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A Review of Research on Addition and Subtraction. Working Paper No DOCADIENIIRESUME ED 223 447 SE 039, 613 AUTHOR' Carpenter, Thomas P.; And Others TITLE A Review of Re'search on Addition andSubtraction. Working Paper No. 330.-Report from the Program on Student Diversity and Classroom Processes. -INSTITUTION Wisconsin Center for Education Research, Madison. SPONS AGENCY National Inst. of Education (ED), Washington, DC. PUB DATE Sep 82 GRANT NIE-G-81-0009 NOTE 186p. PUB TYPE Information Analyses (070) EDRS PRICE MF01/PC08 Plus Postage. DESCRIPTORS *Addition; *Basic Skills; Cognitive Processes; Computation; Educational Research; Elementary Education; *Elementary School Mathematics; *Learning Theories; Literature Reviews; Mathematical Concepts; Mathematics Instructioh; Problem Solving; *SubtraCtion IDENTIFIERS *Mathematics Education Research; Word PrOblems ABSTRACT This material is designed to examine the research on how children acquire basic addition and subtractionconcepts and skills. Two major lines of theories of the development ofbasic number concepts, called logical concept and quantificationskill approaches, are identified. Mjcir recurring issues in thedevelopment of early number concepts are also discussed.This is followed by examination of studies specifically related to addition and subtraction: It is noted that by kindergarten mostchildren understand the rudiments of these operations, but a complete understanding develops over a protracted span of years.The direction 4of initial and ongoing ,research is discussed,and word problem studies and investigations on children's Solutions ofsymbolic addition and subtraction problems are viewed to be concernedwith both problem difficulty and Solution processes.The review suggests that a great-deal is known both about children'sknowledge of addition and subtraction and how they solve addition andsubtraction problems. It also indicates that there is notyet a clear picture of how to apply such new information to design doreeffective instruction; (MP) *********************************************************************** Reproductions supplled by EDRS are the best that can be made from the original document. *************R********************************************************* U S DEPAPTNEENT OE EDUCATION NAIiNa. 11,, PERMISSION TO REPRODULE THIS MATERIAL HA', BEEN UFIANTED BY TH1,1.1,1h a I, ,N4L HL E 1N NT! tOppli Working Paper No. 330 A REVIEW OF RESEARCH ON ADDITION AND SUBTRACTION by Thomas P. Carpenter, GlendonBlume, James Hiebert, ConstanceMartin Anick, David Pimm Report from the Program on Student Diversity andClassroom Processes Wisconsin Center for EducationResearch The University of Wisconsin Madison, Wisconsin September 1982 N) The research reported in this paper was funded by the Wisconsin Center for Education Research which is supported in part by a grant from the National Institute of Education (Grant No. NIE-0-81-0009). The opinions xpressed in this paper do not necessarily reflect the position, policy, or endorsement of the National Institute of Education. ii 4 Wisconsin Center for EducationResearch MISSION STATEMENT The mission of the Wisconsin Centerfor Education Research is to understand, and to help educatorsdeal with, diversity 'among students. The Center pursues its mission byconducting and synthesizing research, developingstrategies and materials, and disseminating knowledge bearing uponthe education of individuals and diverse groups bf studentsin elementary,and secondary schools.' Specifically, the Centerinvestigates diversity as a basic fact of human nature,through studies of learning and development diversity as a central challenge foreducational techniques, through studies of classroom processes diversity as a key-issue in relations between individuals and institutions, through studiesof school processes diversity as a fundamental question in American social thought, through studies ofsocial policy related to education The Wisconsin Center for EducationResearch is a noninstruc- tional department of the University ofWisconsin-Madison School of Education. The Center is supported primarily with funds from the National Institute of Education. Table of Contents Page vii List of Tables ix Abstract 1 Introduction 2 The Development of Basic NumberConceptq 4 Logical Concept Theories 10 Quantification Skill Theories 22 Recurring Issues 23 Counting or Subitizing: Which Comes First9 27 Conservation of Number: How Does it Develop? Early Number Operations: How Do They Affect 32 Later Mathematics Performance? 37 Basic Addition and SubtractionConcepts 41 Symbolic Problems 41 Relative Difficulty of BasicFacts 44 Structural Variables 49 Symmetric Forms 49 Other Variables 49 Horizonta//Vertical Format 50 Existence of a Solution 51 Presence of Aids 52 Strategies of Solution 52 Research Paradigms 60 Results of Latency Studies 70 Error Analysis 72 Direct Observation ofStrategiel 83 Word Problems 84 Factors Telated to Word Problems Knowledge of Basic Facts andPerformances on 84 Verbal Problems Presence of Aids 86 91 Syntax Variables 98 Semantic Variables 98 The Semantic Classificationof Word Problem Types 106 Strategies 114 Problem Structure and SolutionProcess 126 ,Development of Addition andSubtraction Processes 127 Level of Abstraction 132 Choice of Strategy .11 t o 0 Table of Contents (Continued) ; Page Relationship Between Dimensions of Development 134 A Model of Verbal Problem Solving 137 Discussion 141 References 145 oe.) I List of Tables Page Table 1 and Placeholder Position 1 The Effects of Operation Difficulty of Open Additionand Subtraction on the 45 Sentences 102 2 Classification of Word Problems Problem and Structure 116 3 Relation of Strategy to , 122 Problems 4 Relative Difficulty of Word 125 Problems 5 Combine and-Join Addition Separate and JoinSubtraction 6 Results for Selected 136 Problems , 1 4 Abstract The purOse of this paper is to reviewthe research on how children acquire basic addition and subtraction conceptsand skills. The re- view starts with an analysis of majortheories of the development of basic number concepts. Two major lines of theories areidentified: logical concept theories and quantificationskill theories. The most notable aa(ample of a logical concept theory isrepresented by tht work of Piaget while the work of Gelman and Klahrand Wallace provide the most comprehensive examples of quantificationskill theories. Ginsburg's work attempts to build a bridge between the twoapproaches.Major recurring issues in the development of earlynumber concepts are also discussed. These include the relation between countingand subitiLing, how number conservation develops, and howearly number operations affect the acquisition of addition and subtractionskills. The second section of the paper examinesresearch on basic concepts spe- cifically related to addition and subtraction.By kindergarten most children understand, that joining an element to a setincreases its numerosity and removing an element decreasesits numerosity. However, subtraction, a complete understandingof basic properties of addition and like ittversion and compensation, develop over aprotracted span of years. Most early research on children'ssolutions of symbolic addition and subtraction problems was limited to problemdifficulty. Much of the initial research focused on the relativedifficulty of different basic number facts. More recently, the emphasis has shifted to ananalysis of children's solution processes. Using a variety of paradigms, a number of solution strategies based oncounting have been identified. Research on word problems has also beenconcerned with both problem difficulty and solution processes. Most of the problem difficulty re- search has focused on syntax variables. Current research on children's solution processes has attempted to make a connectionbetween the seman- tic structure of problems and the processeschildren use to solve them. A mounting body of evidence clearlydemonstrates that young children can solve a variety ofaddition and subtraction problems by physically modeling the action or relationships described inthe problems. These direct modeling strategies gradually give way to moreabstract counting representations that retain certain characteristicsof the initial modeling strategies. Several simulation models have been developed to account for the internal cognitivemechanisms that are required for the observed patterns of development. The review suggests that a great dealis known about both children's knowledge of addition and subtraction andhow they solve addition and subtraction problems, but also indicates thatthere is not yet a clear picture of how to apply these insights todesign more effective instruc- tion. INTRODUCTION The learning of basic addition andsubtraction concepts and skills is a major objective of primary schoolmathematics instruction. Because of the central place it occupies in themathematics curriculum, there is an extensive body of research onthe teaching and learning of addition and subtraction that dates back to the turnof the century. In the last few years, there has been a resurgence ofinterest in the study of addi- tion and subtraction as researchers have foundthat the techniques of cognitive ptlyhology provide new insights intothe processes that chil- dren use in solving addition and subtractionproblems. The purpose of this paper is to review thisgraring body of research on addition and subtraction. The paper is divided into six major parts. To provide some backgrouLd, the first sectionbriefly summarizes some of the major.lines of inquiry on the developmentof basic number concepts. The second section
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