DOCUMENT. RESUME ED 159 062 SE 024 921 AUTHOR Lesh, RiChard, Ed.; Mierkiewicz, Diane, Ed. TITLE Recent Research Concerning the Development of, Spatial and Geometric Concepts. INSTITUTION. ERIC Information' Analysis Center for Science, Mathematics, and Environmental Education, Columbus, ,Ohio.; Georgia Univ., Athens. Georgia Center for the Study of.Leaining-jand Teaching Mathematics. SPONS AGENCY __National Inst. of, Education (DHEW), Washington, D. PUB DAVE May 78 NOTE. 336p. AVAILABLE FROM Information Reference Center.(ERIC/IRC), The Ohio State University, 1200 Chambers Rd., 3rd Floor, Columbus. Ohio 43212 ($6.00) EDRS PRICE MF-$0.W.1 HC-$18.07 Plus Postage. DESCRIPTORS *Cogni'..ive Development; *Curriculum Development; *Educational Research; *Elementary School Mathematics; Elementary Secondary Education; *Goomotric Concepts; *Geometry;. Secondary School Mathematics; Transformations (Mathematics) IDENTIFIERS Piagetian Levels; *Spatial Concepts ABSTRACT ,These 14 research reports are grouped into three broad categories based on the Piagetian level concerned. The articles, concerning preoperational concepts focus on problems such as: (1) finding an appropriate mathematical description of some of the primitive mathematical concepts;(2). the role of. '!activities" in early concept acquisition; (3)the similarities'anddifferences between Perceptual and conceptual processes;(4) the relation between memory improvement and improved operational ability; and(5) variables responsible for the large gap between visual-visual and haptic-visual discrimin ation. The articles concerning the transitional phases between-concrete and formal operations,focus on elementary transformation geometry concepts and emphasize the role that figurative content can play in influencing the difficulty of Piaget-type tasks. Tyo articles in this group focus on'affine transformations, similarities, or.projections. The articles-dealing with older subjects or formal operational concepts investigate topics such as:(1) the nature of specific mathematical concepts as they .relate to the cognitive structure of the learner; (2) understanding of frames of reference by preservice teacher education students as determined by Piagetian tasks; and (3). the bias for upright figures that students exhibit in forming concepts., (MP) **************1******************************************************** * Reproductions supplied by EDRS are the best that can be made * * from the original document. * *********************************************************************** OF HEALTH. U.S. DEPARTMENT EDUCATION II WELFARE NATIONAL INSTITUTEOF EDUCATION HAS BEEN REPRO. THIS DOCUMENT RECEIVED FROM DUCED EXACTLY AS ORGANIZATION ORIGIN THE PERSON OR ATING IT POINTS OFVIEW OR OPINIONS NECESSARILY REPRE RECENT RESEARCH STATED 00 NOTNATIONAL INSTI tUTEOF SENT OFFICIAL OR POLICY EDUCATION POSITION CONCERNING THE DEVELOPMENT OF SPATIAL AND GEOMETRICCONCEPTS Richard Lesh, Editor Diane Mierkiewicz,Technical Editor 4 ERIC Clearinghouse forScience, Mathematics, and EnvironmentalEducation College of-Education The Ohio StateUniversity 1200 Chambers Road, ThirdFloor Columbus, Ohio 43212 May, 1978 it This publication was prepared pursuant to a contract with the National Institute.of Education, United StatesDepartment of Health, Education, and Welfare. Contractors undertaking such projects under. Government sponsorship are encouraged to express freely their judgment in prOfessional and technical matters. Points'of view or opinions do not, therefore, necessarily represent 'National Institute Of Edutation position or policy. MATHEMATICS EDUCATION REPORTS The Mathematics Education. Reports series makes available recent analyses and syntheses of research and development efforts in mathematics educati.on. As a part Or this series, we are pleased to make available this collection of research reports in the area of spatial and geometric concepts. The thirteen studies reported here were coordinated by. the Geometry Working Group of the Georgia Center for the Study of Learning and Teaching Mathematics. Other MatheMatics Education Reports make available information concerning mathematics education documents analyzed at the ERIC Information Analysis Center or Science, Mathematics, and Environmental Educations These reports fall into three broad categories. Research reviews summarize and analyze recent research in specific areas of .mathematics education. Resource guides identify and analyze materizas 0 and references for use by mathematics teachers at all levels. Special °bibliographies announce the availablility of-documents and review the literature in selected interest areas of mathematics education. Priorities for the development of future Mathematics Education Reports are established by the advisory board of the Center, in cooperation with the National Council of Teachers of Mathematics, the Special Interest Group for Research in Mathematics Education, and other professional groups in mathematics education. Individtiqlcommentson past Reports and , suggestions for future Reports are always welcome, Jon L. Higgins Associate Director for Mathematics Education - Th ditors would like to give Ape.cial thanks to the following people who served as referees and'reviewers for the articlesappearing in this monograph, and as consultants when the projects were in the planning stages. Alan Hoffer Mathematics Department University of Oregon Les Steffe 9Department of Mathematics Education University of Georgia Art Coxford School of Mathematics EducatiOn University of Michigan Paul Traf ton Department of Mathematics Education National College of Education The editors would also like to thank themany other members-of ov,.. space and geometry research group who attended planning conferences and completed projects that,do not appear in thiS monograph. The editors believe that.the existence of a successful, non-funded, multi-institu- tional, cooperative research project is in itself a significant research innovation in mathematics education. Richard Lesh Diane Mierkiewicz Editor Technjcal Editor v /vs FOreward c . The Georgia Center for the. Study of Learning and Teaching Mathematics is a consortium of investigators dedicated to the improvementof the mathematical education'of children. The consortium is currently organized into three Projects and nine Working Groups, listed below. Each Project and Working Group has general Projects Conceptual Development Problem Solving Teaching Strategies and of Mathematics Leaders Leslie P. Steffe Larry Hatfield Thomas. J. Cooney Applications Combinations of Analyzing Teacher Heuristics Behavior Richard Lesh J. Philip Smith "Thomas J. Cooney ah co Models for Task Variables Protocols Development a) Learning Mathematics William Geeslin Gerald Kulm Kenneth Retzer Number and Measurement Teaching Strategies o. John DosSey 3.4 Grayson Wheatley 00 IP Rational Numbers Thomes.Kieren 0 Space and Geometry' . Izzie Weinzweig and specific goals.. The general goals are commonand constitute the goals of the Center as an organization. These goals were established as a result of a meeting of the Working Group Leaders in. October of 1976. Specific goals are emerging for each Working Group. The goals of the Center are listed below,. (1) Establish and maintain a lansortium of investigatorsdedicated to the improvement of mathematical education in the elementary'and secondaryschools through disciplined` inquiry. (2) Identify fundamental problems and issues within mathematics education. (3) Conduct studies in mathematics education which contributesignificantly. to the organization of a particular field ofinvestigation or to the resolution of fundamental problems and issues. vii gr (4) Promote interinstitutional cooperation which most efficiently utilizes available human resourc6s to address fundamental problems and issues in mathe- matics education. (5) PrOmote domMunication'among investigators in mathematics education and allied disciplines through publication of Center studies. (6) Translate knowledge in mathematics education into a form useable by eahool practitioners. This monograph represents a significant step toward realization of Goals 1-5 abov,e. The investigators-of the Space and Geometry,,Working Group are to be congra- tulated for their intensive efforts during the time the "--studies were in progress. A great deal of effort has been also expended by the faculty and students at Northwestern University in preparation of the monograph. It is only through such dedicated efforts that coordinated research can be conducted. The investigators of the Center view the utilization by school practitioners of knowledge generated through research efforts as both desirable and realistic. A next immediate step to be taken after the publication of 'a monograph such as this one is to generatg interpretative reports of the theory and research. The Center Policy Board urges any indiv:TA-1.Y.1:-.:erested in such interpretatiire reports to col;aborate with the Center in preparation of the documents. The Policy Board of the Center would like to thank- ERIC/SMEAC of The Ohio State UniversityAor publishing this, and various other, monographs of the Center., This cooperation has greatly facilitated ongoing studies and has made the monograph published available to, any interested individual. Other monographs and working papers are available and may be obtained by writing any member of the Policy Board. The-Po Policy -Board Leslie P. Steffe, Director Thomas J. Cooney Larry L. Hatfield Jeremy Kilpatrick viii 7 -4 Contents Introduction