Mathematics Teacher Noticing

Seeing Through Teachers' Eyes

Miriam Sherin, Vicki Jacobs, Randy Philipp, Miriam Sherin, Vicki Jacobs, Randy Philipp

• Publisher: Routledge
• Year: 2011
• eBook ISBN: 9781136838255
• Edition: 1
• Topic: Education
• Subtopic: Education Teaching Methods

Mathematics Teacher Noticing is the first book to examine research on the particular type of noticing done by teachers---how teachers pay attention to and make sense of what happens in the complexity of instructional situations. In the midst of all that is happening in a classroom, where do mathematics teachers look, what do they see, and what sense do they make of it? This groundbreaking collection begins with an overview of the construct of noticing and the various historical, theoretical, and methodological perspectives on teacher noticing. It then focuses on studies of mathematics teacher noticing in the context of teaching and learning and concludes by suggesting links to other constructs integral to teaching. By collecting the work of leaders in the field in one volume, the editors present the current state of research and provide ideas for how future work could further the field.

SECTION III
Studies of Mathematics Teacher Noticing

7
DECIDING HOW TO RESPOND ON THE BASIS OF CHILDREN’S UNDERSTANDINGS ¹

Victoria R. Jacobs, Lisa L. C. Lamb, Randolph A. Philipp, and Bonnie P. Schappelle

The unraveling of the math lesson is a continuously reinvented process, with dozens of decision points at which the teacher moves on to the next activity format, which has only just emerged as a likely follow-on exercise, or switches to another exercise as a result of the drift of pupils’ oral response, the level of pupils’ task engagement, the time remaining until recess or the end of the period, or more likely, all these factors. This continuous readjustment results from what Lévi Strauss (1962) has called, felicitously, “engaging in a dialogue with the situation” as that situation unfolds. To tinker well here seems to depend on how quickly and accurately the teacher can read the situation.

Huberman (1993, pp. 15–16)

We appreciate Huberman’s depiction of teaching as a fluid process requiring extensive and critical decision making on the basis of reading a situation in a specific moment (see also Franke, Kazemi, & Battey, 2007; Lampert, 2001; McDonald, 1992; Schoenfeld, 1998; Wells, 1999). Although the craft of teaching involves much more, we have chosen to focus on understanding this in-the-moment decision making both because of the centrality of this skill in effective teaching and because this expertise is so challenging to develop. In mathematics education, a particular type of in-the-moment instructional decision making has been emphasized—decision making in which children’s thinking is central.

“Sizing up students’ ideas and responding” has been identified as one of the core activities of teaching (Ball, Lubienski, & Mewborn, 2001, p. 453), and instruction that builds on children’s mathematical thinking has been endorsed in many reform documents (National Council of Teachers of Mathematics [NCTM], 2000; National Research Council [NRC], 2001). This focus has been informed by the extensive and growing research base on children’s mathematical thinking and development (Lester, 2007; NRC, 2001), and instruction that builds on children’s ways of thinking has been linked to rich instructional environments and documented gains in student achievement (Bobis et al., 2005; Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Jacobs, Franke, Carpenter, Levi, & Battey, 2007; Sowder, 2007; Wilson & Berne, 1999). In addition, focusing on the thinking of children can provide a constant source of professional development for teachers throughout their careers because they can continue to learn from their students’ thinking on a daily basis, even after formal professional development support ends (Franke, Carpenter, Levi, & Fennema, 2001).

Despite these documented benefits for both students and teachers, creating instruction that builds on children’s thinking has proven challenging. In this chapter, we use the construct of noticing to begin to unpack this practice and, in particular, the in-the-moment decision making that occurs, many times a day, when a child shares a verbal or written strategy explanation and the teacher needs to respond.

Noticing

For many years, psychologists have studied how individuals notice or attend to stimuli in their environments, and, more recently, researchers have been describing the distinct patterns of noticing particular to professions (see, e.g., Goodwin, 1994; Mason, 2002; Stevens & Hall, 1998). Those studying expert/novice differences have also acknowledged these professional patterns of noticing by confirming that experts in a field are more likely than novices to focus on and remember noteworthy aspects of complex situations that are relevant to future decision making (for a summary, see NRC, 2000). Mathematics educators have shown interest in the noticing construct as a way to understand how teachers make sense of complex classrooms in which attending and responding to everything is impossible, and they have defined noticing in a multitude of ways (as reflected in the chapters in this volume). Some have addressed solely where prospective and practicing teachers focus their attention (Star, Lynch, & Perova, this volume, Chapter 8; Star & Strickland, 2007), whereas others have also considered how teachers reason about what they see (Sherin 2007; Sherin & Han, 2004; van Es & Sherin, 2008), including their abilities to reflect on teaching strategies and consider alternatives (Santagata, this volume, chapter 10; Santagata, Zannoni, & Stigler, 2007).

This growing body of work on mathematics teacher noticing has underscored the idea that teachers see classrooms through different lenses and that understanding these lenses can be helpful in scaffolding teachers’ abilities to notice in particular ways. We applaud these researchers’ attention to the important role that noticing plays in teaching, and we build on their work by selecting a particular focus for noticing—children’s mathematical thinking—and a particular slice of teaching—the hidden practice of in-the-moment decision making when teachers must respond to children’s verbal or written strategy explanations. This type of in-the-moment decision making is in contrast to the long-term decision making (or planning) that teachers do before or after school when children are not present. Specifically, we want to understand not only how teachers detect children’s ideas that are embedded in comments, questions, notations, and actions but also how teachers make sense of what they observe in meaningful ways and use it in deciding how to respond. Thus, we are less interested in identifying the variety of what teachers notice and more interested in how and the extent to which teachers notice children’s mathematical thinking. As such, we found merit in investigating a specialized type of mathematics teacher noticing that we call professional noticing of children’s mathematical thinking. We conceptualize this expertise as a set of three interrelated skills: (a) attending to children’s strategies, (b) interpreting children’s understandings, and (c) deciding how to respond on the basis of children’s understandings (Jacobs, Lamb, & Philipp, 2010).

In this chapter, we have chosen to focus on the third component skill, deciding how to respond. Note that this skill reflects intended responding, not the actual execution of the response. We recognize that intended responding is not necessarily executed as planned, but we argue that teachers are not likely to respond on the basis of children’s understandings without purposeful intention to do so. We are not looking for teachers to propose any particular responses (that is, there is no checklist of desired moves) but are instead interested in whether their decision making draws on and is consistent with the specifics of children’s thinking in a given situation and the research on children’s mathematical development (see also Jacobs & Philipp, 2010).

Other researchers have also included issues related to responding in their conceptualizations of noticing (see, e.g., Erickson, this volume, chapter 2; Santagata, this volume, chapter 10; Santagata et al., 2007), but we recognize that many may view decision making about how to respond as something that occurs after noticing. Both perspectives have advantages, but we argue for its inclusion as part of noticing given that deciding how to respond is both temporally and conceptually linked to the other two component skills of professional noticing of children’s mathematical thinking (attending to children’s strategies and interpreting children’s understandings) during teachers’ in-the-moment decision making. First, when a child offers a verbal or written strategy explanation, implementation of the three component skills must occur almost simultaneously—as if constituting a single, integrated teaching move—before the teacher responds. Second, expertise in deciding how to respond is nested within expertise in attending to children’s strategies and interpreting children’s understandings. In other words, teachers can decide how to respond on the basis of children’s understandings only if they also have attended to children’s strategies and interpreted the understandings reflected in those strategies. Thus, these three component skills are inextricably intertwined. Finally, the work of teaching orients teachers to constantly consider their next moves (Schoenfeld, 1998; Sherin, 2001); thus, the skills of attending to children’s strategies and interpreting children’s understandings are not ends in themselves but are instead starting points for making effective instructional responses. By integrating teachers’ reasoning about how to respond into the construct of professional noticing of children’s mathematical thinking, we ensure that this ultimate goal of purposeful responding remains visible.

In this chapter, we characterize the component skill of deciding how to respond by investigating the expertise of four groups of participants with different amounts of experience with children’s mathematical thinking. We also explore the specific connection between participants’ expertise in deciding how to respond and their expertise in attending to children’s strategies. Others have underscored the symbiotic relationship between the focus of attention and subsequent decision making. For example, Erickson (this volume, chapter 2) has argued that the selective attention of teachers is opportunistic in that they judiciously direct their attention to what is necessary to take action. Similarly, Sassi (2001), drawing on Aristotle’s notion of practical judgment, argued that “learning to deliberate about the actions one should take is inseparable from cultivating perception of the salient features of one’s situation” (p. 15). Thus we provide evidence for not only the developmental patterns of these two skills but also their connection.

Methods

The data were drawn from a cross-sectional study entitled “Studying Teachers’ Evolving Perspectives” (STEP), in which we collected data on the professional noticing of teachers engaged in sustained professional development focused on children’s mathematical thinking.

Participants

The 131 participants included three groups of practicing K–3 teachers and one group of prospective teachers who were just beginning their studies to become elementary school teachers (see Table 7.1).

Participant groups differed in their experience with children’s mathematical thinking. Specifically, Prospective Teachers, by virtue of their lack of teaching experience and professional development, had the least experience with children’s thinking, followed by Initial Participants, who had teaching experience but no sustained professional development, and then by Advancing Participants, who had teaching experience and 2 years of professional development. Emerging T...

Table of contents

• Studies in Mathematical Thinking and Learning
• CONTENTS
• FIGURES
• TABLES
• LIST OF CONTRIBUTORS
• Foreword
• PREFACE
• ACKNOWLEDGMENTS
• SECTION I Introduction
• SECTION II Foundations of Teacher Noticing
• SECTION III Studies of Mathematics Teacher Noticing
• SECTION IV Conclusion
• AUTHOR INDEX
• SUBJECT INDEX