By Charlotte Dunlap, Megan Webster, Kara Jackson, and Paul Cobb

Student success on tougher assessments from the Common Core and rigorous mathematics standards requires school leaders to have a firm grasp of those standards of the instruction needed to prepare students.

Math_1508_DunlapSuccessfully implementing the more rigorous Common Core State Standards in mathematics requires a significant shift in what students and teachers need to know and be able to do. Typical math instruction focuses on modeling and practicing mathematical procedures, often without attention to building students’ conceptual understanding of mathematical relationships (Hiebert et al., 2005). The Common Core math standards aim to increase students’ depth of understanding but on fewer topics in mathematics in order to develop “conceptual understanding, procedural skills and fluency, and application with equal intensity (Common Core State Standards Initiative, 2014). In order to develop the sophisticated reasoning skills required by the mathematics standards, teachers need to provide students with opportunities to make sense of problems, develop strategies for solving them, and explain their thinking and reasoning to others (Ball, 1991; Boaler & Humphreys, 2005).

We know from research on how math teachers learn that teachers will need sustained support to develop the instructional practices that will support students to achieve the learning goals set out in the Common Core (Little, 1993). Furthermore, school leaders will need to develop an understanding of what is required for students and teachers in order to inform their work as instructional leaders.

Based on our work and that of others over the past seven years with large, urban districts pursuing ambitious reform in middle grades mathematics, it seems clear that school leaders are more likely to be effective when they have a strong understanding of what students need to understand, know, and be able to do (Carver, Steele, & Herbel-Eisenmann, 2010; Sparks, 2002). We suggest a design for professional development for school leaders that helps them better understand Common Core-aligned assessments and instruction that will best support success on those tests.

As part of a district-research partnership, we worked with district leaders in two districts to design a series of three two-hour professional development sessions to support middle school principals to develop a vision for instruction that would undergird their leadership. One district was in a state that had recently transitioned to the Common Core The other district was in Texas, which has not adopted the Common Core but did implement more rigorous standards and state assessments in mathematics. Here, we present the framework for those sessions. Similar to the professional development designs described by Nelson (1999) and Carver, Steele, and Herbel-Eisenmann (2010), we used a backward design approach (Wiggins & McTighe, 2005), in which we first established what students need to know and be able to do to be successful with the more rigorous end-of-year assessments. From there, we consider the implications of these learning demands for instruction and school leadership.

#1. Establish a firm understanding of what students need to know and be able to do on more rigorous assessments.

We began our first session by asking principals to solve two state math assessment items that tested the same mathematical standard. However, one item was representative of how the standard was tested on the old assessment, while the other was representative of how that standard would be tested on the new state assessment. We did not initially reveal this distinction to the principals.

For example, in our work with principals in the Texas district, we used the two 6th-grade items shown in Figure 1.

DunlapFig1The two items ostensibly test the same mathematical standard: Solve real-world and mathematical problems involving area, surface area, and volume. However, Item A (from the former state assessment) primarily requires students to recall and accurately use a procedure for finding the volume of a rectangular prism. Item B (from the new state assessment) requires students to first make sense of the mathematical relationships in the problem, including concepts of rate in addition to volume.

Then, they need to use what they know about the quantity of water in the pool and the rate at which it is being filled before constructing a procedure to solve the problem. In other words, Item B requires students to reason mathematically in order to decide which procedures to carry out. Moreover, Item B allows for multiple solution strategies — students may calculate the amount of time it will take for the pool to be completely full (600 minutes), and then figure out 3/4 of this quantity. Other students might calculate 3/4 of the volume of the pool (18,000 gallons) and then use the fill rate to find how much time would be required to fill that many gallons. More generally, the two contrasting items capture the shift in what is increasingly required of students across the U.S., with Item A representing what is currently required of students in most math classrooms and Item B representing what is required by the Common Core and other more rigorous standards and assessments.

After principals worked through each task, we asked them to share their solution strategies, which we recorded on a whiteboard. Once the principals described their strategies, we asked them to discuss, first in small groups then as a whole group, what they needed to understand, know, and be able to in order to solve each problem. Through a discussion led by a district leader and members of the research team, the principals articulated the distinction described above — namely, that while the first task required them to recall and apply a formula, the second task required them to consider context, make sense of mathematical relationships, and select or create procedures that would enable them to arrive at a solution. The discussion helped principals appreciate the rigorous mathematical reasoning required of students. The problem did not explicitly call for a specific procedure; rather, students needed to figure it out themselves.

Having established these distinctions between Item A and Item B, we then discussed how Item A was representative of earlier state assessments, and how Item B represented what would be tested under the more rigorous standards.

#2. Develop a vision of classroom instruction that will help students succeed under the more rigorous standards and assessments.

Our next step in the series of professional development sessions was to consider the implications for classroom instruction. We asked principals to consider what a 6th-grade math classroom might look like if the teacher’s lessons were targeted to support students’ conceptual understanding of the big ideas in math — such as understanding volume conceptually — as called for in Item B. Principals noted that students who were only exposed to problems like Item A, which explicitly called for specific procedures, were not being prepared to make sense of a novel scenario like Item B, which requires students to think about several mathematical relationships. In order to solve tasks like Item B, students would need to regularly understand and apply mathematical concepts. This meant teachers needed to select tasks for classroom instruction that were non-routine, required students to grapple with relationships represented in a context and make decisions about which procedures would be appropriate to use to solve the problems and why they were appropriate.

We also focused on the role of classroom discussions. Research on students’ mathematics learning suggests that supporting students as they develop a conceptual understanding of mathematical ideas requires providing regular opportunities to explain their reasoning and to make connections between different solution strategies (Smith et al., 2009). However, most principals were not accustomed to seeing rich mathematical discussions, so it was challenging for many of them to envision such discussions and how they might support student learning. Therefore, in the second professional development session, we helped principals develop a clearer vision of such discussions by sharing carefully selected classroom video clips of mathematics lessons. Similar to our approach with the assessment items, we used contrasting cases to help principals identify key characteristics of high-quality discussions.

Having solved and discussed a rigorous middle-grades task (similar to Item B), we watched a video of a classroom discussion in which students presented their answer to the class and talked through the steps of their solution but were not pressed to explain why they did what they did nor to connect their thinking to others’ solutions. We focused principals’ attention on whether they thought students learned through this kind of discussion — particularly students who were in the “audience” as their peers presented. Principals agreed that students may not have been learning mathematics through this kind of discussion.

We then watched a video clip of another classroom discussion around the same task. However, in this video, the teacher pressed students to explain their reasoning and to connect their strategies to those of their classmates. Again, we asked principals to reflect on the mathematics students might have learned through this kind of discussion. Principals agreed that it was much more likely that both presenting and listening students had opportunity to deepen their understanding of the key mathematical ideas, since they were consistently pressed to explain their reasoning and to make connections between solutions.

Having established a shared vision of a high-quality discussion with the principals in the session, we then focused on the specific actions teachers took to develop students’ mathematical reasoning. Next, we framed those teacher actions as practices that principals could look for in classrooms. This fit well with the district’s priorities, since classroom discussion in mathematics was a key indicator of teachers’ practice on the evaluation rubrics principals were required to use by their state.

#3. Translating the vision into goals for teachers’ learning.

Our goal in these sessions was to support principals in developing a shared vision of instruction that leads to student success on more rigorous state assessments. Structuring the discussion of instruction around illustrative assessment items and classroom video helped principals identify characteristics of rigorous instruction they could look for in their own schools. To help principals see these practices as goals for their teachers’ learning, we assigned homework between sessions: We asked principals to pay special attention to the math tasks and discussions in their schools. In the third session, we asked principals to share their classroom observations with colleagues and to compare the shared vision of rigorous instruction we had developed in our professional development sessions with the instructional practices typical in their schools. In the discussion, principals identified the significant challenge that the Common Core math standards posed for teachers as well as students. For example, implementing problem-solving tasks (like Item B) and learning how to facilitate whole-class discussions in which students explain their reasoning and connect their thinking to others required significant changes to teachers’ existing practices. Principals recognized that their teachers needed ongoing support if they were to develop new practices that would support students to succeed on rigorous assessments. These new practices represented goals for teachers’ learning. Against these goals for teachers’ learning, principals could begin to consider how they might draw on existing district and campus resources to support such teacher learning.

Conclusion

The analysis of math assessment items might seem, at first glance, an oddly specific focus for an initial school leader professional development session. However, it provided a concrete way to ground school leaders’ understandings of what the new standards and assessments require of students. Further, understanding the effects of more rigorous standards and assessments on students allowed for more discussions about the nature of classroom instruction that would help students learn mathematics in meaningful and lasting ways. School leaders were then better positioned to identify clear goals for teachers’ practices — and thus consider how they might organize supports for teacher learning. More rigorous assessments suggest ambitious goals for student learning. As such, the analysis of assessment items can generate critical discussions of academic rigor, instructional practices, professional development, and instructional leadership. The deeper the school leaders’ understanding of assessment standards, the more flexible and focused they will be with their implementation of supports.

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CHARLOTTE DUNLAP (charlotte.j.munoz@vanderbilt.edu‎) is a doctoral candidate in the Department of Teaching and Learning at Vanderbilt University, Nashville, Tenn. MEGAN WEBSTER is a doctoral candidate at McGill University, Montreal, Quebec. KARA JACKSON is an assistant professor at the University of Washington, Seattle, Wash. PAUL COBB is a professor of mathematics education at Vanderbilt University, Nashville, Tenn.