2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 5, pp. 73-80. Prague: PME. 5 - 73

LEARNING MATHEMATICS FOR TEACHING Nanette Seago Lynn Goldsmith

WestEd EDC

The identification of mathematical knowledge for teaching (Ball & Bass, 2000), leads to questions about how to promote and assess it. What kinds of professional development experiences might provide teachers with the opportunity to develop mathematical knowledge for teaching? What does it mean to learn it? In this paper, we address these questions with illustrations of teachers learning various aspects of mathematical knowledge for teaching. These teachers were participants in Turning to the Evidence1, a study investigating the role of practice-based professional development centered in the use of classroom artifacts (e.g., student work and classroom video) and its effect on teacher learning2.

MATHEMATICAL KNOWLEDGE FOR TEACHING Mathematical knowledge for teaching focuses attention on the considerable mathematical demands that are placed on classroom teachers (Ball & Bass, 2000; Ball, Bass, & Hill, 2004). Building on Shulman’s (1986) notion of pedagogical content knowledge, characterized as “bundled” mathematical, pedagogical, and cognitive/developmental knowledge which can help teachers anticipate and address typical issues of students’ learning mathematics, Ball and Bass posit an “unbundled,” complementary mathematical knowledge that teachers must call upon as needed in the course of classroom practice to effectively engage students in learning.

Although pedagogical content knowledge provides a certain anticipatory resource for teachers, it sometimes falls short in the dynamic interplay of content with pedagogy in teachers’ real-time problem solving. . . . [A]s they meet novel situations in teaching, teachers must bring to bear considerations of content, students, learning, and pedagogy. They must reason, and often cannot simply reach into a repertoire of strategies and answers. . . .It is what it takes mathematically to manage these routine and nonroutine problems that has preoccupied our interest. . . . It is to this kind of pedagogically useful mathematical understanding that we attend to in our work. (Ball & Bass, 2000, p. 88-9, italics in original)

They also argue that the kind of pedagogically useful mathematical knowledge and understanding differs in a number of ways from the mathematical knowledge and 1 The Turning to the Evidence project is supported by the National Science Foundation under grant no. REC-0231892. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation. 2 We would like to acknowledge Mark Driscoll, Johannah Nikula, Zuzka Blasi, Daniel Heck and Joe Maxwell for their contributions to the ideas discussed in this paper. In addition, we would like to thank Judith Mumme for her contributions to the development of this work.

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understanding required in other disciplines that use mathematics. For example, Ball and Bass argue that the mathematics needed for teaching requires an unpacking of fundamental ideas, while the mathematician seeks their elegant compression. They have identified several core activities of mathematics teaching such as: (1) figuring out what students understand; (2) analyzing methods and solutions different from one’s own, determining their adequacy, and comparing them; (3) unpacking familiar mathematical ideas, procedures, and principles; and (4) choosing representations to effectively convey mathematical ideas.

While these core activities can serve as productive arenas for the investigation of teachers’ learning of mathematics knowledge for teaching, they raise questions about how to promote this kind of learning in teachers. What kinds of experiences might provide teachers with the opportunity to further develop this specialized kind of mathematical content? What does it mean to learn it?

Several research projects have demonstrated that practice-based (Smith, 2001) professional development projects that utilize artifacts of practice, such as classroom video and student work, are effective tools in efforts to increase teachers’ opportunities to learn mathematics knowledge for teaching (Borko, 2004). By bringing the everyday work of teaching into the professional development setting, these tools enable teachers to unpack the mathematics in classroom activities, examine instructional strategies and student learning, and discuss ideas for improvement (Ball & Cohen, 1999; Driscoll et al., 2001; Kazemi & Franke, 2004; Schifter et al., 1999a, 1999b; Seago et al, 2004).

In this paper, we highlight different aspects of mathematical knowledge for teaching (MKT) that we have seen teachers develop through their engagement in professional development centered around the use of classroom artifacts. The teachers described in this paper were participants in the Turning to the Evidence Project, a study investigating the role of classroom artifacts in teacher learning (Seago & Goldsmith, 2005). The professional development focused on a number of the core activities described by Ball and Bass, such as: unpacking familiar mathematical ideas, analyzing alternative methods and solutions, and choosing and using mathematical representations. We present data collected about three participants in this professional development to show how each developed their thinking in one of these areas of MKT

THE TURNING TO THE EVIDENCE STUDY Seventy-four middle and high school teachers participated in this study; 49 teachers enrolled in one of four professional development seminars and 25 served as comparison teachers for pre- and post-program assessments. Two of the seminars used materials from the Fostering Algebraic Thinking Toolkit (Driscoll, et al., 2001) and two used modules from Learning and Teaching Linear Functions: VideoCases for Mathematics Professional Development (Seago et al., 2004). In all,

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18 teachers participated in Fostering Algebraic Thinking (AT) seminars and 31 in Learning and Teaching Linear Functions (LF) groups.

The two sets of materials within this study are both grounded in situative perspectives on learning. A key principle of situative perspectives is that the activities and contexts in which people learn become a fundamental part of what they learn (Greeno, 2003; Greeno, Collins, & Resnick, 1996). Although this principle suggests that teachers’ own classrooms are powerful contexts for their learning, it does not assert that professional development activities should occur only in classrooms (Putnam & Borko, 2000). Both LF and AT use artifacts (e.g., videotapes, samples of student work) to bring teachers’ classrooms into the professional development setting, enabling teachers to examine instructional strategies and student learning, and to discuss ideas for improvement (Ball & Cohen, 1999).

The professional development for both programs is structured in similar ways—both include work on the mathematical task prior to examining and discussing the classroom artifact with the goal to help teachers learn to more deeply focus their attention on students’ mathematical thinking and to connect seminar work to their own practice. The materials differ in two ways: (1) the mathematical focus (algebraic habits of mind versus unpacking linearity, and (2) the kinds of artifacts used (written artifacts of student problem-solving versus video episodes of classroom discussions).

Data sources

The project collected information from several data streams, including a background questionnaire completed at the beginning of the project by all participants, pre- and post-program assessments of mathematics knowledge for teaching (Math Survey) and analysis of classroom artifacts (Artifact Analysis) using paper-and-pencil instruments.

The Math Survey includes both multiple choice and open-response items focusing on understanding algebra (with a particular emphasis on linearity). In constructing the survey, we drew heavily on items from the SII database and also included items used to assess teachers’ learning in California Mathematics Professional Development Institutes (Hill & Ball, 2004). The Artifact Analysis was a two-part instrument designed for this project. The first involved viewing and answering a series of increasingly specific questions about a short video segment of a class discussion that centered on students’ presentation of generalizations of a linear relationship set in a geometric context. The second part asked teachers to comment on three pieces of written student work for the same problem. This measure replicated the professional development work of both projects in that it used both types of artifacts—video and student work. In addition, videotapes of all professional development seminars were collected and transcribed for analysis in order to study teachers’ professional development experiences.

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LEARNING MATHEMATICAL KNOWLEDGE FOR TEACHING: What do teachers know that they didn’t know before?

In this paper we draw from the larger study of 49 teachers to examine what kinds of MKT learning are possible when teachers participate in practice-based professional development that utilizes artifacts of practice. The teacher learning cases included in this paper illustrate what teachers can learn by examining a few individual teachers across the data sources, creating learning stories with corresponding evidence. In doing so, we illustrate the learning of specific knowledge such as: learning to analyze various methods for solving problems, learning to unpack mathematics, and learning to use mathematical representations. We do not make claims about what all teachers in the study learned, but rather about what it looks like when a teacher learns about a particular aspect of MKT. We focus primarily upon three teachers: Trevor, Charles, and Laura in our illustrations of the learning of MKT. We primarily chose these three teachers because they each showed gains in different types of learning. Additionally, we chose these teachers based on how they scored on the pre-program Math Survey. We chose one teacher that scored high (Charles) and two that scored low (Trevor and Laura), because we found their initial scores and their gains in mathematical knowledge for teaching interesting. Learning to unpack the mathematics Trevor has a bachelor’s degree in mathematics and taught high school during the time he was involved with the Video Cases seminars. In the background questionnaire, he wrote that he had signed up for the seminar in hopes of “finding alternative approaches to teaching algebraic thinking and computational skills.” Trevor showed 17% improvement on the Math Survey (MS). On his post-program MS, Trevor not only answered more problems than he had on the pre-program MS (from 27 to 35), but he also showed significant improvement in problems that required him to employ conceptual understanding of y-intercept, as well as geometric representations of linear growth. Three problems he got wrong in the pre-program MS that he answered correctly in the post-program MS were word problems that involved translating a linear relationship into a formula while recognizing the starting point (y-intercept), the slope, and the variable. The three problems all began with a statement about the initial condition or starting point. This is especially interesting given the fact that Trevor’s undergraduate degree is in mathematics. Trevor’s learning to unpack the mathematics of y = mx + b, specifically of the meaning of y-intercept (b), shows up again in his participation in the seminars. In the eighth seminar, the group revisits a video segment from the first session in which a student (James) explains his recursive thinking—focusing on the growth of the pattern and not attending to the starting point. As was common during the beginning seminars, Trevor did not say much during the video portion within each session. His involvement primarily showed up within the portion of the sessions that worked on

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the mathematical task. During the eighth session however, Trevor discusses James’ thinking:

“He’d gotten so deep into the table that he forgot the part that starts the table going.. . . if you just looked at lines 2, 3, and 4, you’ll see a change taking place for sure, but that doesn’t give you your initial conditions because you’re seeing the changes taking place as you march down the table.”

After some discussion by the group about how to build upon James’ idea and help him to see the initial starting point, Trevor suggests “b + mx” as an alternative order. This shows Trevor has learned to appreciate the conceptual difference that putting the “b” first might mean as he unpacks the parts of a linear equation. Trevor recognizes his own learning to unpack the mathematics as he replies to a question about his own learning by saying that the “practice of looking at geometric representations of linear growth has developed in me to extract both the initial conditions and the constant of growth.” Trevor reported that he intends to utilize his learning within his teaching practice because he will “invest more preparation time looking for opportunities for student algorithmic development” and “it influenced me to try to get a depth of understanding in what I’m presenting so that I can anticipate the struggles my students are going through before they get there, and be able to help them through it when they do get there.” Learning to choose and use mathematical representations Laura taught 6th grade during the time she was involved with the Video Cases seminars. She signed up for the LF seminars “in order to learn new teaching skills in hopes of becoming a better teacher”. While she entered into the professional development for pedagogical learning, she in fact gained mathematical knowledge for teaching specifically within the domain of choosing representations to effectively convey mathematical ideas. Overall, Laura improved 20% from on the MS. She was able to answer 7 more questions on her post-program MS than on her pre-MS. She displayed more use of mathematical representations in solving the problems—she used tables, geometric models, arithmetic expressions and some algebraic notation. Though she showed signs of struggles with algebraic notation, she showed a marked increase in willingness to solve the problems. The same theme emerged from her pre-program to post-program Artifact Analysis (AA)—she used more mathematical representations and displayed mathematical persistence and engagement with the problem—noting at one point a pattern she found as “cool.” In fact, in the pre-AA she commented mostly with general and non-mathematical statements. For example, Laura went from initially noticing management and social issues in the video to a more mathematically focused attention—from how many students were in the class and on-task to noticing that the students were looking at the “arms” of the figure. In addition, she improved in her ability to analyze student thinking more accurately and specifically. On the student work, she went from commenting on neatness and drawings to commenting on

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whether or not formulas worked—from “the picture in 3D is much more accurate. I also noticed that I can read their writing,” to “the formula will not work. You must know the previous answer. The drawing is more accurate.” On another student’s work, she initially commented on the amount of written explanation, but in the post-AA she notices mathematical inconsistencies. At the end of the seminar when asked if she thought that participating in the seminars influenced how she thought about math, she said it is harder than she thought it was—it is not so cut and dry. Laura stated that she believed it helped her to become a better teacher, but “I still have a long way to go—I would have to do more of this to really feel comfortable teaching it.” Learning to figure out what students understand Charles taught 6th grade during the time he was involved with AT seminars. He signed up for the AT seminars in order to improve his own algebra as well as his teaching of the subject. By the end of his participation, he displayed an improved ability to figure out what students understand. He improved upon his Math Survey by 11%, even though he reported that he was tired and “gave up more quickly” on the post-program version. Upon close examination of his pre and post-program MS, Charles appears to have made progress in his ability to reason from and with algebraic notation—he moved from using numeric substitution methods for checking accuracy of expressions, to reasoning about the geometric representation with algebraic notation. For example, one problem involved correctly choosing an algebraic representation of a student’s verbally explained method. In the pre-program MS, he chose a correct mathematically equivalent formula for the problem, but it did not accurately represent the student’s method. In the post-program MS, Charles correctly chose the algebraic formula that represented the student’s method. In addition, his work on the problems provided glimpses into his method for checking student’s accuracy—he moved from a substitution strategy in the pre-program version to using algebraic notation in reasoning about student’s methods in the post-program version. Charles increased the mathematical specificity of his analysis of student thinking—in both the video and the written student work sections of the Artifact Analysis. Overall, Charles shifted from a focus on communication and explanation in the pre video analysis to a more mathematical focus on reasoning, and generalization. He displayed an increased ability to follow the mathematical logic of the students in a more precise analytical way. For example, in the pre-program AA, Charles interpreted one student’s work by stating “ you can’t really tell what this student is thinking from the examples given; mixes written explanation with formula.” In examining the same student’s work on the post-program AA, Charles was much more precise in his analysis of the student’s thinking and much better at figuring out what students understand—he was able to break down the mathematical logic, noting places of confusion and shifting from a deficit view of the student’s thinking to suggesting there exists a basis for teaching “to what they know.”

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When asked what he learned by participating in the seminars, Charles stated that he learned “skills like generalizing, patterns, breaking apart and chunking.” These mathematical processes allowed him to improve in the mathematical knowledge for teaching of figuring out what students understand.

CONCLUSION This paper has examined various data from three teachers to illustrate what mathematical knowledge for teaching teachers might learn from artifact-based professional development. By carefully examining the various sources of data and utilizing Ball and Bass’ definition of mathematical knowledge for teaching, we find evidence of gains in various domains of mathematics knowledge for teaching. Trevor, a mathematics major, came into the seminars with a lot of mathematical background and conventional, compressed knowledge—yet he scored low on the pre-program Math Survey. His improvement came on problems that required him to employ conceptual and unpacked understanding of y-intercept (starting point), as well as geometric representations of linear growth. He learned to unpack and conceptualize a familiar mathematical idea. Laura increased in her ability to use various mathematical tools and representations. This gained knowledge showed up in her increased ability to analyze student ideas more accurately and specifically using various mathematical representations. Charles increased in his ability to reason through student ideas—to figure out what students understand. In the student work, Charles moves from a focus on how the student communicates (written explanation and visual models) to specific and detailed mathematical analysis of the student’s logic—including interpretation and error analysis. These cases provide an additional angle from which to view the practice-based theory of mathematical knowledge for teaching (Ball, Bass, & Hill, 2004). This work adds an interpretive frame for examining the construct of mathematical knowledge for teaching and what it might mean to learn it, by examining it from a different mathematical topic area (algebra), grade level (middle and high school teachers) and context—two practice-based professional development programs and their relationship to supporting teacher learning of mathematics for teaching. References Ball, D.L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning

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