The Art of Problem Posing
By Stephen I. Brown and Marion I. Walter
Lawrence Erlbaum Associates
Department of Mathematics
California State University, Stanislaus
When I picked up The Art of Problem Posing by Stephen Brown and Marion Walter, my first thought was “Just what is problem posing, and what does it have to do with problem solving?”
Problem Posing is much more than a new approach to problem solving in the teaching of mathematics. Brown and Walter have integrated problem solving into a much broader solution strategy. They actively encourage students to re-examine the problem itself, in order to derive closely related and potentially insightful questions. They believe it is impossible to solve a problem without posing new problems in the very process of solving it (p. 2). The authors have taken a friendly, inviting tone, and a try-it-yourself attitude permeates their exposition.
This book would be a helpful guide for faculty in mathematics, science, and other disciplines involving related material, and as a text for mathematics education for students teaching at or above the junior high level. Anyone looking beyond the traditional classroom will find a very different approach to teaching problem solving. Problem Posing also contains many ideas on incorporating more written assignments into the curriculum. Although the problem solving strategies discussed in this book could be applied to almost any science course with minor modifications, the examples presented are strictly mathematical. The level of mathematics in Problem Posing should be comprehensible to all readers who are comfortable with high school level mathematics.
Brown and Walter begin by observing that in a traditional classroom teachers present problems, and students develop solutions. They argue that teachers should move away from this rigid teaching format and shift their students’ focus away from the equally rigid notions of the right way and the right answer. The authors state that the book represents an effort to understand three issues:
- What problem posing consists of and why it is important.
- What strategies exist for engaging in and improving problem posing.
- How problem posing relates to problem solving.
Chapter Two, Two Problem Posing Perspectives: Accepting and Challenging, begins with examples of problem posing. Readers are asked to respond to the question “What are some answers to ?” (p. 12)
Most people will immediately apply their own context to this question and provide answers such as x = 3, y = 4, and z = 5, but the authors astutely point out that this question is not clearly defined because it contains an equation. Perhaps irrational numbers or imaginary numbers may comprise acceptable solutions, but there is no way to tell from the question itself. The example is used to clarify and then modify the original equation to generate new questions to consider. This step is what Brown and Walter define as problem posing.
Chapter 3, The First Phase of Problem Posing: Accepting, explains the preliminary step of problem posing, “sticking to the given.” (p. 19) The danger with students brainstorming questions is that they may move the class discussion far away from the original problem. Brown and Walter assert that everyone must therefore agree on a set of assumptions in order to frame the discussion. One advantage of this problem posing strategy is that a much broader range of ideas will be discussed than in a traditional classroom. I personally feel this format would be especially helpful to students reluctant to speak out in class. These students would feel more at ease asking a question than answering one. An important point the authors stress is that students also need to reflect on a problem’s relevance, and I encounter this issue in my classes too. Students rarely consider the significance of a problem while seeking a solution, because they are exclusively focused on finding that solution.
Brown and Walter’s “What-If-Not” problem posing strategy is presented in Chapter 4, The Second Phase of Problem Posing: “What-If-Not.” This strategy has several levels. Level I is the listing of attributes, and the authors repeatedly emphasize that obvious, silly, and seemingly irrelevant attributes can often lead to deep mathematical questions.
Level II is the What-If-Not step, where alternatives to the listed attributes are given. The example of the Pythagorean Theorem is quite detailed. Ten attributes of the Pythagorean theorem are listed, and at least two alternatives for each attribute are given.
Level III is the posing of new questions inspired by these alternatives. A key idea from the examples is that problems in one area of mathematics can imply simpler problems in another. This is a useful idea to stress in a classroom where students feel more comfortable with geometry than algebra, or vice versa. Although numbering the levels of their problem posing strategy would seem to reflect linear thinking, Brown and Walter stress that their scheme is non-linear. They also suggest that teachers not adopt their scheme in a mechanical manner, but rather modify it to fit their own needs.
Chapter 5, The “What-If-Not Strategy in Action,” provides some examples of the What-If-Not strategy in its entirety. One example analyzes the famous Fibonacci Sequence. While many readers will be familiar with this sequence, Brown and Walter present some interesting properties that most readers have not seen before. I often felt that some problems in this book were not precisely stated, but that it was done intentionally. The authors feel that a precise statement of the problem would somewhat stifle the brainstorming of new ideas.
My only criticism of Problem Posing occurs in this chapter. When Brown and Walter have previously proposed questions, which extend well beyond their exposition, they have either explicitly stated the answer or provided the reader with footnoted references. But here they ask several intriguing questions about a modified Fibonacci sequence without any guidance on how to answer them. Moreover, the amount of mathematical explanation varies from example to example, some done in great detail, others left for the reader to verify.
The use of technology in problem posing is briefly addressed in this chapter. The authors cite an article by Eric Knuth, who uses Geometer’s Sketchpad computer software in conjunction with this problem solving strategy. Brown and Walter astutely warn that such software can only generate conjectures, not proofs.
The authors have presented numerous examples of their problem posing strategy in action, but clearly building an entire course around it represents a far greater pedagogical commitment. Brown and Walter have included a catalog description of a course they taught using their strategy, along with their grading policy. A key aspect of their course is that students are required to be both authors and editorial board members of a class mathematics journal. The authors warn that some students will feel uncomfortable criticizing the work of their peers, but Brown and Walter provide useful tips on explaining the value and purpose of constructive peer criticism to students. Brown and Walter have also included samples of the written work created by their students. They also provide details for an interesting variation of the editorial board experience, where the catalog description of the course and the format of the class journals are based on the Talmud. Either format would be an excellent way for students to develop skills in written mathematical communication.
This book draws heavily from Brown and Walter’s experiences teaching courses on problem solving at the Harvard Graduate School of Education in the 1960s. Subsequent variations of their course were taught at many other universities, and the insights gained were incorporated into this third edition. The authors interviewed Deborah Moore-Russo, who taught a course using the editorial board scheme at the University of Buffalo. The excerpts from the interview are quite illuminating, as they provide comments and feedback from an educator who has implemented many of their ideas into her classroom.
The final chapter includes a short discussion of how problem posing may mitigate many students’ math anxiety, which is an issue in every mathematics classroom. Brown and Walter point out that merely asking a question does not invite a judgment of its correctness, and they discuss how their approach reduces the differences between students’ abilities. Finally, they conclude with suggestions for further research inspired by their pedagogy.
I found many useful ideas in this book that I will definitely incorporate into my own teaching. Although a teacher may not adopt all the concepts in The Art of Problem Posing for their own classrooms, I believe they will nonetheless discover numerous ways to enhance their own teaching.
Posted November 17, 2006.
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