Problem Solving: A Handbook for Elementary School Teachers

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    Krulik, Stephen; Rudnick, Jesse A.Problem Solving: A Handbook for Elementary SchoolTeachers.


    249p.; Drawings may not reproduce well.Allyn & Bacon/Logwood Division, 160 Gould Street,Needham Heights, MA 02194-2310 ($35.95, 20% off 10 ormore).

    Guides - Classroom Use - Guides (For Teachers) (052)

    MF01 Plus Postage. PC Not Available from EDRS.*Cognitive Development; Curriculum Development;Decision Making; Elementary Education; *ElementarySchool Mathematics; Expert Systems; *Heuristics;Instructional Materials; Learning Activities;Learning Strategies; Logical Thinking; MathematicalApplications; Mathematics Materials; MathematicsSkills; *Problem Sets; *Problem Solving

    This book combines suggestions for the teaching ofproblem solving with activities and carefully discussed non-routineproblems which students should find interesting as they gain valuableexperience in problem solving. The over 300 activities and problemshave been gleaned from a variety of sources and have been classroomtested by practicing teachers. Topics included are an explanation ofproblem solving and heuristics, the pedagogy of problem solving,strategy games, and non-routine problems. An extensive rumber ofproblems and strategy game boards are given. A 50-item bibliographyof problem solving resources is included. (Author/MVL)

    * Reproductions supplied by EDRS are the best that can be made* from the original document.








    Stephen Krulik

    Jesse A. Rudnick

    Temple University

    Allyn and Bacon, Inc.

    Boston London Sydney Toronto

  • Copyright 0 1988 by Allyn and Bacon, Inc.A Division of Simon & Schuster7 Wells AvenueNewton, Massachusetts 02159

    All rights reserved. No part of the material protected by this copyright notice may be reproducedor utilized in any form or by any means, electronic or mechanical, including photocopying, recording,or by any information storage and retrieval system, without written permission from the copyrightowner.

    The reproducible masters contained within may be reproduced for use with this book, providedsuch reproductions bear copyright notice, but may not be reproduced in any other form for anyother purpose without permission from the copyright owner.

    Lir..Aary of Congress Cataloging-in-Publication Data

    Krulik, Stephen.Problem solving.

    Bibliography: p.1. Problem solvingStudy and teaching. 2. Mathematics

    Study and teaching (Elementary) I. Rudnick, Jesse A.B. Title.QA63.K77 1988 370.15 87-17483ISBN 0-205-11132-7

    Printed in the United States of America

    10 9 8 7 6 5 4 3 2 1 91 90 89 88 87



  • Preface vii

    CHAPTER ONE An Introduction to Problem Solving 1

    What Is a Problem? 2What Is Problem Solving? 3Why Teach Problem solving? 4When Do We Teach Problem Solving? 5What Makes a Good Problem Solver? 6What Makes a Good Problem? 6What Makes a Good Teacher of Problem Solving? 15

    CHAPTER TWO A Workable Set of Heuristics 17

    What Are Heuristics? 18A Set of Heuristics to Use 18Applying the Heuristics 29

    CHAPTER THREE The Pedagogy of Problem Solving 35


  • Contents

    SECTION A A Collection of Strategy Gaines 77

    SECTION B A Collection of Non-Routine Problems 93

    SECTION C A Bibliography of Problem-Solving Resources 153

    SECTION D Masters for Selected Problems (Problem Cards) 157

    SECTION E Masters for Strategy Game Boards 307



  • Preface.1During the past decade, problem solving has become a major focusof the school mathematics curriculum. As we enter the era oftechnology, it is more important than ever that our students learnhow to successfully resolve a problem situation.

    This book is designed to help you, the elementary schoolteacherwhether you are a novice or experiencedto teach problemsolving. Ever since mathematics has been considered a schoolsubject, the teaching of problem solving has been an enigma tomathematics teachers at all levels, whose frustrating efforts to teachstudents to become better problem solvers seem to have had littleeffect. The teaching of problem solving must begin when the childfirst enters school. Even the youngest of children face problemsdaily.

    This book combines suggestions for the teaching of problemsolving with activities and carefully discussed non-routine problemswhich your students will find interesting as they gain valuableexperience in problem solving. The activities and problems havebeen gleaned from a variety of sources and have been classroom-tested by practicing teachers. We believe that this is the first timesuch an expansive set of problems has appeared in a single resource,specifically designed for the elementary school.

    Problem solving is now considered to be a basic skill ofmathematics education. However, we suggest that it is more thana single skill; rather, it is a group of discrete skills. Thus, in thechapter on pedagogy, the subskills of problem solving are



  • Preface

    enumerated and then integrated into a teachable process. Thechapter is highlighted by a flowchart that guides students throughthis vital process. Although there are many publications that dealwith the problem-solving process, we believe that this is the firstone that focuses on these subskills.

    We are confident that this book will prove to be a valuable assetin your efforts to teach problem solving.

    S.K. and J.R.




    An Introductionto Problem Solving

    1 0

  • Chapter One


    A major difficulty in discussing problem solving seems to be a lackof any clear-cut agreement as to what co'.istitutes a "problem." Aproblem is a situation, quantitative or otherwise, that confronts anindividual or group of individuals, that requires resolution, and forwhich the individual sees no apparent path to obtaining the solution.The key to this definition is the phrase "no apparent path." Aschildren purse: their mathematical training, what were problemsat an early age become exercises and are eventually reduced to merequestions. We distinguish between these three commonly usedterms as follows:

    (a) question: a situation that can be resolved by recall frommemory.

    (b) exercise: a situation that involves drill and practice to re-inforce a previously learned skill or algorithm.

    (c) problem: a situation that requires thought and a synthesisof previously learned knowledge to resolve.

    In addition, a problem must be perceived as such by the stu-dent, regardless of the reason, in order to be considered a problemby him or her. If the student refuses to accept the challenge, thenat that time it is not a problem for that student. Thus, a problemmust satisfy the following three criteria, illustrated in Figure 1-1:

    1. Acceptance: The individual accepts the problem. There is apersonal involvement, which may be due to any of a varietyof reasons, including internal motivation, external motiva-tion (peer, parent, and/or teacher pressure), or simply thedesire to experience the enjoyment of solving a problem.

    2. Blockage: The individual's initial attempts at solution arefruitless. His or her habitual responses and patterns of attackdo not work.

    3. Exploration: The personal involvement identified in (1)forces the individual to explore new methods of attack.





    Figure 1-1



  • An Introduction to Problem Solving

    The existence of a problem implies that the individual is con-fronted by something he or she does not recognize, and to whichhe or she cannot merely apply a model. A situation will no longerbe considered a problem once it can be easily solved by algorithmsthat have been previously learned.

    A word about textbook problems

    Although most mathematics textbooks contain sections labeled"word problems," many of these "problems" should not really beconsidered as problems. In many cases, a model solution has alreadybeen presented in class by the teacher. The student merely appliesthis model to the series of similar exercises in order to solve them.Essentially the student is practicing an algorithm, a technique thatapplies to a single class of "problems" and that guarantees successif mechanical errors are avoided. Few of these so-called problemsrequire higher-order thought by the students. Yet the first time astudent sees these "word problems" they could be problems to himor her, if presented in a non-algorithmic fashion. In many cases, thevery placement of these exercises prevents them from being realproblems, since they either follow an algorithmic development de-signed specifically for their solution, or are headed by such state-ments as "Word Problems: Practice in Division by 4."

    We consider these word problems to be "exercises" or "routineproblems." This is not to say that we advocate removing them fromthe textbook. They do serve a purpose, and for this purpose theyshould be retained. They provide exposure to problem situations,practice in the use of the algorithm, and drill in the associated math-ematical processes. However, a teacher should not think that stu-dents who have been solving these exercises through use of acarefully developed model or algorithm have been exposed to prob-lem solving.


    Problem solving is a process. It is the means by which an individualuses previously acquired knowledge, skills, and understanding tosatisfy the demands of an unfamiliar situation. The process beginswith the initial confrontation and concludes when an answer hasbeen obtained and considered with regard to the initial conditions.The student must synthesize what he or she has learned, and applyit to the new and different situation.


  • Chapter One

    Some educators assume that expertise in problem solving de-velops incidentally as one solves many problems. While this maybe true in part, we feel that problem solving must be considered asa distinct body of knowledge and the process should be taught assuch.

    The goal of school mathematics can be divided into severalparts, two of which are (1) attaining information and facts, and(2) acqu!ring the ability to use information and facts. This abilityto use informaticA and facts is an essential part of the problem-solving process. In effect, problem solving requires analysis andsynthesis.


    In dealing with the issue of why we should teach problem solving,we must first consider the larger question: Why teach mathematics?Mathematics is fundamental to everyday life- All of our studentswill face problems, quantitative or otherwise, every day of their lives.Rarely, if ever, can these problem( be resolved by merely referringto an arithmetic fact or a previously learned algorithm. The words"Add me!" or "Multiply me!" never appear in a store window.Problem solving provides the link iietween factsand algorithms and the reallife problem situations we all face. For most people, mathematics isproblem solving!

    In spite of the obvious relationship between mathematics ofthe classroom and the quantitative situations in life, we know thatchildren of all ages see little connection between what happens inschool and what happens in real life. An emphasis on problemsolving in the classroom can lessen the gap between the real worldand the classroom world and thus set a positive mood in t'leclassroom

    In many mathematics classes, students do not see any connec-tions among the various ideas taught during the year. Most regardeach topic as a separate entity. Problem 8c/tying shows the inter-connections among mathematical ideas. Problems are never solvedin a vacuum, but are related in some way to something seen beforeor to so. !thing learned earlier. Thus, good problems can be III:0dto review past mathematical ideas, as well as to sow seeds for idea:ito be presented at a future time.

    Problem solving is more exciting, more challenging, and moreinteresting to children than barren exercises. If we examine studentperformance in the classroom, we recognize the obvious fact thatsuccess leads to persistence and continuation of a task; failure leads


  • An Introduction to Problem Solving

    to avoidance. It is this continuance that we constantly strive for inmathematics. The greater the involvement, the better the end prod-uct. Thus, a carefully selected sequence of problem-solving activitiesthat yield success will stimulate students, leading them to a morepositive attitude toward mathematics in gene. '1. and problem solv-ing in particular.

    Finally, problem solving permits students to learn and to prac-tice heuiistic thinking. A careful selection of problems is a majorvehicle by which we provide a "sharpening" of problem-solving skillsand strategies so necessary in real life.


    Problem solving is a skill everyone uses all their lives. The initialteaching and learning of the problem-solving process must beginas soon as the child enters school, and continue throughout hisor her entire school experience. The elementary school teacherhas the responsibility for beginning this instruction and thuslaying the foundation for the chili's future problem-solvingexperiences.

    Since the process of problem solving is a teachable skill, whendo we teach it? What does it replace? Where does it fit into the day-to-do schedule?

    ,xperiences in problem solving are alwaya at hand. All otheractivities are subordinate. Thus, the teaching of problem solvingsb;uld be continuous. Discussion of problems, proposed solu-tions, methods of attacking problems, etc., should be consideredat all times. Think how poorly students would perform in otherskill areas, such as fractions, if they were taught these skills in oneor two weeks of concentrated work and then the skills were neverused again.

    Naturally there will be times when studies of algorithmic skillsand drill and practice sessions will be called for. We insist thatstudents be able to add, subtract, multiply, and divide. Problemsolving is no a substitute for these computational skills. However,these times will permit the delay necessary for the incubation periodrequired by many problems, which need time to "set." By allowingtime between formal problem-solving sessions, you permit studentsto become familiar with the problem-solving process slowly and overa longer period of time. This is important, since the emphasis is onthe process and not merely on obtaining an answer. The develop-ment of the process takes times


  • Chapter One


    Although we cannot easily determine what it is that makes somestudents good problem solvers, there are certain common charac-teristics exhibited by good problem solvers. For instance, good prob-lem solvers know the anatomy ofa problem. They know that a problemcontains facts, a question, and a setting. They also know that mostproblems (with the exception of some word problems in textbooks)contain distractors, which they can recognize and eliminate.

    Good problem solvers have a desire to solve problems. Problemsinterest them; they offer a challenge. Much...


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