DOCUMENT RESUME ED 416 083 SE 061 120 AUTHOR Pehkonen, Erkki, Ed. TITLE Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (21st, Lahti, Finland, July 14-19, 1997). Volume 2. INSTITUTION International Group for the Psychology of Mathematics Education. ISSN ISSN-0771-100X PUB DATE 1997-00-00 NOTE 337p.; For Volumes 1-4, see SE 061 119-122. PUB TYPE Collected Works Proceedings (021) EDRS PRICE MF01/PC14 Plus Postage. DESCRIPTORS Communications; *Educational Change; *Educational Technology; Elementary Secondary Education; Foreign Countries; Higher Education; *Mathematical Concepts; Mathematics Achievement; *Mathematics Education; Mathematics Skills; Number Concepts IDENTIFIERS *Psychology of Mathematics Education ABSTRACT The second volume of the proceedings of 21st annual meeting of the International Group for the Psychology of Mathematics Education contains the following papers: (1) "The Dilemma of Transparency: Seeing and Seeing through Talk in the Mathematics Classroom"(J. Adler); (2) "Abstraction is Hard in Computer-Science Too"(D. Aharoni and U. Leron); (3) "Constructing Purpose in Mathematical Activity"(J. Ainley); (4) "Effective Teachers of Numeracy in UK Primary Schools: Teachers' Beliefs, Practices and Pupils' Learning"(M. Askew, M. Brown, V. Rhodes, D. Wiliam and D. Johnson); (5) "Can the Average Student Learn Analysis?"(R. R. Baldino, A. Buttner Ciani and A. Carolina Leal); (6) "Cognitive Units, Connections and Mathematical Proof"(T. Barnard and D. Tall); (7) "Subjective Elements in Children's Comparison of Probabilities" (M.J. Canizares, C. Batanero, L. Serrano and J.J. Ortiz); (8) "Reunitizing Hundredths: Prototypic and Nonprototypic Representations" (A.R. Baturo, and J. Cooper); (9) "Students' Perceptions of the Purposes of Mathematical Activities"(A. Bell, R. Phillips, A. Shannon, and M. Swan); (10) "Stereotypes of Literal Symbol Use in Senior School Algebra" (L. Bills); (11) "Approaching Theoretical Knowledge through Voices and Echoes: A Vygotskian Perspective" (P. Boero, B. Pedemonte, and E. Robotti); (12) "The Transition from Arithmetic To Algebra: Initial Understanding of Equals, Operations and Variable" (T.J. Cooper, G. M. Boulton-Lewis, B. Atweh, H. Pillay, L. Wilss & S. Y. Mutch); (13) "Exploring Imagery in P, M and E"(C. Breen); (14) "Teachers' Framework for Understanding Children's Mathematical Thinking" (G.W. Bright, A.H. Bowman and N.N. Vacc); (15) "The Story of Sarah: Seeing the General in the Particular?" (L. Brown, and A. Coles);(16) "Effective Teachers of Numeracy in UK Primary School: Teachers' Content Knowledge and Pupils' Learning"(M. Brown, M. Askew, V. Rhodes, D. Wiliam and D. Johnson);(17) "Metaphorical Thinking and Applied Problem Solving: Implications for Mathematics Learning" (S. Carreira);(18) "Algebra as Language in Use: A Study with 11-12 Year Olds using Graphic Calculators" (T.E.A. Cedillo);(19) "Emergence of Novel Problem Solving Activity" (V. Cifarelli);(20) "NESB Migrant Students Studying Mathematics: Vietnamese Students in Melbourne and Sydney" (P.C. Clarkson and L. Dawe); (21) "Young Children's Concepts of Shape" (D.H. Clements, J. Sarama +++++ ED416083 HasMulti-page SFR--- Level =l +++++ (22) "Learning fromDistributed Theories of and S. Swaminathan); Indonesian Student Teacher Intelligence" (P. Cobb);(23) "Australian and Classic Ratio Task" (J.Conroy Beliefs about Mathematicsand Performance on a Arithmetic To Algebra: ACognitive (24) "The Transition from and B. Perry); Atweh, H. Pillay, L.Wilss Perspective" (G.M.Boulton-Lewis, T.J. Cooper, B. Intelligent Tutoring Systems:An Example and Mutch);(25) "A New Approach for Activities" (G. Bueno and C.A.Cuevas);(26) "Early for Statistical Children: The Development of AlgebraicRepresentation Among 6-13 year-old and F. Jorge-Tarcisio Da Importance of DidacticContract" (L.A.P. Brito in MathematicalProblem-Solving" (V.A. Rocha);(27) "The Affective Domain (28) "Creating a SharedContext: The Use of a DeBellis and G.A. Goldin); K. McClain, and Multimedia Case in a TeacherDevelopment Course" (H.M. Doerr, Frame To InterpretStudents' J. Bowers); (29)"Triple Approach: A Theoretical Drouhard and C. Sackur);(30) "When Does a Way of Activity in Algebra" (T.-P (31) "Development Working Become a Methodology"(J. Duff in and A. Simpson); Problem Posing" (L.D.English); (32) "A Close Look of Seventh-Grade Students' (R. Mathematics-Classroom-Situation Casesin Teacher Education" at the Use of Strategies in AdvancedAlgebraic Even and Z. Markovits);(33) "Action-Based "Tacit Mechanism ofCombinatorial Problem Solving" (P.L.Ferrari); (34) (35) "Shadows on Proof" (F. Intuitions" (E. Fischbeinand A. Grossman); Lines--What is the Problem? Furinghetti and D. Paola);(36) "Perpendicular of Knowledge on How ToCope with Students' Pre-service Teachers' Lack Schemata in Difficulties" (H. Gal & S.Vinner);(37) "Actions and Invariant Problems" (J.A. Garcia-Cruzand A. Martinon); (38) "A Linear Generalizing Study" (R. Garuti); Classroom Discussion and anHistorical Dialogue: A Case Social Structure inDeveloping a Critical Social (39) "The Importance of (40) "Meaning of Proofsin Psychology of MathematicsEducation" (P. Gates); Godino and A.M. Recio); and(41) "A Visual Mathematics Education" (J.D. (F.V. Gomes and C. Presentation of a LongitudinalStudy: Design and Analysis" Hoyles). (ASK) ******************************************************************************** Reproductions supplied by EDRS arethe best that can bemade from the original document. ******************************************************************************** Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education ERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL edited by HAS BE N GRANTED Y Erkki Pehkonen THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement UCATIONAL RESOURCES INFORMATION CENTER (ERIC) This document has been reproduced as received from the person or organization E lginating it. Minor changes have been made to improve reproduction quality. Points of view or opinions stated in this document do not necessarily represent July 14-19, 1997 official OERI position or policy. Lahti, Finland University of Helsinki Volume 2 Lahti Research and Training Centre 1997 2 EST COPY AVMLABLE 21st Conference of the International Group for the Psychology of Mathematics Education University of Helsinki Lahti Research and Trainin Centre BEST COPYAVAILABLE Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education Volume 2 Editor. Erkki Pehkonen Department of Teacher Education University of Helsinki P.O. Box 38 (Ratakatu 6A) FIN-00014 Helsinki Finland Fax: 358-9-191-8073 Email: EPehkonen @bulsa.helsinki.fi Copyright © 1997 left to the Authors All rights reserved ISSN 0771-100X Logo: The logo of the PME 21 Conference has been designed by Paivi Susiluoto. Printed by Gummerus Jyvaskyla, Finland 4 VOLUME 2 Table of contents Research Reports Adler ill 2-1 The dilemma of transparency: seeing and seeing through talk in the mathematics classroom Aharoni Dan & Leron Uri 2-9 Abstraction is hard in computer-science too Ainley Janet 2-17 Constructing purpose in mathematical activity Askew Mike, Brown Margaret, Rhodes Valerie, Wiliam Dylan & Johnson 2-25 David Effective teachers of numeracy in UK primary schools: teachers' beliefs, practices and pupils' learning Baldino Roberto Ribeiro, Ciani Andreia Batter & Leal Ana Carolina 2-33 Can the average student learn analysis? Barnard Tony & Tall David 2-41 Cognitive units, connections and mathematical proof Canizares M. Jesus, Batanero Carmen, Serrano Luis & Ortiz J. Jesus 2-49 Subjective elements in children's comparison of probabilities Baturo Annette R. & Cooper Tom J. 2-57 Reunitising hundredths: prototypic and non prototypic representations Bell Alan, Phillips Richard, Shannon Ann & Swan Malcolm 2-65 Students' perceptions of the purposes of mathematical activities Bills Liz 2-73 Stereotypes of literal symbol use in senior school algebra Boero Paolo, Pedemonte Bettina & Robotti Elisabetta 2-81 Approaching theoretical knowledge through voices and echoes: a Vygotskian perspective Cooper T.J., Boulton-Lewis G. M., Atweh B., Pillay H., Wilss L. & Mutch S. 2-89 The transition from arithmetic to algebra: initial understanding of equals, operations and variable Breen Chris 2-97 Exploring imagery in P, M and E Bright George W., Bowman Anita H. & Vacc Nancy Nesbitt 2-105 Teachers' framework for understanding children's mathematical thinking Brown Laurinda & Coles Alf 2-113 The story of Sarah: seeing the general in the particular? Brown Margaret, Askew Mike, Rhodes Valerie, Wiliam Dylan & Johnson 2-121 David Effective teachers of numeracy in UK primary school: teachers' content knowledge and pupils' learning Carreira Susana 2-129 Metaphorical thinking and applied problem solving: implications for mathematics learning 1 Cedillo Tenoch E. Avalos 2437 Algebra as language in use: a study with 11-12 year olds using graphic calculators Cifarelli Victor 2-145 Emergence of novel problem solving activity Clarkson Philip C. & Dawe Lloyd 2-153 NESB migrant students studying mathematics: Vietnamese students in Melbourne and Sydney Clements Douglas H., Sarama Julie & Swaminathan Sudha 2-161 Young children's concepts of shape Cobb Paul 2-169 Learning from distributed theories of intelligence Conroy John & Perry Bob 2-177 Australian and Indonesian student teacher beliefs about