Teaching Problems and the Problems of Teaching
reviewed by Cindy Hmelo-Silver - 2003
Teaching is a dynamic and complex cognitive and social activity. Even more so when teaching by having students learn through solving problems. Teaching Problems and the Problems of Teaching by Magdalene Lampert addresses this complexity and the problems that a teacher faces in dealing with what occurs in such a classroom. Her goal in this book is to inform debates about how to improve teaching through an adequate understanding of the problems of practice (Bereiter & Scardamalia, 1989) . She accomplished this by making the problems of practice explicit. In this book, Lampert brings the reader into her thinking as she embraces the problems and complexities of practice head-on. This book reports on Lampert’s reflective analysis of her teaching over the course of a year through analyses of classroom transcripts, student artifacts, and her teaching journals.
The book starts out by explaining the rationale for using problems with teaching—an approach consistent with current reforms in mathematics education. Her goal for the students is to help them learn the mathematical concepts as well as the practice of mathematics, and she argues this is best accomplished by teaching with problems. Lampert defines teaching as a practice of structuring study activities in a problem space defined by relationships among the teacher, the learners, and the content. Studying is defined differently when discussing teaching and learning in problem-solving contexts. Studying here refers to any practices that students engage in for learning and includes activities such as reading carefully, inquiring, and examining closely, borrowing from Bereiter and Scardamalia’s (1989) definition of intentional learning. This definition is important for the student-centered classroom that is described in the book. The role of the teacher in this environment is thus to provide opportunities for and to support students’ intentional learning activities. Teaching events are considered as contextualized patterns of action occurring over time, with social groups, and particular content. Lampert’s research program involves using her teaching to develop a model that can take into account the overlapping temporal, social, and intellectual complexities of the problems of teaching practice. To do this, she has collected voluminous records of what occurred in her elementary school mathematics classroom for an entire year. Throughout the book, she uses the metaphor of a photographic lens to examine her teaching practice—sometimes zooming in on the relationship of her teaching with particular children and content on a particular day and at other times, zooming out to examine how concepts unfold over time as the children are learning through problem solving.
Many of the chapters in the book use lenses of different focal length to examine the problems that Lampert faces in her teaching. She focuses on analyses of her actions in addressing common problems of teaching from beginning the school year to ending it, from dealing with individual student learning to covering content in ways that are connected across lessons as well as trying to help students understand that they can be “people who study in school.” The analyses are supported with rich examples of student dialogue, work, and excerpts from Lampert’s teaching journals.
The book presents its first analysis with the problems involved in starting the school year. Chapter 4 discusses the problems in creating a classroom culture in which students are encouraged to make their reasoning public and available for discussion. This approach honors student thinking and takes advantage of the resource that students’ thinking provides for teaching. This required establishing several kinds of routines that would afford support for students' independent and collaborative activities and for introducing students to the notion that they were responsible for judging their own work and the contributions of others. To accomplish this, Lampert thought a great deal about how to establish discourse structures that would have students judging the mathematical legitimacy of their work. She established rules for small group interaction that encouraged both groups and individuals to have responsibility for their own thinking. She also structured the students’ notebooks as a way for her to communicate with individual students. Both the notebooks and classroom discourse used similar language to talk about mathematical reasoning. Lampert’s classes begin by posting the problem of the day. There is a parallel structure to the students’ notebooks, which begin by writing this problem. Students are encouraged to experiment, conjecture, and reason about the problem, and the use of these particular terms are deliberate. These linguistic routines help scaffold the students’ mathematical inquiry. The notebooks are places for students to explain their reasoning about the problem. They provide a window into the students thinking for the teacher as she reads and responds to the notebooks. As well, they provide information about the intellectual resources and misunderstanding that instruction can build upon. Thus the notebook becomes a shared referent for classroom mathematics discussions. The teacher responds to all student ideas in this classroom by discussing the reasoning that students used to construct their solutions. In this chapter, as in most, Lampert summarizes the classroom dilemmas that she must address—here those are the problems involved in changing the classroom culture and in identifying the affordances and constraints that the physical, temporal, technological, and material resources of the classroom provide.
In Chapter 5, the issues in preparing to teach a lesson are laid out. Preparation for a student-centered lesson is quite different from preparing for a more teacher-centered lesson. One must prepare for a general terrain that might be covered and develop problems that allow the students to explore the terrain. She zooms out again later to consider these issues as the book discusses how to teach to connect lessons and cover the whole curriculum in a way that preserves relationships among content. In this chapter, Lampert is zoomed in on the problems she faces in teaching a single lesson—these involve issues in preparing the lesson, structuring and monitoring the students’ independent work, and leading the whole class discussion. In preparing the lesson, she wrestles with how she can connect particular students with particular mathematical content, and this involves trying to find a problem that will locate the students’ thinking in the appropriate mathematical space. Once a problem is chosen, there are further problems that teachers must address—the symbols and representations to use, particular number combinations that might raise interesting issues, and how to make connections to earlier lessons. Choosing an appropriate problem also means knowing her students as learners, a theme that Lampert returns to frequently. Once a problem is chosen, she needs to anticipate some of the reasoning strategies that students might use and whether it will be both engaging and appropriate. Her thinking about choosing problems is very much situated in the fifth grade class that she is teaching. In other chapters she zooms in on individual students working independently and collaboratively as well as discussing the problems in teaching the whole class at once.
Chapter 7 focuses on teaching the whole class and includes the analysis of a 10 min portion of transcript that is included in the appendix. This illustrates how she can build on students’ conceptions to help them connect symbols, words, and mathematical ideas. As well, she shows how representations can be used to help students bridge their everyday and mathematical thinking. The importance of representations is discussed in several chapters, and representations are often used to provide evidence of the depth of student thinking as well as to provide a sense of coherence across problems and topics. In this portion of transcript, Lampert also illustrates how students participate in public sense-making that analyzes how they have come up with different solutions to problems rather than focusing on right or wrong answers. As in many of the chapters, several problems in leading the whole class discussion are identified—creating visual representations, deciding who to call on, simultaneously teaching individual students and engaging the whole class, keeping a discussion on track while honoring individual student contributions, and maintaining the agenda. The discussions of connecting content and covering the curriculum are important reading for anyone thinking about how to teach in a student-centered manner. Zooming out to these larger issues makes it clear how important it is to consider both levels, as particular content might not be covered when expected. However, when curriculum is considered as a system of intertwined problems and content, then content is covered from multiple angles. Here the teacher’s job, Lampert argues, is to help the students see the connections among the different parts of the mathematical terrain.
Chapter 10 “Teaching students to be people who study in school” (p. 265) is particularly interesting. This chapter focuses on students' need to develop a sense of themselves as learners for whom it is socially acceptable to be engaged in intellectual pursuits. Case studies illustrate how she supports this development. The first case is Richard. Richard was working up to being able to take a risk by making his mathematical reasoning public as he interpreted the problem of the day. Lampert used three strategies to help build Richard’s confidence and trust. First, she responded to his reasoning in his notebook. Second, she rearranged his physical location in relation to other students, and third noting his progress in communications with his parents. Richard was regrouped with other students so that he was in a position in which he needed to be the one to explain things to others in his group.
This book ends with a discussion of an elaborated model of teaching practice. This model focuses on the dynamic relationships among teacher, students, and content and on the work of teaching as one of structuring these changing relations. Although this review provides only a small flavor of the richness of Lampert’s analyses, her work should prove invaluable to those trying to understand student-centered, problem-centered teaching practice. By using the metaphor of zooming in and out, and focusing on the problems that are involved in practice at each level, Lampert makes an important contribution to our understanding of new models of learning and teaching. If there is any limitation in the book, it is that it is very much context bound. One wonders how these ideas would translate to other content domains. But this rich description of the context is also one of the book’s strength; it is because of her deep knowledge of the domain that Lampert can be so reflective on the nature of her teaching, her students, and the subject matter that her students' study.
Bereiter, C., & Scardamalia, M. (1989). Intentional
learning as a goal of instruction. Pp. 361-392 in L.
B. Resnick (Ed.), Knowing learning, and instruction:
Essays in honor of Robert Glaser. Hillsdale, NJ:
Bereiter, C., & Scardamalia, M. (1989). Intentional learning as a goal of instruction. Pp. 361-392 in L. B. Resnick (Ed.), Knowing learning, and instruction: Essays in honor of Robert Glaser. Hillsdale, NJ: Erlbaum.