# Theoretical Background

The study we propose is a multi-method piece, using qualitative research to explore sociocultural aspects of mathematics communication, and using quantitative measures of mathematics and literacy performance to discern relative effectiveness of teachers' instructional strategies that attend to the literacy demands of the curricula. Three larger bodies of research inform this work. The first of these is sociocultural research on literacy, including that which has taken place in mathematics classes; the second is the multiple framings of research on teacher development; and the third is the body of research on students' algebraic reasoning. These bodies of research intersect at our interest in situating mathematics and literacy instruction in urban settings, and in understanding the development of teachers' learning within this setting.

Literacy

Many studies during the 1970s and 1980s on the teaching of reading in secondary school were concerned with learners’ cognitive processes and teachers’ instructional approaches in a variety of subject-area classes (Alvermann & Moore, 1991). This work provided insights about how to facilitate learners’ comprehension and knowledge acquisition related to key concepts in specific subject areas. However, teachers did not implement these recommendations on a dependable basis, largely because of complexities related to the structures of secondary schooling and school culture (Hinchman & Moje, 1998; O’Brien, Stewart, & Moje, 1995). More recent research has attended to the complex intersections of adolescent learners, texts, and contexts (Chandler, 2000; Hinchman, Payne-Bourcy, Thomas, & Chandler-Olcott, in press; Hinchman & Young, 2001; Hinchman & Zalewski, 1996; Moje, Dillon, & O’Brien, 2000).

Literacy has come to be seen as multifaceted, involving reading, writing, speaking, and listening, as well as other performative acts. Enactment of multiple literacies occurs in certain social settings and happens in certain ways, about certain topics, for certain purposes (Hicks, 1995/96). These multiple literacies include all the discursive competencies by which we navigate, shape, and are shaped by our multiple, competing social existences (Street, 1995). Central to much of this work is the idea of discourse, articulated by Gee (1989), who defines the term as “a socially accepted association among ways of using language, of thinking, and of acting, that can be used to identify oneself as a member of a socially meaningful group” (p. 18).

Like other domains of study, mathematics classes at the secondary level require teachers and students to use various kinds of literacies and to participate in various discourse communities specific to the domain (Hinchman & Young, 2001; Hinchman & Zalewski, 2000). Some of these literacy practices have been studied from a cognitive perspective (Friel, Curcio, & Bright, 2001; MacGregor & Price, 1999; Mosenthal & Kirsch, 1993; Muth, 1991). Other recent studies have used a sociocultural frame (Atweh, 1993; Lerman, 2001; Sturtevant, Duling, & Hall, 2001) because it accounts for aspects of learning mathematics in complex classroom contexts that a focus on thinking processes alone may not.

Begun in the late 1980’s, the line of work pursued by Siegel, Borasi, and their colleagues (Borasi & Siegel, 2000; Borasi, Siegel, Fonzi, & Smith, 1998; Siegel, Borasi, & Fonzi, 1998; Siegel & Fonzi, 1995) represents a particularly important contribution to research on literacy and mathematics. Several of their studies analyzed what kinds of literacy practices “counted” in social interactions in several different mathematics classes. Their work suggested the importance of defining reading broadly in mathematics and of embedding reading, writing, and oral language deeply within inquiry cycles—findings on which we hope to build. There are important differences, however, between their work and this proposed study. Although they used sociocultural theory to frame some of their results, their research was also grounded in transactional and functional perspectives on literacy that can be less sensitive to cultural and linguistic diversity and that tend not to position mathematics learners as members of discipline-specific discourse communities. In addition, their inquiries took place in different settings than the proposed study. One of their sites was located in a city about the same size as the one where we propose to work, but the setting was an alternative school, where teachers had smaller class sizes, a less heterogeneous student population, and greater freedom related to curriculum and assessment than our teacher collaborators experience. The other research site was a middle school in a rural-suburban area. No studies thus far have investigated the literacy demands inherent in enacting reform-based mathematics curricula in typical urban settings, nor do we know enough about how such enactments are constructed from teachers’ and students’ perspectives. By understanding more about the literacy practices privileged by such curricula, and individuals’ stances toward those practices, mathematics and literacy educators can do a better job of helping teachers in mediating students’ understandings and contributing to ongoing curriculum reform.

Teacher Development

Research on teaching, teaching practices and teacher development has gone through different periods of emphasis and reform during the last sixty years. Koehler and Grouws (1992) discussed research on teaching from the perspective of complexity and categorized research work into four levels of complexity that “reflect changes and progress in research on teaching” (p. 115). Research at the first level examines a particular aspect of teaching in isolation. Studies at the second level “usually involve multiple classroom observations that provide extensive detail concerning instruction in mathematics” (p. 116). Research at the third level incorporates learning characteristics and broadens learner outcomes to include attitudes as well as achievement. Koehler and Grouws categorized studies into the fourth level of complexity when the research had a strong theoretical foundation and involved many factors, most notably pairing research on teaching with research on learning.

Current research on teaching and teacher development falls into this fourth category. Research on teacher development indicates that teachers need opportunities to observe and reflect on students’ mathematical thinking and on how effective teachers build on students’ thinking (Ball, 1993; Simon, 1995). However, researchers have only begun to document effective ways for teachers to investigate student learning and incorporate this into their ideas of teaching (cf., Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996). Efforts to address such aspects of teacher change have described a shift towards strategies by which teachers are encouraged to become reflective about their practice (Cooney & Krainer, 1996; Simon, 1995). Cooney (1999) has highlighted the importance of examining the contexts through which teachers develop and use their knowledge. One aspect of understanding the development of teachers’ knowledge involves understanding how teachers deal with the complexity of teaching. Lampert (2001) noted, “One reason teaching is a complex practice is that many of the problems a teacher must address to get students to learn occur simultaneously, not one after another” (p. 2). Doerr and Lesh (in press) argued that “the essence of the development of teachers’ knowledge … is in the creation and continued refinement of sophisticated models, or ways of interpreting, the situations of teaching, learning and problem solving.” What we need to understand is how teachers interpret the complexity of teaching, particularly when using conceptually rich, reform-based mathematics curricula.

School-university collaborations provide one important context for teacher development, and for anchoring research into this development (Chandler-Olcott, 2001a; Cochran-Smith & Lytle, 1993). Involving varieties of negotiations and compromises regarding insights, understandings, and theories, such projects allow teachers and researchers to construct models that are reflect the day-to-day decision-making of teachers in complex classroom settings (Chandler & the Mapleton Teacher-Research Group, 1999; Chandler-Olcott, 2001b; Hinchman, Boljonis, Haun, Heinrich, Molinari, Ryan, & Woodward, 1999). Engaging teachers in discussing case studies of practice has been found to promote reflectiveness about the teachers’ own practice (Lundeberg, Levin, & Harrington, 1999). Multimedia case studies have been found to help teachers analyze and reflect on their own practice through analyzing and reflecting on the practice of case study teachers (Masingila & Doerr, in press). Masingila and Doerr found that a multimedia case study enabled teachers to delve more deeply into issues revealing the complexities of teaching and assisted them in understanding how to use student thinking in teaching mathematics. Case studies can provide sites for investigating relevant issues and enabling teachers to reflect on the complexities of their own teaching practice. Analyzing mathematical tasks can also be a powerful tool in assisting teachers in understanding the mathematical thinking required in the task and the difficulties faced by students in achieving this thinking (Stein, Grover & Henningsen, 1996; Stein, Smith, Henningsen & Silver, 2000). Through supported reflection on teaching and analysis of teaching tasks, teachers can better understand how to mediate their students' learning.

The Teaching and Learning of Algebra

The past three decades have generated a considerable body of research on the learning of algebra, but often the teacher is absent from the picture. The research on algebra learning has tended to focus on the algebraic nature of the mathematical tasks, the development of algebraic ideas by the learner and, more recently, on the influence of technology on learning. The research on the learning of algebra (Bednarz, Kieran, & Lee, 1996; Usiskin, 1988) has suggested four conceptualizations of algebra: namely, as (a) generalized arithmetic, as (b) a means to solve certain problems, as (c) a study of relationships, and as (d) structure. This framework has provided the research community with an extensive knowledge base on the nature and scope of students' conceptions and misconceptions about key algebraic concepts such as variable, equivalence and linearity (e.g. MacGregor & Stacey, 1993, 1997, 1998; Sfard & Linchevski, 1994). This research has led to an increased emphasis on graphical and numerical representations, along with the symbolic; on understanding how the symbols of algebra can be used for recording and communicating ideas; and on the interpretation and representation of meaningful phenomena (Confrey & Doerr, 1996; Brenner et al, 1997).

Only rarely, however, have teachers and the nature and development of their knowledge and teaching practice been the focus of study on the learning of algebra. Using Shulman’s (1986) pedagogical content knowledge framework, researchers have investigated the conceptions and misconceptions that secondary teachers have about the concepts of functions and variable. Collectively, this work would suggest that teachers’ knowledge about functions tends to be instrumental, rather than relational or conceptual (Even, 1993; Even & Tirosh, 1995; Haimes, 1996; Hitt, 1994). There is no evidence that would suggest that teachers see the concept of function and the many contexts in which it occurs as an integrating theme for algebra instruction across the curriculum, as is envisioned in the NCTM curriculum standards. Teachers would appear to hold a view of algebra that gives preference to the symbol-based representations and procedures, rather than visual and graphical representations and problem-based contexts (Borba & Confrey, 1996; Nathan & Koedinger, 2000). In his study of teachers' conceptions of algebra, Gadanidis (2001) found that the teachers saw the formula for area as the key piece of algebra when teaching students a technology-based lesson on maximizing area through an interactive exploration of graphs. This suggests a diminished value that is placed on the graphical representation and on the use of meaningful contexts, but also a disconnection between those activities and the development of student's algebraic thinking. As a consequence, when teachers tend to teach the rules for manipulating algebraic expressions, students do not seem to be able to use symbolic expressions as tools for meaningful mathematical communication (Kieran & Sfard, 1999).

By focusing our attention on algebra as a tool for meaningful mathematical communication, we are shifting our perspective from the cognitive conceptualization of algebra to a view of algebra as a language for mathematical communication (c.f., Verschaffel, Greer, & de Corte, 2000; Sutherland, Rojano, Bell & Lins, 2001). That is, algebra can be seen as text (in the broadest possible sense of that word) and the teaching and learning of algebra can be seen as the participation in a discourse community or community of practice (Gee, 1996; Lave & Wenger, 1991). Teachers then need to consider how to help students to acquire the ways of thinking, acting, writing, reading, and speaking that characterize membership in that community.