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Article Excerpt
In a workshop on their new instructional materials, 30 intermediate and middle grade teachers worked in small groups on a staircase task to determine the number of cubes needed to build the 100th staircase if the first staircase was one cube, the second three cubes, the third six cubes, and so on. When they finished, Sue, the facilitator, asked participants to share how they thought about the problem. Elise came to the overhead and showed her table of values. As she did, she said, "I noticed that my table had 1 then 1 + 2, then 1 + 2 + 3, then 1 + 2 + 3 + 4 and I remember from one of my reviews of high school math that to find the total of these I use n(n + 1) divided by 2. But, I am not an algebra teacher so...." Sue asked if anyone had questions. A teacher asked what n stood for and Elise showed how to plug values into the formula to get the total number of cubes. The group applauded and Sue asked if another person might share. Mario said he also got n(n + 1) divided by 2 and illustrated his approach using the model of the staircase to fit a second staircase of the same size on top to make a rectangle. Pointing to the dimensions of the rectangle he said, "n is the number of cubes along the bottom and the n + 1 is the number of cubes along the side. I divided by two because my staircase was half of the rectangle." Teachers signaled how much they liked the use of a visual model with applause and a buzz of praises. The facilitator prompted for another way and Christa showed her approach next. She noticed that if n was the number of cubes in the last column, then the total cubes for one staircase was n + (n - 1) + (n - 2) + (n - 3).... She said the total for any staircase was the number of cubes in the last column plus the previous total. "But I ran out of time trying to figure out how to write it." The sharing continued and each presenter was applauded. Sue continued to ask, "Is there another way?" until no one else volunteered. Sue then said, "Okay, let's look at one more problem from the chapter before breaking for lunch."
This depiction of professional development (PD) illustrates common practices in PD. The facilitator elicits teachers' varying solutions and invites teachers to question one another in a supportive learning environment (Desimone, 2009; Loucks-Horsley et al., 2003). As professional development leaders ourselves, we have often facilitated such discussions. We use this example as a means to begin a conversation that we, as researchers and facilitators of PD, believe is imperative to advance leaders' learning. In the scenario, although the leader provided a rich task and encouraged teachers to share solutions, the mathematical purpose for teacher learning was not made evident, neither was how teachers' mathematical learning might be useful for supporting students. In advancing teachers' mathematical learning, a leader (2) may need to "slow down" teachers' conversation to explicitly engage the group in mathematical ideas. For example, a leader might ask, "What might be gained from 'seeing' the solution in the visual model? Were there multiple ways to see the pattern in the model and what are the implications of seeing the pattern in different ways for representing a solution symbolically?" Leaders can use these kinds of questions to explore how mathematical models can be used to represent different, albeit equivalent, expressions or to recognize alternative expressions that stem from defining the variable differently. For intermediate and middle grade teachers working with students who are learning about variables and characterizing patterns using words and symbols, the discussion relating the model to the symbolic notion would support teachers who remembered a formula to understand the mathematical reasoning and the use of multiple representations in algebraic thinking.
Because facilitation moves in the scenario focused on displaying the different ways in which teachers solved the task and providing an open forum to ask questions, it was not clear what teachers were to learn. Although there may be many worthwhile purposes to pursue based on the methods and solutions teachers developed, part of the work of leading is deciding on a purpose that is mathematically worthwhile and relevant for a particular group of teachers. Although teachers can be enthusiastically engaged when doing and sharing different mathematical solutions, we contend that more can, and should, be made explicit about the purposes for doing mathematics in PD and the mathematical knowledge teachers need to develop that would benefit their work with students. In this article, we use our study of leader learning to contribute to the limited body of research on leaders' skills and understandings necessary to support teachers' mathematical learning.
The Leader's Role in Professional Development
Leaders of professional development are central to providing opportunities for teachers to gain new understandings of subject matter. The United States has supported numerous initiatives and committed billions of dollars to bolster teacher learning (Birman et al., 2007). From evaluations of the National Science Foundation's Local Systemic Change Initiative projects, we know that a high quality leader makes a difference in the effectiveness of supporting teacher learning in PD (Banilower, Boyd, Pasley, & Weiss, 2006). Yet, little attention has been given to what or how PD leaders learn (Ball & Cohen, 1999; Elliott, 2005; Even, Robinson, & Carmeli, 2003). What leaders of PD need to know and be able to do in their practice is underdefined and understudied, so much so that Even's recent international review of the literature on leader practice focused instead on the missing literature (Even, 2008). Compounding the issue that there is limited research on what leaders need to learn to improve teachers' ability to teach mathematics effectively is the fact that all states are required to provide teachers with high quality learning opportunities (Borko, 2004; U.S. Department of Education, 2001). Reaching all teachers in a substantive way taxes the PD infrastructure beyond its present capabilities. As a result, more teachers are being asked to serve as leaders (Lord & Miller, 2000). The research community has lagged behind in providing insights on how to best support these new leaders as they facilitate teacher learning. Filling the knowledge gap in the research on leading PD is an urgent issue if teacher learning is to be improved and adequately addressed (Lord & Miller, 2000).
The purpose of this article is to share what we have learned about supporting leader learning through a series of seminars aimed at developing leaders' knowledge and skills for cultivating mathematically rich learning opportunities for teachers. In particular, we discuss how our frameworks conceptualizing this work have evolved based on the analyses of data collected during the first 2 years of a 5-year project--Researching Mathematics Leader Learning (RMLL). (3) Our research and development work focuses on one aspect of mathematics PD, when teachers are engaged in solving, discussing, and sharing mathematical work. Although mathematics PD may include other activities, we specifically focus on how leaders learn to attend to doing mathematics with teachers because it is a primary time during PD that teachers may be developing deeper understandings of mathematics. To support their learning about cultivating rich teacher learning environments, leaders explored two frameworks: sociomathematical norms (norms for mathematical reasoning) and a set of practices for orchestrating productive mathematical discussions. The staff of RMLL created and facilitated seminars as learning opportunities for leaders, studied what and how leaders learned about facilitation, and investigated how leaders facilitated PD in their schools and districts.
In this article, we share our developing understandings of what is involved in the work of leading PD and how our frameworks and designs for supporting leaders need to change to better prepare leaders. The research questions investigated are as follows:
Research Question 1: How did RMLL frameworks help leaders make sense of the work of facilitation related to mathematical reasoning in PD?
Research Question 2: How did leaders use these frameworks to support the negotiation of mathematical reasoning in PD?
We report here two central findings associated with our research questions. First, leaders responded positively to using the frameworks as tools for learning to lead mathematically rich discussions. Moreover, the leaders' discussions of these frameworks provided insights on the tensions they experienced in working with adult learners. Second, leaders recognized the importance of having a purpose when facilitating mathematical tasks and were challenged to specify and realize the implications of such purposes in PD. Based on these two findings, we offer an expansion and specification of our frameworks for leader practice in ways that link particular sociomathematical norms for explanation and practices for orchestrating discussions to the purposeful development of teachers' specialized knowledge of mathematics (Ball, Thames, & Phelps, 2008). Before discussing the two findings below, we review the frameworks for leader learning we used in the design of RMLL seminars.
Frameworks for Leader Practice
We drew on classroom research to create frameworks for leaders to use to support teachers' mathematical understandings in PD. Because research on PD suggests that teachers' mathematical conversations tend to move away from mathematics to focus on other (albeit pressing) pedagogical issues (Hill & Ball, 2004; Wilson, 2003; Wilson & Berne, 1999), we looked to classroom research for ways that teachers keep mathematics central in discourse. From this research, we were drawn to ways that Cobb and colleagues supported teachers in advancing mathematical learning by attending to the sociomathematical norms of classroom practice. Their framework, distinguishing social from sociomathematical norms, suggested that learning opportunities are guided by patterns of interaction, both explicit and implicit, that establish how a group works with each other and accomplishes mathematics (Yackel & Cobb, 1996). A second framework adapted from classroom research was Stein, Engle, Smith, and Hughes's (2008) set of practices for orchestrating productive mathematical discussions. These practices support leaders' ability to focus conversations on important mathematical ideas. When conceptualizing RMLL seminars, we saw these two frameworks as guiding what kinds of mathematical explanations should be normative in PD (sociomathematical norms) and how leaders might facilitate these discussions (practices for orchestrating discussion). We used these two frameworks in our seminars to develop leaders' understandings and skills for facilitating mathematical work in PD. We elaborate on the frameworks below.
Social and Sociomathematical Norms
In classrooms, sociomathematical norms guide the nature of the mathematical work that gets accomplished (Kazemi & Stipek, 2001; Yackel & Cobb, 1996). Several factors shape the development of these norms including the students, the specific contexts, the mathematical content, and what is valued and defined as competent participation in a mathematics class (Lampert, 2001). How and what students share of their mathematical thinking is negotiated between the teacher and the students, guiding the nature of mathematical discourse in a classroom. One could imagine a classroom with a norm for privatization of mathematical thinking, where students do not share. Other classrooms may have multiple solutions being shared and a teacher who negotiates with students (a) how and what ideas are explained...
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