Tools for Rethinking Classroom Participation in

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Classroom Participation as Opportunity to Learn

Tools for Rethinking Classroom Participation in Secondary Mathematics Laurie H. Rubel and Anders J. Stachelek City University of New York

In this article, we share our experiences in mathematics teacher education around professional development for teachers with a focus on student participation as an opportunity to learn. We describe a process through which teacher educators can support teachers in increasing and improving classroom participation opportunities for their students. We present Lesson Activity Bars and Difference in Participation Proportion, complementary tools that quantify and represent student participation in the mathematics classroom. We demonstrate the effectiveness of these tools in supporting teacher growth in the context of a professional growth project for teachers in urban secondary schools. In general, the teachers in this project increased the amounts of active participation they made available to their students. The cases of two teachers are analyzed in detail, using Clarke and Hollingsworth’s (2002) Interconnected Model of Professional Growth, to add depth and nuance to our understanding of processes of teacher growth around increasing student participation opportunities in the mathematics classroom. Keywords: Equity; Participation; Urban schools

Introduction In this article, we report research on a professional growth project (PGP) for secondary mathematics teachers in urban schools. The participating teachers taught in schools in underserved areas in a large city, a school context in which deficit beliefs about low-income Black and Latino/a students are prevalent (Davis & Martin, 2008). These deficit beliefs influence teachers’ expectations of their students and typically manifest in instruction through a foregrounding of student compliance, skill remediation, and procedural fluency (Cross, 2009; Kitchen & Berk, 2016). Because instruction with these emphases usually does not engage students, any weak achievement then reinforces underlying deficit views

about students, their families, and their communities. The PGP was designed to disrupt this cycle through a focus on improving specific instructional practices around the theme of “centering” (Tate, 1994) instruction on students. Our focus is on the PGP’s approach to supporting mathematics teachers in increasing and diversifying classroom participation opportunities, which we describe in further detail below.

Perspectives Participation as Opportunity to Learn Teachers create classroom participation opportunities in correspondence with their perspectives on learning and beliefs about mathematics (Boaler & Greeno, 2000; Cross, 2009). For example, a direct instruction model of learning premises that students learn best through direct instruction, listening, watching, and then practicing (Munter, Stein, & Smith, 2015). In contrast, a dialogic model of learning favors the development of understanding through connected processes of problem solving, communication, and reflection (Munter et al., 2015). Researchers have found that the dialogic-based approach is more effective than direct instruction (Boaler & Greeno, 2000; Lampert, 1990; Marshman & Brown, 2014; Nathan & Knuth, 2003; Sfard, 2008), particularly for students from underserved groups (Diversity in Mathematics Education [DiME], 2007; Franke, Kazemi, & Battey, 2007; Moschkovich, 2013; Silver, Smith, & Nelson, 1995; Zahner, Velazquez, Moschkovich, Vahey, & Lara-Meloy, 2012). In a dialogic-based approach, participation opportunities provided by teachers are an integral aspect of an opportunity to learn (Cobb, Gresalfi, & Hodge, 2009; DiME, 2007; Gresalfi, Martin, Hand, & Greeno, 2009; Wager, 2014). Since participation can be uneven (Esmonde & Langer-Osuna, 2013), specific pedagogical strategies that are known to foster equitable participation are crucial (Hand, Kirtley, & Matassa, 2015). One such strategy is Complex Instruction’s multidimensionality (Cohen & Lotan, 1995), which has been shown to be a critical feature of equity pedagogy in effective urban high schools (Boaler & Staples, 2008). Multidimensionality premises that all students have strengths and providing more ways for students to participate in classroom learning affords more students greater access to success (Boaler & Staples, 2014; Horn, 2012).

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Laurie H. Rubel and Anders J. Stachelek Participation can be considered only in concert with the lesson’s mathematical task because it is the nature of the task that invites dialogical forms and categories of participation. For instance, cognitively demanding tasks (Stein & Smith, 1998) foster multidimensionality in the way they are oriented around mathematical investigations, discussions, and reflections (Doerr & English, 2006; Horn, 2012). This is in contrast to procedural tasks that are naturally suited to a narrower set of participation categories, such as listening, copying, and practicing. In a classroom that emphasizes multidimensionality, teachers provide mathematical tasks that allow students to participate in mathematics in a variety of ways and that lead them to construct connected knowing, such as contributing to mathematical discussions, investigating patterns, communicating ideas, posing questions, generating alternative strategies, or using technology to explore mathematical concepts.

Teacher Growth In this article, we study teacher growth around considering student participation in connection with their opportunities to learn. We conceptualize growth using Clarke and Hollingsworth’s (2002) Interconnected Model of Professional Growth (IMPG, Figure 1), which comprises four domains. The external domain includes professional development, feedback, and ideas from colleagues or resources, whereas the domain of practice represents a

teacher’s instructional practices. The personal domain consists of beliefs, knowledge, and attitudes, and the domain of consequence indicates a teacher’s goals for students. The domains are connected by cyclical processes of enactment and reflection. The envelopment of these four domains within a “change environment” stresses that aspects of the school context can facilitate or constrict teacher growth. A “change sequence” is evidenced by changes in at least two domains as well as the enactive or reflective links that connect them and are typically reinforced by the model’s inherent cyclicality (Witterholt, Goedhart, Suhre, & van Streun, 2012). Goldsmith, Doerr, and Lewis (2014) frame their research synthesis about mathematics teacher growth using the IMPG, highlighting the model’s recursivity, and they use the IMPG to show that teacher growth in one domain can further growth in others. Here, we use the IMPG to study the process of teacher growth in attending to and improving their students’ participation opportunities.

Overview of Professional Growth Project and Research Methods The PGP focused on the theme of “centering” (Tate, 1994) instruction on students, which we operationalized according to four dimensions: (a) holding high expectations for Black and Latino/a students as learners of mathematics (Martin, 2007); (b) selecting cognitively demanding mathematical tasks (Stein, Grover, & Henningsen, 1996);

Figure 1. Interconnected Model of Professional Growth (Clarke & Hollingsworth, 2002, p. 951).

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Participants The first author recruited teachers through a local teacher network by advertising the theme of centering instruction on students, and then she observed participants during the semester prior to the PGP. After this, the PGP was initiated with a 40-hour summer institute and continued with 2-hour monthly meetings through the school year (an overview is provided in Appendix A). Participants were observed again in the fall semester of the following school year. Here, we focus on the eight teachers who participated across all three data collection years and analyze in more depth the growth of two of the teachers in the project. Seven participants identified as women (four White, two Black, and one Latina) and one as a White man. Four had entered teaching through alternative certification programs and four through traditional pathways. The most experienced teacher had 12 years of experience, the least experienced had 2 years of experience, and the median number of years of experience was 5. Seven of the participants remained at the same schools across the PGP’s 3 consecutive years, and one teacher changed schools each year. Their geographically diverse schools predominantly served African American and Latino/a students (63–100%, median 94%) from low-income families 1 2

Classroom Participation as Opportunity to Learn (68–95% free or reduced lunch eligibility, median 84%). We present results for all eight teachers but highlight Lucy and Teresa1 among those who showed the greatest and the least changes by the chosen measures.

Research Methods Our research question is, How can professional development that emphasizes attention to multidimensionality in the design of mathematics instruction support teachers in changing the participation opportunities and structures they afford to the students in their mathematics classrooms? We visited teachers two times before the PGP in spring 2012; five times during 2012–2013; and two times the following year, in fall 2013, for a total of 70 lesson observations.2 We scheduled these visits in advance and asked teachers to conduct their classes as usual, with the exception of not administering tests. We wrote field notes during each lesson observation and converted those into detailed, time-indexed narratives. We used the time-indexed narratives to separate each lesson into non-overlapping activities, whereby an activity is bounded by the teacher signaling a change in seating or grouping configuration or a change in what students are being asked to do (Stapleton, LeFloch, Bacevich, & Ketchie, 2004). Each activity has a corresponding participation structure (Erickson & Schulz, 1981) determined by what the students are asked to do by the teacher, using categories and subcategories shown in Table 1 (adapted from Stodolsky, 1988, and Weiss, Pasley, Smith, Banilower, & Heck, 2003). Representations of participation. We represent a lesson’s chronological arrangement of activities and corresponding participation structures using Lesson Activity Bars (LABs) (similar to techniques of lesson representation by Stigler, Fernandez, & Yoshida, 1996, and Zahner et al., 2012). For example, Figure 2 shows a lesson that begins with 10 minutes of listening to a teacher, is interrupted by 5 minutes of school announcements, progresses to 15 minutes of whole-class discussion, and concludes with 15 minutes of practicing. To holistically and quantitatively capture a lesson’s participation opportunities, we measure lessons in terms of the difference in their proportions of active or passive participation (Difference in Participation Proportion [DPP]), a measure that ranges from -1, signifying that a lesson includes only passive participation structures, to 1, signifying that a lesson includes only active ones (Rubel & Monroe, 2012).

Teachers’ names are pseudonyms. Two observations were cancelled because of unanticipated school closures.

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Laurie H. Rubel and Anders J. Stachelek Table 1 Examples of Student Participation Structures Category Listen. . .

Passive

Category

To teacher

Discuss. . .

To guest speaker

Active In whole class format In small group or pair format

To video Copy, check. . .

Copy text from board

Reflect. . .

Check answers against solutions

Reflect on activities or problems Prepare a written report Write a description of procedure

Practice. . .

Complete worksheets

Investigate. . .

Work with manipulatives

Answer textbook exercises

Play a game

Test preparation

Follow specific instructions in an investigation

Recall of information from previous lessons

Record, represent, and/or analyze data Recognize patterns, cycles, or trends Evaluate the validity of a mathematical argument Provide an informal justification Provide a formal proof

Other. . .

Nonmathematical but at teacher’s discretion

Technology. . .

Develop conceptual understanding Collect or analyze data

Each activity’s participation structure is categorized as active or passive (see Table 1). Listening to a teacher presentation, practicing procedures, or copying information positions students as passive, whereas participating in a discussion, gathering or analyzing data, or providing a proof positions students as active learners. Whereas an individual student might be more engaged in completing a worksheet (categorized as passive) than in a whole-class discussion (categorized as active), our categorization as active or passive refers to how the designated participa-

tion structure positions students in the intended activity and not to the level of their engagement. The DPP for the lesson shown in Figure 2, which contains 40 minutes of instruction in a 45-minute class period, 15 of which are in an active participation structure, is (15–25)/40, or -0.25. Two or three researchers observed 46 of the 70 observed lessons and completed the activity coding immediately after each observation, resolving discrepancies through discussion. For the 24 lessons that were observed by a single researcher, a similar double coding

Figure 2. Lesson Activity Bar (LAB) for sample lesson, DPP -0.25.

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and discussion process was conducted using the detailed narrative descriptions. For the case studies, we used lesson narratives as evidence of professional experimentation (domain of practice) and drew on evidence from teachers’ written reflections and transcripts of talk from interviews and PGP sessions (Table 2) for personal domain and domain of consequence, as well as links connecting the four domains. We analyzed these data sources using Dedoose (2015) first by identifying segments of text in relation to the IMPG domains. For these coded segments, we created root and, in some cases, further child codes in additional, multiple iterations (see Appendix B). This coding process allowed for exploration of teacher growth over time. Introduction of perspectives about participation with teachers. During the PGP’s summer institute, teachers explored a growing pattern problem, working in groups

(like the problem in Zazkis, Liljedahl, and Chernoff, 2008). Teachers used the figure’s geometry to generalize the pattern in multiple ways, generating a set of equivalent algebraic expressions, and then used equivalence to explore algebraic structure as well as connections between geometry and algebra. After this session, the first author prompted teachers to reflect on the ways this task had invited multidimensional mathematical ­participation. Teachers generated a set of verbs, including “listen, practice, taking notes, writing, copying, discuss, investigate, ask a question, present, collaborate, reflect, solving problems, reading, interpreting, and using technology” (Field notes, 8/20/12). When “think” was suggested, it was recorded alongside the longer list, with a rhetorical question by the facilitator as to whether we can separate thinking or learning from the other listed forms of participation. Next, the first author shared the idea of representing lessons with a LAB tool, with different colors ­representing

Table 2 Data Sources by Category, IMPG Domain, Teacher, and Dates Narratives of Lesson Observations Domain of practice Lucy

Teresa

2/2/12

3/15/12

2/3/12

3/16/12

10/312

10/11/12

11/7/12

11/8/12

12/12/12

11/20/12

2/27/13

3/4/13

Interviews Personal domain Domain of consequence Lucy

Teresa

2/2/12

3/15/12

Written Reflections Personal domain Domain of consequence Lucy

Teresa

Group Meetings External domain Both 1/25/12

10/11/12

8/19/12

11/8/12

8/27/12

10/3/12

8/19/12

9/19/12

8/25/12

10/17/12

12/16/12

11/7/12 1/16/13 2/6/13

2/27/13 3/20/13

3/4/13

3/5/13

4/9/13

3/6/13 4/10/13

5/15/13

10/11/13

10/9/13

12/4/13

12/10/13

10/11/13

5/17/13

6/14/13

5/15/13

6/19/13

6/21/13

6/12/13

5/23/13

10/9/13

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Laurie H. Rubel and Anders J. Stachelek various participation structures. This author facilitated a discussion with the teachers that included the following questions:

Suppose we represented each of the participation verbs from the above list with a unique color and that we made color-coded LABs for your lessons. Are your lessons dominated by a specific color? Are any colors absent? Do your lessons use the same set of colors in a consistent color pattern?

These questions prompted a discussion in which teachers shared their anticipations of and reflections about patterns in their LABs. We then shared anonymized LABs from their observed lessons from the previous spring (two are in Figure 3). Teachers seemed interested in these LABs; one inquired aloud as to whether there is an ideal color pattern. We directed teachers to consider the relationship between a lesson’s task and its potential to support multidimensional participation. They noted how lessons with low-demand tasks were dominated by passive participation structures. Several teachers said that they wanted to view the LABs again, so the following day, we collectively viewed the LABs a second time. Many teachers opted to receive their own individual LABs.

Results Table 3 lists teachers according to change in mean DPP from before the PGP (spring 2012) to afterward (fall 2013).

These results show a general trend of increasing DPP values, suggesting that the PGP supported teachers in changing the kinds of classroom participation opportunities they afford. For example, four teachers (Lucy, #2, #3, and to a lesser extent #5) taught lessons dominated by passive participation structures prior to the PGP; three teachers (#4, #6, and Teresa) balanced active with passive; and the lessons of one teacher (#8) already consisted primarily of active participation structures. After the PGP, five teachers (Lucy, #2, #4, #6, and #8) taught lessons consisting nearly entirely of active participation structures; two teachers (#3 and #5) vacillated between ­primarily passive and primarily active; and Teresa’s lessons were consistently slightly dominated by passive participation structures. We selected Lucy and Teresa for case studies because Lucy had the largest change in mean DPP and Teresa’s DPP means remained consistent and unchanged.

Case Studies Case 1: Lucy Lucy was 38 years old and in her second year of teaching; she had entered teaching as a second career through a graduate program in mathematics education. She identified as White, and English was her first language. She had lived in her school’s neighborhood decades earlier and had recently returned. This part of the city now had a high rate of “foreign born population” (47%), and Mexico was the modal country of origin. Her school

Figure 3. Lesson Activity Bars (LABs) shown during the PGP, DPP -0.63 and DPP 0.88.4

3 4

Passive = 40 minutes, Active = 10 minutes: DPP = (10 – 40)/50 = -0.6 Passive = 3 minutes, Active = 47 minutes: DPP = (47 – 3)/50 = 0.88

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Table 3 DPP Values by Teacher Over Time (70 observations) Lessons Teacher Lucy #2 #3

Prior Spring 2012 (2 lessons each) #1

During Fall 2012 (3 lessons each)

During Spring 2013 (2 lessons each)

After Fall 2013 (2 lessons each)

#2

#3

#4

#5

#6

#7

#8

#9





0.533

0.933

0.800

0.533

0.567

0.571

0.741

1.000

0.767 0.000

0.625

0.750

0.552

0.857









0.500

0.625

0.500

0.083



n/a

0.628

0.545

0.636

0.318

0.500

-0.524

0.783



0.439

0.585

0.610

0.026

0.683

0.650

0.850

0.378

0.244

-0.089



0.244

0.362

-0.447



0.447

0.600

-0.064

-0.192

0.544

0.698

0.543 #4

0.440

0.065

#5

0.046

– 1.000

#6

0.111

0.156

0.733 n/a



0.822

0.911

0.511 Teresa #8

0.018 0.875







0.483

0.333

0.579

0.250

1.000

0.875

0.022 0.040 0.458

consisted of grades 9–12; it had opened in 2009 and predominantly served students from low-income families from this neighborhood (81% identified as Hispanic, 83% were eligible for free or reduced lunch [New York State Education Department, 2013]). Her school describes itself as “committed to teaching students of all abilities in the same classes” (Inside Schools, 2012). Before PGP. Lucy taught 11th-grade algebra during all three years and stated that she joined the PGP because of an interest in better engaging students who are “used to failure” by “mak(ing) the curriculum . . . more reflective of them” (Meeting, 1/25/12). Her school progressed students to Algebra II regardless of their performance on the state’s Algebra I test, and Lucy’s class was divided among students with and without that prior success. Perhaps for this reason, Lucy’s central goal was to prepare students for state exams and to do so in terms of covering the breadth of required topics. She articulated tensions around the “inquiry-based” approach advocated by her school. When asked about her thoughts on the best way to teach mathematics, Lucy responded: I think I’ve kind of changed my mind and then changed it back. I guess it depends on the kid. Like the school’s focus . . . is inquiry-based learning where kids discover stuff. And I think that’s good for some kids, and I do think that it’s





0.193

0.123

0.708

0.750

helpful to see where things come from rather than just being straight out told . . . but then as I’m working with kids, like in theory, that sounds wonderful; and that’s what we learn in college and all that stuff. But in practice, I’m noticing that a lot of kids, be it because they’ve never really had inquiry-based instruction before in the younger grades, or you know, a lot of them are just lazy and they just want to know, a lot of them just really can’t grasp the whole concept of discovery before school development, that I’m still kind of on the fence. Like some kids I think that direct instruction is the way to go: “These are the steps you need to do. Let’s practice those steps.” (Interview, 2/2/12)

Lucy’s observed lessons nearly entirely comprised passive participation structures. For example, in a lesson on factoring quadratics (see the LAB representation of this lesson in Figure 4), Lucy presented students with a product-sum numerical method and an area model method multiplication (see Mark, Goldenberg, Fries, Kang, & Cordner, 2014). Students were directed to copy notes about these methods and then to individually practice factoring, with the lesson closing by the bell. Lucy rotated around the room, supporting individual students by demonstrating one of the methods. Students asked few questions, and any question was answered quickly

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Figure 4. LAB for Lucy’s 2/2/12 lesson (DPP = -15).

and directly by the teacher and not extended into a discussion. A student asked how the area method would be adapted if the polynomial expression had four terms instead of three. Instead of pursuing this question and highlighting this “what if?” generalization question, Lucy answered by presenting a trivial example of a polynomial expression with four terms, whose like-terms could be combined to result in a trinomial. During PGP (2012–2013). During the summer institute, Lucy was among the teachers who opted to view their LABs in addition to viewing the collective set. She reflected that she had previously planned her lessons according to mathematical content, to what she wanted students to know without considering how they would come to learn it. She highlighted that the LABs and discussion around participation prompted her to consider how to plan her lessons in terms of planning for multidimensional mathematics instruction that attends to diversifying participation opportunities and structures. In summarizing her next steps, she indicated that she had “never paid much attention to how much time students were spending” in a particular participation structure and that “By determining how I want students to process information (i.e., peer-to-peer discussions, practicing, investigating, etc.), I can better tailor the day’s lesson/ task” (Reflection, 8/27/12). As part of her thinking about engaging students, Lucy articulated new goals around fostering mathematical discussions and planned to achieve these goals by contextualizing mathematics in relevant content. She noted that topics like “poverty and racism” or “things that they could really connect with” served to prompt “fierce debates in class” (Meeting, 9/19/2012). She began her Algebra II course with a data analysis unit and with a plan to focus on social justice issues that directly impact her students, like “rate of asthmatics in Brooklyn, immigration, the path of neighborhood gentrification in New York City, 5

etc.” (Reflection, 8/19/12). Lucy reflected that this unit was especially successful in terms of getting to know the students and that the students seemed to enjoy it as well. She explained: The other Algebra II teachers thought I was crazy for doing the statistics so long, but I really enjoyed it because the kids really connected and the kids got into it. . . . But I really enjoyed doing that and I think they enjoyed it. It really got them fired up and maybe looking at math in a different way. Until we hit the algebra. That’s where I struggle on and I would love to figure out how to take some of that kind of momentum and bring it into the algebraic component, which is most of the curriculum. (Interview, 5/15/13)

We note here that Lucy’s decision to support more classroom discussion was accompanied by her emphasis on data and statistics topics through social justice issues, which contrasts with the previous year’s observed lessons that focused on decontextualized algebraic expressions. She was pleased with the ensuing intensity of student engagement but unsure how to connect issues that she saw as relevant or interesting to students to algebra topics in the curriculum. Lucy indicated that she was setting higher expectations for her students by using more challenging tasks that she described as a “gigantic jump from where [they] were” (Interview, 2/27/2013). She characterized her instruction as more emphatic of mathematical discourse. She reflected on the outcome of this experimentation: I’m working on getting the kids to think like mathematicians and all of sudden, like in the last two weeks or so, I hear kids saying “Well, what if this . . .” or “Suppose that . . .” or “Is this always true?” And I’m like, “Wow!” They’re really

Passive = 53 minutes, Active = 0 minutes: DPP = (0–53) / 53 = -1

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Classroom Participation as Opportunity to Learn pushing the boundaries of the problems that are in front of them and these concepts, to seeing how they can extend them on our own instead of just waiting for the next thing to come and the next thing to come and the next thing to come. (Meeting, 3/6/2013)

Expecting students to “think like mathematicians” contrasts with her prior characterizations of her students as incapable, lazy, or in need of direct instruction. Lucy’s efforts to expand the kinds of participation opportunities initiated her interest in facilitating more and better mathematical discussions, and those discussions, in turn, allowed her greater access to student thinking (Doerr & English, 2006; Smith & Stein 2011). Beyond PGP. Lucy’s observed lessons continued to be dominated by active participation structures like mathematical investigations, discussions, and reflections. For example, as shown in Figure 5, students conducted an experiment to generate data, represent their datasets, and interpret results (using a task described in Rossman, Chance, and Lock, 2001). Lucy then facilitated a discussion to draw out student reasoning and concluded the lesson with written reflection (Observation, 10/11/13). Lucy reflected that she was holding higher expectations for students in terms of mathematical content, and students seemed more engaged in response to those higher expectations (Interview, 10/11/13). Case Summary. Our observations before the PGP demonstrated a dominance of passive participation opportunities in Lucy’s lessons, reflecting a “received knowing” model of learning. Our analysis shows a change sequence in that her experience in the PGP prompted her to experiment with participation in her classroom (Figure 6, Link 1). Her reflections about this experimentation led to changes in how she talked about her students and about mathematics and to revised goals for her students (Links

4, 8). These changes cycled to further experimentation in terms of contextualizing tasks in students’ experiences (Link 5) and organizing lessons that reflected a “connected knowing” model of learning, including more opportunities for discussion and investigation (Link 9). Her reflection about her professional experimentation reinforced changes in the way she spoke about her students and about instruction (Link 4).

Case 2: Teresa Teresa was 25 years old and was in her second year of teaching. Like Lucy, she had entered teaching through a graduate program in mathematics education. She identified as Mexican American, with Spanish as her first language. Her school was a grades 5–12 charter school that opened in 2006 in another low-income area with a high rate of “foreign born” population but with the Dominican Republic as the modal country of origin (92% of students identified as Hispanic, 95% as eligible for free or reduced lunch [New York State Education Department, 2013]). Like Lucy, she lived near her school. Teresa’s school was characterized by traditional teaching, strict discipline, and tracking policies (Inside Schools, 2012). Before PGP. Teresa indicated that she became a teacher because of the potential to impact society (Interview, 3/15/12). She taught seventh grade and identified with her students and positioned herself as a family authority figure by using Spanish to manage classroom behavior, explaining: . . . it just happens naturally. I feel like in my own upbringing, when you were scolded to in Spanish, it was like, “Oh, you’d better stop.”. . . And it’s the kids are all—they’re, like, “Uh-oh. She’s speaking Spanish.” So I was, like, “cállate!” They react to it because I think they’re dealing with it the same— it’s a cultural thing. (Interview, 3/15/12)

Figure 5. LAB for Lucy’s 10/11/13 lesson (DPP = 0.5716).

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Passive = 12 minutes, Active = 44 minutes: DPP = (44–12) / 50 = 0.571

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Figure 6. IMPG model of Lucy’s growth.

Accentuating her familial stance were Teresa’s regular references to her students as “lovelies” and her bustling around the classroom catering to their needs adorned in an apron holding supplies. Teresa’s school advocated a constellation of pedagogical techniques from Teach Like a Champion (Lemov, 2010). For example, her school expected an “I Do, We Do, You Do” structure to every lesson (Interview, 5/23/13): that is, for the teacher to demonstrate a skill in a whole-class presentation, facilitate guided practice, and then assign

similar problems for independent practice. Teresa used this structure and peppered her instructions with phrases like “Get in ready position” and “Track the speaker.” She told her class that “before we can get to big stuff, we have to do baby stuff,” and she emphasized completion time by using a large timer on her SMART Board. As an example, the lesson shown in Figure 7 (Observation, 3/16/12) contained many (14) activities, but most of the activities asked students to copy or to practice. Even when a student asked a potentially generative question (“Do we ever find the variable?” which could have led to

Figure 7. LAB for Teresa’s 3/16/12 lesson (DPP = -0.4837). 7

Passive = 86 minutes, Active = 30 minutes: DPP = (30−86)/116 = -0.483

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a discussion about the distinction between an expression and an equation), Teresa answered “Sometimes” and did not pursue the question, despite its potential to deepen mathematical understanding or access student thinking. During PGP (2012–2013). At the end of the summer institute, Teresa was reassigned to teach high school Algebra I. She indicated an interest in deemphasizing the duration of practicing in her lessons, noting the relationship between participation opportunities and task selection. She wrote: Moving forward I want to make sure that my students do more than just practice skills that I have already modeled for them. . . . I am aware now how my students’ low procedural fluency led me to require students to practice skills. . . . I learned to create what I considered effective tasks for students to practice procedural skills. . . . Now, I am going to commit to developing my ability to create higher-level thinking tasks that require students to truly do mathematics. (Reflection, 8/25/12)

She noted the contrast with her prior belief that she needed to “teach the kids the steps and let them practice the steps” in a sequence of “walk, run, jog, fly,” by starting with basic examples and progressing to more advanced ones (Interview, 11/8/12). Referring to the colorful LABs shown to the teachers in the summer institute, she acknowledged that “if you think about that rainbow of different types of activities, like mine’s practice, practice, practice . . . and I’m just kind of, like, ok, how can I still expose them to the content but it’s not just practice?” (Meeting, 9/19/12). She felt that effecting this change would be challenging but was optimistic: “I remember I left last time from our PD, and I was like, ‘I can do this’” (Interview, 10/11/12).

that’s what my kids deserve. (Interview, 5/23/13, emphasis added)

This contrasts with her prior statement about choosing mathematics because it is straightforward; her talk about mathematics now reflected more of an interest in challenging herself and her students to justify mathematical ideas and relationships. The changes in Teresa’s beliefs about instruction and salient outcomes for students conflicted with her school’s prescribed techniques. She noted that the expectations from her school seemed oriented around compliance rather than participation: They’re copying it. They’re doing it, but they’re not necessarily understanding. And then that becomes clear. I’m always very aware, especially when I have DC (department chair) or admin – observations, and I’ll say “track this speaker” . . . ’cause we have to do that with Teach like a Champion. And then to them, they are looking like, “Oh, these kids are so engaged.” . . . And I’m like . . . They were just behaving, not thinking. (Interview, 3/4/13)

Here, Teresa shares her insight that “tracking the speaker” implies compliance, which can be misread as engagement. Teresa indicated that because of this insight, she wanted to organize her lessons around mathematical discussions. And yet, she noted the difficulty in shifting this aspect of her teaching: I realize that I am not creating space for us to discuss the math because I am not sure how to facilitate conversations. I am not sure what they are going to say, how they are going to think, how they are going to interpret the problem so when I teach something new that I have never taught, I don’t create the space to get to know that information. . . . That’s bad, real bad. Ay dios! (Reflection, 3/5/13)

Teresa’s statements of identification with and care for her students continued. She expressed affection for them, such as “I love these kids” (Interview, 11/8/12) alongside her commitment to them, such as “I just want to be a good teacher for them” (Interview, 10/11/12). She directly attributed her desire to change her instruction away from an emphasis on practicing skills to her commitment to equity for her students:

She attributed the challenge in including more discussion to a lack of expertise in facilitating those discussions and in anticipating student thinking, especially when teaching a topic for the first time.

My biggest take-away from this is gonna be that I believe that math isn’t just about solving problems on a worksheet independently. . . . I remember there was a point where it’s . . . I’m gonna do it every time that they . . . visit . . . but then I was, like, I’m gonna do this every single time because

Beyond PGP. Teresa’s school required that students retake Algebra I to achieve “college-ready” test scores and assigned Teresa to teach these students. Her observed lessons continued to be organized around low-demand tasks and include significant durations of passive participation structures (e.g., Figure 8). Even though her DPPs

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Laurie H. Rubel and Anders J. Stachelek

Figure 8. LAB for Teresa’s 10/9/13 lesson (DPP = -0.0648).

remained roughly consistent in value, Teresa felt that she was doing things in new ways. For example, she explained that she was striving to organize her lessons to contain less practicing and instead include more discussions, valuing the goal of conceptual understanding and not only procedural fluency. She explained, “I know in the very beginning before our training, I would have given equations out and just have them practice. . . . So this is something different; I’m having them talk about it” (Interview, 10/9/13, emphasis added).

Case Summary. Findings evidence change sequences for Teresa in terms of reflection about the PGP and its impact on her thinking about effective instruction (Figure 9, Link 2). She recognized her tendency to emphasize practicing (Link 4) and stated a desire to shift participation to be more multidimensional and to reconcile her beliefs in her students’ capabilities with her instruction (Links 2, 4). In response to changes in her thinking about instruction and reflection on her practice, Teresa changed the outcomes she deemed salient for students (Links 6, 8) to include

Figure 9. IMPG model of Teresa’s growth.

8

Passive = 25 minutes, Active = 22 minutes: DPP = (22–25)/47 = -0.064

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20 goals around conceptual understanding. Teresa’s school, however, as the change environment, directed students to repeat courses toward raising test scores, favored mastery over content exposure, and dictated lesson structures and procedures to teachers. This lack of support for dialogicbased teaching seemed to constrain Teresa in terms of professional experimentation.

Discussion Most teachers in the study showed increases in DPP (see Table 3). Our interpretation of this growth is that the PGP’s focus on broadening student participation and its representational tools supported teachers in making these changes. One explanation for the ready uptake of the LABs by teachers could be that unitizing lessons this way is congruous with the way teachers already think about lesson planning (Stodolsky, 1988). The breadth of this PGP creates difficulties in precisely pinpointing the factors that engaged teachers toward this growth. Although we have argued here that the focus on student participation accompanied by LAB tools supported teachers in expanding the participation opportunities in their lessons, we acknowledge that the underlying importance of participation was grounded in the PGP's broader focus on centering instruction on urban youth. Perhaps the coupling of this broader theme with the representational tools that readily connect to teachers’ everyday practice supported the teacher growth. Through the cases of Lucy and Teresa, we have charted two potential courses of teacher growth initiated by the PGP’s perspectives on participation and supported by its representational tools. Lucy’s change in DPP mean values was the largest in the group as her lessons took on longer durations of active participation structures like discussions or investigations. Although the DPP measurement categorizes these participation structures as active, it does not measure the quality of these activities. A further step would be to complement the DPP measurement with other tools that can measure and support teachers in maintaining the academic rigor, for instance, of a mathematical discussion (e.g., Boston, 2012) or the distribution of participation in discussions across individual students (e.g., Shah et al., 2016). Lucy’s choice to emphasize data and statistics topics, as a result of her belief that these topics would better engage her students, is salient here, and we do not know if or how she tailored this growth around student participation in her teaching of algebra. On the other end of the growth spectrum, Teresa seemingly showed the smallest change in terms of DPP measures. Our analysis shows how Teresa’s school acted as a constraining change environment, with its emphasis

Classroom Participation as Opportunity to Learn on test preparation and rote learning. We argue that this change environment constrained Teresa from changing her task selection toward cognitively demanding tasks and thereby also from changing the nature of participation opportunities. It is possible that Teresa needed more time or support for enactments related to her various reflections to manifest in classroom observations, or that the study’s two observations after the PGP were too sparse to notice changes in her practice. It is also likely that the measures used in the study did not effectively capture ways in which Teresa was experimenting with her practice. This limitation speaks more generally to challenges for research on teacher education whereby changes indicated by teachers about their own practice are not observable by the outside viewers or research instruments. Our professional development project leads us to suggest that teacher education and development programs focused on student participation opportunities can hold potential to support teacher growth with designing and enacting dialogic mathematics instruction. But this focus alone may not be sufficient to produce the kind of changes in mathematics instruction the field has been advocating. An array of studies have demonstrated the importance of the quality of mathematics tasks teachers use in their classroom that have led to research studies about the impact of teacher development on teachers’ ability to select and maintain high-level tasks (e.g., Boston & Smith, 2009). Further research could investigate how a dual focus on student participation opportunities and multidimensionality of mathematical tasks might support growth around task selection with attention to broadening student participation structures. Similarly, a focus on student participation opportunities and multidimensionality of tasks might be a springboard for teacher growth around effective facilitation of mathematical discussions. Although we premise that active participation is more beneficial for learning, there might be a cut-value for limited durations of passive participation (listening and practicing) in support of learning, but this value has not yet been empirically established. Finally, although the LABs and DPP measurement were designed as research instruments, their use here shows a more general potential for the repurposing of research tools as reflection tools for teachers. To conclude, direct instruction was prevalent among the participating teachers in our project before the PGP, and participation opportunities were dominated by passive structures like listening, copying, and practicing. We argue that the PGP’s focus on centering instruction on students, its perspective on participation opportunities afforded to students, and its LAB and DPP tools played a role in facilitating teacher growth around classroom

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Laurie H. Rubel and Anders J. Stachelek participation. This approach and these tools seem especially useful for school contexts dominated by passive classroom participation opportunities, and they hold promise for mathematics teacher education and professional development of teachers who work in these school settings.

and Teacher Education, 18(8), 947–967. doi:10.1016/S0742-051X(02)00053-7 Cobb, P., Gresalfi, M., & Hodge, L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education, 40(1), 40–68. Retrieved from http://www.jstor.org /stable/40539320

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Laurie H. Rubel and Anders J. Stachelek Smith, M.S., & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics. Smith, M., Stein, M. K., Arbaugh, F., Brown, C., & Mossgrove, J. (2004). Characterizing the cognitive demands of mathematical tasks: A task-sorting activity. In G. Bright & R. Rubenstein (Eds.), Professional development guidebook for perspectives on the teaching of mathematics: Companion to the sixty-sixth yearbook. Reston, VA: National Council of Teachers of Mathematics. Stapleton, J., LeFloch, K., Bacevich, A., & Ketchie, B. (2004). Researching education as it happens: Using classroom observations to generate quantifiable data. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA. Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488. doi:10.3102/00028312033002455

in Mathematics Education, 45(3), 312–350. doi:10.5951/jresematheduc.45.3.0312 Weiss, I., Pasley, J. D., Smith, P. S., Banilower, E. R., & Heck, D. J. (2003). Looking inside the classroom: A study of K–12 mathematics and science education in the United States. Chapel Hill, NC: Horizon Research. Witterholt, M., Goedhart, M., Suhre, C., & van Streun, A. (2012). The interconnected model of professional growth as a means to assess the development of a mathematics teacher. Teaching and Teacher Education, 28(5), 661–674. doi:10.1016/j.tate .2012.01.003 Zahner, W., Velazquez, G., Moschkovich, J., Vahey, P., and Lara-Meloy, T. (2012). Mathematics teaching practices with technology that support conceptual understanding for Latino/a students. The Journal of Mathematical Behavior, 31(4), 431–446. doi:10.1016/j.jmathb.2012.06.002 Zazkis, R., Liljedahl, P., & Chernoff, E. J. (2008). The role of examples in forming and refuting generalizations. ZDM, 40(1), 131–141. doi:10.1007/s11858-0070065-9

Authors’ Note

Stein, M. K., & Smith, M. S. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School 3(5), 344–350. Retrieved from http://www.jstor.org /stable/41180423 Stigler, J. W., Fernandez, C., & Yoshida, M. (1996). Mathematics in Japanese and American elementary classrooms. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematics learning. Mahwah, NJ: Lawrence Erlbaum Associates. Stodolsky, S. (1988). The subject matters: Classroom activity in mathematics and social studies. Chicago, IL: University of Chicago Press. Tate, W. (1994). Race, retrenchment, and the reform of school mathematics. The Phi Delta Kappan, 75(6), 477–484. Retrieved from http://www.jstor.org /stable/20405144

This material is based on work supported by the National Science Foundation under Grant No. 0742614. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Special thanks to Scott Monroe and Haiwen Chu for their collaboration on this project.

Authors Laurie H. Rubel, Brooklyn College of the City University of New York, 2900 Bedford Ave, Brooklyn, NY 11210; [email protected] Anders J. Stachelek, Hostos Community College, 500 Grand Concourse, Bronx, NY 10451; [email protected]

Wager, A. (2014). Noticing children’s participation: Insights into teacher positionality toward equitable mathematics pedagogy. Journal for Research

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Appendix A Contents and Resources of Professional Development Meetings Month

Meeting topic

Associated resource

September

Levels of cognitive demand

Task sorting activity described in Smith, Stein, Arbaugh, Brown, & Mossgrove (2004)

October

Share out of community walks in school neighborhoods

Protocol for studying community

November

Improving questioning

Boaler & Humphreys (2005)

December

Launching a mathematical task

Talk by K. Jackson, reading: Jackson, Shahan, Gibbons, & Cobb (2012)

January

Preparing for teacher presentation at national conference

February

Racial microaggressions

Talk by Battey, about Battey and Leyva (2015)

March

Community-based mathematics

Talk by Remillard and Lim, about Ebby et al. (2011)

April

Teacher presentations on classroom discourse

Stein and Smith (2011)

May

Share out of classroom observation data & next steps

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Appendix B Sample Overview of Codes in Dedoose by IMPG Domain* IMPG Domain Personal domain

Root Code Reflections about instruction

Child Codes Difference of abilities in class Connecting mathematics to students’ experiences Question or identified struggle Decisions (explanations, goals, grouping, reflection)

Reflections about students

Expectations Ideas about student learning Student attributes Student thinking or progress Engagement Identification with students

Domain of Consequence

Within class outcomes

Engagement Equitable class environment Curriculum coverage

Preparation for...

Future learning Standardized testing Careers

External Domain

PGP Individual visits

Lesson debrief about student participation Lesson debrief about mathematics Lesson debrief about connects to PGP meetings

PGP Group meetings

Monthly meetings Summer institute

Other

Feedback from school administration

*Domain of practice analyzed through DPP rather than qualitative coding.

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