Symposium Overview and Objectives/ Educational Significance/ Papers/ References
Symposium Overview and Objectives
Calls for teacher quality and improved U.S. student achievement in mathematics reinforce the need for continued research and theoretical work to support programmatic and policy shifts in the preparation and continuing professional development of teachers. Students' performance in algebra is particularly worrisome (Blume & Heckman, 2000; 2002) . Today's accountability-oriented policy environment produces pressure both for the improvement of students' performance in mathematics, and for the intensified efforts to explain variation in their performance. In light of current recommendations for the mathematical preparation of teachers (Conference Board of the Mathematical Sciences, 2001) , as well as growing attention among researchers to the mathematical knowledge used in teaching (e.g., Ball & Bass, 2000a, 2000b) we address the topic of the mathematical knowledge for teaching secondary school algebra.
The claim that teachers' knowledge plays a significant role in what students learn seems self-evident, and assumptions about the truth of this claim underlie policy decisions as well as many research studies. Empirical evidence that justifies this claim, however, is scant. We still know little about either what knowledge is important for teaching particular mathematics or the mechanisms by which teacher knowledge translates into student learning. Studies of teacher qualifications, as a proxy for teacher knowledge, have yielded puzzling results, with indications that taking mathematics courses is important, but only to a point; and that methods courses can be equally important (Wilson, Floden, & Ferrini-Mundy, 2002) . The difficulty of finding appropriate and theoretically justified proxies for teacher knowledge has led to more direct studies of what teachers know and how they use their knowledge in practice. The research reported here is in the latter category, with a specific focus on school algebra.
We work from the assumption that teachers have knowledge about algebra that they use in practice, but that may or may not have been directly taught in their own mathematics educations. We assume that at least three kinds of knowledge may be useful to teachers of algebra: knowledge that parallels what they are teaching their own students; advanced knowledge of mathematics that follows or underlies the mathematics they are teaching; and mathematical knowledge that is specific to teaching, not specifically taught in algebra classes or in more advanced mathematics coursework. In our framework and empirical investigations, we explain and explore each of these aspects of teachers' mathematical knowledge.
The objectives of this symposium are to introduce the framework we have developed over the last two years; to illustrate its use in three empirical studies of teacher knowledge: an analysis of algebra textbooks, an interview study with practicing teachers, and an analysis of video of algebra teaching; and to elicit feedback and input from those who attend the session. The framework itself is the result of an iterative process of theorizing and data analysis, in which we have used both logical analysis and empirical evidence to build the framework recursively. The papers in the symposium explain and illustrate this process.
Educational Significance
This symposium will provide a forum for presenting our initial work on this very important topic, and for getting extensive feedback from other researchers. In our ongoing research program, we are using the framework and empirical results to develop items that will be used in research measuring teachers' knowledge of algebra for teaching and its impact on students. Our work contributes to the pressing need to improve the mathematical education of teachers in the area of algebra, and provides a foundation for ongoing research that can be used to develop more effective courses, professional development, and assessments of teacher knowledge.
Papers
Knowledge for Teaching School Algebra: Challenges in Developing an Analytic Framework, Joan Ferrini-Mundy, Robert Floden, Raven McCrory, Gail Burrill, Dara Sandow
Usiskin (2000) suggests that mathematical knowledge for teaching is a kind of applied mathematics that needs to be taught in its own right, yet we do not have a theoretically justified basis for defining applied mathematics for teaching algebra. Usiskin and colleagues (Usiskin, Peressini, Marchisotto, & Stanley, 2002) textbook for prospective secondary mathematics teachers addresses connections across topics as well as specific topics of importance for teaching. Their book begins to define applied mathematics for teaching at a summative level, across the many domains of mathematics for which secondary teachers are responsible. Ball and colleagues (Ball & Bass, 2003; Ball, Lubienski, & Mewborn, 2001; Hill, Rowan, & Ball, 2004; Hill, Schilling, & Ball, 2004) have studied mathematical knowledge for teaching elementary school mathematics. Their research reveals different ways of knowing arithmetic and elementary geometry that impact teachers' effectiveness. Ma (1998) contrasts teachers in China with teachers in the US, and finds that US teachers lack a "profound understanding of fundamental mathematics" that Chinese teachers have and apparently use effectively in practice.
Although current research is making progress toward understanding and assessing teacher knowledge in some areas as outlined above, the specific domain of school algebra has not been addressed. In our research, we build on the work cited above to develop a framework for studying and understanding knowledge for algebra teaching. Our framework started with three overarching categories that constitute ways of using knowledge: decompressing or unpacking (Ball & Bass, 2000a) , trimming (Ferrini-Mundy, Burrill, Floden, & Sandow, 2003) , and bridging (Ferrini-Mundy et al., 2003) . From these three categories, we have built an extensive framework that we are now using to develop items for assessing and understanding teacher knowledge.
Within school algebra, we focus on two primary topics: (a) expressions, equations, and inequalities, and (b) functions and their properties, linear and nonlinear. The framework consists of a three dimensional matrix. One dimension of the matrix consists of categories of knowledge for teaching: knowledge of school algebra, advanced mathematical knowledge, and teaching knowledge. The second dimension is the mathematical topics (a) and (b) above. The third dimension specifies four aspects of mathematics within each topical area: core concepts and procedures, representations, applications, and reasoning and proof. We thus have a 3x2x4 matrix. We use the three overarching categories - decompressing, trimming, and bridging - to understand what a teacher would do with particular instances of knowledge in cells of the matrix.
One of the purposes of our ongoing research is to understand better how the three categories of teacher knowledge specified in our first dimension relate to each other. Are they distinct? In what ways do teachers differ across these three? Do differences in teacher knowledge across these three categories have an impact on student achievement?
In our paper, we explain the matrix and give examples of items we are using to assess teacher knowledge in individual cells. We will seek input and discussion from the audience in our ongoing research to develop a solid theoretical and practical framework for studying knowledge of algebra for teaching.
Curriculum analysis: Approaches to school algebra in secondary curricula, Sharon L. Senk, Gail Burrill, Dana Olanoff (not available online)
At present there is considerable interest among the scholarly community in three areas of research: (a) how teachers' mathematical knowledge develops and how it is used in teaching, (b) the effects of textbooks and other instructional materials on students' learning, and (c) the teaching and learning of algebra. The study described in this article is an analysis of secondary school textbooks, and builds on research in each of these areas. Specifically, it addresses the following questions:
1. How does the treatment of key topics in algebra vary among U. S. texts available for teaching algebra?
2. What does the treatment of these key topics suggest about the knowledge needed for teaching algebra?
The work reported in this paper describes three aspects of the treatment of expressions, equations, and inequalities in four textbook series, including the introduction to these concepts, linear equations in one variable, and logarithmic expressions and equations. The analysis describes numerous differences in scope, sequence, and development of concepts. Based on the analysis, five components of a framework for knowledge for teaching algebra are hypothesized: core content knowledge, knowledge of mathematical reasoning, knowledge of representations, knowledge of applications and contexts, and curricular knowledge. These categories of knowledge are likely not to be discrete; so knowledge of connections may be considered a sixth component of knowledge for teaching algebra.
Introducing Solving Equations: Teachers and the One Variable First [Algebra] Curriculum, Robin Marcus and Dan Chazan
In the context of school algebra, the word "equation" is commonly used. Yet, curricula differ in the sorts of equations students meet and the order in which they are introduced. A traditional approach, for example, first introduces students to equations of one variable. After an extended time on linear equations in one variable, students meet linear equations in two variables and then other equations of two variables. By contrast, a functions-based approach might start with expressions as representations of functions in one variable, and address equations in one and two variables later.
For the teacher, a key question is being able to put oneself in students' shoes. Given the curriculum that they have taught with, what do they think equations are? What sorts of equations will ask them to generalize or expand their notion of what an equation is? From our perspective these questions ask a teacher to unpack and decompress their understandings of equations. They must look back at content that they already know and use distinctions that may not be important for them as expert users of algebra in order to have criteria and distinctions to understand the experience of their students and the impact of the curriculum on students' understandings.
In an attempt to surface issues of unpacking and decompression on the part of algebra teachers, our interview protocol focused on teachers' thinking about equations and expressions. First we asked questions that gave us a sense of teachers' perspectives on variables, equations, and functions; then we selected follow up questions that would be challenging for teachers given their perspective. For example, teachers were given a set of equations and asked to put them in the order they would be taught. Follow up questions asked them to compare and contrast pairs of equations and to assess appropriate and inappropriate solution strategies for these equations.
We interviewed 16 middle and high school teachers who were using five different textbooks and curricula. Since our sample was small, and not representative, what we report is not conclusive findings, but rather suggestions derived from the data that have helped develop and refine our framework.
For example, among teachers who begin with equations in one variable and focus on solving as isolating the variable, there is a quite natural transition to solving an equation in two or more variables for a particular variable. However, there is the potential that solving then loses its close connection to solutions. What does it mean to "check" after solving? How do you make the connection that the graph of an equation in two variables represents the solution set of that equation? How do you explain that x 2 = 2 x is not solvable (you can't isolate x), yet it has 3 real solutions? How do you explain that 3(x - 4) = 6x - 8 and ? [(2x - 5) 2 ] = 2x - 5 are not solvable (you can't isolate x), yet they have infinitely many solutions -- one all real numbers, the other an infinite subset of the real numbers? We explored these and other issues in our interviews.
In this paper, we give examples from interviews of instances of and differences in teacher knowledge. We illustrate how we used these examples in developing our framework.
Using Videos of Teaching to Study Teacher Knowledge, Raven McCrory
An essential way to understand the knowledge teachers use in practice is to study their practice (Ball & Bass, 2003) . Studying practice is problematic, though, as a means of identifying teacher knowledge, because knowledge itself is not visible. What we see when we study practice is teachers' actions, what they do and say. Sometimes their actions include specific bits of knowledge, but for the most part, their knowledge is implicit. Although it is impossible to "see" knowledge, it is feasible to extrapolate from what happens in an actual class to what mathematical knowledge is in play or could be useful given the circumstances. We use classroom episodes to explore the mathematics students and teachers use in practice, and what knowledge could support that mathematics. In this study, as part of our effort to develop our framework, we sought video of algebra teaching and used it to investigate teachers' knowledge. We began with the overarching categories for our framework -- compressing, trimming, and bridging -- and looked in classrooms for evidence of these kinds of activities, or uses, of knowledge.
Using video from the Videocases for Mathematics Professional Development (Mumme & Seago, 2002) , the Annenberg/CPB Teaching Math Video Library (Annenberg/CPB Math and Science Project & WGBH Boston, 1996) , and the Third International Math and Science Study (Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1998) public release video, we analyzed five sessions in which algebra was taught. In this paper, we will report on a flow chart we developed as an aid to hypothesizing about teacher knowledge, and we will discuss specific episodes of teaching that were particularly useful in furthering our research.
For example, in two sessions, different teachers taught the same lesson and, although both teachers knew the "school algebra" they were teaching, one had conceptions or understandings of algebra that led her to treat the topic in a different way. In terms of our framework, her "knowledge of advanced mathematics" stood out. A core concept for her was equivalence, which she used to make connections across representations of mathematical ideas. Equivalence itself was something she wanted her students to learn about implicitly. The other teacher was more focused on engaging students in activity that led to a particular outcome without attention to the mathematical concept (equivalence) that gave coherence to the activity. We hypothesize that, in terms of our framework, we will be able to devise an assessment that could measure the difference in these two teachers with respect to the specific topic of equivalent representations of functions.
In our paper, we discuss examples of teacher knowledge and illustrate how they contributed to and are reflected in our framework. We also discuss methodological issues of studying knowledge in practice.
References
Ball, D. L., & Bass, H. (2000a). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics. Westport, CT: Ablex.
Ball, D. L., & Bass, H. (2000b). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. C. Phillips (Ed.), Constructivism in Education. Chicago: University of Chicago Press.
Blume, G. W., & Heckman, D. S. (2000). Algebra and functions. In E. A. Silver & P. A. Kenney (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 269-306). Reston, VA: National Council of Teachers of Mathematics.
Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers (Vol. 11). Providence, RI: American Mathematical Society.
Wilson, S. M., Floden, R. E., & Ferrini-Mundy, J. (2002). Teacher preparation research: An insider's view from the outside. Journal of Teacher Education, 53(3), 190-204.