The Art of Problem Posing

Stephen I. Brown, Marion I. Walter

• Publisher: Routledge
• Year: 2012
• Print ISBN: 9780805802573
• eBook ISBN: 9781136464645
• Topic: Didattica
• Subtopic: Didattica generale

Updated and expanded, this second edition satisfies the same philosophical objective as the first -- to show the importance of problem posing. Although interest in mathematical problem solving increased during the past decade, problem posing remained relatively ignored. The Art of Problem Posing draws attention to this equally important act and is the innovator in the field.

Special features include:

* an exploration ofthe logical relationship between problem posing and problem solving

* a special chapter devoted to teaching problem posing as a separate course

* sketches, drawings, diagrams, and cartoons that illustrate the schemes proposed

* a special section on writing in mathematics

THE FAR SIDE COPYRIGHT 1988 UNIVERSAL PRESS SYNDICATE.

Reprinted with permission. All rights reserved.

1

Introduction

Where do problems come from, and what do we do with them once we have them? The impression we get in much of schooling is that they come from textbooks or from teachers, and that the obvious task of the student is to solve them. Schematically, we have the following model:

The purpose of this book is to encourage a shift of control from “others” to oneself in the posing of problems, and to suggest a broader conception of what can be done with problems as well.¹ Why, however, would anyone be interested in problem posing in the first place? A partial answer is that problem posing can help students to see a standard topic in a new light and provide them with a deeper understanding of it as well. It can also encourage the creation of new ideas derived from any given topic. Although our focus is on the field of mathematics, the strategies we discuss can be applied to activities as diverse as trying to create something humorous (as in The Far Side cartoon), attempting to understand the significance of the theory of evolution, or searching for the design of a new type of car bumper.

Have you ever thought, for example, of designing car bumpers that make use of liquid or that are magnitized, or shaped like a football, or capable of inflation upon impact? Or, have you thought of the possibility that they may be made of glass, the fragility of which might discourage people from driving recklessly or relying so heavily on the use of the automobile?

In addition to teaching explicit strategies for problem generation, there is an underlying attitude towards “coming to know” something that we would like to encourage. Coming to know something is not a “spectator sport,” though standard textbooks, especially in mathematics, and traditional modes of instruction may give that impression. To say, rather, that coming to know something is a participant sport is to commit ourselves to a point of view requiring that we operate on and even modify the things we are trying to understand.² This attitude is central to the problem posing activities we shall develop in this book.

Our strategy for presenting ideas is generally an inductive one. Whenever possible, we attempt first to expose some problem posing issue through an activity that gets at it in an implicit and playful manner. After there has been some immersion in an activity we turn towards a reflection on its significance. We believe that it is necessary first to get “caught up in” (and sometimes even “caught” in the sense of “trapped by”) the activity in order to appreciate what it is we are after. Such a point of view requires both patience and also an inclination to recover quickly from any embarrassment that results from being “caught.” We believe the rewards will be worth the effort.

One way of gaining an appreciation for the importance of problem posing is to relate it to problem solving—a topic that has gained widespread acceptance (or rejuvenation, depending on your point of view). Problem posing is deeply embedded in the activity of problem solving in two very different ways. First of all, it is impossible to solve any novel problem without first reconstructing the task by posing new problem(s) in the very process of solving. Asking questions like, “What is this problem really saying?” or “What if I shift my focus from what seems to be an obvious component of this problem to a part that seems remote?” propels us to generate new problems in an effort to “crack” the original one.

Secondly, it is frequently the case that after we have supposedly solved a problem, we do not fully understand the significance of what we have done, unless we begin to generate and try to analyze a completely new set of problems. You have probably had the experience of solving some problem (perhaps of a practical, non-mathematical nature) only to remark, “That was very clever, but what have I really done?” These matters are discussed with examples in chapter 6.

Often our formal education suppresses the relationship between the asking of questions and the coming up with answers. In a book which calls for a new perception of education, D. Bob Gowin comments:

Recently a teacher was overheard to announce: “When I want your questions, I'll give them to you.” … Much of school practice consists of giving definite, almost concrete answers. Perhaps boredom sets in as answers are given to questions that were never asked.³

More than boredom is at stake, however, when we are robbed of the opportunity of asking questions. The asking of questions or the posing of problems is a much more significant task than we are usually led to believe. The point is made rather poignantly in the story of Gertrude Stein's response to Alice B. Toklas, on Gertrude's death bed. Alice, awaiting Gertrude's legacy of wisdom, asked, “The answers Gertrude, what are the answers?”—Whereupon Gertrude allegedly responded, “The questions, what are the questions?”

The centrality of problem posing or question asking is picked up by Stephen Toulmin in his effort to understand how disciplines are subdivided within the sciences. What distinguishes atomic physics from molecular biology, for example? He points out that our first inclination to look for differences in the specific content is mistaken, for specific theories and concepts are transitory and certainly change over time. On the other hand, Toulmin comments:

If we mark sciences off from one another…by their respective “domains,” even these domains have to be identified not by the types of objects with which they deal, but rather by the questions which arise about them.… Any particular type of object will fall in the domain of (say) “biochemistry,” only in so far as it is a topic for correspondingly “biochemical” questions.⁴

An even deeper appreciation for the role of problem generation in literature is expressed by Mr. Lurie to his son, in Chaim Potok's novel In the Beginning:

I want to tell you something my brother David, may he rest in peace, once said to me. He said it is as important to learn the important questions as it is the important answers. It is especially important to learn the questions to which there may not be good answers.⁵

Indeed, we need to find out why some questions may not have good answers. For example, the questions might seem foolish or meaningless; or it might be that the questions are fundamental human questions that each of us might fight a lifetime to try to understand; it may be that they are unanswerable questions because they are undecidable; it might also be, however, that our perspective on a problem is too rigid and we are blinded in our ability to see how a question might bear on a situation.

The history of every discipline—including mathematics—lends credence to the belief not only that it may be hard to distinguish good questions from bad ones in some absolute sense, but that very talented people may not be capable of seeing the difference even for a period of centuries. For a very long time, people tried to prove Euclid's fifth postulate:

Through a given external point, there is exactly one line parallel to a given line.

It was only during the past century that we began to realize that the difficulty in answering the question lay in the assumptions behind the question itself. The implicit question was:

How can you prove the parallel postulate from the other postulates or axioms?

It took hundreds of years to appreciate that the “how” was an unrecognized monster. If you delete the “how,” the question is answerable (in the negative it turns out); if you do not do so, the question destroys itself as is the case with the question, “When did you stop beating your spouse”?

So far, we have tried to point out some intimate connections between the asking and answering of questions, and between the posing and solving of problems. There is a sense, however, in which problem generation is an important activity to pursue, even in the absence of uncovering “the right” solution, question or questions. In fact, there are many situations for which the concept of a right question is inappropriate. Imagine being given a situation in which no problem has been posed at all. A reasonable task might be to generate a problem or to ask a question, not for the purpose of solving the original situation (a linguistically peculiar formulation), but in order to create a problem that derives from the situation.

Suppose, for example, that you are given a sugar cube or the statement, “A number has exactly three factors.” Strictly speaking, there is no problem in either case. Yet there is an infinite number of questions we can ask about either of the situations—some more meaningful than others, some more significant than others. However, it is rarely possible to tell in the absence of considerable reflection what questions or problems are meaningful or significant. We hope to persuade you in much of what follows, that things like significance and meaningfulness are as much a function of the ingenuity and the playfulness we bring to a situation as they are a function of the questions we ask. Frequently, even a slight turn of phrase will transform a situation that appears dull into one that “glitters.”

There is another reason for asking interesting questions about a situation besides wanting to engage in creative activity or searching for answers: In looking at what happens when we pose new questions concerning an object—like a sugar cube or a statement that a number has exactly three different factors—we begin to “see” the object as we had never seen it before. We gain a deeper understanding of what constitutes a sugar cube or what it means for a number to be divisible by three factors. We shall further discuss issues of meaning and significance in chapter 3.

Sometimes the questions we ask about a phenomenon or a situation keep our perceptions of it intact. Sometimes, however, we end up drastically revising what we start with in the modifications we make. It is ironic that we sometimes come to “see” what is staring us in the face only after we “destroy” it in some way—at least mentally. The strategies behind such radical reconstruction of an object are discussed in chapters 4 and 5.

It is no great secret that many people have a considerable fear of mathematics or at least a wish to establish a healthy distance from it. There are many reasons for this attitude, some of which derive from an education which focuses on “right” answers. People tend to view a situation or even a problem as something that is given and that must be responded to in a small number of ways. Frequently people fear that they will be stuck or will not be able to come up with what they perceive to be the right way of doing things.

Problem posing, however, can create a totally new orientation towards the issue of who is in charge and what has to be learned. Given a situation in which one is asked to generate problems or ask questions—in which it is even permissible to modify the original thing—there is no right question to ask at all. Instead, there are an infinite number of questions and/or modifications and, as we implied earlier, even they cannot easily be ranked in an a priori way.

Thus we can break the “right way” syndrome by engaging in problem generation. In addition, we may very well have the beginnings of a mechanism for confronting the rather widespread feelings of mathematical anxiety.

This book then represents an effort on our part to try to understand:

1. What problem posing consists of and why it is important.

2. What strategies exist for engaging in and improving problem posing.

3. How problem posing relates to problem solving.

While problem posing is a necessary ingredient of problem solving, it takes years for an individual—and perhaps centuries for the species—to gain the wisdom and courage to do both of these well. No single book can provide a panacea for improving problem posing and problem solving. However this book offers a first step for those who would like to learn to enhance their inclination to pose problems. While this book does touch upon problem solving, it does so primarily as it relates to problem posing.

AUDIENCE

The Art of Problem Posing is written for a wide audience. It is intended for college math students, interested laypersons, present and future teachers of mathematics in middle school, in secondary school, and in higher levels of education. It also has implications for curriculum writers and for those who wish to do research on the power of problem posing and its relationship to a host of variables ranging from fear of mathematics to new strategies for teaching mathematics. We hope that it suggests directions for educators in fields other than mathematics as...

Table of contents

• Front Cover
• Half Title
• Title Page
• Copyright
• Contents
• Acknowledgments: Second Edition
• 1  Introduction
• 2  Two Problem Posing Perspectives:
• 3  The First Phase of Problem Posing: Accepting
• 4  The Second Phase of Problem Posing: “What-If-Not”
• 5  The “What-If-Not” Strategy in Action
• 6  Some Natural Links Between Problem Posing and Problem Solving
• 7  In the Classroom: Student as Author and Critic
• 8  Conclusion
• Bibliography
• Author Index
• Subject Index