Teachers’ Ways of Listening and Responding to Students’ Emerging Mathematical Models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students’ thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a welldeveloped schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed. ZDM-Classification: I20, M13 More than a decade of research on teachers’ professional development would suggest that, among other things, effective teachers need to attend to students’ ways of thinking about mathematical tasks (Borko & Putnam, 1996; Franke & Kazemi, 2001; Schifter & Fosnot, 1993; Simon & Schifter, 1991). Understanding how students might approach a mathematical task and how their ideas might develop would seem to provide a basis for teachers to interact with students in ways that could promote student learning. However, much of the research on teachers’ understandings of students' ways of thinking has focused on tasks in elementary mathematics, including important ideas in numeracy, rational numbers, and geometry (e.g., Ball, 1993; Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996; Jacobsen & Lehrer, 2000). Relatively little research has been done on teachers’ understanding of students’ ways of thinking about tasks in areas of secondary mathematics, such as functions, algebraic equations, Euclidean geometry, and data analysis (e.g., Heid, Blume, Zbiek, & Edwards, 1999). In addition, recent research has pointed to some of the challenges and difficulties inherent in the tasks of listening and responding to students' thinking (Chamberlin, 2003; Even & Wallach, 2003; Morgan & Watson, 2002; Peressini & Knuth, 1998; Schifter, 2001). Collectively, this research suggests a need to examine secondary teachers' ways of listening and responding to students' emerging mathematical models and the nature of the difficulties they encounter in so doing. In this paper, these issues are addressed within the context of students' engagement with a modeling task related to exponential growth. 1. Theoretical Framework Recent research on the nature of teachers' knowledge has emphasized that teaching is a complex and ill-structured knowledge domain (Davis & Simmt, 2003; Lampert, 2001). Other researchers have focused on the large extent to which knowledge is situated and grounded in the particularities of the contexts and constraints of practice (Borko, Mayfield, Marion, Flexer, & Cumbo, 1997; Lave & Wenger, 1991; Leinhardt, 1990). Taken together, these two perspectives suggest that understanding teachers' knowledge means understanding how teachers interpret the complexity and uncertainty of the situated practical problems of the classroom, and understanding how those interpretations influence their decisions and actions in the classroom. Expertise in complex and ill-structured domains requires the flexible use of cognitive structures to accommodate partial information, changing or unclear goals, multiple perspectives and uncertain consequences (Feltovich, Spiro, & Coulson, 1997; Spiro, Coulson, Feltovich, & Anderson, 1988). This implies that expertise in teaching is not a uniform, consistent or fixed set of constructs, but rather it is highly variable and needs to be understood as knowledge that develops along multiple dimensions in varying contexts for particular purposes (Doerr & Lesh, 2003; Lesh & Doerr, 2000). In this paper, I describe three dimensions of the knowledge needed to teach mathematics through modeling in order to illuminate teachers' ways of listening and responding to the development of students' mathematical models. The context for this study is a model-eliciting task for exponential growth (more fully described below and in Appendix A). Model-eliciting tasks (Lesh, Hoover, Hole, Kelly, & Post, 2000; Lesh, Cramer, Doerr, Post, & Zawojewski, 2003) are those in which students' thought processes are explicitly revealed through the descriptions, explanations, justifications, and representations that are produced as they engage with the task and present their end products. Like other tasks which promote highlevel mathematical thinking (Henningsen & Stein, Analyses ZDM 2006 Vol. 38 (3) 256 1997; Stein, Grover, & Henningsen, 1996), modeleliciting tasks provide students with opportunities to reveal how they are thinking about the situation by representing their ideas. Students' solutions will show the kinds of mathematical quantities, the relations among those quantities, and the operations and patterns that the students were thinking about. This is often referred to as the Documentation Principle (Lesh et al., 2003). Another important characteristic of model-eliciting tasks is that they are such that the students will be able to judge for themselves when their responses are good enough; this is known as the Self-Evaluation Principle. The students should not need to refer to an external authority (such as the teacher) to know when they have a satisfactory solution to the task. These task features have important implications for teachers' approaches to task implementation. Modeling tasks that are designed to engage students in high-level mathematical reasoning are more complex and more time consuming than routine mathematical tasks. These tasks are much more open to various implementation factors that could increase their complexity and the uncertainty of how events might unfold in the classroom. Situated within the context of a model-eliciting task, this study is framed around three dimensions of teachers' knowledge: (1) an understanding of the multiple ways that students' thinking might develop, (2) ways of listening to that development and (3) ways of responding with pedagogical strategies that will support that development. The first dimension, teachers' understanding of the landscape of the development of students' ideas, has been the subject of much research (particularly at the elementary level). Knowing this landscape is not the same as understanding one particular area of the terrain (one way of thinking) or one particular pathway through the terrain (one learning trajectory). Rather, when using model-eliciting tasks, teachers are faced with the challenge of understanding the multiple ways that students might have of interpreting a problem situation and the multiple paths they might take for refining and revising their ideas. The knowledge needed for teaching includes seeing the multiple ways that students might interpret a situation and understanding that their ideas might be revised along various dimensions (while not being tested or refined along other dimensions). The teacher, in turn, needs to respond in ways that will support students' conceptual development towards more refined, more generalized, more flexible, and more integrated ways of thinking (Doerr & Lesh, 2003; Lesh & Doerr, 2000). A theoretical basis for the development of students' concepts of exponential functions is provided by Confrey and Smith's (1994, 1995) descriptions of the role of multiplicative structures. These researchers have argued that the origin of the multiplicative world of the exponential function is in the actions of splitting, rather than in repeated addition. Their "splitting conjecture" suggests that seeing the constancy of successive ratios is as essential to an understanding of exponential functions as is the constancy of first order differences in linear functions. This assumption leads to an increased instructional emphasis on the role of tables in understanding exponential functions and for the need for students to coordinate their understanding of the additive change in the independent variable with the multiplicative change in the dependent variable as co-varying quantities (Confrey & Doerr, 1996; Doerr, 2000). Particular examples of “splitting” include the doubling of bacteria, the half-lives of radioactive decay, tree diagrams, paper folding and geometric similarity (Confrey & Smith, 1994, 1995). The the modeleliciting task for the students in this study is finding the closed form of a function to describe a doubling pattern of multiplicative growth. This study focuses on how the teachers see and interpret students' conceptual development as they engage with the task. A second dimension of teachers' knowledge consists of the ways that teachers listen to the expression of student ideas. In his studies of teaching practices, Davis (1996, 1997) provided a framework that made three distinctions in the ways of listening that he found among the teachers. An evaluative way of listening focuses on hearing students' answers for the purpose of determining correctness. With an interpretive orientation to listening, teachers endeavor to make sense of students' responses, requiring more elaborate answers and explanations. Teachers with a hermeneutic way of listening to students' ideas engage in negotiating meaning with students and are participants who revise their own knowledge. The difficulties in listening to students with an interpretive or hermeneutic orientation are illuminated by the work of several researchers at the secondary level. In their study of teachers learning to use interviews to understand students' mathematical reasoning, Heid and colleagues (Heid et al., 1999) found that the teachers tended to use the interviews to help students arrive at correct answers, rather than to probe students' thinking. The teachers had difficulty focusing on deep understanding of students' ideas and often noted what they expected to hear. Similarly, Morgan and Watson (2002) 1 This is the term used by Confrey and Smith. ZDM 2006 Vol. 38 (3) Analyses 257 found that teachers tended to have difficulty in understanding and valuing student solutions that deviated from the response that the teacher was anticipating. Even and Wallach (2003) offer two perspectives on these difficulties. One perspective sees the difficulties as deficiencies or obstacles that can be overcome through experience or participation in formal professional development activities. The other perspective suggests that the intrinsic nature of hearing students is such that one "always hears through various personal factors; that it is unrealistic to expect an accurate teacher understanding of what students are saying and doing." (p. 321, emphasis in the original). How a teacher hears students is not an obstacle to be overcome, but rather hearing students is always influenced by the teacher's knowledge and understandings and the specificity of the students and the context. When struggling with the difficulties inherent in listening to students' ways of thinking, teachers are faced with the challenge of responding in appropriate and effective ways to what they do see and interpret in students' activity. This leads to the third dimension of teacher knowledge that is relevant for this study, namely, the knowledge of pedagogical strategies that will support students' conceptual development. Current reform-based rhetoric that would exhort teachers to listen and to "not tell" does little to provide insight into what teachers could do and when and why (Chazan & Ball, 1999), while simultaneously eroding teachers' sense of efficacy when left with little sense of a new role (Smith, 1996). Furthermore, such exhortations tend to leave the notions of listening and responding as unproblematic. In responding to the multiplicity of conceptual developments that may be taking place in her classroom, the teacher needs to choose various strategies to further that development. Such strategies could include the use of appropriate representations and the connections among those representations, a repertoire of probing questions, or insightful ways of using computational technologies. The ways in which a teacher might respond to students' mathematical activity is, of course, dependent on what it is that the teacher sees and interprets in that activity. It is precisely a teacher’s perceptions and interpretations of classroom situations that influence when and why as well as what the teacher does. The underlying conceptualization of teachers' knowledge is that of expertise that varies along the three interdependent dimensions of knowledge described above. The central questions for this study are: • How do teachers see and interpret students' ways of thinking about exponential functions? • How do teachers' interpretations of students' thinking influence their actions in the classroom? To investigate these questions, I examined the practices of four experienced teachers when teaching a model-eliciting task for exponential growth. 2. Description of the Study This particular case study is part of a larger research project on the development of effective pedagogical strategies for teaching modeling tasks in technology-enhanced environments. The overall project includes modeling tasks that are intended to elicit the development of students' models (or conceptual systems) of linear change, exponential growth and decay, and periodic functions. The mathematical task that the students worked on is a well-known problem in exponential growth and served as the introductory lesson for the larger unit on exponential growth and decay. The task posed to the students was to investigate the doubling pattern of pennies on a checkerboard when one penny is placed on the first square, two pennies on the second square, four on the third square, and so on. This simple recursive doubling pattern is easy for the students to see. What is considerably more difficult for the students, despite their familiarity with the algebra of exponents and exponential functions from previous coursework, is to move from a recursive view of the function to an explicit form of a function that expresses the number of pennies on each square as a function of the number of the square. The task also included an optional exploration that asked students to determine on which square the height of the pennies would reach the top of their classroom, the top of a mountain, and the distance to moon. (See Appendix for the full text of the task.)