategies. Indicators of pedagogical content knowledge specific to the use of video include, for example, facilitator(s)’ selection of video clips appropriate to the teachers’ need and interests; questions about video clips that encourage teachers to think deeply about instructional practices; questions about video clips that encourage teachers to think deeply about students’ mathematical ideas; and teachers’ careful unpacking of students’ mathematical ideas and reasoning. Three members of the research team independently rated video from the PD workshops and then met to compare ratings, clarify or revise definitions of categories and indicators, and re-rate when necessary. They came to consensus for all rating decisions, frequently re-watching portions of video together and discussing segments until agreement was reached. Once the three researchers were confident that they were consistent in their ratings (after rating approximately one third of the workshops), the remaining videos were rated independently by two members of the team, who then met to discuss and reconcile their ratings. They recorded the set of ratings for each category in tables, organized by workshop and by TL. The three authors then examined the tables for patterns in the data. Using the Lemonade problem workshop tables we identified the major patterns of results within each category. We then selected a subset of indicators in each category that were representative of these patterns, and we created a second set of (more manageable) tables that included only these representative indicators. We repeated this process for the Fuel Gauge problem workshops, compared the sets of indicators, and created a final set of representative indicators. To further reduce the data, for each TL we computed the average rating on each indicator, across all of their workshops in each of the PSC cycles. We then computed the overall average rating across all TLs on each indicator. These tables are presented and discussed in the results section. After determining the final set of representative indicators and creating the final set of tables, we used vignette analyses to create detailed descriptions that illustrate TLs’ facilitation of discussions during PSC workshops that would be rated highly on the indicators in each category. The vignettes are intended to reconstruct and authentically represent the events, people, and activities under consideration (Erickson, 1986; LeCompte & Schensul, 1999; Miles & Huberman, 1994). To create the vignettes, the authors examined videotaped records and field notes from several highly rated PSC workshops and selected the activities and conversations that were most representative of high ratings on indicators in each category. We then constructed vignettes (one vignette for each category) to depict the nature of the events and how TLs thought about them,

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Table 4 Workshop culture: average ratingsa for Lemonade Workshops 1–3. Indicator

1 2 3 4 5 a

Teacher leader

Climate of respect for experiences, ideas and contributions Collaborative working relationship between TL and participants Collegial working relationships among participants Active participation encouraged and valued Participants demonstrated willingness to share ideas and take intellectual risks

Avg

Jr

M

R

5.00 5.00 5.00 5.00 4.50

4.50 4.44 4.39 3.67 4.22

4.00 4.25 3.00 3.75 3.00

4.50 4.58 4.13 4.14 3.94

Based on 5-point Likert scale from 1 (not at all) to 5 (to a great extent).

drawing from the videotapes, field notes, and interviews. Vignettes are written in the present tense and set in italics. Interpretive commentary is interwoven using regular font. 4.4. Analysis of TL interviews Three sets of interviews with each TL served as secondary data sources for this paper: initial interviews and interviews following completion of the Lemonade problem and Fuel Gauge problem PSC cycles. The initial interviews, conducted with TLs prior to their participation in the summer leadership academy, focused on their conceptions of the PSC, their ideas about the role of teacher leader, and what they hoped to learn about facilitation from participating in the project. Post-cycle interviews asked TLs to reflect on their experiences facilitating the set of workshops, new insights and understandings about facilitating PD, and the support we provided in the Instructional Support Meetings to help them prepare for facilitating the workshops. All interviews were audiotaped and transcribed. We read each of the transcripts, identified instances where TLs talked about topics related to the four PDOP categories that were the focus of our workshop analysis, and then selected representative comments illustrating the TLs’ own interpretations of their workshops. 5. Results and discussion: patterns of enactment across workshops and across teacher leaders This section of the paper addresses patterns in the TLs’ enactment of the Problem-Solving Cycle. We focus on the three categories of ratings most central to the goals and design of the PSC: workshop culture, specialized content knowledge, and pedagogical content knowledge. For each category, we highlight specific characteristics of the PSC that the TLs seemed to take up easily, and characteristics that they found more problematic. We also present a vignette depicting facilitation that was rated highly on indicators in the category. 5.1. Workshop culture The first category addresses the workshop culture created by the TLs. This category includes indicators related to the nature and extent of participants’ engagement in the PSC. Specifically, we considered the degree to which the TLs fostered a climate of respect and positive working relationships and encouraged active participation and a willingness to share ideas throughout the workshop. We paid close attention to interactions among the participants to see if they seemed engaged and comfortable sharing ideas about the content and their instruction; and the extent to which they took intellectual risks such as bringing up something they did not understand. These indicators applied to all workshops in each cycle; in other words, we rated workshop culture when the teachers solved the selected PSC problem and when they discussed video. In all cases, we noted the degree to which their discussions were respectful, inclusive, and reflected collegial working relationships. Reviewing the Lemonade workshops, we found that in general all three TLs were successful in establishing a trusting, productive culture (Table 4). Their workshops were characterized by a climate of respect for participating teachers’ experiences and ideas. The TLs facilitated the teachers’ engagement as members of a professional learning community and encouraged them to contribute actively to the group’s exploration of mathematics, student thinking, and instructional practices. Researchers rated indicators of the collaborative working relationships between each TL and the teachers in his or her group as consistently very high. Interview data suggest the TLs also felt positive about the sense of community in their workshops. As Mandy explained, “I already knew the teachers, so the comfort level and things like that were pretty much set.” Jordan agreed, noting that “the group came together rather quickly.” Robert’s Lemonade workshops were rated somewhat lower than those of Mandy and Jordan on most indicators of workshop culture. Researchers noted that in Robert’s group the teachers’ engagement in and contributions to discussions were sometimes uneven. However, in reviewing interview transcripts from the two teachers in Robert’s group, we found their comments consistently very positive; they had high praise for their experiences in the PSC and for Robert as a facilitator. As was true for the Lemonade cycle, all eight TLs were generally successful in establishing productive cultures within their Fuel Gauge cycle workshops (Table 5). Researchers consistently gave the workshops high ratings on indicators of collegial working relationships among participants and climate of respect for experiences, ideas, and contributions. Again there were some notable individual differences. Both Jordan and Mandy worked with a different group of teachers for the Fuel Gauge

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Table 5 Workshop culture: average ratingsa for Fuel Gauge Workshops 1–3. Indicator

1 2 3 4 5 a

Climate of respect for experiences, ideas and contributions Collaborative working relationship between TL and participants Collegial working relationships among participants Active participation encouraged and valued Participants demonstrated willingness to share ideas and take intellectual risks

Teacher leader

Avg

Jr

M

R

Cn

Cr

Kt

K&J

4.50 4.00 4.67 4.00 3.83

4.33 4.67 4.50 4.17 3.83

4.33 4.50 4.00 4.00 3.67

4.33 3.00 4.67 3.83 3.50

4.50 4.17 4.33 4.00 3.50

4.83 5.00 5.00 4.67 4.17

5.00 5.00 5.00 4.83 4.83

4.55 4.33 4.60 4.21 3.90

Based on 5-point Likert scale from 1 (not at all) to 5 (to a great extent).

cycle (compared to the Lemonade cycle), and they expressed during interviews that they found it more difficult to conduct PD with these new groups. Their decrease in comfort with the participating teachers may explain why Jordan and Mandy’s Fuel Gauge workshops received lower ratings on several indicators of culture compared to their Lemonade workshops, particularly on indicators such as the collaborative working relationship between the TL and participants. Of the workshops led by TLs new to the project during the second year, the ones co-facilitated by Kyla and Jason received the highest ratings. These two teacher leaders were very aware of the importance of building community and thoughtful about ways to encourage participation by all their teachers. Kyla explained, “In the beginning, we had our group of teachers that tended to speak up and share their ideas. . .. We messed around with the grouping a bit as we went through the year, trying to find out which mix was best for . . . getting everyone to share.” Looking back at the end of the year, her co-facilitator Jason reflected, “We’ve always been a group that doesn’t like to share in a big group setting. . .. This year we actually broke down those barriers, and now we’re more open about discussing things.” Candace, unlike the other TLs, was not the chair of the mathematics department and had no formal leadership role in her school prior to her involvement in the PSC. Her limited experience in leadership positions may help explain why her workshops were rated somewhat lower than those of the other TLs.

5.1.1. Workshop culture vignette: Jason and Kyla’s Fuel Gauge Workshop 1 The vignette below highlights Jason and Kyla’s facilitation of Workshop 1 of the Fuel Gauge cycle. The manner in which they introduced their group of teachers to the Fuel Gauge task exemplifies their on-going effort to create an inclusive culture by framing questions that have personal relevance to each teacher. Prior to having the teachers work through the task in small groups, Jason and Kyla asked them to individually consider how they and their students would approach the problem as well as the difficulties their students might encounter when solving it. This facilitation strategy helped to encourage the teachers to carefully reflect on the problem and to actively participate in the whole group conversation that followed. Jason and Kyla distribute copies of the Fuel Gauge problem. Jason tells the teachers, “After you read the problem, think about how you would solve it. You do not have to actually solve it, just think about how you would. Also, think about the issues or difficulties your students might encounter when solving this problem.” Teachers spend a few minutes silently reading the problem and considering the questions that Jason posed. When Jason brings the teachers back together for a full group conversation he reminds them, “Think about your particular group of students. If you teach sixth grade, what difficulties might they experience when solving the problem? If you teach eighth grade, what might your students experience?” These questions incite a lively discussion, with most teachers in the group participating. They share ideas about how their students might struggle including misunderstanding the context of the problem, having difficulty distinguishing between the amount of miles and the amount of gas, reading the lengthy text, and doing the mathematics using incorrect strategies. Jason and Kyla’s technique of delving into the task by asking a question about each participant’s students is likely to build community by engaging all participants in a relevant and safe manner. This strategy invites participants to contribute to the professional learning community and take ownership of issues raised in the PD. Jason and Kyla were pleased with the community they saw developing in their math department over the course of the iPSC project. Jason explained in an interview that the PSC workshops enabled teachers in his school to more actively share their ideas. “That was probably the most awesome thing that happened. We brought sharing out of people, and then it got better as we went on. There was more openness as we went on.”

5.1.2. General patterns in workshop culture Overall, the five indicators of workshop culture received very high ratings across TLs and across the two PSC cycles. The TLs seemed to have little difficulty garnering a climate of respect and promoting collaborative, collegial working relationships within their groups. Ratings of participants’ willingness to share ideas and take intellectual risks were somewhat lower, perhaps because this aspect of community is more challenging to develop and thus takes more time to establish.

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159

Table 6 Specialized content knowledge: average ratingsa for Lemonade Workshop 1. Indicator

1 2 3 4 5 6 7 8 a b

Teacher leader

Teachers generate and analyze ways to solve task Teachers generate/analyze reasoning used to arrive at correct/incorrect solutions Discussion of various solution strategies Discussion of relationships among solution strategies Discussion of affordances and constraints of various solution strategies Discussion of various mathematical representations Discussion of relationships among representations Discussion of affordances and constraints of various representations

Avg b

Jr

M

R

5.00 4.00 4.00 3.00 3.00 2.50 1.50 1.50

4.33 3.67 4.67 3.67 4.33 4.33 4.33 3.67

– – – – – – – –

4.67 3.84 4.34 3.34 3.67 3.42 2.92 2.59

Based on 5-point Likert scale from 1 (not at all) to 5 (to a great extent). Robert’s Lemonade Workshop 1 was not filmed.

5.2. Specialized content knowledge Indicators of specialized content knowledge (SCK) denote the extent to which the TLs engaged teachers in productive discussions about the mathematical knowledge needed to teach a lesson using the focal PSC problem. In keeping with directions on the PDOP, we rated this category only for workshops where increasing participants’ SCK was a key purpose or a vehicle for accomplishing other stated purposes of the session. Thus, Tables 6 and 7 include ratings only for PSC Workshop 1 (of the Lemonade and Fuel Gauge cycles, respectively). PSC mathematics problems, such as the Lemonade and Fuel Gauge problems, are intentionally selected to meet a variety of criteria, including the potential to be solved using multiple representations and strategies (Koellner et al., 2007). Consistent with goals of the PSC, the eight indicators we chose to include in these tables present data on characteristics of the workshop discussions related to representations and solution strategies. Representations refer to pictures, diagrams, manipulatives, and other models used to represent mathematical ideas or procedures. By different solution strategies, we mean different mathematical approaches to solving a problem (LMT, 2006). For both representations and solution strategies we considered the extent that TLs (a) generated and discussed multiple examples, (b) discussed the mathematical relationships among the examples, and (c) examined the mathematical affordances and constraints of the various examples. We were able to rate only the first Lemonade workshop for Mandy and Jordan, as Robert’s first workshop was not videotaped because of logistical circumstances. Both Mandy and Jordan facilitated discussions about a variety of possible solution strategies for the Lemonade problem, and elicited reasoning that could be used to arrive at correct and incorrect answers. In almost all cases, indicators related to solution strategies (Table 6, indicators 1–5) were rated higher than indicators related to representations (Table 6, indicators 6–8). Mandy and Jordan also were better able to foster discussions about various solution strategies and mathematical representations (Table 6, indicators 1–3, 6) than about either mathematical relationships among solution strategies and representations, or their affordances and constraints (Table 6, indicators 4, 5, 7, 8). The general pattern of ratings for the first workshop in the Fuel Gauge cycle was similar to that seen in Mandy’s and Jordan’s first Lemonade cycle workshops. In all seven Fuel Gauge Workshop 1, the TLs were able to facilitate discussions to explore the variety of representations and solution strategies that could be used in solving the Fuel Gauge problem (Table 7, indicators 1–3, 6). Again, for both solution strategies and representations, TLs had more difficulty engaging teachers in discussions to address either the affordances and constraints of the examples they generated, or the relationships among them (Table 7, indicators 4, 5, 7, 8). And, in general, discussions were rated higher with respect to solution strategies (Table 7, indicators 1–5) than mathematical representations (Table 7, indicators 6–8). As was true for the other rating categories, neither Jordan nor Mandy demonstrated the ability to foster stronger SCK in their Fuel Gauge Workshop 1 compared to their Lemonade Workshop 1. In fact, Mandy’s Fuel Gauge Workshop 1, although Table 7 Specialized content knowledge: average ratingsa for Fuel Gauge Workshop 1. Indicator

1 2 3 4 5 6 7 8 a

Teachers generate and analyze ways to solve task Teachers generate/analyze reasoning used to arrive at correct/incorrect solutions Discussion of various solution strategies Discussion of relationships among solution strategies Discussion of affordances and constraints of various solution strategies Discussion of various mathematical representations Discussion of relationships among representations Discussion of affordances and constraints of various representations Based on 5-point Likert scale from 1 (not at all) to 5 (to a great extent).

Teacher leader

Avg

Jr

M

R

Cn

Cr

Kt

K&J

3.50 2.50 4.00 2.50 1.50 2.50 1.50 1.50

4.50 3.50 4.50 2.50 2.50 3.00 1.50 2.50

3.00 2.50 3.00 2.00 3.00 3.00 2.00 3.00

4.00 1.50 3.50 2.00 1.50 3.50 2.50 3.00

3.50 2.50 3.50 2.50 1.50 2.50 2.00 1.50

4.50 3.50 4.00 3.00 2.50 3.50 2.50 3.00

4.50 4.00 4.50 3.00 2.50 3.50 2.50 3.00

3.93 2.86 3.86 2.50 2.14 3.07 2.07 2.50

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again rated very highly on the first four indicators, was rated lower than her Lemonade Workshop 1 on indicators related to relationships and affordances and constraints. We speculate that these findings may be partially attributed to characteristics of the Lemonade and Fuel Gauge problems. These two problems, like all PSC problems, can be solved using multiple representations and multiple solution strategies. However, the Lemonade problem lends itself to a wider variety of representations and solution strategies (correct and incorrect) than does the Fuel Gauge problem.

5.2.1. Specialized content knowledge vignette: Mandy’s Lemonade Workshop 1 Throughout Workshop 1 of the Lemonade cycle, Mandy was able to lead fruitful discussions around a variety of mathematical solution strategies. She seamlessly interwove related topics, such as how the teachers themselves solved the Lemonade problem, what their answers mean in the context of the problem, how their students might approach the task, and instructional moves that could be beneficial such as using manipulatives or other representations. These types of conversations are consistent with the goal of supporting the development of teachers’ specialized content knowledge, in preparation for them to teach the task. “First I am going to have you solve the Lemonade problem and think about ways your students might solve it too,” Mandy tells the teachers as she begins her workshop. The teachers work on the problem individually and then share their ideas. Krista notes, “I would use a part-whole fraction to solve the problem.” Ryan adds, “I used a calculator to divide the fractions and find the percent to determine which lemonade recipe had the stronger flavor.” Another teacher, Mary, explains, “I used a part-whole fraction too because I knew that I could compare them and use a decimal or percent to find the solution. I did that because I am comfortable with that as a teacher.” Mandy asks, “Are there any other strategies that a sixth grader might come up with, perhaps even a strategy that might not be successful?” Mary responds, “I could see them looking at the first pitcher and maybe saying that they are the same. Because they would say that the first pitcher is 2 to 3. And then looking at the other pitcher and saying they just added one more water and one more lemonade, so they are still the same.” Mandy points to the green and yellow tiles she has strategically placed on the teachers’ desks and asks Mary to represent this way of thinking. As Mary begins arranging the tiles, Krista comments, “I think the kids might see that the one extra water and the one extra lemonade would cancel each other out.” Mandy begins using chart paper to record all of the strategies the group has discussed thus far, and to ensure that everyone has a thorough understanding of each strategy. Mandy states, “The first method that all of you used was a fraction representation that was part-to-whole, 2/5 compared to 3/7. Then you also did this as a percent, 40% compared to 42.8%. And then the ratio 2:3 compared to 3:4.” Mary notices that there is an additional strategy students could use. She explains, “I was thinking that they might say that pitcher B would be more concentrated because they might look at ¾ and say that was 75% and then 2/3 is 67%.” Mandy presses Mary to talk more about this strategy and she continues, “Because those are benchmark fractions. They might say one has 75% lemonade and the other has 67%.” Mandy questions the group, “What do those percents mean in this situation?” Krista says she doesn’t know. Ryan thinks for a moment and then answers, “For every 100 parts water you get 75 parts of lemonade mix.” He then adds, “So, one cup water is ¾ cup of lemonade mix.” Mandy reiterates, “That is like finding the unit rate for one cup of water.” As this snapshot of a portion of the workshop illustrates, Mandy’s teachers were able to generate a range of solution strategies fairly quickly. However, they did not necessarily have a deep understanding of each strategy, and they struggled to interpret what the ratios meant in the context of the problem. Through skillful questioning and probing, Mandy encouraged the teachers to examine the meaning of the different ratios, and to think carefully about how to accurately guide their students, including anticipating misconceptions and demonstrating why the right answers were in fact accurate.

5.2.2. General patterns of SCK in generating solution strategies In general, workshops were rated highly with respect to the SCK that teachers exhibited in generating and discussing different ways to solve the PSC tasks. Most TLs saw Workshop 1 as an opportunity to discuss solution strategies. For instance, Kaitlyn explained, “During the first workshop we look at a mathematical situation [PSC task] that can be solved from many different angles or ways. And we discuss how we solved it and how we think kids will approach it. We look at the various possible strategies in solving it and we analyze them.” Discussions of solution strategies were rated higher than discussions of mathematical representations. Also, overall, average ratings were consistently lower on indicators for the Fuel Gauge problem as compared to the Lemonade problem. Most workshops were rated higher on the three indicators addressing the degree to which teachers generated and discussed multiple representations and solution strategies, compared to the five indicators that entail analyzing reasoning, discussing relationships among representations or solution strategies, and discussing affordances and constraints of representations or solution strategies. Making comparisons among examples (of solution strategies or representations) is likely to be a more cognitively complex task than exploring individual examples.

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Table 8 Pedagogical content knowledge: average ratingsa for Lemonade Workshops 2 and 3. Indicator

1 2 3

Video clips are accessible and relevant to the teachers Video clips are appropriate with respect to the level of trust within the community Questions about video clips encourage teachers to think deeply about students’ mathematical ideas and reasoning Does the TL frame discussions and use prompts in ways designed to foster development of KCS? Do the teachers engage in careful unpacking and deep analysis of students’ mathematical ideas and reasoning? Questions about video clips encourage teachers to think deeply about instructional practices Does the TL frame discussions and use prompts in ways designed to foster development of KCT? Do the teachers engage in careful unpacking and deep analysis of instructional practices?

4 5 6 7 8 a

Teacher leader

Avg

Jr

M

R

4.00 3.00 4.00

4.75 5.00 4.75

3.75 4.00 2.25

4.17 4.00 3.67

3.50

4.50

3.00

3.67

3.00

4.50

3.00

3.50

2.50

4.50

2.00

3.00

–

3.75

2.00

2.88

–

3.75

2.00

2.88

Based on 5-point Likert scale from 1 (not at all) to 5 (to a great extent).

5.3. Pedagogical content knowledge Indicators of pedagogical content knowledge (PCK) provide a measure of the extent to which TLs used strategies designed to deepen teachers’ knowledge of content and students (KCS) and knowledge of content and teaching (KCT), particularly with respect to problem-based teaching. KCS includes, for example, engaging in analysis of students’ mathematical ideas and reasoning. KCT includes engaging in analysis of teachers’ instructional practices. Because these knowledge domains are not the focus of PSC Workshop 1, Tables 8 and 9 include only data from Workshops 2 and 3, in which groups view and discuss video clips of their lessons on the PSC problem. As explained above, to take into account the PSC model’s emphasis on using video to situate PD in participating teachers’ classrooms, we adapted the PDOP to include indicators of TLs’ skill in selecting appropriate video clips and framing the viewing and discussion of these clips during their workshops. Four of the indicators included in Tables 8 and 9, which we developed specifically for this analysis, focus on TLs’ use of video (Tables 8 and 9; indicators 1–3, 6). The other four indicators, taken from the PDOP, address their skills in fostering the development of KCS and KCT (Tables 8 and 9; indicators 4, 5, 7, 8). As Table 8 indicates, overall the Lemonade workshops were rated highly with respect to relevance and appropriateness of video clips selected by TLs from their teachers’ PSC lessons. This was true despite the fact that in their interviews, all three TLs commented on the difficult and time-consuming nature of picking good video clips. Jordan explained that when planning a workshop, “You have to find the clip that interests you and then develop the guiding question.. . . And it’s not easy finding the clip.” They also sometimes felt limited by the videotapes from which they could choose. Robert noted that it was challenging “getting enough from a videotape to give you a couple of really good situations to discuss, as opposed to having a videotape that’s been made and chosen (by someone outside the PD group).. . . Sometimes it’s hard finding the examples you’re looking for.”

Table 9 Pedagogical content knowledge: average ratingsa for Fuel Gauge Workshops 2 and 3. Indicator

1 2 3 4 5 6 7 8 a

Video clips are accessible and relevant to the teachers Video clips are appropriate with respect to the level of trust within the community Questions about video clips encourage teachers to think deeply about students’ mathematical ideas and reasoning Does the TL frame discussions and use prompts in ways designed to foster development of KCS? Do the teachers engage in careful unpacking and deep analysis of students’ mathematical ideas and reasoning? Questions about video clips encourage teachers to think deeply about instructional practices Does the TL frame discussions and use prompts in ways designed to foster development of KCT? Do the teachers engage in careful unpacking and deep analysis of instructional practices? Based on 5-point Likert scale from 1 (not at all) to 5 (to a great extent).

Teacher leader

Avg

Jr

M

R

Cn

Cr

Kt

K&J

4.00 3.75

4.00 4.75

3.75 4.00

3.50 4.50

5.00 4.50

4.75 4.25

5.00 4.75

4.29 4.36

1.00

4.75

3.25

1.50

3.50

4.00

3.75

3.11

–

4.25

2.50

1.50

3.75

4.00

4.50

3.42

–

4.25

2.50

1.50

3.25

4.25

4.25

3.33

2.50

5.00

2.00

2.00

4.00

4.25

4.00

3.39

3.00

4.50

3.25

1.50

3.50

3.75

4.25

3.39

2.00

4.00

2.75

2.50

2.75

3.75

4.00

3.11

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The TLs were more successful in selecting video clips to use as springboards for discussions of the Lemonade problem than in actually facilitating the discussions to foster teachers’ development of KCS and KCT. Also, discussions about students’ mathematical ideas and reasoning (Table 8, indicators 4, 5) were stronger than discussions about instructional practices (Table 8, indicators 7, 8). Perhaps because the Lemonade Cycle was the first set of PSC workshops that TLs conducted, they seemed to be more tentative about focusing on the practice of teaching than on student reasoning. TLs may have been less comfortable pushing teachers to critique their instructional practices than to deeply analyze their students’ mathematical reasoning. As was true for the Lemonade workshops, the video clips that TLs selected for Fuel Gauge Workshops 2 and 3 were rated as accessible and relevant to the teachers with whom they were working and appropriate with respect to the level of trust within the community (Table 9, indicators 1, 2). Also similarly to the Lemonade workshops, TLs’ ability to select appropriate video clips was generally stronger than their ability to engage teachers in thinking deeply about instructional practices and students’ mathematical ideas and reasoning. Many of the TLs realized that facilitating these types of discussions was an area where they could benefit from additional support. For example, Candace noted that she would like “more help asking the right question [when showing video]. In one workshop, I had some questions up [on the Smartboard] and that brought better discussion.” Unlike the Lemonade workshops, there was not a large difference between discussions of instructional practices and student thinking in terms of the quality of TLs’ questions or the engagement of teachers. This lack of a difference supports our conjecture that initially TLs may have been hesitant to press the teachers’ analysis of their own instructional practices. We also hypothesize that, over time, conversations about instruction and student reasoning became more intertwined as teachers felt more comfortable talking about these topics and recognized the degree to which they are interrelated. Mandy’s facilitation of discussions remained stronger than Jordan’s and Robert’s across both PSC cycles, as indicated by their ratings on all six discussion-related indicators. There also were individual differences in the quality of discussions facilitated by the TLs who joined the project during the second summer institute. As was true for Workshop Culture, Candace’s workshops were rated somewhat lower than those of other TLs. Kyla and Jason’s workshops, like Mandy’s, were rated higher than others on most PCK indicators. These individual differences may be due to variations in skills and abilities of the TLs, the nature of their PD groups, or some combination of factors. 5.3.1. Pedagogical content knowledge vignette: Mandy’s Fuel Gauge Workshop 3 In her third Fuel Gauge workshop, Mandy selected a video clip in which a small group of students were discussing their solution to the problem. In her facilitation of a conversation about the clip, Mandy clearly promoted the development of teachers’ pedagogical content knowledge. Throughout the conversation, she skillfully negotiated two related goals: helping teachers to identify and understand the mathematical misconceptions underlying the student’s error, and encouraging them to identify pedagogical practices that would help the students understand and correct the error. Prior to watching the video clip, Mandy shares with the teachers the discussion questions she has prepared: “What are the students’ mathematical misconceptions? What are their understandings? What are the teacher’s instructional moves? Where do we go from here?” Teachers then watch the video clip in which a student explains that her strategy involves dividing 600 by 120, and arriving at the answer 50. The teachers quickly recognize that not only has the student incorrectly divided 600 by 120 rather than 120 by 600, but she has done the division inaccurately. Mandy pushes the group to continue to unpack the student’s thinking: “Can we do something on a big sheet of paper, just so it’s clear for everyone what’s going on?” She places a large sheet of paper in the center of the table and the teachers begin to collaboratively recreate the student’s strategy. As they are working, the teachers discuss how the student has set up the ratio in the problem and what the numbers in the ratio represent. One teacher suggests that the student has found “the inverse of what she wanted, which is 1/5 of the tank.” Another teacher provides a different way of interpreting the 5, suggesting that the student may have been trying to determine the number of trips between the two farms that can be taken on a full tank of gas, “5 times back and forth between Stan and Louisa on one full tank.” Mandy asks the teacher to repeat this idea and she elaborates, “She can go 1,2,3,4,5–five one way trips, 2 ½ round trips–if she had a full tank of gas.” Mandy then encourages the group to think about the pedagogical implications: “What are our instructional moves? What would we do as the teacher?” One teacher suggests, “I would just ask them, ‘What does the 600 mean? What does the 120 mean? If you are dividing 600 by 120, what are you dividing?’ And hopefully they would see that if they are dividing a whole tank by the number of miles in one route, their answer is how many trips on that route they can make with a whole tank.” Mandy selected a video clip involving a student error that challenged teachers both mathematically as well as pedagogically. In facilitating the discussion she made sure that the teachers understood the meaning of the numbers in the student’s strategy as well as the error she made in setting up the ratio, prior to having them consider the instructional implications. Mandy’s focus on both understanding students’ thinking and considering possible pedagogical moves to make in response to students’ misconceptions is typical of her facilitation of video-based discussions. 5.3.2. General patterns of PCK in selecting video and facilitating discussions The TLs were generally successful in selecting video clips that were relevant to the teachers and appropriate with respect to the level of trust within their groups. They were less successful, however, in facilitating discussions to deeply analyze

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instructional practices or student thinking. During our Instructional Support Meetings, we focused heavily on the importance of selecting video and formulating questions to launch meaningful discussions; and the data may reflect the TLs’ extensive efforts to find appropriate video clips, both during the Instructional Support Meetings and on their own time. Navigating these discussions during real time in PD workshops is likely to be a more challenging skill than either selecting video clips or developing launching questions, and thus may require additional time and support for TLs to develop. Individual differences among TLs also come into play in this regard, with some likely needing more support than others.

6. General discussion and implications It is important to note that in the iPSC study, the research team worked with a limited number of teacher leaders for an extended period of time to help them learn to facilitate the Problem-Solving Cycle. Our findings should be considered tentative, and not necessarily replicable or applicable to all PD contexts. At the same time, these analyses offer a number of insights regarding novice leaders’ facilitation of the mathematics PD workshops. They also suggest several issues for the field to take up as we continue to develop PD models and consider how to prepare and support PD facilitators to lead them. We begin exploring these issues by revisiting our research questions. Our first research question asked to what extent the teacher leaders enacted the PSC with integrity to its key characteristics. Overall, our ratings of their workshops indicate that TLs were able to successfully facilitate the PSC with groups of teachers in their schools and maintain integrity to the key characteristics of the PSC. The TLs led workshops that closely matched the broad outlines and intentions of the PSC model, including a focus on the relevant goals and activities within PSC Workshops 1–3. The TLs navigated significant logistical obstacles including recruiting teachers to consistently attend workshops, finding time and space to meet within their school, and incorporating the use of video. Meeting these pragmatic challenges ensured that they could establish and maintain a situated learning environment, a professional community, the use of classroom video, and an ongoing sustainable structure for the PD. Although these findings address one central aspect of scalability, namely that PD leaders other than the program designers can learn to facilitate the PSC with integrity to its core characteristics, additional research is needed to determine if the leadership preparation and support can be accomplished in a way that is less labor intensive and includes a larger number of teacher leaders. Our second research question asked which characteristics of the PSC the TLs enacted particularly well and which characteristics were more problematic to enact. The importance of this question became clear early in the analysis as we noticed that some of the PSC characteristics were addressed well by virtually all TLs, whereas other characteristics were notably difficult to enact. In this section we highlight both sets of characteristics and suggest possible reasons for these differences in ease of enactment.

6.1. Characteristics more easily enacted The TLs were consistent in their ability to create a climate of respect and trust in their workshops and to establish collaborative working relationships among their teacher participants. A central component of the PSC model involves developing and maintaining a professional learning community in which teachers are comfortable working together to improve their teaching. In designing the model we were mindful that some PD programs have found teachers unwilling to share video from their own classrooms with their colleagues (Grossman, Wineburg, & Woolworth, 2001; Sherin & Han, 2002). For teachers to be willing to expose their instructional practices in this way, they must feel part of a safe and supportive professional environment and be confident that showing their videos will provide productive learning opportunities for themselves and their colleagues. In our own experiences with developing and leading the PSC, participants formed a supportive community and engaged in increasingly reflective and productive conversations around video from one another’s classrooms (Borko et al., 2008). As a result of these experiences, in both our initial meetings with the TLs and the Facilitator’s Guide we provided concrete suggestions for establishing and maintaining community. It seems likely that explicit attention to community contributed to the TLs’ success in enacting this key PD practice beginning with first year of their participation. Another area where TLs consistently received high ratings was the selection of video clips. One question frequently asked about the PSC is whether TLs new to this PD model will be able to select video clips that can foster rich discussions about important ideas relevant to classroom practice. Two sets of concerns underlie this question. First, prior to their involvement in the PSC, teachers are likely to have had little experience discussing classroom video. Second, because of the PSC’s focus on using video from participating teachers’ classrooms, the set of video recordings from which TLs have to choose is limited. Because the selection of appropriate clips is critical to the success of the PSC, the Facilitator’s Guide details key characteristics of “rich” video clips and provides examples of the types of clips that we found to be successful in fostering productive discussions in our previous research. In addition, during the Instructional Support Meetings the TLs were given time to watch video on their own and with colleagues, select potential clips, and share those clips with the group. These multiple supports—particularly the time during Instructional Support Meetings to select potential video clips and receive feedback about their selections—likely contributed to the TLs’ successful enactment of this feature of the PSC.

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6.2. Characteristics more difficult to enact While there were notable individual differences among TLs, they generally had more difficulty supporting deep analysis in discussions to foster both SCK and aspects of PCK including knowledge of content and students (KCS) and knowledge of content and instruction (KCT). In this section, we posit some ideas as to why novice facilitators may have found these characteristics more difficult to enact, despite the extensive preparation and support they received as part of the iPSC project. We also suggest possible modifications to the summer academies and Instructional Support Meetings to provide additional support related to these characteristics. With respect to promoting the development of teachers’ specialized content knowledge, TLs were generally able to engage their groups in discussions focused on multiple mathematical representations and multiple solution strategies for a particular problem. However, they were less successful at engaging teachers in discussions about the relationships, affordances, and constraints of representations and solution strategies. The three indicators on which the TLs were rated more highly—generating and analyzing ways to solve the task, discussions of various representations, and discussions of various solution strategies—are similar to what Kazemi and Stipek (2001) referred to as “low-press exchanges” in the elementary mathematics classrooms they studied, exchanges such as solving open-ended problems in groups and sharing solution strategies. In contrast, the five indicators that entail analyzing reasoning, discussing relationships among representations or solution strategies, and discussing affordances and constraints of representations or solution strategies are similar to the “high-press exchanges” they described. Kazemi and Stipek argued that high-press exchanges such as collaboratively examining the relationships among multiple strategies and using mathematical argumentation to arrive at shared understandings are necessary for promoting development of students’ mathematical ideas. They also posited that for teachers to foster high-press exchanges in elementary mathematics classrooms, they must have a deep understanding of the nature of conceptual thinking. In a similar vein, in their review of PD for technology-enhanced inquiry science, Gerard et al. (2011) noted that PD leaders’ ability to support teachers in “distinguishing” and “integrating” ideas, rather than simply “eliciting” and “adding” new ideas, was a key factor distinguishing programs that influenced teachers’ use of technology to improve students’ inquiry learning experiences. Similarly, we suggest that guiding discussions in PD workshops to focus on reasoning and relationships requires a deeper conceptual understanding of mathematics than does facilitating discussions that do not go beyond identifying and describing mathematical representations and solution strategies. Our findings suggest that teacher leaders would benefit from additional types of support than we provided in the summer academies and Instructional Support Meetings to help them develop the specialized knowledge of mathematics needed to engage teachers in reasoning about similarities and differences among mathematical representations and solution strategies. For example, although we included extensive opportunities for the TLs to solve and analyze the PSC tasks and plan their Workshop 1, we did not create opportunities for them to rehearse these workshops and receive feedback. In addition, the TLs may have benefitted from working in small groups to examine a set of possible PSC tasks, with each group analyzing a different task and preparing to lead a discussion of that task, and then facilitating the discussion and receiving feedback from their peers and the research team. Promoting the development of teachers’ pedagogical content knowledge is another area in which the TLs sometimes had difficulty. They were successful at selecting video clips to use as springboards for discussions to foster PCK and at developing questions to launch the discussions. However, they were less capable of orchestrating discussions that maintained a deep level of analysis relating to student reasoning and instructional practices. The Instructional Support Meetings did include extended opportunities for the TLs to anticipate the types of conversations their selected clips would support and to plan accordingly. Also, the Facilitator’s Guide offers suggestions for leading discussions and provides detailed case illustrations. It is likely that developing this skill is a process that requires extensive practice with more hands-on support, perhaps including more rehearsals, role-plays, and coaching. The TLs may have benefited from more frequent opportunities to practice facilitation, for example by taking turns leading discussions using their selected video clips during the Instructional Support Meetings and then receiving feedback from their peers and the research team. Our final interviews with the mathematics leaders provided some support for this conjecture. For example, when asked how the Instructional Support Meetings might have better supported their facilitation, Kyla suggested “spending more time choosing video, and taking clips and actually looking at them together as a group . . . and pulling out questions together. Doing that more as a group could have been beneficial.” Smith and Stein (2011) noted that facilitating discussions with students that build on their thinking and use their responses to advance the mathematical understanding of the class involves skillful improvisation. To orchestrate conversations that extend students’ mathematical reasoning, teachers must be able to “diagnose students’ thinking on the fly and to quickly devise responses that will guide students to the correct mathematical understanding.” They identified five practices designed to moderate the degree of improvisation required by the teacher during a discussion by “shifting some of the decision making to the planning phase of the lesson” (p. 7): anticipating, monitoring, selecting, sequencing, and connecting. Facilitating discussions among teachers in PD workshops—whether those discussions address mathematics, student thinking, or instructional practices—is similarly demanding and also requires skillful improvisation. As van Es (2010) noted, facilitators must be able to “notice ‘teachers’ thinking’ in the midst of facilitating a video club discussion” (p. 12). To do this, they must have ideas about particular events in the video clips that are important to discuss and then listen carefully so they can build on teachers’ comments in ways that enable them to explore these events. As we work to detail the knowledge that

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PD leaders need to orchestrate productive discussions with teachers, it may be helpful to consider how the five practices Smith and Stein (2011) recommended for classroom teachers are relevant for professional development facilitators.

7. Conclusion Scaling professional development requires, as a key consideration, developing a large number of skillful and capable PD facilitators. Given the current demand for extensive, high-quality PD, there is an urgent need to identify the types of knowledge and skills these facilitators draw on, and to determine how they can best be supported to develop and expand their relevant knowledge and skills (Marrongelle et al., 2013). Although the iPSC project involved a relatively small number of PD leaders, the analyses of their facilitation efforts as reported in this paper moves us one step forward in achieving these goals. To further this work and to facilitate the sharing of ideas across research teams, we conjecture there is an area of professional knowledge pertaining to leaders of effective PD that needs to be labeled, defined, and systematically investigated (Stein & Nelson, 2003). Within the field of mathematics education, building on the work of Ball et al. (2008), we propose to give this knowledge the initial label of Mathematical Knowledge for Professional Development (MKPD). We anticipate that, similar to the construct Mathematical Knowledge for Teaching, MKPD has multiple domains that must be identified and disentangled. As a starting point, we posit that MKPD may include specialized content knowledge and pedagogical content knowledge (comprising both knowledge of content and students and knowledge of content and teaching) but from a PD leader perspective. The SCK that facilitators need is likely to entail a deep understanding of the range of potential solution strategies and representations specific to a given mathematical context; the mathematical relationships within these sets of solution strategies and representations; and the mathematical affordances and constraints of each strategy and representation. The PCK that facilitators need likely includes the ability to engage teachers in the interpretation of students’ mathematical ideas and the purposeful analysis of instructional practices. An additional domain that we have found to be critical to the effective implementation of the PSC is the knowledge needed to establish and maintain a professional mathematics learning community, which we label mathematics learning community knowledge. We speculate that this knowledge includes the ability to create a culture of respect, establish group norms, and foster active participation in which teachers share ideas and take intellectual risks. Learning community knowledge is likely to be an important and relevant domain for most, if not all, mathematics PD endeavors. Our conjectures about the nature of MKPD, and the specifics of what its various components may entail, are based on our analyses of the teacher leaders’ facilitation of the PSC PD and in many ways parallel the three categories of facilitation practices that we highlighted in the results section. Although the logic here may appear to be somewhat circular, our argument is that these practices are associated with highly related knowledge constructs, and it is the interrelationship of both knowledge and practice that ultimately impacts the quality of facilitation of a given workshop. We believe that these three possible domains of MKPD—specialized content knowledge, pedagogical content knowledge, and learning community knowledge—go beyond and look different than the knowledge that a typical mathematics classroom teacher holds. Because PD leaders are expected to promote the development of teachers’ knowledge in these domains, they must hold a deeper and more sophisticated knowledge of mathematics than their colleagues, just as teachers must hold a deeper and more sophisticated knowledge than their students. In addition, PD leaders should be knowledgeable about how to work productively with adult learners, and construct environments for teachers to collaborate about relevant topics. These ideas are compatible with the work of Elliott et al. (2009), who have begun to identify features of math leaders’ SCK and the nature of sociomathematical norms productive for teacher learning. It seems clear from our work, and the work of others, that focused and sustained support for facilitators is necessary to promote the development of MKPD. Novice PD leaders are likely to require increased knowledge of how to improvise skillfully (Smith & Stein, 2011) and orchestrate conversations that elicit “high-press exchanges” (Kazemi & Stipek, 2001) where teachers deeply examine central mathematical relationships, instructional practices, and student thinking. In the present study novice facilitators enacted some characteristics of effective PD more readily than others. These findings highlight several questions about the development of MKPD that warrant further investigation. For example, can PD leaders develop their knowledge in particular domains of MKPD earlier and with greater ease than in other domains? Are certain leadership-preparation and -support activities better suited to promoting facilitators’ MKPD focused on specialized content knowledge, whereas other types of activities are better suited to promoting MKPD focused on pedagogical content knowledge? What factors differentiate PD leaders and their ability to develop MKPD? Additional empirical and theoretical investigations are needed to further elucidate MKPD and to generate the experiences PD leaders need in order to take on the important role of driving change in the teaching profession.

Acknowledgments The professional development project discussed in this article was supported by National Science Foundation (NSF) Award number DRL 0732212. The views shared here are ours and do not necessarily represent those of NSF. We wish to acknowledge Erin Baldinger and Sara Kate Selling for their efforts in rating and analyzing video. We also extend our gratitude

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to the district mathematics coordinators, the teacher leaders, and all other teachers who participated in this project and continue to use the PSC model of mathematics professional development in their schools.

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