DOCUMENT PESUME ED 301 459 SE 050 179 AUTHOR Krulik, Stephen; Rudnick, Jesse A. TITLE Problem Solving: A Handbook for Elementary School Teachers. REPORT NO ISBN-0-205-11132-7 PUB DATE 88 NOTE 249p.; Drawings may not reproduce well. AVAILABLE FROMAllyn & Bacon/Logwood Division, 160 Gould Street, Needham Heights, MA 02194-2310 ($35.95, 20% off 10 or more). PUB TYPE Guides - Classroom Use - Guides (For Teachers) (052) EDRS PRICE MF01 Plus Postage. PC Not Available from EDRS. DESCRIPTORS *Cognitive Development; Curriculum Development; Decision Making; Elementary Education; *Elementary School Mathematics; Expert Systems; *Heuristics; Instructional Materials; Learning Activities; Learning Strategies; Logical Thinking; Mathematical Applications; Mathematics Materials; Mathematics Skills; *Problem Sets; *Problem Solving ABSTRACT This book combines suggestions for the teaching of problem solving with activities and carefully discussed non-routine problems which students should find interesting as they gain valuable experience in problem solving. The over 300 activities and problems have been gleaned from a variety of sources and have been classroom tested by practicing teachers. Topics included are an explanation of problem solving and heuristics, the pedagogy of problem solving, strategy games, and non-routine problems. An extensive rumber of problems and strategy game boards are given. A 50-item bibliography of problem solving resources is included. (Author/MVL) * Reproductions supplied by EDRS are the best that can be made * * from the original document. * PROBLEM SOLVING BEST COPYAVAILABLE 3 PROBLEM SOLVING A HANDBOOK FOR ELEMENTARY SCHOOL TEACHERS 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 & Schuster 7 Wells Avenue Newton, Massachusetts 02159 All rights reserved. No part of the material protected bythis copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, includingphotocopying, recording, or by any information storage and retrieval system, without written permissionfrom the copyright owner. The reproducible masters contained withinmay be reproduced for use with this book, provided such reproductions bear copyright notice, butmay not be reproduced in any other form for any other purpose without permission from the copyrightowner. 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 87-17483 ISBN 0-205-11132-7 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 919089 88 87 5 L Preface vii CHAPTER ONE An Introduction to Problem Solving 1 What Is a Problem? 2 What Is Problem Solving? 3 Why Teach Problem solving? 4 When Do We Teach Problem Solving? 5 What Makes a Good Problem Solver? 6 What Makes a Good Problem? 6 What Makes a Good Teacher of Problem Solving? 15 CHAPTER TWO A Workable Set of Heuristics 17 What Are Heuristics? 18 A Set of Heuristics to Use 18 Applying the Heuristics 29 CHAPTER THREE The Pedagogy of Problem Solving 35 v 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 (ProblemCards) 157 SECTION E Masters for Strategy Game Boards 307 vi 7 .1Preface During the past decade, problem solving has become a major focus of the school mathematics curriculum. As we enter the era of technology, it is more important than ever that our students learn how to successfully resolve a problem situation. This book is designed to help you, the elementary school teacherwhether you are a novice or experiencedto teach problem solving. Ever since mathematics has been considered a school subject, the teaching of problem solving has been an enigma to mathematics teachers at all levels, whose frustrating efforts to teach students to become better problem solvers seem to have had little effect. The teaching of problem solving must begin when the child first enters school. Even the youngest of children face problems daily. This book combines suggestions for the teaching of problem solving with activities and carefully discussed non-routine problems which your students will find interesting as they gain valuable experience in problem solving. The activities and problems have been gleaned from a variety of sources and have been classroom- tested by practicing teachers. We believe that this is the first time such 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 of mathematics education. However, we suggest that it is more than a single skill; rather, it is a group of discrete skills. Thus, in the chapter on pedagogy, the subskills of problem solving are vii S Preface enumerated and then integrated intoa teachable process. The chapter is highlighted by a flowchart that guides students through this vital process. Although thereare many publications that deal with the problem-solving process,we believe that this is the first one that focuses on these subskills. We are confident that this book willprove to be a valuable asset in your efforts to teach problem solving. S.K. and J.R. viii 9 CHAPTER ONE An Introduction to Problem Solving 1 0 Chapter One WHAT IS A PROBLEM? A major difficulty in discussing problem solvingseems to be a lack of any clear-cut agreementas to what co'.istitutes a "problem." A problem is a situation, quantitativeor otherwise, that confronts an individual or group of individuals, thatrequires resolution, and for which the individual seesno apparent path to obtaining the solution. The key to this definition is the phrase"no apparent path." As children purse: their mathematical training,what were problems at an early age become exercises andare eventually reduced to mere questions. We distinguish between these threecommonly used terms as follows: (a) question: a situation thatcan be resolved by recall from memory. (b) exercise: a situation that involvesdrill and practice to re- inforce a previously learned skillor algorithm. (c) problem:a situation that requires thought and a synthesis of previously learned knowledge to resolve. In addition, a problem must be perceivedas such by the stu- dent, regardless of the reason, in orderto be considered a problem by him or her. If the student refusesto accept the challenge, then at that time it is not a problem for that student. Thus,a problem must satisfy the following three criteria, illustratedin Figure 1-1: 1. Acceptance: The individual accepts theproblem. There is a personal involvement, whichmay be due to any of a variety of reasons, including internal motivation,external motiva- tion (peer, parent, and/or teacher pressure),or simply the desire to experience the enjoyment ofsolving a problem. 2. Blockage:The individual's initial attempts at solutionare fruitless. His or her habitualresponses and patterns of attack do not work. 3. Exploration: The personal involvementidentified in (1) forces the individual to explorenew methods of attack. L O C K Acceptance Blockage Exploration Figure 1-1 2 11 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 which he or she cannot merely apply a model. A situation will no longer be considered a problem once it can be easily solved by algorithms that 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 be considered as problems. In many cases, a model solution has already been presented in class by the teacher. The student merely applies this model to the series of similar exercises in order to solve them. Essentially the student is practicing an algorithm, a technique that applies to a single class of "problems" and that guarantees success if mechanical errors are avoided. Few of these so-called problems require higher-order thought by the students. Yet the first time a student sees these "word problems" they could be problems to him or her, if presented in a non-algorithmic fashion. In many cases, the very placement of these exercises prevents them from being real problems, 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 "routine problems." This is not to say that we advocate removing them from the textbook. They do serve a purpose, and for this purpose they should 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 a carefully developed model or algorithm have been exposed to prob- lem solving. WHAT IS PROBLEM SOLVING? Problem solving is a process. It is the means by which an individual uses previously acquired knowledge, skills, and understanding to satisfy the demands of an unfamiliar situation. The process begins with the initial confrontation and concludes when an answer has been obtained and considered with regard to the initial conditions. The student must synthesize what he or she has learned, and apply it to the new and different situation. 3 Chapter One Some educators assume that expertise in problemsolving de- velops incidentally as one solvesmany problems. While this may be true in part, we feel that problem solvingmust be considered as a distinct body of knowledge and theprocess should be taught as such. The goal of school mathematicscan be divided into several parts, two of which are (1) attaining information andfacts, and (2) acqu!ring the ability touse information and facts.