Problem Solving: A Handbook for Senior High School Teachers.

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    SE 050 180

    Krulik, Stephen; Rudnick, Jesse A.Problem Solving: A Handbook for Senior High SchoolTeachers.ISBN-0-205-11788-089

    233p.; Drawings and some small print may notreproduce well.

    Allyn & Bacon/Logwood Division, 160 Gould Street,Needham Heights, MA 02194-2310 ($34.95, 20% off 10 ormore).

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

    MFO1 Plus Postage. PC Not Available from EDRS.Educational Games; Educational Resources;*Heuristics; High Schools; Instructional Improvement;*Instructional Materials; Mathematics Education;*Mathematics Instruction; *Problem Sets; *ProblemSolving; *Secondary School Mathematics

    The teaching of problem solving begins the moment achild first enters school and the senior high school plays a majorrole in the development of this skill since a number of studentsterminate their formal education at the end of this period. This bookcombines suggestions for the teaching of problem solving withactivities, problems, and strategy games that students findinteresting as they gain valuable experiences in problem solving.Over 120 classroom-tested problems are included. Discussions in thisvolume include a definition of problem solving, heuristics, and howto teach problem solving. Also provided are collections of strategygames and nonroutine problems, including 35 reproducible blacklinemasters for selected problems and game boards; and a bibliography of51 resources on problem solving. (CW)

    * 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

    Boston London Sydney Toronto


  • Copyright CD 1989 by Allyn and BaconA Division of Simon & Schuster160 Gould StreetNeedham Heights, Massachusetts 02194

    All rights reserved. No part of the material protected by this copyright notice may bereproduced or utilized in any form or by any means, electronic or mechanical, includingphotocopying, recording, or by any information storage and retrieval system, withoutwritten permission from the copyright owner. The masters in Sections D and E may bereproduced for use with this book, provided such reproductions bear copyright notice,but may not be reproduced in any other form or for any other purpose without permissionfrom the copyright owner.

    Library of Congress Cataloging-in-Publication Data

    Krulik, Stephen.Problem solving : a handbook for senior high school teachers /

    Stephen Krulik, Jesse Rudnick.p. cm.

    Bibliography: p.ISBN 0-205-11788-0 : $32.951. Problem solving. I. Rudnick, Jesse A. II. Title.

    QA63.K775 1989 88-23324153.4'3dc19 CIP

    Printed in the United States of America

    10 9 8 7 6 5 4 3 2 1 92 91 90 89 88


  • Contentsimisramm

    Preface v

    CHAPTER ONE An Introduction to Problem Solving 1

    What Is a Problem? 3What Is Problem Solving? 5Why a Special Emphasis on Problem Solving? 5When Do We Teach Problem Solving? 6What Makes a Good Problem Solver? 7What Makes a Good Problem? 8What Makes a Good Teacher of Problem Solving? 20

    CHAPTER TWO A Workable Set of Heuris" -s 21

    What Are Heuristics? 23A Set of Heuristics to Use 24Applying the Heuristics 38

    CHAPTER THREE The Pedagogy of Problem Solving 47

    1. Create a Problem-Solving Environment in theClassroom 49


  • Contents

    2. Encourage Your Students to Solve Problems 493. Teach Students How to Read a Problem 514. Require Your Students to Create Their Own

    Problems 585. Have Your Students Work Together in Small

    Groups 606. Encourage the Use of Drawings 637. Have Your Students Flowchart Their Own Problem-

    Solving Process 658. Suggest Alternatives When the Present Approach

    Has Apparently Yielded All PossibleIns nnation 67

    9. Raise Creative, Constructive Questions 7410. Emphasize Creativity of Thought and

    Imagination 7511. Apply the Power of Algebra 7812. Emphasize Estimation 7913. Utilize Calculators 8414. Capitalize on the Microcomputer 8715. Use Strategy Games in Class 89

    SECTION A A Collection of Strategy Games 93

    SECTION B A Collection of Non-Routine Problems 115

    SECTION C A Bibliography of Problem-Solving Resources 189

    SECTION D Masters for Selected Problems 195

    SECTION E Masters for Strategy Game Boards 253


  • Preface

    During the past decade, problem solving has become a major focusof the school mathematics curriculum. As we enter the era of tech-nology, it is more important than ever that our students learn howto succeed in resolving problem situations.

    This book is designed to help you, the senior high school math-ematics teacherwhether you are a novice or experiencedto teachproblem solving. Although the teaching of problem solving beginswhen a child first enters school, the senior high school must playa major role in the development of this skill. Indeed, a Significantnumber of students terminate their formal education at the end ofsenior high school and thus are dependent upon their elementaryand secondary school training to cope with the many problems theyface every day. Traditionally, the senior high school mathematicsprogram has been oriented toward preparing students to enter col-leges or other institutions of higher education. The content hastherefore concentrated on the skills and concepts of algebra, ge-ometry, functions, and other mathematical topics. Problem solvinghas never really been the major focus of these programs, althoughit should have been according to the Agenda for Action published bythe National Council of Teachers of Mathematics.

    This book combines suggestions for the teaching of problemsolving with activities, carefully discussed non-routine problems,and s 1- negy games your students will find interesting as they gainvaluable experiences in problem solving. The activities, problems,and games have been gleaned from a variety of sources and have



  • Preface

    been classroom-tested by practicing teachers. We believe that thisis the first timer such an extensive set of problems has appeared ina single resource that is specifically Cesigned for the senior highschool.

    Problem solving is now considered to be a basic skill of math-ematics education, but we suggest that it is more than a single skill.Rather, it is a group of discrete skills. In the chapter on pedagogy,the subskills of problem solving are enumerated and then integratedinto a teachable process. The chapter features a flowchart that guidesstudents through this vital process. Although there are many pub-lications that deal with the problem-solving process, we believe thatthis is the first one 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 Introductiun to Problem Solving


    Until very recently, a major difficulty in discussing problem solvingwas a lack of any dear-cut agreement as to what constituted a "prob-lem." This has finally been resolved; most mathematics educatorsaccept the following definition of a problem:

    Definition A problem is a situation, quantitative or otherwise, thatconfronts an individual or group of individuals, that re-quires resolution, and for which the individual sees noapparent path to the solution.

    The key to this definition is the phrase "no apparent path." Asch;laren pursue their mathematical training, what were problemsat an early age become exercises and eventually reduce to mere ques-tions. We distinguish between these three commonly used terms asfollows:

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

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

    (c) problem: a situation that requires analysis and synthesis ofpreviously 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 refures to accept the challenge, it isnot a problem for that student at that time. Thus, a problem mustsatisfy the following three criteria, illustrated in Figure 1-1.

    1. Acceptance:

    2. Blockage:

    3. Exploration:

    The individual accepts the problem. There ispersonal involvement, which may be due to anyof a variety of reasons, including internal mo-tivation, external motivation (peer, parent, and/or teacher pressure), or simply the desire to ex-perience the enjoyment of solving a problem.The individual's initial attempts at solution arefruitless. His or her habitual responses and pat-terns of attack do not work.The personal involvement identified in (1) forcesthe individual to explore new methods of attack.



  • B




    Chapter One


    Figure 1-1

    A word about textbook "r:oblems"


    The heading "problem" implies that the individual is being con-fronted by something he or she does not recognize. A situation willno longer be considered a problem once it has been modeled or caneasily be solved by applying algorithms that have been previouslylearned.

    While all mathematics textbooks contain sections labeled "wordproblems," many of these. cannot really be considered as problems.In most cases, a model has been developed and a general solutionpresented in class by the teacher. Following this presentation, thestudent merely applies the model solution to the subsequent seriesof exercises in order to solve them. These exercises, except for achange in the numbers and the cast of characters, all fit the samemodel. Essentially, the student is practicing an algorithma tech-nique that applies to a single class of "problems" and that guaranteessuccess if mechanical errors are avoided. Under this format, few ofthe so-called problems require higher-order thought by the stu-dents. Yet the first time a student sees these "word problems," theycould be problems to him or her if they are presented in a non-algorithmic fashion. In many cases, the very placement of theseexercises within the text prevents them from being real problems,since they either follow the algorithm designed specifically for theirsolution or are headed by such statements as "Pi oblem Solving:Time, Rate, and Distance." We consider these sections of the text-book to contain "exercises." Some authors refer to them as "routineproblems." We do not advocate removing them from the textbooks,because they do serve a purpose; they provide exposure to problemsituations, practice in the use of the algorithm, and drill in the as-sociated mathematical processes. However, a teacher should realizethat students who have been solving these exercises through theuse of a carefully developed model or algorithm have not been in-volved in problem solving. In fact, as George Polya stated, solving



  • An Introduction to Problem Solving

    "the routine problem has practically no chance to contribute to themental development of the student."*


    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.

    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 that the process should be taughtas such.

    There are many goals for school mathematics. Two of these arethe attainment of information and facts and the ability to use in-formation and facts. The latter ability is an essential part of the prob-lem-solving process. In effect, problem solving requires analysis andsynthesis. To succeed in problem solving is to learn how to learn.


    We believe that a major task of the senior high school mathematicsteacher is to provide students with the skills, concepts, and under-standing of algebra, geometry, and functions that make up the stan-dard college preparatory program. However, the usual treatment ofthis material does not provide the student with adequate problem-solving experiences. The major emphasis isupon attaining skills andconcepts (power within the subject). Little time is devoted to thedevelopment of the open-ended thought process that is problemsolving. Achievement in algebra and geometry by itself does notguarantee success in problem solving. Students enrolled in a collegepreparatory program, as well as those pursuing alternate programs,

    * George Polya. "On Teaching Problem Solving," in The Role of Ariomatics and ProblemSolving in Mathematics. New York: Ginn, 1966, p. 126.


    i 04

  • Clapter One

    are required to resolve problems, quantitative or otherwise, everyday of their lives. Rarely, if ever, can these problems be resolvedby merely referring to a mathematical fact or a previously learnedalgorithm. The words "Solve me!", "Factor me!", or "Find my area!"never appear in a store window. Problem solving is the link betweenfacts and algorithms and the rcal-life problem situations we all face. Formost people, mathematics is problem solving!

    In spite of the relationship between the mathematics of theclassroom and the quantitative situations in life, we know that stu-dents see little connection between what happens in school andwhat happens in real life. An emphasis on problem solving in the..1....aroom can lessen the gap between the real world and the class-room world and thus set a more positive mood in the classroom.

    In many mathematics classes, students do not even see anyconnections among the various ideas taught within a single year.Most regard each topic as a sepante entity. Problem solving showsan interconnection between mathematical ideas. Problems are neversolved in a vacuum; they are related in some way to something seenbefore or to something learned earlier. Thus, good problems can beused to review past mathematical ideas, as well as to sow the seedsfor ideas to be presented at a future time.

    Problem solving is more exciting, more challenging, and moreinteresting to students than are I irren exercises. If we examine stu-dent performance in the classroom, we recognize the obvious factthat success leads to persistence and continuation of a task, whilefailure leads to avoidance. It is this continuation that we constantlystrive for in mathematics. The greater the involvement, the betterthe end product. Thus, a carefully selected sequence of problem-solving activities that yield success will stimulate students, leadingthem to a more positive attitude towards mathematics in generaland problem solving in particular.

    Finally, problem solving is an integral part of the larger area ofcritical thinking, which is a universall...


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