[Maths-Education] Conference Announcement - ALGEBRA/Int'l <fwd>
Laurinda Brown
Laurinda.Brown@bristol.ac.uk
Wed, 20 Dec 2000 10:30:29 +0000
--- Begin Forwarded Message ---
Date: Tue, 19 Dec 2000 17:26:14 -0800
From: Rosamund Sutherland <ros.sutherland@bristol.ac.uk>
Subject: Conference Announcement - ALGEBRA/Int'l <fwd>
Sender: Ros.Sutherland@bristol.ac.uk
CALL FOR PAPERS by January 31, 2001
The Twelfth ICMI Study: THE FUTURE OF THE TEACHING AND LEARNING OF ALGEBRA
University of Melbourne, 10-14 December 2001
Details may be found at the web site
http://www.edfac.unimelb.edu.au/DSME/icmi-algebra/
The Discussion Document appears below and can be seen at the website.
Study Secretary:
Dr Helen Chick,
Department of Science and Mathematics Education,
University of Melbourne, Victoria 3010, Australia.
Tel: +61 3 8344 8538 Fax: +61 3 8344 8739 <h.chick@edfac.unimelb.edu.au>
Program Chair:
Professor Kaye Stacey,
Department of Science and Mathematics Education,
University of Melbourne, Victoria 3010, Australia
<k.stacey@edfac.unimelb.edu.au>
____________________________
Discussion Document for the Twelfth ICMI Study
The Future of the Teaching and Learning of Algebra
INTRODUCTION
This document introduces a new ICMI study entitled The Future of the=20
Teaching and Learning of Algebra, to be held at the University of=20
Melbourne (Australia) in December 2001. The intention is that the=20
word 'Algebra' will be interpreted broadly to encompass the diversity=20
of definitions around the world, extending beyond the standard=20
curriculum in some countries. It will include, for example, algebra=20
as a language for generalisation, abstraction and proof; algebra as a=20
tool for problem solving through equation solving or graphing; for=20
modelling with functions; and the way algebraic symbols and ideas are=20
used in other parts of mathematics and other subjects. The principal=20
interest of many participants is likely to be related to secondary=20
school mathematics (ages 11 - 18) and algebra with real variables,=20
but the study is also concerned with tertiary algebra (e.g. linear=20
algebra and abstract algebra) and with algebra and its precursors for=20
young children.
There are many reasons why it is timely to focus on the future of the=20
teaching and learning of algebra. We are at a critical point when it=20
is desirable to take stock of what has been achieved and to look=20
forward to what should be done and what can be done. In many=20
countries, increasing numbers of students are now receiving secondary=20
education and this is causing every part of the mathematics=20
curriculum to be scrutinised. For algebra, perhaps more than other=20
parts of mathematics, concerns of equity and of relevance arise. As=20
the language of higher mathematics, algebra is a gateway to future=20
study and mathematically significant ideas, but it is often a wall=20
that blocks the paths of many. Should algebra be made more accessible=20
to more students by changing the amount or nature of what is taught?=20
Many countries have already embarked on such changes, hoping to=20
increase access and success. Alternatively, are these changes=20
necessary: is algebra truly useful for the majority of people and,=20
even if it is, will it be useful in the future?
An algebra curriculum that serves its students well in the coming=20
century may look very different from an ideal curriculum from some=20
years ago. The increased availability of computers and calculators=20
will change what mathematics is useful as well as changing how=20
mathematics is done. At the same time as challenging the content of=20
what is taught, the technological revolution is also providing rich=20
prospects for teaching and is offering students new paths to=20
understanding. In the past two decades, a substantial body of=20
research on the learning and teaching of many aspects of algebra has=20
been established and there have been many experiments with adapting=20
curricula and teaching methods. There is therefore a strong=20
scientific basis upon which to build this study.
Outline of the program
The study has two aims: to make a synthesis of current thinking and=20
lessons from the past which will help set directions for future work=20
in the field, and to suggest guidelines for advancing the teaching=20
and learning of algebra. Following the pattern of previous ICMI=20
studies, this study will have two components: an invited study=20
conference and a study volume to appear in the ICMI Study Series,=20
which will share the findings with a broad international audience. A=20
report will also be made at ICME-10 in 2004. The study conference=20
program will therefore contain plenary and sub-plenary lectures,=20
working groups and panels. At least two panels are planned. One will=20
attempt to make explicit some perspectives on algebra, algebra=20
activity, algebraic thinking or algebraic understanding. A second=20
aims to highlight the significant differences in algebra education=20
around the world and identify the main strands in the goals, content=20
and teaching methods of this worldwide enterprise. A major part of=20
the working time will be spent in working groups addressing different=20
aspects of the study problem. Working groups are likely to be=20
established to correspond with each of the sections listed below.
WHY ALGEBRA?
The technological future of a modern society depends in large part on=20
the mathematical literacy of its citizens and this is reflected in=20
the worldwide trend towards mass secondary education. For an=20
individual, algebra is a gateway to much of higher education and=20
therefore to many fields of employment. Educators also argue that=20
algebra is part of cultural heritage and is needed for informed and=20
critical citizenship. However, for many, algebra acts more like a=20
wall than a gateway, presenting an obstacle that they find too=20
difficult to cross. This section of the study is concerned with the=20
significance of algebra for the broad population of secondary school=20
students, recognising that regional and cultural differences may=20
impact upon the answers in interesting ways. It addresses questions=20
such as:
=85 Should algebra be taught to all? There has been a call for algebra=20
for all secondary students, but what aspects of algebra are of value=20
to all? What should comprise a minimal curriculum? How do answers to=20
these questions relate to regional or cultural differences?
=85 What do we expect of an algebra-literate individual? What are the=20
values of algebra learning for the individual, especially in view of=20
increasingly powerful computing capabilities? Access to higher=20
learning and employment are two values, but what are the more=20
immediate values and how can they be achieved?
=85 How can we reshape the algebra curriculum so that it has more=20
immediate value to individuals? Can we identify explicit examples in=20
contexts meaningful to students in which algebraic ideas have clear,=20
unambiguous value? Are there undesirable consequences of such=20
orientations to algebra?
=85 How can we reshape the algebra curriculum so that specific=20
difficult ideas are more easily accessed?
APPROACHES TO ALGEBRA
Recent research has focused on a number of approaches for developing=20
meaning for the objects and processes of algebra. These approaches=20
include, but are not limited to, problem-solving approaches,=20
functional approaches, generalisation approaches, language-based=20
approaches, and so on. Problem-solving approaches tend to emphasise=20
an analysis of problems in terms of equations and a view of letters=20
as unknowns. Functional approaches support a different set of=20
meanings for the objects of algebra; for instance, the use of=20
expressions to represent relationships and an interpretation of=20
letters in terms of quantities that vary. A somewhat different=20
perspective is encouraged by generalisation approaches that stress=20
expressions of generality to represent geometric patterns, numerical=20
sequences, or the rules governing numerical relationships-such=20
approaches often serving as a basis for exploring underlying=20
numerical structure, predicting, justifying and proving. Some algebra=20
curricula develop student algebraic thinking exclusively along the=20
lines of one such approach throughout the several grades of secondary=20
school; others attempt to combine facets of several approaches.
Synthesising the experience with and research on the use of various=20
approaches in the teaching/learning of algebra leads to questions=20
such as the following:
=85 What does each of these various teaching approaches mean?
=85 What are the algebraic meanings supported by each?
=85 What are the epistemological obstacles inherent in each?
=85 Which important aspects of algebra are favoured/neglected in each appro=
ach?
=85 What are the difficulties encountered by students in extending the=20
meanings that are developed by each of these approaches to include=20
the meanings inherent in other approaches?
LANGUAGE ASPECTS OF ALGEBRA
This section considers theoretical and applied aspects of the=20
languages and notations of algebra, in relation to teaching and=20
learning. The evolution of algebra cannot be separated from the=20
evolution of its language and notations. Historically the=20
introduction of good notations has had enormous impact upon the=20
development of algebra but a good notation for science may not be a=20
good notation for learning. With new computer technology we are now=20
seeing a flowering of new quasi-algebraic notations, which may offer,=20
support or eventually enforce new notations. However, current=20
theories of mathematics teaching and learning do not seem adequate to=20
deal with learning about notation. It is therefore timely to focus=20
on algebraic notations asking questions such as:
=85 How do theories of mathematics teaching and learning embrace the=20
linguistic aspects of algebra and what can we propose to better take=20
into account these aspects?
=85 Algebra is not a language but it has a language and the two cannot=20
be dissociated. What does it mean to talk about algebra as a language=20
and what are the implications of such a perspective?
=85 There is a wide range of theories of how mathematical concepts are=20
learned and taught (in particular the constructivist theories) but=20
learning a language is not just a matter of learning concepts. How do=20
acknowledged theories of mathematics learning and teaching embrace=20
the non-conceptual aspects of learning the language of algebra and=20
what can we propose to better take these aspects into account?
=85 Would some changes of algebraic language contribute to the=20
development of algebraic thinking, communication and understanding?
=85 Is it feasible and desirable to remove some of the ambiguities that=20
are present in standard mathematical symbolism, for example in the=20
use of the equals sign?
=85 Should some effort be made in the teaching of mathematics to=20
explain and bridge differences in notation between algebra as it is=20
taught in mathematics courses and algebra as it is used in other=20
disciplines?
=85 What are the characteristics of good notation? What does=20
mathematics education research have to say on this? Are some=20
notational choices better for science but others better for learning?
TEACHING AND LEARNING WITH COMPUTER ALGEBRA SYSTEMS
The advent of affordable computer systems and calculators that can=20
perform symbolic calculations may lead to far-reaching changes in=20
mathematics curricula and in mathematics teaching. This section=20
addresses questions that arise from the increasing accessibility of=20
computer symbolic manipulation. Answers to these questions will draw=20
upon established research on the teaching and learning of algebra as=20
well as reporting on recent experimental work. They may suggest new=20
directions for research, including:
=85 For which students and when is it appropriate to introduce the use=20
of a computer algebra system? When do the advantages of using such a=20
system outweigh the effort that must be put into learning to use it?=20
Are there activities using such systems that can be profitably=20
undertaken by younger students?
=85 What algebraic insights and 'symbol sense' does the user of a=20
computer algebra system need and what insights does the use of the=20
systems bring?
=85 A strength of computer algebra systems is that they support=20
multiple representations of mathematical concepts. How can this be=20
used well? Might it be over-used?
=85 What are the relationships and interactions between different=20
approaches and philosophies of mathematics teaching with the use of=20
computer algebra systems?
=85 Students using different computational tools solve problems and=20
think about concepts differently. Teachers have more options for how=20
they teach. What impact does this have on teaching and learning?=20
Which types of system support which kinds of learning? Can these=20
differences be characterised theoretically?
=85 What should an algebra curriculum look like in a country where=20
computer algebra systems are freely available? What 'by hand' skills=20
should be retained?
TECHNOLOGICAL ENVIRONMENTS
Recent research, curriculum development, and classroom practice have=20
incorporated a number of technologies to help students develop=20
meaning for various algebraic objects, ideas and processes. These=20
include, but are not limited to, function graphers, spreadsheets,=20
programming languages, one-line programming on calculators, and other=20
specific computer software environments. [Here, we exclude computer=20
algebra systems that are treated elsewhere.] In an attempt to=20
characterise recent research and experience, this section will=20
explore which aspects of specific computer/calculator environments=20
are related to which kinds of algebra learning. This question will be=20
explored in depth for specific examples of such technology, by=20
addressing questions such as the following:
=85 For a given technological environment, what are the implicit=20
assumptions regarding the underlying core aspects of algebra?
=85 Which important aspects of algebra are and are not touched upon by=20
this environment?
=85 What kinds of algebra learning does this environment promote?
=85 What particular limitations are associated with the use of this=20
environment and how can such limitations be dealt with?
=85 To what extent ought the goals of algebra education be affected by=20
the availability of this technology?
=85 To which aspects of algebra learning does this particular=20
technology make a distinctive, unique contribution?
=85 Are there documented long-term consequences of embedding this=20
particular technology in an algebra curriculum, and if so, what are=20
they?
Submissions for this section should include discussion of as many of=20
the above sub-questions as possible, but with particular attention=20
paid to the first two items above.
ALGEBRA WITH REAL DATA
Modelling the behaviour of real things with algebraic functions is=20
fundamental to applications of mathematics. Using real data to teach=20
about functions is therefore important in the curriculum, and can=20
also be highly motivating for students. Moreover, new devices (such=20
as data loggers) and new communications technologies (such as the=20
internet) provide new opportunities for bringing real data into the=20
classroom. Questions such as the following arise:
=85 What new opportunities for using real data have proved to be=20
successful and how do they relate to research on students' learning=20
of functions and other algebraic concepts?
=85 What are the strengths and weaknesses of using real data and how=20
are these best managed in the classroom and in the curriculum?
=85 A commitment to using real data may lead to significant changes in=20
curriculum content and sequence, for example by giving prominence to=20
the exponential function over the quadratic. What changes may be=20
required and what are their consequences?
=85 Interpreting real data can lead students and teachers to question=20
why the world is as it is. What is the role of algebra education in=20
the development of critical thinking about social issues such as=20
economics, health and environment?
USING THE HISTORY OF ALGEBRA
The history of algebra has been used extensively to identify=20
epistemological obstacles in the learning of algebra and to=20
characterise ruptures in the development of algebraic notions.=20
Drawing on the history (or histories) of algebra from around the=20
world, this section aims to analyse significant contributions and the=20
value of these previous uses and also to reflect on possible avenues=20
for research based on new areas, including:
=85 The history of symbolism; that is, the history of ways of=20
representing quantities and operations in calculations;
=85 The history of methods for solving problems;
=85 The history of methods for solving equations;
=85 The history of the interactions of algebra with other mathematical=20
domains (such as geometry); and
=85 The development of the idea of algebraic structures.
EARLY ALGEBRA EDUCATION
This section encompasses two different readings of the title, being=20
concerned with both the algebra education for young children - say=20
age 6 and above-and also the initial steps in more formal algebra=20
education, which happens in some countries when students are about 12=20
years old. An ongoing concern is the relationship between arithmetic=20
and algebra. Previous research has documented ways in which students'=20
limited arithmetical experience can constitute an obstacle to the=20
learning of algebra, so that an earlier start might reduce the=20
problem; approaches have been proposed to achieve that. On the other=20
hand, a much favoured approach to initial algebra education is based=20
on the view of school algebra as generalised arithmetic, in which=20
case an earlier start may not be appropriate. The general point here=20
is that different views on the relationship between arithmetic and=20
algebra will probably result in different views on algebra education,=20
and this most important fact is a central concern in this section.=20
The interest in algebra education for students at an early age is=20
recent, and so there are as yet only a few studies in this area. It=20
is important that answers to the following questions be thoroughly=20
research-based:
=85 How early is "early algebra" and what are the advantages and=20
disadvantages of an early start? How do the answers to these=20
questions link to views on cognitive development and on learning, and=20
on cultural and educational traditions?
=85 What aspects of algebra and algebraic thinking should be part of an=20
early algebra education? Since the symbolic aspect of algebra is so=20
essential, its early introduction may be beneficial, but is an=20
awareness of algebra as a method to solve problems (for example) more=20
important?
=85 What are the consequences of an early start to algebra for teachers=20
and teacher education?
TERTIARY ALGEBRA
Problems exist in the teaching and learning of tertiary algebra=20
courses such as abstract algebra, linear algebra, and number theory.=20
Some are similar to the problems of secondary algebra: students'=20
difficulties with abstraction, concerns of relevance, what to do with=20
computing technology, etc. Other problems such as proof-making or=20
seeing the objects of calculus as algebraic objects seem particular=20
to the tertiary level. The questions below are concerned with these=20
issues of learning and teaching and also with the specific question=20
of education for prospective teachers.
=85 What are the contributions of tertiary algebra courses to the=20
education of prospective secondary mathematics teachers? How do=20
secondary teachers perceive the value of their tertiary algebra=20
courses to their teaching experience?
=85 Secondary algebra has been well researched, and specific obstacles=20
have been found in making the transition from arithmetic thinking to=20
algebraic thinking. Do tertiary level students similarly experience=20
obstacles in making the transition from secondary-level algebraic=20
thinking to that required for the tertiary level?
=85 Why are certain types of definitions difficult for students? For=20
example, why are definitions given in terms of properties to be=20
satisfied (for example, subspaces and group automorphisms) so=20
difficult for students? How can this problem be addressed?
=85 There are specific questions about specific aspects of specific=20
courses in algebra; for example, why do students who seem competent=20
in R^n have difficulty with more concrete questions in R^2 and R^3?=20
How can such questions be resolved?
=85 How does symbolic logic (through statements, connectives,=20
quantifiers, qualified statements, and arguments) affect students'=20
proof-making and their view of the value of proof-making?
=85 Secondary school algebra seems to lead more directly to applied=20
mathematical modelling at the tertiary level, rather than to abstract=20
algebra. What is going on here?
=85 Should secondary students learn more about algebraic structure?
HOW TO PARTICIPATE
The study conference will be held at the University of Melbourne from=20
December 10 to December 14, 2001. As is the normal practice for ICMI=20
studies, participation in the study conference is by invitation,=20
given on the basis of papers submitted. A submitted paper may address=20
issues from a number of sections above but it should identify one=20
section as the primary focus. The pre-proceedings will contain the=20
submissions of all participants and will form the basis for the=20
scientific work of the study conference. The study volume, published=20
after the conference, will contain selected revised contributions and=20
reports. Submissions should pay particular attention to implications=20
for the future of the teaching and learning of algebra. The work may=20
report the results of individual studies (completed or in progress),=20
or offer well-argued opinions. Survey and overview articles are=20
especially welcome.
Submissions are invited from all interested who will be able to make=20
a sound contribution to a scientific meeting. New researchers in the=20
field are especially encouraged to submit, as are those with=20
significant responsibility for curriculum development and=20
implementation. The study conference is a fine opportunity for=20
international exchange, so participants from countries=20
under-represented in mathematics education research meetings are very=20
welcome to submit. We hope that interaction in this study of=20
mathematics teachers from the early years to tertiary levels,=20
mathematics educators and mathematicians will produce new insights=20
and guidelines for future work.
Submissions should be a paper 5 to 8 pages in length and should reach=20
the Program Chair at the address below by January 31, 2001. Camera=20
ready copy for the pre-proceedings is required. All submissions must=20
be in English, the language of the study conference. Further=20
technical details about the format of submissions will be available=20
on the study website (see below), which will be progressively updated=20
with all study and travel information. The combined fee for=20
registration and college accommodation is expected to be less than=20
US$500.
The members of the International Program Committee are: Program Chair=20
Kaye Stacey (Australia), Dave Carlson (USA), Jean-Phillipe Drouhard=20
(France), Desmond Fearnley-Sander (Australia), Toshiakira Fujii=20
(Japan), Carolyn Kieran (Canada), Barry Kissane (Australia), Romulo=20
Lins (Brazil), Teresa Rojano (Mexico), Luis Puig (Spain), Rosamund=20
Sutherland (UK), Bernard Hodgson (ex-officio, ICMI). Helen Chick=20
(Australia) is the conference secretary.
For further information:
Study Secretary:
Dr Helen Chick, Department of Science and Mathematics Education,=20
University of Melbourne, Victoria 3010, Australia. Tel: +61 3 8344=20
8538 Fax: +61 3 8344 8739 <h.chick@edfac.unimelb.edu.au>
Program Chair:
Professor Kaye Stacey, Department of Science and Mathematics=20
Education, University of Melbourne, Victoria 3010, Australia
<k.stacey@edfac.unimelb.edu.au>
Website:
http://www.edfac.unimelb.edu.au/DSME/icmi-algebra/
**********************************************************
--=20
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
Fax: (618) 453-4244
E-mail: jbecker@siu.edu
--- End Forwarded Message ---
Rosamund Sutherland
Graduate School of Education
35 Berkeley Square
Bristol BS8 1JA
Tel 01179 287105
Fax 01179 251537
ros.sutherland@bristol.ac.uk
--- End Forwarded Message ---
----------------------
Laurinda Brown
Laurinda.Brown@bristol.ac.uk
0117-9287019