The Twelfth ICMI Study
Discussion Document
Overview This document introduces a new ICMI study entitled The Future of the Teaching and Learning of Algebra, to be held at the University of Melbourne (Australia) in December 2001. The themes of the study are likely to be influenced by the main sections of the document, which are indicated below. 

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Introduction There are many reasons why it is timely to focus on the future of the teaching and learning of algebra. We are at a critical point when it is desirable to take stock of what has been achieved and to look forward to what should be done and what can be done. In many countries, increasing numbers of students are now receiving secondary education and this is causing every part of the mathematics curriculum to be scrutinised. For algebra, perhaps more than other parts of mathematics, concerns of equity and of relevance arise. As the language of higher mathematics, algebra is a gateway to future study and mathematically significant ideas, but it is often a wall that blocks the paths of many. Should algebra be made more accessible to more students by changing the amount or nature of what is taught? Many countries have already embarked on such changes, hoping to increase access and success. Alternatively, are these changes necessary: is algebra truly useful for the majority of people and, even if it is, will it be useful in the future? An algebra curriculum that serves its students well in the coming century may look very different from an ideal curriculum from some years ago. The increased availability of computers and calculators will change what mathematics is useful as well as changing how mathematics is done. At the same time as challenging the content of what is taught, the technological revolution is also providing rich prospects for teaching and is offering students new paths to understanding. In the past two decades, a substantial body of research on the learning and teaching of many aspects of algebra has been established and there have been many experiments with adapting curricula and teaching methods. There is therefore a strong 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 lessons from the past which will help set directions for future work in the field, and to suggest guidelines for advancing the teaching and learning of algebra. Following the pattern of previous ICMI studies, this study will have two components: an invited study conference and a study volume to appear in the ICMI Study Series, which will share the findings with a broad international audience. A report will also be made at ICME10 in 2004. The study conference program will therefore contain plenary and subplenary lectures, working groups and panels. At least two panels are planned. One will attempt to make explicit some perspectives on algebra, algebra activity, algebraic thinking or algebraic understanding. A second aims to highlight the significant differences in algebra education around the world and identify the main strands in the goals, content and teaching methods of this worldwide enterprise. A major part of the working time will be spent in working groups addressing different aspects of the study problem. Working groups are likely to be established to correspond with each of the sections listed below.  
Why algebra? The technological future of a modern society depends in large part on the mathematical literacy of its citizens and this is reflected in the worldwide trend towards mass secondary education. For an individual, algebra is a gateway to much of higher education and therefore to many fields of employment. Educators also argue that algebra is part of cultural heritage and is needed for informed and critical citizenship. However, for many, algebra acts more like a wall than a gateway, presenting an obstacle that they find too difficult to cross. This section of the study is concerned with the significance of algebra for the broad population of secondary school students, recognising that regional and cultural differences may impact upon the answers in interesting ways. It addresses questions such as:


Approaches to algebra 

Recent research has focused on a number of approaches for developing meaning for the objects and processes of algebra. These approaches include, but are not limited to, problemsolving approaches, functional approaches, generalisation approaches, languagebased approaches, and so on. Problemsolving approaches tend to emphasise an analysis of problems in terms of equations and a view of letters as unknowns. Functional approaches support a different set of meanings for the objects of algebra; for instance, the use of expressions to represent relationships and an interpretation of letters in terms of quantities that vary. A somewhat different perspective is encouraged by generalisation approaches that stress expressions of generality to represent geometric patterns, numerical sequences, or the rules governing numerical relationships, such approaches often serving as a basis for exploring underlying numerical structure, predicting, justifying and proving. Some algebra curricula develop student algebraic thinking exclusively along the lines of one such approach throughout the several grades of secondary school; others attempt to combine facets of several approaches. Synthesising the experience with and research on the use of various approaches in the teaching/learning of algebra leads to questions such as the following:


Language aspects of algebra 

This section considers theoretical and applied aspects of the languages and notations of algebra, in relation to teaching and learning. The evolution of algebra cannot be separated from the evolution of its language and notations. Historically the introduction of good notations has had enormous impact upon the development of algebra but a good notation for science may not be a good notation for learning. With new computer technology we are now seeing a flowering of new quasialgebraic notations, which may offer, support or eventually enforce new notations. However, current theories of mathematics teaching and learning do not seem adequate to deal with learning about notation. It is therefore timely to focus on algebraic notations asking questions such as:


Teaching and learning with Computer Algebra Systems 

The advent of affordable computer systems and calculators that can perform symbolic calculations may lead to farreaching changes in mathematics curricula and in mathematics teaching. This section addresses questions that arise from the increasing accessibility of computer symbolic manipulation. Answers to these questions will draw upon established research on the teaching and learning of algebra as well as reporting on recent experimental work. They may suggest new directions for research, including:


Technological environments 

Recent research, curriculum development, and classroom practice have incorporated a number of technologies to help students develop meaning for various algebraic objects, ideas and processes. These include, but are not limited to, function graphers, spreadsheets, programming languages, oneline programming on calculators, and other specific computer software environments. [Here, we exclude computer algebra systems that are treated elsewhere.] In an attempt to characterise recent research and experience, this section will explore which aspects of specific computer/calculator environments are related to which kinds of algebra learning. This question will be explored in depth for specific examples of such technology, by addressing questions such as the following:
Submissions for this section should include discussion of as many of the above subquestions as possible, but with particular attention paid to the first two items above. 

Algebra with real data 

Modelling the behaviour of real things with algebraic functions is fundamental to applications of mathematics. Using real data to teach about functions is therefore important in the curriculum, and can also be highly motivating for students. Moreover, new devices (such as data loggers) and new communications technologies (such as the internet) provide new opportunities for bringing real data into the classroom. Questions such as the following arise:
The history of algebra has been used extensively to identify epistemological obstacles in the learning of algebra and to characterise ruptures in the development of algebraic notions. Drawing on the history (or histories) of algebra from around the world, this section aims to analyse significant contributions and the value of these previous uses and also to reflect on possible avenues for research based on new areas, including:


Early algebra education 

This section encompasses two different readings of the title, being concerned with both the algebra education for young children  say age 6 and above  and also the initial steps in more formal algebra education, which happens in some countries when students are about 12 years old. An ongoing concern is the relationship between arithmetic and algebra. Previous research has documented ways in which students' limited arithmetical experience can constitute an obstacle to the learning of algebra, so that an earlier start might reduce the problem; approaches have been proposed to achieve that. On the other hand, a much favoured approach to initial algebra education is based on the view of school algebra as generalised arithmetic, in which case an earlier start may not be appropriate. The general point here is that different views on the relationship between arithmetic and algebra will probably result in different views on algebra education, and this most important fact is a central concern in this section. The interest in algebra education for students at an early age is recent, and so there are as yet only a few studies in this area. It is important that answers to the following questions be thoroughly researchbased:


Problems exist in the teaching and learning of tertiary algebra courses such as abstract algebra, linear algebra, and number theory. Some are similar to the problems of secondary algebra: students' difficulties with abstraction, concerns of relevance, what to do with computing technology, etc. Other problems such as proofmaking or seeing the objects of calculus as algebraic objects seem particular to the tertiary level. The questions below are concerned with these issues of learning and teaching and also with the specific question of education for prospective teachers.

For information about this page, contact: Kaye Stacey
Contact Email Address: k.stacey@unimelb.edu.au
Department Homepage: http://extranet.edfac.unimelb.edu.au/DSME/icmialgebra/index.shtml
Faculty Homepage: www.edfac.unimelb.edu.au/
Last modified: Fri 21 September 2012
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