Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12

A Resource for the Mathematics Teacher

Alfred S. Posamentier, Stephen Krulik

• Publisher: Corwin
• Year: 2008
• Print ISBN: 9781412959704, 9781412959698
• eBook ISBN: 9781452295572
• Edition: 2
• Topic: Education
• Subtopic: Teaching Mathematics

"The authors have provided a unique, strategy-focused resource supported by a wealth of engaging examples that mathematics teachers can readily use to help students develop a more purposeful, systematic, and successful approach to problem solving."
—Howard W. Smith, Superintendent
Public Schools of the Tarrytowns, Sleepy Hollow, NY

"Helps both new and veteran teachers better understand the nature of problem solving as a critical mathematics process. The authors present in very simple terms the strategies that are the backbone of mathematics instruction. This indispensable material is useful at all levels, from basic stages to advanced student work to the development of top problem solvers."
—Daniel Jaye, Principal
Bergen County Academies, Hackensack, NJ

Help students become skilled and confident problem solvers!

Demonstrating there is always more than one approach to solving a problem, well-known authors and educators Alfred S. Posamentier and Stephen Krulik present ten basic strategies that are effective for finding solutions to a wide range of mathematics problems. These tried-and-true methods—including working backwards, finding a pattern, adopting a different point of view, solving a simpler analogous problem, and making a visual representation—make problem solving easier, neater, and more understandable for students as well as teachers.
Providing numerous sample problems that illustrate how mathematics teachers and specialists can incorporate these techniques into their mathematics curriculum, this updated edition also includes:

• A variety of new problems that show how to use the strategies
• References to current NCTM standards
• Solutions to the problems in each chapter
• Extensive discussions of the empowering strategies used to solve sample problems

The second edition of Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6–12 helps teachers develop students? creative problem-solving skills for success in and out of school.

1

Introduction to
Problem-Solving Strategies

Before we can discuss what problem solving is, we must first come to grips with what is meant by a problem. In essence, a problem is a situation that confronts a person, that requires resolution, and for which the path to the solution is not immediately known. In everyday life, a problem can manifest itself as anything from a simple personal problem, such as the best strategy for crossing the street (usually done without much “thinking”), to a more complex problem, such as how to assemble a new bicycle. Of course, crossing the street may not be a simple problem in some situations. For example, Americans become radically aware of what is usually a subconscious behavior pattern while visiting a country such as England, where their usual strategy for safely crossing the street just will not work. The reverse is also true; the British experience similar feelings when visiting the European continent, where traffic is oriented differently than that in Britain. These everyday situations are usually resolved “subconsciously,” without our taking formal note of the procedures by which we found the solution. A consciousness of everyday problem-solving methods and strategies usually becomes more evident when we travel outside of our usual cultural surroundings. There the usual way of life and habitual behaviors may not fit or may not work. We may have to consciously adapt other methods to achieve our goals.

Much of what we do is based on our prior experiences. As a result, the level of sophistication with which we attack these problems will vary with the individual. Whether the problems we face in everyday life involve selecting a daily wardrobe, relating to friends or acquaintances, or dealing with professional issues or personal finances, we pretty much function automatically, without considering the method or strategy that best suits the situation. We go about addressing life’s challenges with an algorithmic-like approach and can easily become a bit frustrated if that approach suddenly doesn’t fit. In these situations, we are required to find a solution to the problem. That is, we must search our previous experiences to find a way we solved an analogous problem in the past. We could also reach into our bag of problem-solving tools and see what works.

When students encounter problems in their everyday school lives, their approach is not much different. They tend to tackle problems based on their previous experiences. These experiences can range from recognizing a “problem” as very similar to one previously solved to taking on a homework exercise similar to exercises presented in class that day. The student is not doing any problem solving—rather, he or she is merely mimicking (or practicing) the earlier encountered situations. This is the behavior seen in a vast majority of classrooms. In a certain sense, repetition of a “skill” is useful in attaining the skill. This can also hold true for attaining problem-solving skills. Hence, we provide ample examples to practice the strategy applications in a variety of contexts.

This sort of approach to dealing with what are often seen as artificial situations, created especially for the mathematics class, does not directly address the idea of problem solving as a process to be studied for its own sake, and not merely as a facilitator. People do not solve “age problems,” “motion problems,” “mixture problems,” and so on in their real lives. Historically, we always have considered the study of mathematics topically. Without a conscious effort by educators, this will clearly continue to be the case. We might rearrange the topics in the syllabus in various orders, but it will still be the topics themselves that link the courses together rather than the mathematical procedures involved, and this is not the way that most people think! Reasoning involves a broad spectrum of thinking. We hope to encourage this thinking here.

We believe that there can be great benefits to students in a mathematics class (as well as a spin-off effect in their everyday lives) by considering problem solving as an end in itself and not merely as a means to an end. Problem solving can be the vehicle used to introduce our students to the beauty that is inherent in mathematics, but it can also be the unifying thread that ties their mathematics experiences together into a meaningful whole. One immediate goal is to have our students become familiar with numerous problem-solving strategies and to practice using them. We expect this procedure will begin to show itself in the way students approach problems and ultimately solve them. Enough practice of this kind should, for the most part, make a longer-range goal attainable, namely, that students naturally come to use these same problem-solving strategies not only to solve mathematical problems but also to resolve problems in everyday life. This transfer of learning (back and forth) can be best realized by introducing problem-solving strategies in both mathematical and real-life situations concomitantly. This is a rather large order and an ambitious goal as well. Changing an instructional program by relinquishing some of its time-honored emphasis on isolated topics and concepts, and devoting the time to a procedural approach, requires a great deal of teacher “buy-in” to succeed. This must begin by convincing the teachers that the end results will prepare a more able student for this era, where the ability to think becomes more and more important as we continue to develop and make use of sophisticated technology.

When we study the history of mathematics, we find breakthroughs that, although simple to understand, often elicit the reaction, “Oh, I would never have thought about that approach.” Analogously, when clever solutions to certain problems are found and presented as “tricks,” they have the same effect as the great breakthroughs in the history of mathematics. We must avoid this sort of rendition and make clever solutions part of an attainable problem-solving strategy knowledge base that is constantly reinforced throughout the regular instructional program.

You should be aware that, in the past few decades, there has been much talk about problem solving. While many new thrusts in mathematics last a few years, then disappear leaving some traces behind to enrich our curriculum, the problem-solving movement has endured for more than a quarter of a century and shows no sign of abatement. If anything, it shows signs of growing stronger. The National Council of Teachers of Mathematics (NCTM), in its Agenda for Action (1980), firmly stated that “problem solving must be the focus of the (mathematics) curriculum.” In their widely accepted Curriculum and Evaluation Standards for School Mathematics (1989), the NCTM offered a series of process Standards, in addition to the more traditional content Standards. Two of these four Standards (referred to as the “Process Standards”), Problem Solving and Reasoning, were for students in all grades, K through 12. In their Principles and Standards for School Mathematics (2000), the NCTM continued this emphasis on problem solving throughout the grades as a major thrust of mathematics teaching. All these documents have played a major role in generating the general acceptance of problem solving as a major curricular thrust. Everyone seems to agree that problem solving and reasoning are, and must be, an integral part of any good instructional program. In an effort to emphasize this study of problem solving and reasoning in mathematics curricula, most states are now including problem-solving skills on their statewide tests. Teachers sometimes ask, “I...

Table of contents

• Cover
• Title Page
• Copyright
• Contents
• Preface
• Acknowledgments
• About the Authors
• Dedication
• 1. Introduction to Problem-Solving Strategies
• 2. Working Backwards
• 3. Finding a Pattern
• 4. Adopting a Different Point of View
• 5. Solving a Simpler Analogous Problem
• 6. Considering Extreme Cases
• 7. Making a Drawing (Visual Representation)
• 8. Intelligent Guessing and Testing (Including Approximation)
• 9. Accounting for All Possibilities
• 10. Organizing Data
• 11. Logical Reasoning
• Afterword
• Sources for Problems
• Readings on: Problem Solving