ERIC ED416083: Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (21st, Lahti, Finland, July 14-19, 1997). Volume 2.

338 Pages·1997·4.1 MB·English

by  ERIC

DOCUMENT RESUME
SE 061 120
ED 416 083
Pehkonen, Erkki, Ed.
AUTHOR
Proceedings of the Conference of the International Group for
TITLE
the Psychology of Mathematics Education (21st, Lahti,
Finland, July 14-19, 1997). Volume 2.
International Group for the Psychology of Mathematics
INSTITUTION
Education.
ISSN-0771-100X
ISSN
1997-00-00
PUB DATE
337p.; For Volumes 1-4, see SE 061 119-122.
NOTE
Proceedings (021)
Collected Works
PUB TYPE
MF01/PC14 Plus Postage.
EDRS PRICE
Communications; *Educational Change; *Educational
DESCRIPTORS
Technology; Elementary Secondary Education; Foreign
Countries; Higher Education; *Mathematical Concepts;
Mathematics Achievement; *Mathematics Education; Mathematics
Skills; Number Concepts
*Psychology of Mathematics Education
IDENTIFIERS
ABSTRACT
The second volume of the proceedings of 21st annual meeting
of the International Group for the Psychology of Mathematics Education
"The Dilemma of Transparency: Seeing and
contains the following papers:
(1)
Seeing through Talk in the Mathematics Classroom"
(J. Adler);
(2)
(D. Aharoni and U. Leron);
"Abstraction is Hard in Computer-Science Too"
(3)
"Effective
(J. Ainley);
"Constructing Purpose in Mathematical Activity"
(4)
Teachers of Numeracy in UK Primary Schools: Teachers' Beliefs, Practices and
(M. Askew, M. Brown, V. Rhodes, D. Wiliam and D. Johnson);
Pupils' Learning"
(R. R. Baldino, A. Buttner
"Can the Average Student Learn Analysis?"
(5)
"Cognitive Units, Connections and
Ciani and A. Carolina Leal);
(6)
"Subjective Elements in
(T. Barnard and D. Tall);
Mathematical Proof"
(7)
Children's Comparison of Probabilities" (M.J. Canizares, C. Batanero, L.
"Reunitizing Hundredths: Prototypic and
Serrano and J.J. Ortiz);
(8)
Nonprototypic Representations" (A.R. Baturo, and J. Cooper);
"Students'
(9)
Perceptions of the Purposes of Mathematical Activities"
(A. Bell, R.
"Stereotypes of Literal Symbol Use
Phillips, A. Shannon, and M. Swan);
(10)
"Approaching Theoretical Knowledge
in Senior School Algebra" (L. Bills);
(11)
through Voices and Echoes: A Vygotskian Perspective" (P. Boero, B. Pedemonte,
"The Transition from Arithmetic To Algebra: Initial
and E. Robotti);
(12)
Understanding of Equals, Operations and Variable" (T.J. Cooper, G. M.
Boulton-Lewis, B. Atweh, H. Pillay, L. Wilss & S. Y. Mutch); (13) "Exploring
"Teachers' Framework for
Imagery in P, M and E"
(C. Breen);
(14)
Understanding Children's Mathematical Thinking" (G.W. Bright, A.H. Bowman and
"The Story of Sarah: Seeing the General in the Particular?"
N.N. Vacc);
(15)
(16) "Effective Teachers of Numeracy in UK Primary
(L. Brown, and A. Coles);
(M. Brown, M.
School: Teachers' Content Knowledge and Pupils' Learning"
"Metaphorical Thinking and
Askew, V. Rhodes, D. Wiliam and D. Johnson);
(17)
Applied Problem Solving: Implications for Mathematics Learning"
(S.
(18) "Algebra as Language in Use: A Study with 11-12 Year Olds
Carreira);
"Emergence of Novel Problem
using Graphic Calculators" (T.E.A. Cedillo);
(19)
(20) "NESB Migrant Students Studying
Solving Activity" (V. Cifarelli);
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
Multi-page SFR--- Level =l +++++
+++++ ED416083 Has
Distributed Theories of
(22) "Learning from
and S. Swaminathan);
Indonesian Student Teacher
(23) "Australian and
Intelligence" (P. Cobb);
Conroy
Classic Ratio Task" (J.
and Performance on a
Beliefs about Mathematics
Cognitive
Arithmetic To Algebra: A
"The Transition from
(24)
and B. Perry);
Wilss
Atweh, H. Pillay, L.
Boulton-Lewis, T.J. Cooper, B.
Perspective" (G.M.
An Example
Intelligent Tutoring Systems:
(25) "A New Approach for
and Mutch);
(26) "Early
Cuevas);
Activities" (G. Bueno and C.A.
for Statistical
Children: The
Representation Among 6-13 year-old
Development of Algebraic
and F. Jorge-Tarcisio Da
Contract" (L.A.P. Brito
Importance of Didactic
Problem-Solving" (V.A.
in Mathematical
(27) "The Affective Domain
Rocha);
Context: The Use of a
(28) "Creating a Shared
DeBellis and G.A. Goldin);
K. McClain, and
Development Course" (H.M. Doerr,
Multimedia Case in a Teacher
Students'
Frame To Interpret
"Triple Approach: A Theoretical
J. Bowers); (29)
(30) "When Does a Way of
Drouhard and C. Sackur);
Activity in Algebra" (T.-P
(31) "Development
(J. Duff in and A. Simpson);
Working Become a Methodology"
English); (32) "A Close Look
Problem Posing" (L.D.
of Seventh-Grade Students'
(R.
in Teacher Education"
Mathematics-Classroom-Situation Cases
at the Use of
Algebraic
Strategies in Advanced
(33) "Action-Based
Even and Z. Markovits);
Combinatorial
"Tacit Mechanism of
Ferrari); (34)
Problem Solving" (P.L.
(35) "Shadows on Proof"
(F.
and A. Grossman);
Intuitions" (E. Fischbein
Lines--What is the Problem?
(36) "Perpendicular
Furinghetti and D. Paola);
Cope with Students'
of Knowledge on How To
Pre-service Teachers' Lack
Schemata in
(37) "Actions and Invariant
Vinner);
Difficulties" (H. Gal & S.
and A. Martinon); (38) "A
Problems" (J.A. Garcia-Cruz
Linear Generalizing
Study" (R. Garuti);
Historical Dialogue: A Case
Classroom Discussion and an
Developing a Critical Social
Social Structure in
(39) "The Importance of
in
(40) "Meaning of Proofs
Education" (P. Gates);
Psychology of Mathematics
(41) "A Visual
Godino and A.M. Recio); and
Mathematics Education" (J.D.
(F.V. Gomes and C.
Study: Design and Analysis"
Presentation of a Longitudinal
(ASK)
Hoyles).
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Proceedings
of the
21st Conference
of the
International Group for the
Psychology of Mathematics Education
ERMISSION TO REPRODUCE AND
edited by
DISSEMINATE THIS MATERIAL
Erkki Pehkonen
HAS BE N GRANTED Y
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
E
received from the person or organization
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
EST COPY AVMLABLE
2
21st Conference
of the
International Group for the
Psychology of Mathematics Education
University of Helsinki
Lahti Research and Trainin Centre
BEST COPY
AVAILABLE
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: [phone]
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
2-17
Ainley Janet
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?
2-41
Barnard Tony & Tall David
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
2-57
Baturo Annette R. & Cooper Tom J.
Reunitising hundredths: prototypic and non prototypic representations
2-65
Bell Alan, Phillips Richard, Shannon Ann & Swan Malcolm
Students' perceptions of the purposes of mathematical activities
2-73
Bills Liz
Stereotypes of literal symbol use in senior school algebra
2-81
Boero Paolo, Pedemonte Bettina & Robotti Elisabetta
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
2-97
Breen Chris
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
2-113
Brown Laurinda & Coles Alf
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
2-129
Carreira Susana
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 mathematics and
performance on a classic ratio task
Boulton-Lewis G.M., Cooper T. J., Atweh B., Pillay H., Wilss L. & Mutch S.
2-185
The transition from arithmetic to algebra: a cognitive perspective
Bueno Graciela & Cuevas Carlos A.
2-193
A new approach for intelligent tutoring systems: an example for statistical
activities
Brito Lima Anna Paula & Da Rocha Falcao Jorge Tarcisio
2-201
Early development of algebraic representation among 6-13 year-old children: the
importance of didactic contract
DeBellis Valerie A. & Goldin Gerald A.
2-209
The affective domain in mathematical problem-solving
Doerr Helen M., McClain Kay & Bowers Janet
2-217
Creating a shared context: the use of a multimedia case in a teacher development
course
Drouhard Jean - Philippe & Sackur Catherine
2-225
Triple approach: a theoretical frame to interpret students' activity in algebra
Duffin Janet & Simpson Adrian
2-233
When does a way of working become a methodology?
English Lyn D.
2-241
Development of seventh-grade students' problem posing
Even Ruhama & Markovits Zvia
2-249
A close look at the use of mathematics-classroom-situation cases in teacher
education
Ferrari Pier Luigi
2-257
Action-based strategies in advanced algebraic problem solving
Fischbein Efraim & Grossman Aline
2-265
Tacit mechanism of combitonatorial intuitions
Furinghetti Fulvia & Paola Domingo
2-273
Shadows on proof
Gal Hagar & Vinner Shlomo
2-281
Perpendicular lines
what is the problem?
Garcia-Cruz, Juan Antonio & Martinon Antonio
2-289
Actions and invariant schemata in linear generalising problems
Garuti Rossella
2-297
A classroom discussion and an historical dialogue: a case study
2-iv
2-305
Gates Peter
The importance of social structure in developing a critical social psychology of
mathematics education
2-313
Godino Juan D. & Recio Angel M.
Meaning of proofs in mathematics education
2-321
Gomes Ferreira Veronica & Hoyles Celia
A visual presentation of a longitudinal study: design and analysis
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2-v
RESEARCH REPORTS
8
THE DILEMMA OF TRANSPARENCY:
SEEING AND SEEING THROUGH TALK IN THE MATHEMATICS CLASSROOM
Jill Adler, Witwatersrand University
In this paper, talk is understood as a tool and resource for mathematical learning in school.
As a resource it needs to be seen (be visible) to be used, and as a tool it needs to be seen
through Ito be invisible) to provide access to mathematical learning. This paper argues that
the dual function of visibility and invisibility of talk in mathematics classrooms creates
dilemmas for teachers. An analytic narrative vignette drawn from a secondary mathematics
classroom in South Africa illustrates the 'dilemma of transparency' that mathematics
teachers face, particularly if they are teaching multilingual classes.
INTRODUCTION
The paper draws from a study of South African secondary mathematics teachers'
knowledge of their practices in their multilingual classrooms (Adler, 1996a). In initial
interviews, English-speaking teachers whose 'whites only' classrooms had recently and
rapidly become racially integrated argued the benefit to all learners of explicit mathematics
language teaching (Adler, 1995). This implies that language itself, and particularly talk,
becomes the object of attention in the mathematics class and a resource in the teaching-
learning process. Now that their classes included pupils whose main language was not
English, it became obvious to these teachers that they needed to be more explicit about
instructions for tasks, as well as mathematical terms and the expression of ideas.
In follow-up workshops in the study, Helen specifically problematised the issue of explicit
language teaching. She has tried to develop mathematical language teaching as part of her
practice in her multilingual classroom. However, as she sees and reflects on her teaching
she begins to question what this means in practice and whether and how explicit
is
mathematics language teaching actually helps. And we are alerted to a dilemma: There
always the problem in explicit language teaching of 'going on too long', of focusing too
much on what is said and how it is said. Yet explicit mathematics language teaching
appears to be a primary condition for access to mathematics, particularly for those pupils
whose main language is not English or for those pupils less familiar with educated discourse.
This paper argues that Lave and Wenger's idea that access to a practice requires its
resources to be 'transparent', while not usually applied to language as a resource, nor to
learning in school, is useful and illuminating here. Explicit mathematics language teaching,
where teachers attend to pupils' verbal expressions as a public resource for whole class
teaching, offers possibilities for enhancing access to mathematics, especially'in multilingual
classrooms. However, such practices easily slip into possibilities for alienation through a
shift of attention off the mathematical problem and onto language per se. Teachers'
decision-making at critical moments, while always a reflection of both their personal identity
and their teaching context, requires the ability to shift focus off and then back onto the
mathematical problem. The challenge, of course, is when and how such shifts are best for
whom and for what. These assertions will be instantiated and illuminated through an
analytic narrative vignette (Erickson, 1986) based on an episode in Helen's multilingual Std
9 (Grade 11) trigonometry class.
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