Investigations into Assessment in Mathematics Education: An ICMI Study

272 Pages·1993·25.522 MB·English

by  Mogens Niss (auth.), Mogens Niss (eds.)

Investigations into Assessment in Mathematics Education

Investigations into
Assessment in
Mathematics Education
An ICMI Study
edited by
Mogens Niss
Roskilde University, Denmark
~.
"
Springer-Science+Business Media, B.V.
ISBN 978-90-481-4232-3 ISBN 978-94-017-1974-2 (eBook)
DOI 10.1007/978-94-017 -197 4-2
03-0298-150 ts
Printed on acid-free paper
All Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
MOGENS NISS
Assessment in Mathematics Education and its Effects:
An Introduction 1
JEREMY KILPATRICK
The Chain and the Arrow:
From the History of Mathematics Assessment 31
GEOFFREY HOWSON
The Relationship Between Assessment, Curriculum and Society 47
JIM RIDGWAY & DON PASSEY
An International View of Mathematics Assessment
- Through a Class, Darkly 57
PETER GALBRAITH
Paradigms, Problems and Assessment:
Some Ideological Implications 73
DAVID WHEELER
Epistemological Issues and Challenges to Assessment:
What is Mathematical Knowledge? 87
THOMAS A. ROMBERG
How One Comes to Know:
Models and Theories of the Learning of Mathematics 97
ANTOINE BODIN
What Does to Assess Mean?
The Case of Assessing Mathematical Knowledge 113
STIEG MELLIN-OLSEN
A Critical View of Assessment in Mathematics Education:
Where is the Student as a Subject? 143
HERBERT P. GINSBURG & SUSAN F. JACOBS & LUZ S. LOPEZ
Assessing Mathematical Thinking and Learning Potential
in Primary Grade Children 157
BENGT JOHANSSON
Diagnostic Assessment in Arithmetic 169
JOHN IZARD
Challenges to the Improvement of Assessment Practice 185
MALCOLM SWAN
Improving the Design and Balance of Mathematical Assessment 195
DEREK FOXMAN
The Assessment of Performance Unit's
Monitoring Surveys 1978-1987 217
DAVID F. ROBITAILLE & J. STUART DONN
TIMSS: The Third International Mathematics and Science Study 229
GIlA HANNA
The Validity of International Performance Comparisons 245
NORMAN L. WEBB
Visualizing a Theory of the Assessment
of Students' Knowledge of Mathematics 253
Index 265
MOGENS NISS
ASSESSMENT IN MATHEMATICS EDUCATION
AND ITS EFFECTS: AN INTRODUCTION
1. THE PRESENT BOOK AS AN ICMI STUDY
The present book, Investigations into Assessment in Mathematics Education,
is one of two studies resulting from an ICMI Study Conference on
Assessment in Mathematics Education and Its Effects. The book which is
published in the series of ICMI Studies under the general editorship of the
President and Secretary of ICMI is closely related to another study
resulting from the same conference: Cases of Assessment in Mathematics
Education (Niss, 1992). The two books, although originating from the same
sources and having the same editor, emphasize different aspects of
assessment in mathematics education and can be read independently of one
another. While the present book is devoted to critically analyzing general
and principal aspects of assessment, the other study presents and discusses
cases of assessment that have been implemented in actual practice.
Naturally, the content of either book is enriched by the materials and
perspectives provided by the other one.
In order to put this book and its background into context, the nature and
scope of the ICMI studies are outlined briefly below.
Since 1986 the International Commission on Mathematical Instruction
(ICMI) has been engaged in publishing a series of studies on essential
topics and key issues in mathematics education. Previously, the following
studies have been published (all by Cambridge University Press): School
Mathematics in the 1990s (Howson & Wilson, 1986), The Influence of
Computers and Informatics on Mathematics and Its Teaching (Churchhouse
et aI., 1986), Mathematics as a Service Subject (Howson, et aI., 1988), The
PopUlarization of Mathematics (Howson, et aI. 1990), Mathematics and
Cognition: A Research Synthesis by the International Group for the Psychology
of Mathematics Education (Nesher & Kilpatrick, 1990).
Depending on the theme under consideration a study may either be
research oriented or action oriented (or both). In either case the aim is to
provide an up-to-date presentation and analysis of the state-of-the-art
concerning a theme, whether by identifying and describing current research
contributions and their findings, or by identifying and discussing crucial,
non-rhetorical issues involving genuine controversies or dilemmas and the
different positions towards them held by various mathematics educators.
M. Niss (ed.},lnvestigations into Assessment in Mathematics Education. 1-30.
© 1993 Kluwer Academic Publishers.
2 MOGENS NISS
In order to provide a platform for producing an ICMI study the
following normal procedure has been adopted (the exception is the study
on cognition). The Executive Committee of ICMI appoints a fairly small,
international Program Committee. Its first task is to write a so-called
Discussion Document that outlines the theme, the aims, and the scope of
the study, and presents the items and issues to be dealt with. The
Discussion Document is published in international journals (including the
official organ of ICMI, L 'Enseignement mathematique) and newsletters with
an invitation to mathematics educators to respond to the Document and to
apply for participation in a so-called Study Conference.
The Study Conference is held with a limited number (50-100) of
individuals and constitutes a working forum of experts and novices with
ideas, experiences and expertise to investigate the theme of the study. This
investigation is guided by the Discussion Document, assisted by working
papers (written by participants), presentations, debates, and group work.
Finally, the study proper is produced and published under the general
editorship of the President and the Secretary of ICMI, and based on the
written materials and the work done at the Study Conference. As every
study is written and edited as an independent publication for a wide
international readership, its nature is not that of a conference proceedings.
In May 1989 the Executive Committee of ICMI appointed the following
international Program Committee:
Claudi Alsina, local organizer, Universitat Politecnica de Catalunya,
Barcelona, Spain;
Desmond Broomes, University of the West Indies, Bridgetown, Barbados;
Hugh Burkhardt, Shell Centre for Mathematical Education, University of
Nottingham, UK;
Mogens Niss, chairman of the Program Committee, Roskilde University,
Denmark;
Thomas A. Romberg, National Center for Research in Mathematical
Sciences Education, University of Wisconsin-Madison, USA;
David Robitaille, University of British Columbia, Vancouver, Canada;
Jl1lianna Szendrei, O.P.I. (National Institute of Education), Budapest,
Hungary.
The Discussion Document was officially published in L 'Enseignement
mathematique, 36, fasc. 1-2, Janvier-juin 1990, 197-206, as well as in a
number of other journals and newsletters.
The Study Conference which was held at Cap Roig, Calonge, Spain,
11-16 April 1991, had 80 contributing participants from 25 different
countries in Europe, North America and the Caribbean, Asia, Oceania,
Mrica and the Middle East.
ASSESSMENT IN MATHEMATICS AND ITS EFFECTS 3
A Note on Terminology
To begin with some terminological clarification may be in order. The field
we shall be dealing with frequently uses terms such as assessment,
evaluation, tests, exams. However, these words and their counterparts in
other languages carry quite different connotations within different
educational systems and contexts. The variation is so large that the same
word often has different meanings to different people. We shall confine
ourselves to making one distinction, namely between assessment and
evaluation. Further terminological questions will be addressed in later
sections of this chapter.
While assessment and evaluation are often used interchangeably we shall
adopt, as was suggested in the Discussion Document for the present study,
the following terminological convention: Assessment in mathematics
education is taken to concern the judging of the mathematical capability,
performance, and achievement - all three notions to be taken in their
broadest sense - of students whether as individuals or in groups, with the
notion of "student" ranging from Kindergarten pupils to Ph.D. students.
Assessment thus addresses the outcome of mathematics teaching at the
student level. Evaluation in mathematics education is taken to be the
judging of educational or instructional systems, in its entirety or in parts,
as far as mathematics teaching is concerned. Evaluation may concern
system components such as curricula, programs, teachers, teacher training,
and specific segments of the educational system such as schools or school
districts etc. So, evaluation addresses mathematics education at the systems
level.
When tests and exams are considered to be ways of judging student
performance they are special forms of assessment and are thus subsumed
under the assessment category. As a contradistinction, when tests and
exams are viewed as being part of the modes of operation of an education
al (sub)system, or when the outcomes of tests and exams are used as
indicators of the quality of such a system, as is the case with international
performance comparisons, exams and tests belong to the realm of
evaluation. This duality shows features of the general relationship between
assessment and evaluation: Assessment items - in particular assessment
results, but also assessment modes - may be involved in the judging of
system aspects, hence they would form part of an evaluation activity. The
converse normally will not hold for evaluation; for instance the appraisal
of teachers will often involve a multitude of components having nothing to
do with assessment of students. So, the relationship between assessment
and evaluation is not a symmetrical one.
In the present study the emphasis will be on assessment as defined above
rather than on evaluation. Due to the duality just mentioned this does not
imply that evaluation issues will not be considered. However, only those