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SE 015 465 AUTHOR TITLE the Content of Arithmetic Included in A DOCUMENT' RESUME ED 070 656 24 SE 015 465 AUTHOR Harvey, John G. TITLE The Content of Arithmetic Included in a Modern Elementary Mathematics Program. INSTITUTION Wisconsin Univ., Madison. Research and Development Center for Cognitive Learning, SPONS AGENCY National Center for Educational Research and Development (DHEW/OE), Washington, D.C. BUREAU NO BR-5-0216 PUB DATE Oct 71 CONTRACT OEC-5-10-154 NOTE 45p.; Working Paper No. 79 EDRS PRICE MF-$0.65 HC-$3.29 DESCRIPTORS *Arithmetic; *Curriculum Development; *Elementary School Mathematics; Geometric Concepts; *Instruction; *Mathematics Education; Number Concepts; Program Descriptions IDENTIFIERS Number Operations ABSTRACT --- Details of arithmetic topics proposed for inclusion in a modern elementary mathematics program and a rationale for the selection of these topics are given. The sequencing of the topics is discussed. (Author/DT) sJ Working Paper No. 19 The Content of 'Arithmetic Included in a Modern Elementary Mathematics Program U.S DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS SEEN REPRO Report from the Project on Individually Guided OUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIG INATING IT POINTS OF VIEW OR OPIN Elementary Mathematics, Phase 2: Analysis IONS STATED 00 NOT NECESSARILY REPRESENT OFFICIAL OFFICE Of LOU Of Mathematics Instruction CATION POSITION OR POLICY V lb. Wisconsin Research and Development CENTER FOR COGNITIVE LEARNING DIE UNIVERSITY Of WISCONSIN Madison, Wisconsin U.S. Office of Education Corder No. C03 Contract OE 5-10-154 Published by the Wisconsin Research and Development Center for Cognitive Learning, supported in part as a research and development center by funds from the United States Office of Education, Department of Health, Educa- tion, and Welfare. The opinions expressed herein do not necessarily re- flect the position or policy of the Office of Education and no official endorsement by the Office of Education should be inferred. Center No. C-03 / Contract OE 5-10-154 Working Paper No. 79 THE CONTENT OF ARITHMETIC INCLUDEDIN A MODERN ELEMENTARY MATHEMATICS PROGRAM by John G. Harvey Report from the Project on Individually Guided ElementaryMathematics Phase Two, Analysis of MathematicsIastruction Thomas A. Romberg and John G. Harvey Principal Investigators James M. Moser Associate Scientist Wisconsin Research and Development Center for'Cognitive Learning The University ,.of Wisconsin Madison, Wisconsin October 1971 NATIONAL EVALUATION COMMITTEE Samuel Brownell Henry Chauncey Elizabeth Koontz Patrick Suppes Professor of Urban Education President Wage and Labor Standards Professor Graduate School Educational Testing Service Administration, U.S. Department of Mathematics Yale University Deportment of labor, Stanford University Washington Launor F. Carter Martin Deutsch Roderick McPhee *Benton J. Underwood Senior Vice President on Director, Institute for President Professor Technology and Development Developmental Studies Punohou School, Honolulu Department of Psychology System Development Corporation N. York Medical College Northwestern University Francis S. Chase Jack Edling G. Wesley Sowards Professor Director, Teaching Research Director, Elementary Education Deportment of Educotiun Division Florida State University University of Chicago Oregon State System of Higher Education RESEARCH AND DEVELOPMENT CENTER POLICY REVIEW BOARD Leonard Bete;lwitz Russell J. Hosier Stephen C. Kleene B. Robert Tabachnick Chairman Profesior, Curriculum Dean, Cullege of Chairman, Deportment Deportment of Psychology and Instruction letters and Science of Curriculum and Instruction Archie A. Buchmiller Clauston Jenkins Donald J. McCarty Henry C. Weinlick Deputy State Superintendent Assistont Director Deon Executive Secretary Deportment of Public Instruction Coordinating Committee for School of Education Wisconsin Education Association Higher Education Robert E. Grinder Herbert J. Klizusmeier Ira Sharkansky M. Crawford Young Chairman Director, R & D Center Associate Professor of Political Associate Dean Deportment of Edur Professor of Educational Science The Graduate School Psychology Psychology EXECUTIVE COMMITTEE Edgar F. Borgatta Robert E. Davidson RussellJ. Hosier Wayne Otto Brittinghom Professor of Assistant Professor, Professor of Curriculum and Professor of Cunkulum and Sociology Educational Psychology Instruction and of Business Instruction (heading) Anne E. Buchanan Frank H. Farley *Herbert J. Klausmeier Robert G. Petzold Project Specialist Associate Professor, Director, R & 0 Center Associate Dean of the School R & 0 Ce.iter Educational Psychology Professor of Educational of Education Psychology Professor of Curriculum and Instruction and of Music Robin S. Chapman Research Associate R & 0 Center FACULTY OF PRINCIPAL INVESTIGATORS Vernon L. Allen Frank H. Farley James Moser RichardI,. Venezky Professor of Psychology Associate Professor of Educational Assistant Professor of Mathematics Assistant Professor of English Psychology Education;Visiting Scholar and of Computer Sciences Ted Czajkowski Lester S. Golub Wayne Otto Alan Voeiker Assistant Professor of Curriculum Lecturer in Curriculum and Professor of Curriculum and Assistont Professor of Curriculum and Instruction Instruction and in English Instruction (Reading) and Instruction Robed E. Davidson John G. Harvey Milton 0. Pella Larry Wilder Assistant Professor of Asiociote Professor of Professor of Curriculum and Assistant Professor of Curriculum Educational Psychology Mathematics and of Curriculurr. Instruction (Science) and Instruction and Instruction Gary A. Davis Herbert J. Klausmeier Thomas A. Romberg Peter Wolff Associate Professor of Director, R & 0 Center Associate Dintetor, R & 0 Center Assistant Professor of Educational Educational Psychology Professor of Educational Professor of Mathematics and of Psychology Psychology Curriculum and Instruction M. Vere DeVault Donald Lange B. Robert Tabachnick Professor of Curriculum and Assistant Professor of Curriculum Chairman, Deptatment Instruction (Mathematics) and Instruction of Curriculum and Instruction MANAGEMENT COUNCIL Herbert J. Klausmeier Thomas A. Romberg James Walter Dan G. Woolpert Director, R & Center Associate Director . Director Director V.A.C. Henan Professor of Dissemination Program _ Operations and Business . Educational Psychology Mary R. Quilling Director t Technical Development Program COMMITTII.CHAIRMAN ii t! STATEMENT OF FOCUS The Wisconsin Research and Development Center for Cognitive Learning focuses on contributing to a better understanding of cognitive learning by children and youth and to the improvement of related educational practices. The strategy for research and development is comprehensive. It includes basic research to generate new knowledge about the conditions and processes of learning and about the processes of instruction, and the subsequent development of research-based instructional materials, many of which are designed for use by teachers and others for use by students. These materials are tested and refined in school settings. Throughout these operations behavioral scientists, curriculum experts, academic scholars, and school people interact, insuring that the results of Center activities are based soundly on knowledge of subject matter and cognitive learning and that they are applied to the improvement of educational practice. This working paper is from the Project on Individually Guided Elementary Mathematics in Program 2. General objectives of the program are to establish rationale and strategy for developing instructional systems, to identify sequences of concepts and cognitive skills, to develop assessment procedures for those concepts and skills, to identify or develop instructional materials associated with the concepts and cognitive skills, and to generate new knowledge about instructional procedures. Contributing to the program objectives, the mathematics project has developed'and tested a televised course in arithmetic for grades 1-6 which provides not only a complete program of instruction for the pupils but also inservice training for teachers. Analysis of Mathematics instruction is currently the only active phase of the mathematics project and has a long-term goal of providing an indi- vidually guided instructional program in elementary mathematics. Preliminary activities include identifying instructional objectives, student activities, teacher activities materials, and assessment procedures for integration into a total mathematics curriculum. The third phase focused on the development of a computer system for managing individually guided instruction in mathematics and on a later extension of the system's applicability. a% TABLE OF CONTENTS Page List of Tables and Figures vii Abstract ix I Introduction 1 II Arithmetic in the Elementary School 3 .III The Arithmetic Content of a Modern Elementary Mathematics Program 7 IV Instructional Sequences for the Arithmetic Content 13 VConclusion 35 References 37 L LIST OF TABLES Table Page 1 Summary of Properties on which the Arithmetics are Different 12 s. 2 Representative.Sequencing of Arithmetic Topics 14 LIST OF FIGURES Figure 1 Sets of Numbers Ordered by Set Inclusion . 9 vii ABSTRACT This paper details the arithmetic topics proposed for inclusion in a modern elementary mathematics program, gives a rationale for the selection of these topics, and discusses the sequencing of'the topics. ix I INTRODUCTION The purpose of the Analysis 'of Mathematics Instruction Project is to generate knowledge about mathematics
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