Harvard Educational Review
  1. Teaching Problems and the Problems of Teaching

    By Magdalene Lampert.

    New Haven, CT: Yale University Press, 2001. 496 pp. $35.00.

    A cacophony continues about standards and accountability in states and districts, but the conversation about teaching practice continues to be muffled. Magdalene Lampert’s Teaching Problems and the Problems of Teaching brings a welcome perspective to the discourse. As an elementary school mathematics teacher and an educational researcher, she is uniquely positioned to describe her everyday work in the "teaching of problems" and the "problems of teaching" that she encounters in this work. Lampert provides a valuable study of the teaching practice of using complex mathematical problems to generate conceptual understanding in her fifth-grade mathematics class. The book benefits from Lampert’s professional hybridization; she brings obvious skill as both a teacher and researcher to this study. Her book offers a close examination of one teacher, herself, with one group of students studying one subject, math, over the course of one academic year, providing an understanding of the implications of teaching with problems.

    Lampert observes that coordinating the teacher’s actions with the students’ actions is the "essence of [teaching] practice" (p. 7). However, she points out that the implications of teaching with problems for this "coordination" are rarely documented. Lampert undertakes this documentation beautifully, demonstrating how incredibly complicated teaching is through vivid, detailed accounts of lessons, complete with transcripts of teacher-student interactions, figures used during the lessons, a replica of the blackboard during various stages of the lesson, and excerpts from her own teaching journal. Those familiar with Lampert’s writing have come to expect her to take on such questions as, "How do teachers manage to teach?" In her introduction, Lampert explains that she wrote the book to "inform debates about [reform] issues with a more adequate understanding of the problems in practice that teachers need to manage in order to teach productively" (p. 8). Her book provides one of the most comprehensive and thoughtful descriptions of teachers’ work in the field.

    Lampert begins with the question, "Understanding Teaching: Why Is It So Hard?" (p. 1), and sets out to understand the problems that an individual teacher must address and to document the ways a teacher manages to teach, given the complicated nature of the enterprise. She explains that in order to study teaching in a way that acknowledges this complexity, researchers need an approach that captures several levels of this practice, such as teaching a lesson, a unit of study, and teaching individual students or groups. Next, she describes what it means to teach with problems. She concludes,

    Learning in my class was a matter of becoming convinced that your strategy and your answer are mathematically legitimate. . . . Studying mathematics this way involves my students in finding out what kind of activity mathematics is; it provides them an opportunity to learn and use the concepts, tools, and procedures that the field has developed. (p. 6)

    In chapter two, Lampert briefly describes her school, classroom, students, and partnership with her co-teacher before introducing the cornerstone of her practice, the "Problem of the Day," which catalyzes each day’s mathematical lesson. This chapter provides a case study of one lesson and introduces the reader to the book’s format; Lampert uses at least one "problem of the day" to illustrate her points so that the structure of the book replicates the structure of her teaching. Lampert introduces the problem. Students work individually, then discuss. To help readers understand her pedagogy, Lampert "zooms in" by featuring students’ responses to her questions during discussion. She then analyzes the challenge she has undertaken: to find a representation of the "multiple levels of teaching action as they occur in different social relationships over time to accomplish multiple goals simultaneously" (pp. 27–28).

    In the third chapter, Lampert explains "why [she] wrote this book and how" (p. 28). She definitively states, "Without a professional discourse about classroom practice, education is in a weak position to improve itself" (p. 30). She provides a model of teaching practice, and elaborates upon the model by describing teaching in terms of "time . . . relationships with social groups . . . connections in content . . . [and] overlapping complexities" (pp. 37-38). She continues by explaining her methodology and data collection strategies. Researchers will find this section quite interesting; classroom teachers will envy the resources and possibilities for copious documentation of practice and student learning.

    Each of the remaining chapters addresses a specific issue of practice. The topics are well known, but it is fascinating how Lampert delves into each one. For example, she begins the chapter about teaching to cover the curriculum by outlining her district’s curricular expectations and the textbook she is given. She provides her curriculum topics chart, a list of curriculum topics ordered by a "range of student activity," and an annual calendar of topics. Then Lampert "uses a wide-angle lens" (p. 220) to give readers some theory about mathematical learning; for example, she describes the "theory of conceptual fields" (p. 221), which holds that students must see relationships among concepts and topics in several types of problems. In addition, Lampert offers specific excerpts from a lesson, explaining why she has designed the problem in this manner. She then refers to her teaching journal and includes the transcript of her lesson with diagrams from the blackboard with accompanying teacher and student text from the lesson. At the end, she offers her "map of the mathematical terrain" (p. 256) for this unit on time-speed-distance, an analysis of the curricular topics addressed, and students’ mathematical modeling.

    The most interesting aspect of this chapter is Lampert’s discussion of the "invisible work" in covering the curriculum: the deliberate, careful work of making connections among the topics that arise within the teaching of problems (p. 259). Lampert says that because teaching with problems is not a linear, topic-by-topic approach, visitors might miss altogether her construction of lessons that allow students to examine "different, but related" topics (p. 260). She emphasizes that her teaching occurs in several "nested time frames" (p. 263); it is the strength of the book that Lampert provides the reader with data, reflections, and student experiences across those days, weeks, and months.

    The final chapter presents an "elaborated model of teaching practice" (pp. 423–448). Lampert builds on the familiar triangle model of teaching practice — the relationships between the teacher, the student, and the content — to address the problems of teaching a whole class over an extended time; the importance of social relationships and their influence of the "social complexities" (p. 426) on practice; and the ways in which time or "temporal complexities" (p. 427) develop and provide a historical context and continuity of events within which a teacher teaches. Lampert’s model portrays teaching practice as an action of "zooming in and out temporally and socially" (p. 430). Lampert illuminates the ways in which teaching mathematics with problems is complicated by the way that classrooms are organized, including the fact that the triangle-model only includes one student and one teacher (p. 424). She also contends that this model assumes classroom interactions are static, rather than social relationships that have a history and a future (p. 425). Next, she includes the influences of the nature of the subject matter on teaching (p. 434). Lampert concludes with a hopeful observation:

    As the relationships in the work of teaching are made more explicit in each elaboration on the model, the problems a teacher faces in practice and the resources available to solve those problems both increase multiplicatively. . . . I hope this book has gone some distance toward showing that such teaching is possible in school classrooms, but also how it is possible to manage the myriad relationships involved in doing it. (p. 448)

    In documenting the teaching of problems and the problems of teaching, Lampert has catapulted the professional discourse of teaching into a place where teaching is described across dimensions of time and relationships, from the minute detail of a teacher-to-student exchange, to the teacher’s personal reflections, to the complexities of whole-class teaching, to the challenges of teaching content across these dimensions. Researchers will marvel at the studious ethnography that has yielded so much detailed data about the events of her classroom. Practitioners will appreciate how this book has captured the teacher’s infinite, split-second decisions and the richness of the exchanges between teacher and students and among students. Parents will wish every teacher had the inclination, time, and resources to document student learning with such precision. All will enjoy looking into a teacher’s mind when she is preparing, teaching, and reflecting upon her practice, and the rare delight of "listening in" as students puzzle over mathematical problems and engage in the very hard work of learning.

    H.G.P.
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    Teaching Problems and the Problems of Teaching
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