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Journal of Mathematical Behavior 33 (2014) 149– 167

Contents lists available at ScienceDirect

The Journal of Mathematical Behavior

journa l h om epa ge: ww w.elsev ier .com/ locate / jmathb

xamining novice teacher leaders’ facilitation of mathematicsrofessional development

ilda Borkoa,1, Karen Koellnerb,∗, Jennifer Jacobsc

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

a r t i c l e i n f o

eywords:athematics professional development

rofessional development leaderseacher leadersacilitating professional developmentnowledge for professional development

a b s t r a c t

This paper reports on novice teacher leaders’ efforts to enact mathematics PD through ananalysis of their facilitation in workshops conducted at their schools. We consider the extentto which teacher leaders facilitated the Problem-Solving Cycle model of PD with integrityto its key characteristics. We examine the characteristics they enacted particularly welland those that were the most problematic to enact. Facilitators were generally successfulwith 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 numberof 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 havelabeled Mathematical Knowledge for Professional Development (MKPD), and we posit somedomains that may comprise this construct.

© 2013 Elsevier Inc. All rights reserved.

. Introduction

Professional development (PD) opportunities for mathematics teachers are widely recognized as a critical componentf efforts to support increased student achievement. A growing body of empirical research on the structure, content, andutcomes of effective PD offers insights into the characteristics of programs that provide high-quality, high-impact learningpportunities (Borko, Jacobs, & Koellner, 2010; Desimone, 2009; Wei, Darling-Hammond, Andree, Richardson, & Orphanos,009). However, having a deep understanding of what PD should look like is only part of the equation. An often overlookedariable is having well-prepared facilitators to ensure the PD’s effectiveness (Katz, Earl, & Ben Jaafar, 2009; Stein, Smith, &ilver, 1999). Presently in the United States, PD facilitators represent a new cadre of prominent players on the educationalcene, and in the field of mathematics most PD facilitators are only in the beginning stages of honing their leadership skillsZaslavsky & Leikin, 2004). In order to promote powerful and lasting change in the teaching profession there is an urgenteed to prepare novice PD facilitators to successfully facilitate newly developed PD models that offer high-quality learning

pportunities for teachers.

Practicing mathematics teachers appear to be an obvious personnel source to assume leadership positions, especiallyor the delivery of local, site-based PD. However, having inservice teachers work with adult learners—typically teachersn 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: hildab@stanford.edu (H. Borko), kkoellne@hunter.cuny.edu (K. Koellner).

1 Tel.: +1 650 723 7640.

732-3123/$ – see front matter © 2013 Elsevier Inc. All rights reserved.ttp://dx.doi.org/10.1016/j.jmathb.2013.11.003

150 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

of teaching mathematics to K–12 students. How can we foster the leadership capacity of a large base of mathematics PDfacilitators to keep up with the demand for widespread, high-quality professional learning opportunities? What sort ofguidance and support do novice facilitators need in order to be successful? Researchers are just beginning to investigatethese 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 willresult in increased student learning of complex subject matter (Even, 2008; LeFevre, 2004). Articulating the practice ofleading high-quality PD is a key component in defining the process through which these programs contribute to advancesin mathematics teacher knowledge, instructional practice, and student learning. Without closer attention to facilitationPD programs, although designed in accordance with general criteria of effectiveness, may, when enacted, fail to produceincreases 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 byschools 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 Phase1, researchers design a PD program and provide initial evidence that it can have a positive impact on teacher learning. InPhase 2 researchers “determine whether the professional development program can be enacted with integrity in differentsettings and by different professional development providers” (p. 9). “Major design activities in Phase 2 include refining aprofessional development program’s tasks and materials for teachers. . ., specifying the role of the facilitator, and developingresources and training for facilitators” (p. 10). Finally, in Phase 3, researchers compare multiple PD programs, investigatingtheir 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 ofresearch on PD programs is Phase 1 research. Phase 2 and Phase 3 investigations are rare; as Borko (2004) noted; herliterature review “did not yield any professional development programs for which there is adequate evidence that they canbe 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 effectivenessof established PD programs. The study discussed in this paper can be considered a Phase 2 investigation, where researchersprovided initial preparation and ongoing support to local providers of PD who, in turn, implemented the PD model withteachers in their schools while researchers documented the process and the impacts. This paper reports on these noviceteacher leaders’ efforts to enact PD, through an analysis of their facilitation skills as they conducted PD workshops in theirschools.

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 processand structure, high-quality PD programs provide opportunities for teachers to participate actively and collaboratively in aprofessional 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 forfacilitators to situate PD in practice (Jacobs, Borko, & Koellner, 2009; Sherin, 2007; Taylor, 2011; van Es & Sherin, 2010;). Trustand respect are also important aspects of PD. Leaders in effective PD programs pay explicit attention to creating a safe andsupportive professional community where teachers are willing to share information often considered “private,” such as videofrom 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 modelinstructional strategies—for example, practices for engaging students in mathematics problem solving and for facilitatingproductive discussions (Clark, Jacobs, Pittman, & Borko, 2005; Stein, Engle, Smith, & Hughes, 2008). These characteristicsensure that participants experience effective instructional strategies as learners and then reflect collaboratively on theirlearning (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 theirstudents’ learning (Borko et al., 2010; Desimone, 2009; Wei et al., 2009). Shulman (1986) introduced a framework for artic-ulating the relevant domains of professional knowledge, a structure expanded upon in the field of mathematics educationby Ball et al. in the Learning Mathematics for Teaching (LMT, 2006) project, through which they identified and elucidatedthe construct “mathematics knowledge for teaching” (MKT)—the mathematical knowledge that teachers must have in orderto 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 con-tent 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 ofstrengthening skills focused on teaching with rich mathematics problems (Lampert, 2001) and eliciting and building on

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H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167 151

tudent thinking (Kazemi & Franke, 2004; Roth et al., 2011). Studies have shown a positive impact on student achievementhen PD leaders forefront the development of MKT (Hill & Ball, 2004; Hill, Rowan, & Ball, 2005).

.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 variety of local contexts and advocate for school and district support of the PD while maintaining integrity to its goals andesign (Cobb & Smith, 2008; Goos, Dole, & Makar, 2007). For site-based PD programs, building the leadership capabilities of

ocal professional developers is critical. PD leaders provide the human resources for building a district’s internal capacity toromote ongoing learning among its teachers. They can foster professional learning communities and collaborative cultures,

ncreasing the human and social capital of schools and districts (Little, 1990). Without strong leadership, professional learningommunities are likely to be ineffective and fail to promote either teacher learning or increased student achievementMcLaughlin & Talbert, 2006).

Despite its importance, developing the knowledge base, experience, and leadership skills of local leaders is often a missingtep in educational reform efforts (Loucks-Horsley et al., 2003; McLaughlin & Talbert, 2006). In the field of mathematicsducation, a small number of researchers are focusing on what PD leaders should know and be able to do, and on how torepare and support them (Elliott et al., 2009; Even, 2008; Koellner et al., 2011). In the project Research on Mathematicseaders’ Learning (RMLL, Elliott et al., 2009) researchers conducted a series of seminars for mathematics PD facilitators,imed at developing knowledge and skills for cultivating mathematically rich learning opportunities for teachers. They aretudying the specialized knowledge that leaders need to facilitate teachers’ learning of MKT. Their assertion, based on initialnalyses, is that leaders must be able to identify mathematics problems and discussion prompts that promote in-depthonversations focused on the mathematics content, support productive social interactions, and orchestrate discussions thatelp teachers unpack their often highly symbolic or incomplete reasoning (Elliott et al., 2009). Findings from RMLL suggesthat to successfully facilitate mathematics PD, leaders require knowledge of mathematics that includes MKT along withnowledge about how to facilitate the development of MKT.

. 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 Professionalevelopment: A Field Study of Preparing Facilitators to Implement the Problem-Solving Cycle (iPSC). The iPSC project inves-

igated the scalability and sustainability of the Problem-Solving Cycle (PSC) model of mathematics PD (described in the nextection), and in particular the degree to which the PSC could be implemented with integrity by novice local facilitators. Thetudy included 2½ years of preparation and support for teacher leaders (TLs) from the participating school district, teachersho volunteered (or were selected by school or district administrators) to facilitate the PSC with the mathematics teachers

n their schools.Our research incorporated video, classroom artifacts, and interviews to document the preparation and support provided

o the TLs; the range and quality of their implementation of the PSC; and the impact of the intervention on TLs, teachers, andtudents. In this paper, we focus on the range and quality of the TLs’ implementation of the PSC. Elsewhere we have writtenbout the preparation and support provided to the TLs and the impact of the intervention (Jacobs, Koellner, & Funderburk,012; 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 oneaching quality as a key factor in impacting student learning. Teaching quality was addressed by preparing TLs to provideD for mathematics teachers, using the Problem-Solving Cycle. The PD of both TLs and teachers was informed by a processf mutual adaptation between the model and the organizational and cultural/linguistic contexts of the district and schoolsBerman & 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 focusn novice teacher leaders’ efforts to enact the PSC model of PD effectively. We address the following research questions:

. To what extent did the teacher leaders enact the Problem-Solving Cycle with integrity to its key characteristics?

. Which characteristics of the PSC did the teacher leaders enact particularly well? Which characteristics were the mostproblematic to enact?

.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.,007; Koellner et al., 2007; Koellner, Schneider, Roberts, Jacobs, & Borko, 2008). The key characteristics of the PSC are derived

rom 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;

152 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

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 instruc-

tional practice.

As implemented in the iPSC project, the PSC entails multiple cycles of three interconnected PD workshops, all organizedaround 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 instructionalpractices. During Workshop 1 of a given cycle, teachers collaboratively solve the selected mathematics task and developplans for teaching it, taking into consideration the needs of their students. The goals of this workshop are to help teachersdevelop 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 highlightkey moments in the instruction and in students’ thinking about the problem. Workshops 2 and 3 of the cycle focus on theteachers’ 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.

H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167 153

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

eachers learn how to elicit and build on student thinking, and to explore a variety of instructional strategies for teachingith rich problems based on targeted learning goals.

.2. Teacher leader preparation

A central goal of the iPSC project was to prepare novice teacher leaders to implement the PSC with integrity to itsey characteristics. By integrity we mean that the teacher leaders adhere to the intended goals and design of the PD;aintaining integrity does not imply rigidly implementing a specific set of activities and procedures (Borko, 2004; LeFevre,

004). Implementing the PSC with integrity entails using a rich mathematics problem as a shared experience; facilitatingroductive discussions about the mathematical content, student thinking and instructional practices; focusing attention onultiple representations and solution strategies; and using video from the teachers’ own classrooms. Based on prior research

n which the developers of the PSC model also served as the PD facilitators, we have documented preliminary evidence ofhe effectiveness of the PSC as well as an emerging understanding of the characteristics of successful facilitation (Borkot al., 2008; Borko, Jacobs, Seago, & Mangram, in press; Clark et al., 2005; Jacobs et al., 2007; Jacobs, Koellner, John, & King,n press; Koellner et al., 2008). This research identified basic supports that novice facilitators of the PSC would be likely toeed in order to (1) create a professional learning community, (2) facilitate mathematics discussions with teachers, and (3)

acilitate 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

SC workshops in their schools. The project provided 2½ years of ongoing, yet gradually decreasing, support for the TLs.his support involved two major components (see Fig. 3): a summer leadership academy and multiple cycles of structureduidance for facilitating the PSC. During Year 1, for the first wave of TLs, the project incorporated one academic semester inhich the TLs participated in a cycle of the PSC facilitated by the research team, as shown in Table 1. (For future waves of

Ls, 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

acilitate PSC workshops. Summer academies were held during each year of the project, and included both new and returningLs. During these academies the TLs reflected on their experiences as PSC “teacher participants” (in Year 1 only), viewed andiscussed selected video clips from prior PSC workshops with respect to key PSC characteristics, developed a general planor implementing the PSC with mathematics teachers in their schools, and participated in PSC simulations (mini-cycles)sing the mathematics problems selected for the upcoming academic year PSC cycles. In initial simulations, members of theesearch team modeled practices central to the successful enactment of PSC workshops. Subsequently TLs planned and thenook turns leading simulations of the various activities that compose the three workshops. These simulations are examples ofhe approximations of practice—“opportunities to rehearse and develop discrete components of complex practices in settingsf reduced complexity” that Grossman and McDonald (2008, p. 190) recommended incorporating into teacher education

rograms.

In addition, members of the research team provided ongoing structured guidance as the TLs facilitated the PSC. Prioro conducting each PSC workshop, TLs attended a full-day Instructional Support Meeting led by the research team. These

able 1PSC project implementation timeline.

Year 1 7 TLs (4 schools) Year 2 5 TLs (3 schools) Year 3 8 TLs (6 schools)

Experience PSCas teachers

SummerAcademy 1

Facilitate PSC Cycle 1 Facilitate PSCCycle 2

Summer Academy 2 Facilitate PSCCycle 3

Facilitate PSCCycle 4

January–May June August–December January–May June August–December January–May2008 2009 2010

154 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

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

Participation dates Teacher leaders Teachers Middle schools

Year 1 (spring–summer 2008) 7 0 4Year 2 (fall 2008–summer 2009) 5 13 3Year 3 (fall 2009–spring 2010) 8 45 6

Totalsa 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 firstInstructional Support Meeting (prior to PSC workshop 1) supported TLs to lead discussions with teachers that would helpthem 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 wereprovided to the TLs. The second and third Instructional Support Meetings (held prior to PSC workshops 2 and 3) supported TLsto 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 toeach school’s context (e.g., cultural/linguistic diversity in the student population, specific workplace norms, constraints ontime and scheduling). The research team remained available to TLs on an as-needed basis throughout the school year toaddress 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 keycharacteristics of the PSC and provide ongoing support for its successful implementation. It provides a description of andrationale for the types of activities that constitute each PSC workshop, describes the various decisions facilitators needto make as they prepare for and conduct each workshop, and includes examples from the research team’s experiences indeveloping the PSC model and conducting PSC workshops. The research team continually made use of the Guide during thesummer academies and Instructional Support Meetings, drawing TLs’ attention to relevant sections as they learned aboutand 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 districtmathematics coordinator, with support from other district administrators, agreed to help recruit mathematics teachersfrom their middle schools to serve as Teacher Leaders and learn to implement the PSC in their own schools. As shown inTable 2, the project began in Spring 2008 with seven TLs from four schools. One school dropped out of the project after thefirst summer academy owing to a change in their local administration. The five TLs from three schools who continued withthe project facilitated two cycles of the PSC during the 2008–2009 academic year (1 cycle per semester). In the followingyear, another school dropped out of the project, but four new schools joined. Thus, eight TLs from six schools participatedin 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 teachingexperience, whether they were the mathematics department chair in the school, and the number of teachers with whomthey worked. Six of the TLs individually facilitated PSC workshops, and two of the TLs co-facilitated.

Table 3Teacher leader demographic information.

Participant (TL) namea Date joined project School name1 Years’ experienceb Department chair

Jordan (J)c January 2008 Pride 10 NoMandy (M)c January 2008 Pride 1 NoRobert (R) January 2008 Champion 32 YesCandace (Cn) June 2009 Woodlawn 13 NoCarla (Cr) June 2009 Torrence Road 15 YesKaitlyn (Kt) June 2009 Four Reed 19 YesKyla (K)d June 2009 Fire Crest 13 NoJason (J)d June 2009 Fire Crest 20 Yes

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

H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167 155

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

.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 workshopsacilitated by the three TLs who participated in the entire 3-year intervention, and the last cycle of workshops facilitatedy the eight TLs who participated in the final year of the intervention. The first PSC cycle used the Lemonade problem, aixture problem adapted from Van de Walle (2007) (Fig. 4). The last PSC cycle used the Fuel Gauge problem, a ratio-and-rate

roblem adapted from Jacob and Fosnot (2008) (Fig. 5). In all cases, the TLs led three workshops per PSC cycle. With twoxceptions (due to last-minute scheduling changes), all of these workshops were videotaped by members of the researcheam. We engaged in a detailed examination of this subset of 28 videotaped workshops to address our research questionsbout adherence of TL workshops to key PSC characteristics, and about characteristics of the PSC model that were particularlyasy or difficult to enact with integrity over time. Interviews with TLs—conducted at the beginning of their participation inhe project and after each PSC cycle—along with field notes from our observations of the workshops, served as secondaryata sources.

.3. Rating and analysis of PSC workshops

In rating the TLs’ videotaped Lemonade and Fuel Gauge workshops, we used an observation protocol adapted from therofessional Development Observation Protocol (PDOP, Banilower & Shimkus, 2004). The PDOP was designed by Horizonesearch, Inc. to evaluate PD sessions using standards for exemplary practice derived from NCTM’s Principles and Standards

or School Mathematics (2000) and NRC’s National Science Education Standards (1996). The PDOP involves holistic ratingsf an entire PD workshop within six major categories: design, implementation, mathematics/science content, exploringedagogy/instructional materials, leadership content, and culture of the session). Each category is composed of a set of morepecific indicators, ranging in number from six to eleven. In addition, each category includes space for the observer to addne or more indicators for attributes of the workshop not captured by the other indicators. Furthermore, the coding manualpecifies that there may be categories that are not applicable to a particular PD session and provides guidelines for observerso use in choosing appropriate categories to rate. Trained observers watch the video of an entire workshop session and thenate the workshop holistically on each indicator using a 5-point Likert scale with possible ratings of 1 (not at all) to 5 (to areat extent), 6 (do not know), and 7 (not applicable). Thus one number on the Likert scale is used to indicate the extent tohich 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 math-matics content category to address TLs’ knowledge of mathematics for teaching, adding indicators related to use of video inhe pedagogy/instructional materials category, eliminating the leadership content section, and eliminating indicators within

emaining categories that the researchers agreed were not relevant to our project. The additional content indicators wereerived from indicators of knowledge of the mathematical terrain in the Mathematical Quality of Instruction instrumenteveloped by the Learning Mathematics for Teaching project (LMT, 2006) and the five practices for orchestrating productive

156 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

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 ourproject.

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 specializedcontent knowledge include, for example, teachers’ discussions of: mathematical skills, procedures, and concepts; a variety ofsolution strategies; relationships among solution strategies; and affordances and constraints of solution strategies. Indicatorsof pedagogical content knowledge specific to the use of video include, for example, facilitator(s)’ selection of video clipsappropriate to the teachers’ need and interests; questions about video clips that encourage teachers to think deeply aboutinstructional practices; questions about video clips that encourage teachers to think deeply about students’ mathematicalideas; 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 ratingdecisions, frequently re-watching portions of video together and discussing segments until agreement was reached. Oncethe three researchers were confident that they were consistent in their ratings (after rating approximately one third of theworkshops), the remaining videos were rated independently by two members of the team, who then met to discuss andreconcile 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 weidentified the major patterns of results within each category. We then selected a subset of indicators in each category thatwere representative of these patterns, and we created a second set of (more manageable) tables that included only theserepresentative 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 ratingon each indicator, across all of their workshops in each of the PSC cycles. We then computed the overall average rating acrossall 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 analysesto create detailed descriptions that illustrate TLs’ facilitation of discussions during PSC workshops that would be rated highlyon 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 thevignettes, the authors examined videotaped records and field notes from several highly rated PSC workshops and selectedthe activities and conversations that were most representative of high ratings on indicators in each category. We thenconstructed vignettes (one vignette for each category) to depict the nature of the events and how TLs thought about them,

H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167 157

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

Indicator Teacher leader Avg

Jr M R

1 Climate of respect for experiences, ideas and contributions 5.00 4.50 4.00 4.502 Collaborative working relationship between TL and participants 5.00 4.44 4.25 4.583 Collegial working relationships among participants 5.00 4.39 3.00 4.134 Active participation encouraged and valued 5.00 3.67 3.75 4.14

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5 Participants demonstrated willingness to share ideas and take intellectual risks 4.50 4.22 3.00 3.94

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

rawing from the videotapes, field notes, and interviews. Vignettes are written in the present tense and set in italics.nterpretive commentary is interwoven using regular font.

.4. Analysis of TL interviews

Three sets of interviews with each TL served as secondary data sources for this paper: initial interviews and interviewsollowing completion of the Lemonade problem and Fuel Gauge problem PSC cycles. The initial interviews, conducted withLs prior to their participation in the summer leadership academy, focused on their conceptions of the PSC, their ideasbout the role of teacher leader, and what they hoped to learn about facilitation from participating in the project. Post-cyclenterviews asked TLs to reflect on their experiences facilitating the set of workshops, new insights and understandings aboutacilitating PD, and the support we provided in the Instructional Support Meetings to help them prepare for facilitating theorkshops. All interviews were audiotaped and transcribed. We read each of the transcripts, identified instances where TLs

alked about topics related to the four PDOP categories that were the focus of our workshop analysis, and then selectedepresentative comments illustrating the TLs’ own interpretations of their workshops.

. 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 threeategories of ratings most central to the goals and design of the PSC: workshop culture, specialized content knowledge, andedagogical content knowledge. For each category, we highlight specific characteristics of the PSC that the TLs seemed toake up easily, and characteristics that they found more problematic. We also present a vignette depicting facilitation thatas rated highly on indicators in the category.

.1. Workshop culture

The first category addresses the workshop culture created by the TLs. This category includes indicators related to theature and extent of participants’ engagement in the PSC. Specifically, we considered the degree to which the TLs fostered alimate of respect and positive working relationships and encouraged active participation and a willingness to share ideashroughout the workshop. We paid close attention to interactions among the participants to see if they seemed engaged andomfortable sharing ideas about the content and their instruction; and the extent to which they took intellectual risks suchs bringing up something they did not understand. These indicators applied to all workshops in each cycle; in other words,e rated workshop culture when the teachers solved the selected PSC problem and when they discussed video. In all cases,e 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, pro-uctive culture (Table 4). Their workshops were characterized by a climate of respect for participating teachers’ experiencesnd ideas. The TLs facilitated the teachers’ engagement as members of a professional learning community and encour-ged them to contribute actively to the group’s exploration of mathematics, student thinking, and instructional practices.esearchers rated indicators of the collaborative working relationships between each TL and the teachers in his or her groups consistently very high. Interview data suggest the TLs also felt positive about the sense of community in their workshops.s Mandy explained, “I already knew the teachers, so the comfort level and things like that were pretty much set.” Jordangreed, noting that “the group came together rather quickly.” Robert’s Lemonade workshops were rated somewhat lowerhan those of Mandy and Jordan on most indicators of workshop culture. Researchers noted that in Robert’s group the tea-hers’ engagement in and contributions to discussions were sometimes uneven. However, in reviewing interview transcriptsrom the two teachers in Robert’s group, we found their comments consistently very positive; they had high praise for theirxperiences 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 theiruel Gauge cycle workshops (Table 5). Researchers consistently gave the workshops high ratings on indicators of collegialorking relationships among participants and climate of respect for experiences, ideas, and contributions. Again there were

ome notable individual differences. Both Jordan and Mandy worked with a different group of teachers for the Fuel Gauge

158 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

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

Indicator Teacher leader Avg

Jr M R Cn Cr Kt K&J

1 Climate of respect for experiences, ideas and contributions 4.50 4.33 4.33 4.33 4.50 4.83 5.00 4.552 Collaborative working relationship between TL and participants 4.00 4.67 4.50 3.00 4.17 5.00 5.00 4.333 Collegial working relationships among participants 4.67 4.50 4.00 4.67 4.33 5.00 5.00 4.604 Active participation encouraged and valued 4.00 4.17 4.00 3.83 4.00 4.67 4.83 4.215 Participants demonstrated willingness to share ideas and take intellectual risks 3.83 3.83 3.67 3.50 3.50 4.17 4.83 3.90

a 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 conductPD with these new groups. Their decrease in comfort with the participating teachers may explain why Jordan and Mandy’sFuel 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 receivedthe highest ratings. These two teacher leaders were very aware of the importance of building community and thoughtfulabout ways to encourage participation by all their teachers. Kyla explained, “In the beginning, we had our group of teachersthat 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-facilitatorJason reflected, “We’ve always been a group that doesn’t like to share in a big group setting. . .. This year we actually brokedown those barriers, and now we’re more open about discussing things.” Candace, unlike the other TLs, was not the chairof the mathematics department and had no formal leadership role in her school prior to her involvement in the PSC. Herlimited experience in leadership positions may help explain why her workshops were rated somewhat lower than those ofthe other TLs.

5.1.1. Workshop culture vignette: Jason and Kyla’s Fuel Gauge Workshop 1The 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 cultureby framing questions that have personal relevance to each teacher. Prior to having the teachers work through the task insmall groups, Jason and Kyla asked them to individually consider how they and their students would approach the problemas well as the difficulties their students might encounter when solving it. This facilitation strategy helped to encourage theteachers 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 abouthow you would solve it. You do not have to actually solve it, just think about how you would. Also, think about the issues ordifficulties your students might encounter when solving this problem.” Teachers spend a few minutes silently reading theproblem 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 particulargroup of students. If you teach sixth grade, what difficulties might they experience when solving the problem? If you teacheighth grade, what might your students experience?” These questions incite a lively discussion, with most teachers in thegroup participating. They share ideas about how their students might struggle including misunderstanding the context ofthe 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 tobuild community by engaging all participants in a relevant and safe manner. This strategy invites participants to contributeto the professional learning community and take ownership of issues raised in the PD. Jason and Kyla were pleased withthe community they saw developing in their math department over the course of the iPSC project. Jason explained in aninterview that the PSC workshops enabled teachers in his school to more actively share their ideas. “That was probably themost awesome thing that happened. We brought sharing out of people, and then it got better as we went on. There wasmore openness as we went on.”

5.1.2. General patterns in workshop cultureOverall, 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 relationshipswithin 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.

H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167 159

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

Indicator Teacher leader Avg

Jr M Rb

1 Teachers generate and analyze ways to solve task 5.00 4.33 – 4.672 Teachers generate/analyze reasoning used to arrive at correct/incorrect solutions 4.00 3.67 – 3.843 Discussion of various solution strategies 4.00 4.67 – 4.344 Discussion of relationships among solution strategies 3.00 3.67 – 3.345 Discussion of affordances and constraints of various solution strategies 3.00 4.33 – 3.676 Discussion of various mathematical representations 2.50 4.33 – 3.427 Discussion of relationships among representations 1.50 4.33 – 2.928 Discussion of affordances and constraints of various representations 1.50 3.67 – 2.59

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.2. Specialized content knowledge

Indicators of specialized content knowledge (SCK) denote the extent to which the TLs engaged teachers in productiveiscussions about the mathematical knowledge needed to teach a lesson using the focal PSC problem. In keeping withirections on the PDOP, we rated this category only for workshops where increasing participants’ SCK was a key purpose or

vehicle for accomplishing other stated purposes of the session. Thus, Tables 6 and 7 include ratings only for PSC Workshop (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 ofriteria, including the potential to be solved using multiple representations and strategies (Koellner et al., 2007). Consistentith goals of the PSC, the eight indicators we chose to include in these tables present data on characteristics of the workshopiscussions related to representations and solution strategies. Representations refer to pictures, diagrams, manipulatives,nd other models used to represent mathematical ideas or procedures. By different solution strategies, we mean differentathematical approaches to solving a problem (LMT, 2006). For both representations and solution strategies we considered

he extent that TLs (a) generated and discussed multiple examples, (b) discussed the mathematical relationships among thexamples, 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 notideotaped because of logistical circumstances. Both Mandy and Jordan facilitated discussions about a variety of possibleolution strategies for the Lemonade problem, and elicited reasoning that could be used to arrive at correct and incorrectnswers. In almost all cases, indicators related to solution strategies (Table 6, indicators 1–5) were rated higher than indicatorselated to representations (Table 6, indicators 6–8). Mandy and Jordan also were better able to foster discussions aboutarious solution strategies and mathematical representations (Table 6, indicators 1–3, 6) than about either mathematicalelationships among solution strategies and representations, or their affordances and constraints (Table 6, indicators 4, 5, 7,).

The general pattern of ratings for the first workshop in the Fuel Gauge cycle was similar to that seen in Mandy’s andordan’s first Lemonade cycle workshops. In all seven Fuel Gauge Workshop 1, the TLs were able to facilitate discussions toxplore the variety of representations and solution strategies that could be used in solving the Fuel Gauge problem (Table 7,ndicators 1–3, 6). Again, for both solution strategies and representations, TLs had more difficulty engaging teachers iniscussions to address either the affordances and constraints of the examples they generated, or the relationships amonghem (Table 7, indicators 4, 5, 7, 8). And, in general, discussions were rated higher with respect to solution strategies (Table 7,

ndicators 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 inheir Fuel Gauge Workshop 1 compared to their Lemonade Workshop 1. In fact, Mandy’s Fuel Gauge Workshop 1, although

able 7pecialized content knowledge: average ratingsa for Fuel Gauge Workshop 1.

Indicator Teacher leader Avg

Jr M R Cn Cr Kt K&J

1 Teachers generate and analyze ways to solve task 3.50 4.50 3.00 4.00 3.50 4.50 4.50 3.932 Teachers generate/analyze reasoning used to arrive at correct/incorrect solutions 2.50 3.50 2.50 1.50 2.50 3.50 4.00 2.863 Discussion of various solution strategies 4.00 4.50 3.00 3.50 3.50 4.00 4.50 3.864 Discussion of relationships among solution strategies 2.50 2.50 2.00 2.00 2.50 3.00 3.00 2.505 Discussion of affordances and constraints of various solution strategies 1.50 2.50 3.00 1.50 1.50 2.50 2.50 2.146 Discussion of various mathematical representations 2.50 3.00 3.00 3.50 2.50 3.50 3.50 3.077 Discussion of relationships among representations 1.50 1.50 2.00 2.50 2.00 2.50 2.50 2.078 Discussion of affordances and constraints of various representations 1.50 2.50 3.00 3.00 1.50 3.00 3.00 2.50

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

160 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

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

5.2.1. Specialized content knowledge vignette: Mandy’s Lemonade Workshop 1Throughout Workshop 1 of the Lemonade cycle, Mandy was able to lead fruitful discussions around a variety of mathemat-

ical solution strategies. She seamlessly interwove related topics, such as how the teachers themselves solved the Lemonadeproblem, what their answers mean in the context of the problem, how their students might approach the task, and instruc-tional moves that could be beneficial such as using manipulatives or other representations. These types of conversations areconsistent with the goal of supporting the development of teachers’ specialized content knowledge, in preparation for themto teach the task.

“First I am going to have you solve the Lemonade problem and think about ways your students might solve it too,” Mandytells 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 fractionsand find the percent to determine which lemonade recipe had the stronger flavor.” Another teacher, Mary, explains, “I useda part-whole fraction too because I knew that I could compare them and use a decimal or percent to find the solution. I didthat 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 mightnot 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 addedone more water and one more lemonade, so they are still the same.” Mandy points to the green and yellow tiles she hasstrategically placed on the teachers’ desks and asks Mary to represent this way of thinking. As Mary begins arranging thetiles, Krista comments, “I think the kids might see that the one extra water and the one extra lemonade would cancel eachother out.”

Mandy begins using chart paper to record all of the strategies the group has discussed thus far, and to ensure that everyone hasa thorough understanding of each strategy. Mandy states, “The first method that all of you used was a fraction representationthat 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 ratio2:3 compared to 3:4.”

Mary notices that there is an additional strategy students could use. She explains, “I was thinking that they might say thatpitcher B would be more concentrated because they might look at ¾ and say that was 75% and then 2/3 is 67%.” Mandypresses Mary to talk more about this strategy and she continues, “Because those are benchmark fractions. They might sayone 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 oflemonade mix.” He then adds, “So, one cup water is ¾ cup of lemonade mix.” Mandy reiterates, “That is like finding the unitrate 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 solutionstrategies fairly quickly. However, they did not necessarily have a deep understanding of each strategy, and they struggled tointerpret what the ratios meant in the context of the problem. Through skillful questioning and probing, Mandy encouragedthe teachers to examine the meaning of the different ratios, and to think carefully about how to accurately guide theirstudents, including anticipating misconceptions and demonstrating why the right answers were in fact accurate.

5.2.2. General patterns of SCK in generating solution strategiesIn 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 manydifferent angles or ways. And we discuss how we solved it and how we think kids will approach it. We look at the variouspossible strategies in solving it and we analyze them.” Discussions of solution strategies were rated higher than discussionsof mathematical representations. Also, overall, average ratings were consistently lower on indicators for the Fuel Gaugeproblem 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 thefive indicators that entail analyzing reasoning, discussing relationships among representations or solution strategies, anddiscussing affordances and constraints of representations or solution strategies. Making comparisons among examples (ofsolution strategies or representations) is likely to be a more cognitively complex task than exploring individual examples.

H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167 161

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

Indicator Teacher leader Avg

Jr M R

1 Video clips are accessible and relevant to the teachers 4.00 4.75 3.75 4.172 Video clips are appropriate with respect to the level of trust within the community 3.00 5.00 4.00 4.003 Questions about video clips encourage teachers to think deeply about students’

mathematical ideas and reasoning4.00 4.75 2.25 3.67

4 Does the TL frame discussions and use prompts in ways designed to fosterdevelopment of KCS?

3.50 4.50 3.00 3.67

5 Do the teachers engage in careful unpacking and deep analysis of students’mathematical ideas and reasoning?

3.00 4.50 3.00 3.50

6 Questions about video clips encourage teachers to think deeply about instructionalpractices

2.50 4.50 2.00 3.00

7 Does the TL frame discussions and use prompts in ways designed to fosterdevelopment of KCT?

– 3.75 2.00 2.88

8 Do the teachers engage in careful unpacking and deep analysis of instructional – 3.75 2.00 2.88

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

.3. Pedagogical content knowledge

Indicators of pedagogical content knowledge (PCK) provide a measure of the extent to which TLs used strategies designedo deepen teachers’ knowledge of content and students (KCS) and knowledge of content and teaching (KCT), particularlyith respect to problem-based teaching. KCS includes, for example, engaging in analysis of students’ mathematical ideas

nd reasoning. KCT includes engaging in analysis of teachers’ instructional practices. Because these knowledge domains areot the focus of PSC Workshop 1, Tables 8 and 9 include only data from Workshops 2 and 3, in which groups view and discussideo 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’lassrooms, we adapted the PDOP to include indicators of TLs’ skill in selecting appropriate video clips and framing theiewing and discussion of these clips during their workshops. Four of the indicators included in Tables 8 and 9, which weeveloped specifically for this analysis, focus on TLs’ use of video (Tables 8 and 9; indicators 1–3, 6). The other four indicators,aken 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 appropriatenessf video clips selected by TLs from their teachers’ PSC lessons. This was true despite the fact that in their interviews, allhree TLs commented on the difficult and time-consuming nature of picking good video clips. Jordan explained that whenlanning a workshop, “You have to find the clip that interests you and then develop the guiding question.. . . And it’s notasy finding the clip.” They also sometimes felt limited by the videotapes from which they could choose. Robert noted thatt was challenging “getting enough from a videotape to give you a couple of really good situations to discuss, as opposed

o having a videotape that’s been made and chosen (by someone outside the PD group).. . . Sometimes it’s hard finding thexamples you’re looking for.”

able 9edagogical content knowledge: average ratingsa for Fuel Gauge Workshops 2 and 3.

Indicator Teacher leader Avg

Jr M R Cn Cr Kt K&J

1 Video clips are accessible and relevant to the teachers 4.00 4.00 3.75 3.50 5.00 4.75 5.00 4.292 Video clips are appropriate with respect to the level of trust within the

community3.75 4.75 4.00 4.50 4.50 4.25 4.75 4.36

3 Questions about video clips encourage teachers to think deeply aboutstudents’ mathematical ideas and reasoning

1.00 4.75 3.25 1.50 3.50 4.00 3.75 3.11

4 Does the TL frame discussions and use prompts in ways designed to fosterdevelopment of KCS?

– 4.25 2.50 1.50 3.75 4.00 4.50 3.42

5 Do the teachers engage in careful unpacking and deep analysis of students’mathematical ideas and reasoning?

– 4.25 2.50 1.50 3.25 4.25 4.25 3.33

6 Questions about video clips encourage teachers to think deeply aboutinstructional practices

2.50 5.00 2.00 2.00 4.00 4.25 4.00 3.39

7 Does the TL frame discussions and use prompts in ways designed to fosterdevelopment of KCT?

3.00 4.50 3.25 1.50 3.50 3.75 4.25 3.39

8 Do the teachers engage in careful unpacking and deep analysis ofinstructional practices?

2.00 4.00 2.75 2.50 2.75 3.75 4.00 3.11

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

162 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

The TLs were more successful in selecting video clips to use as springboards for discussions of the Lemonade problemthan 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, theyseemed to be more tentative about focusing on the practice of teaching than on student reasoning. TLs may have been lesscomfortable pushing teachers to critique their instructional practices than to deeply analyze their students’ mathematicalreasoning.

As was true for the Lemonade workshops, the video clips that TLs selected for Fuel Gauge Workshops 2 and 3 were ratedas accessible and relevant to the teachers with whom they were working and appropriate with respect to the level of trustwithin the community (Table 9, indicators 1, 2). Also similarly to the Lemonade workshops, TLs’ ability to select appropriatevideo clips was generally stronger than their ability to engage teachers in thinking deeply about instructional practices andstudents’ mathematical ideas and reasoning. Many of the TLs realized that facilitating these types of discussions was anarea where they could benefit from additional support. For example, Candace noted that she would like “more help askingthe right question [when showing video]. In one workshop, I had some questions up [on the Smartboard] and that broughtbetter discussion.”

Unlike the Lemonade workshops, there was not a large difference between discussions of instructional practices andstudent thinking in terms of the quality of TLs’ questions or the engagement of teachers. This lack of a difference supportsour 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 asteachers 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 indicatedby their ratings on all six discussion-related indicators. There also were individual differences in the quality of discussionsfacilitated by the TLs who joined the project during the second summer institute. As was true for Workshop Culture, Candace’sworkshops were rated somewhat lower than those of other TLs. Kyla and Jason’s workshops, like Mandy’s, were rated higherthan 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 3In 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 ofteachers’ pedagogical content knowledge. Throughout the conversation, she skillfully negotiated two related goals: helpingteachers to identify and understand the mathematical misconceptions underlying the student’s error, and encouraging themto 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 arethe 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 involvesdividing 600 by 120, and arriving at the answer 50. The teachers quickly recognize that not only has the student incorrectlydivided 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 soit’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 tocollaboratively recreate the student’s strategy. As they are working, the teachers discuss how the student has set up the ratioin the problem and what the numbers in the ratio represent. One teacher suggests that the student has found “the inverse ofwhat she wanted, which is 1/5 of the tank.” Another teacher provides a different way of interpreting the 5, suggesting thatthe student may have been trying to determine the number of trips between the two farms that can be taken on a full tankof gas, “5 times back and forth between Stan and Louisa on one full tank.” Mandy asks the teacher to repeat this idea andshe 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? Whatwould we do as the teacher?” One teacher suggests, “I would just ask them, ‘What does the 600 mean? What does the 120mean? If you are dividing 600 by 120, what are you dividing?’ And hopefully they would see that if they are dividing a wholetank 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 pedagogi-cally. In facilitating the discussion she made sure that the teachers understood the meaning of the numbers in the student’sstrategy 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 responseto students’ misconceptions is typical of her facilitation of video-based discussions.

5.3.2. General patterns of PCK in selecting video and facilitating discussionsThe 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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nstructional practices or student thinking. During our Instructional Support Meetings, we focused heavily on the importancef selecting video and formulating questions to launch meaningful discussions; and the data may reflect the TLs’ extensivefforts to find appropriate video clips, both during the Instructional Support Meetings and on their own time. Navigatinghese discussions during real time in PD workshops is likely to be a more challenging skill than either selecting video clips oreveloping launching questions, and thus may require additional time and support for TLs to develop. Individual differencesmong TLs also come into play in this regard, with some likely needing more support than others.

. 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 forn extended period of time to help them learn to facilitate the Problem-Solving Cycle. Our findings should be consideredentative, and not necessarily replicable or applicable to all PD contexts. At the same time, these analyses offer a numberf insights regarding novice leaders’ facilitation of the mathematics PD workshops. They also suggest several issues for theeld to take up as we continue to develop PD models and consider how to prepare and support PD facilitators to lead them.e 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 charac-

eristics. Overall, our ratings of their workshops indicate that TLs were able to successfully facilitate the PSC with groupsf teachers in their schools and maintain integrity to the key characteristics of the PSC. The TLs led workshops that closelyatched the broad outlines and intentions of the PSC model, including a focus on the relevant goals and activities within

SC Workshops 1–3. The TLs navigated significant logistical obstacles including recruiting teachers to consistently attendorkshops, finding time and space to meet within their school, and incorporating the use of video. Meeting these pragmatic

hallenges ensured that they could establish and maintain a situated learning environment, a professional community, these of classroom video, and an ongoing sustainable structure for the PD. Although these findings address one central aspectf scalability, namely that PD leaders other than the program designers can learn to facilitate the PSC with integrity to its coreharacteristics, additional research is needed to determine if the leadership preparation and support can be accomplishedn 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 charac-eristics were more problematic to enact. The importance of this question became clear early in the analysis as we noticedhat some of the PSC characteristics were addressed well by virtually all TLs, whereas other characteristics were notablyifficult to enact. In this section we highlight both sets of characteristics and suggest possible reasons for these differences

n ease of enactment.

.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 establishollaborative working relationships among their teacher participants. A central component of the PSC model involves devel-ping and maintaining a professional learning community in which teachers are comfortable working together to improveheir teaching. In designing the model we were mindful that some PD programs have found teachers unwilling to shareideo from their own classrooms with their colleagues (Grossman, Wineburg, & Woolworth, 2001; Sherin & Han, 2002).or teachers to be willing to expose their instructional practices in this way, they must feel part of a safe and supportiverofessional environment and be confident that showing their videos will provide productive learning opportunities forhemselves and their colleagues. In our own experiences with developing and leading the PSC, participants formed a sup-ortive community and engaged in increasingly reflective and productive conversations around video from one another’slassrooms (Borko et al., 2008). As a result of these experiences, in both our initial meetings with the TLs and the Facili-ator’s Guide we provided concrete suggestions for establishing and maintaining community. It seems likely that explicitttention to community contributed to the TLs’ success in enacting this key PD practice beginning with first year of theirarticipation.

Another area where TLs consistently received high ratings was the selection of video clips. One question frequentlysked about the PSC is whether TLs new to this PD model will be able to select video clips that can foster rich discus-ions about important ideas relevant to classroom practice. Two sets of concerns underlie this question. First, prior to theirnvolvement in the PSC, teachers are likely to have had little experience discussing classroom video. Second, because ofhe PSC’s focus on using video from participating teachers’ classrooms, the set of video recordings from which TLs haveo choose is limited. Because the selection of appropriate clips is critical to the success of the PSC, the Facilitator’s Guideetails key characteristics of “rich” video clips and provides examples of the types of clips that we found to be successful

n fostering productive discussions in our previous research. In addition, during the Instructional Support Meetings the

Ls were given time to watch video on their own and with colleagues, select potential clips, and share those clips withhe group. These multiple supports—particularly the time during Instructional Support Meetings to select potential videolips and receive feedback about their selections—likely contributed to the TLs’ successful enactment of this feature of theSC.

164 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

6.2. Characteristics more difficult to enact

While there were notable individual differences among TLs, they generally had more difficulty supporting deep analysisin discussions to foster both SCK and aspects of PCK including knowledge of content and students (KCS) and knowledgeof content and instruction (KCT). In this section, we posit some ideas as to why novice facilitators may have found thesecharacteristics 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 additionalsupport related to these characteristics.

With respect to promoting the development of teachers’ specialized content knowledge, TLs were generally able toengage their groups in discussions focused on multiple mathematical representations and multiple solution strategies for aparticular problem. However, they were less successful at engaging teachers in discussions about the relationships, affor-dances, and constraints of representations and solution strategies. The three indicators on which the TLs were rated morehighly—generating and analyzing ways to solve the task, discussions of various representations, and discussions of varioussolution strategies—are similar to what Kazemi and Stipek (2001) referred to as “low-press exchanges” in the elementarymathematics classrooms they studied, exchanges such as solving open-ended problems in groups and sharing solutionstrategies. In contrast, the five indicators that entail analyzing reasoning, discussing relationships among representationsor solution strategies, and discussing affordances and constraints of representations or solution strategies are similar tothe “high-press exchanges” they described. Kazemi and Stipek argued that high-press exchanges such as collaborativelyexamining the relationships among multiple strategies and using mathematical argumentation to arrive at shared under-standings are necessary for promoting development of students’ mathematical ideas. They also posited that for teachers tofoster high-press exchanges in elementary mathematics classrooms, they must have a deep understanding of the nature ofconceptual 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 improvestudents’ inquiry learning experiences.

Similarly, we suggest that guiding discussions in PD workshops to focus on reasoning and relationships requires adeeper conceptual understanding of mathematics than does facilitating discussions that do not go beyond identifying anddescribing mathematical representations and solution strategies. Our findings suggest that teacher leaders would bene-fit from additional types of support than we provided in the summer academies and Instructional Support Meetings tohelp them develop the specialized knowledge of mathematics needed to engage teachers in reasoning about similaritiesand differences among mathematical representations and solution strategies. For example, although we included extensiveopportunities for the TLs to solve and analyze the PSC tasks and plan their Workshop 1, we did not create opportuni-ties for them to rehearse these workshops and receive feedback. In addition, the TLs may have benefitted from workingin small groups to examine a set of possible PSC tasks, with each group analyzing a different task and preparing to leada discussion of that task, and then facilitating the discussion and receiving feedback from their peers and the researchteam.

Promoting the development of teachers’ pedagogical content knowledge is another area in which the TLs sometimes haddifficulty. They were successful at selecting video clips to use as springboards for discussions to foster PCK and at developingquestions to launch the discussions. However, they were less capable of orchestrating discussions that maintained a deeplevel of analysis relating to student reasoning and instructional practices. The Instructional Support Meetings did includeextended opportunities for the TLs to anticipate the types of conversations their selected clips would support and to planaccordingly. 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, perhapsincluding more rehearsals, role-plays, and coaching. The TLs may have benefited from more frequent opportunities to practicefacilitation, for example by taking turns leading discussions using their selected video clips during the Instructional SupportMeetings and then receiving feedback from their peers and the research team. Our final interviews with the mathematicsleaders provided some support for this conjecture. For example, when asked how the Instructional Support Meetings mighthave better supported their facilitation, Kyla suggested “spending more time choosing video, and taking clips and actuallylooking at them together as a group . . . and pulling out questions together. Doing that more as a group could have beenbeneficial.”

Smith and Stein (2011) noted that facilitating discussions with students that build on their thinking and use theirresponses to advance the mathematical understanding of the class involves skillful improvisation. To orchestrate conversa-tions that extend students’ mathematical reasoning, teachers must be able to “diagnose students’ thinking on the fly and toquickly devise responses that will guide students to the correct mathematical understanding.” They identified five practicesdesigned to moderate the degree of improvisation required by the teacher during a discussion by “shifting some of thedecision making to the planning phase of the lesson” (p. 7): anticipating, monitoring, selecting, sequencing, and connect-ing. 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 theycan 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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D leaders need to orchestrate productive discussions with teachers, it may be helpful to consider how the five practicesmith and Stein (2011) recommended for classroom teachers are relevant for professional development facilitators.

. Conclusion

Scaling professional development requires, as a key consideration, developing a large number of skillful and capableD facilitators. Given the current demand for extensive, high-quality PD, there is an urgent need to identify the types ofnowledge and skills these facilitators draw on, and to determine how they can best be supported to develop and expandheir relevant knowledge and skills (Marrongelle et al., 2013). Although the iPSC project involved a relatively small numberf PD leaders, the analyses of their facilitation efforts as reported in this paper moves us one step forward in achieving theseoals.

To further this work and to facilitate the sharing of ideas across research teams, we conjecture there is an area of pro-essional knowledge pertaining to leaders of effective PD that needs to be labeled, defined, and systematically investigatedStein & Nelson, 2003). Within the field of mathematics education, building on the work of Ball et al. (2008), we proposeo give this knowledge the initial label of Mathematical Knowledge for Professional Development (MKPD). We anticipatehat, similar to the construct Mathematical Knowledge for Teaching, MKPD has multiple domains that must be identifiednd disentangled. As a starting point, we posit that MKPD may include specialized content knowledge and pedagogicalontent knowledge (comprising both knowledge of content and students and knowledge of content and teaching) but from

PD leader perspective. The SCK that facilitators need is likely to entail a deep understanding of the range of potentialolution strategies and representations specific to a given mathematical context; the mathematical relationships withinhese sets of solution strategies and representations; and the mathematical affordances and constraints of each strategy andepresentation. The PCK that facilitators need likely includes the ability to engage teachers in the interpretation of students’athematical 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

o establish and maintain a professional mathematics learning community, which we label mathematics learning communitynowledge. We speculate that this knowledge includes the ability to create a culture of respect, establish group norms, andoster active participation in which teachers share ideas and take intellectual risks. Learning community knowledge is likelyo 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 onur analyses of the teacher leaders’ facilitation of the PSC PD and in many ways parallel the three categories of facilitationractices that we highlighted in the results section. Although the logic here may appear to be somewhat circular, ourrgument is that these practices are associated with highly related knowledge constructs, and it is the interrelationship ofoth 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,nd learning community knowledge—go beyond and look different than the knowledge that a typical mathematics classroomeacher holds. Because PD leaders are expected to promote the development of teachers’ knowledge in these domains, they

ust hold a deeper and more sophisticated knowledge of mathematics than their colleagues, just as teachers must hold aeeper and more sophisticated knowledge than their students. In addition, PD leaders should be knowledgeable about howo work productively with adult learners, and construct environments for teachers to collaborate about relevant topics. Thesedeas are compatible with the work of Elliott et al. (2009), who have begun to identify features of math leaders’ SCK and theature of sociomathematical norms productive for teacher learning. It seems clear from our work, and the work of others,hat focused and sustained support for facilitators is necessary to promote the development of MKPD. Novice PD leaders areikely to require increased knowledge of how to improvise skillfully (Smith & Stein, 2011) and orchestrate conversations thatlicit “high-press exchanges” (Kazemi & Stipek, 2001) where teachers deeply examine central mathematical relationships,nstructional practices, and student thinking.

In the present study novice facilitators enacted some characteristics of effective PD more readily than others. Thesendings highlight several questions about the development of MKPD that warrant further investigation. For example, canD leaders develop their knowledge in particular domains of MKPD earlier and with greater ease than in other domains? Areertain leadership-preparation and -support activities better suited to promoting facilitators’ MKPD focused on specializedontent knowledge, whereas other types of activities are better suited to promoting MKPD focused on pedagogical contentnowledge? What factors differentiate PD leaders and their ability to develop MKPD? Additional empirical and theoreticalnvestigations are needed to further elucidate MKPD and to generate the experiences PD leaders need in order to take onhe important role of driving change in the teaching profession.

cknowledgments

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

166 H. Borko et al. / Journal of Mathematical Behavior 33 (2014) 149– 167

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

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