Implications for mathematics education policy of research on algebra learning.

The teaching of algebra in Victorian secondary schools has changed substantially in the last decade. Here we present implications for curriculum policy arising from research into students' algebra learning in Years 7 to 11. Data were collected from approximately 3000 students in 34 schools. Information about programs offered was obtained from teachers, by textbook analysis and by some lesson observation and teaching interventions. Performance varied considerably between schools and classes. Some differences are attributable to teaching methods, the content taught, and the arrangement of the curriculum. Subtle reductions in goals and the isolation of topics in the curriculum were disturbing trends. We discuss findings that have important implications for mathematics education policy Australia-wide.

Over the six years 1991-96, we have studied the learning of school algebra, with particular focus on the causes of common misunderstandings and low attainment. In the course of this research, we tested and talked to students, observed teaching, discussed approaches with teachers, and analysed textbooks and curriculum documents. Readers will find details of the data collection and specific findings in our research reports and articles (see Stacey & MacGregor, 1997c). In this article, we discuss the `broad brush' issues and the implications for curriculum policy that arise from them. We intend that our discussion and analysis will stimulate and inform debate on issues such as whether algebra as currently being taught is achieving its purpose for various groups of students, the levels of achievement that we expect, and the quantity and quality of instruction needed to reach such levels.

The first section of this article places algebra in context in the secondary school mathematics curriculum and summarises our research procedure and findings. The subsequent sections highlight aspects of the findings relating to the issues above.

School algebra and students' achievement

Place of algebra in the mathematics curriculum

Algebra is one of the five strands of content in A national statement on mathematics for Australian schools (Australian Education Council, 1991) and the Curriculum and standards framework: Mathematics (Board of Studies, 1995). As highlighted in the National statement, ideas essential for learning algebra have a place in the primary curriculum, but only in secondary school do students begin formal algebra, which for us is signified by the use of letters to denote unknown or variable quantities. This late introduction reflects the special role of algebra as a gateway to higher mathematics. Algebra is the language of higher mathematics and is also a set of methods to solve problems encountered in professional, rather than everyday, life.

Some algebra is taught to all junior secondary students in normal Victorian (and Australian) schools. This is done for two reasons. First, some familiarity with algebra is considered to be important for informed participation in a democratic society, and therefore all students should learn about its key concepts. Secondly, since algebra is important for further mathematics, on grounds of equity no student should be denied access to it. The inclusion of algebra in a common curriculum for all secondary school students is not universal, however. In the United States, for example, a different curriculum structure means that many students never take the one-year algebra course that is a prerequisite for subsequent mathematics courses. There is at present a campaign for all students to take this course, so that decisions about future access to mathematics are not made so early. In Australia, since all students have the opportunity to learn some algebra in every year level, the questions that must be asked about `algebra for all' are `What type of algebra?' and `How much for whom?'. These are important questions.

In recent years, there have been substantial changes in the way algebra is taught, in Australia and in other countries, at least as indicated in curriculum documents and textbooks. These changes have been stimulated by:

* dissatisfaction with the outcomes of previous algebra teaching methods;

* the desire to meet the needs of the whole age cohort at least up to the end of Year 10;

* the attractiveness of fresh teaching ideas;

* recent rethinking of content (e.g. the role of functions and their representations);

* re-assessment of the goals, given the likely impact of new technologies.

These concerns are reflected in the National statement on mathematics for Australian schools and the Victorian Curriculum and standards framework. (The first author took part in preparing both these documents.) Our research has led us to believe that, although there is much of value in the new approaches, much serious thinking still needs to be done.

Outline of research and findings

Data on students' understanding were obtained from written tests administered to classes by their own teachers as part of the normal process of schooling, and from content-based interviews with individual students withdrawn from class. In the interviews, we listened to students' explanations of what they were trying to do as they worked on the test items, thus gaining valuable insights into their reasoning. As well as collecting cross-sectional and longitudinal data from approximately 3000 students in 34 schools, we observed lessons, talked to teachers about their programs, analysed textbooks, and arranged teaching interventions. Almost all the data were collected from schools in Melbourne, and the textbooks analysed are those popular in Victoria. Reports from teachers and observations of lessons support the assumption that the widely selling texts are a good guide to how algebra is actually taught, for the vast majority of students.

In 1990-93, we investigated students' understanding of fundamentals of algebraic notation, which is the central part of `Expressing generality', one of the three sections of the algebra strand in the National statement. We traced errors to several causes, including (a) students making incorrect analogies between algebraic notation and other symbol systems, (b) misleading teaching materials, and (c) mental models that underlie comprehension (MacGregor & Stacey, 1993a, 1993b, 1997; Stacey & MacGregor, 1994, 1997b). We were able to distinguish errors that can be easily fixed in the classroom from deep-seated and resilient errors, and we suggested strategies for dealing with both categories (MacGregor & Stacey, 1994).

In 1994-96, we looked at the `Equations' section of the algebra strand in the National statement. Algebra is a language for expressing mathematical information, and it is also a set of powerful problem-solving methods. These methods transform the solution of large classes of problems from intellectual challenges or puzzles to routine tasks. We found that, in many schools, most students do not learn to use algebra in this way. They are not learning how to analyse real situations, formulate the critical relationships as one or more algebraic equations and then apply routine techniques to solve them. Some of these deficiencies are linked to particular teaching strategies or recommendations in textbooks (MacGregor & Stacey, 1995a, 1995b, 1996, 1997; Stacey & MacGregor, 1995, 1996).

Summary of policy-level findings

In addition to detailed findings reported elsewhere, our research has produced general findings likely to interest many educators. These findings, summarised below, are the topic of this paper. They are based on data that were collected in a scientific way by testing and interviews, and also on informal and incidental data obtained from teachers and students during the course of the project.

* Algebra teaching ii1 schools has changed substantially. Many aspects of the new ideas are excellent, but others need reconsideration.

* Too few students are achieving the full extent of the goals of the Victorian curriculum statements.

* There is great variation between schools in achievement, the nature of instruction, and the amount of practice work or homework given. These factors seem to be related.

* Many current textbooks in use in Victoria in Years 7-11 show a clear reduction in the goals of algebra teaching. Almost all problems in the algebra sections are easily solved without algebra. Some of the essential features of elementary algebraic reasoning are being sidelined or omitted. (We suspect from other evidence that reduction of goals is also happening in other strands of the mathematics curriculum.)

* There is frequent misjudgement by textbook writers and teachers about what fundamental algebraic concepts students should learn and how these concepts should be presented.

* The mathematics curriculum is becoming highly fragmented. Students are not being given opportunities in the rest of their mathematics to use and practise the algebra they are learning. Consequently skills and concepts have to be continually re-taught.

* The algebra now being taught is not oriented sufficiently to the knowledge that will be needed when algebra computer software is widely available. As one example, textbooks and teachers still tend to emphasise the solving of equations instead of the formulation of equations from problem situations.

Conveying the purpose of algebra

In this section, we discuss how the introductory algebra curriculum has been remodeled to appear more meaningful, but techniques that actually make it useful have simultaneously been withdrawn in some schools.

Linking algebra to real situations

Traditional approaches to algebra emphasised the manipulation of algebraic expressions. This method portrayed algebra as a meaningless activity where letters were moved from one position to another, rather like pieces in a chess game. Many of the new curriculum approaches have tried to change this presentation of algebra. One widespread improvement has been to use real situations, so that the algebraic letters have clear referents. Functions and equations can then be interpreted as saying something about the relationships inherent in real situations -- for example, the rate of increase of rabbits on an island or the distance traveled by a spacecraft in a certain time. Algebra is used to summarise the relationship as an equation, and predictions can be made from it. There are several positive aspects of this approach: use of a real situation; emphasis on a functional relationship between variables; and assuming or showing the functional relationship to be a generalisation on which useful predictions can be based.

Examples are too easy and do not show the power of algebra

Unfortunately the increased relevance of the contexts through which algebra is now being taught is undermined by a reduction of demand. Our analysis of several popular textbook series showed that problems about real situations are often far too simple. They describe situations where using algebra seems a nuisance, rather than a powerful technique. The Year 9 question shown in Figure 1, for example, involves important algebraic concepts (constant, dependent variable, independent variable, equivalent value) but the algebra is used to make a prediction that could have easily been made otherwise. It is very easy to guess that for 2 hours both charges are $12. Solving the Year 10 question demands no knowledge of algebra. A young child can easily work out -- by repeated adding of 3 to 22 -- that you could afford to buy 9 cassettes.

Figure 1 Examples of problems used to demonstrate use of algebra

Year 9 question

At Budrose beach, Jenny and Abdul set up rival sailboard hire businesses. Jenny charges $5 plus $3.50 per hour, while Abdul charges $3 plus $4.50 per hour.

(a) Write down equations for the cost of hiring a sailboard from each business for t hours.

(b) Find the hire cost from each business for:

   (i) 1 hour, (ii) 3 hours, (iii) 8 hours.

(c) For what period of time will both businesses charge the same amount?

(Jacaranda Mathematics 9, 1994, p. 168)

Year 10 question

At a music store I have spent $22 on a compact disc and wish to purchase some blank cassettes which are $3 each, How many blank cassettes could I buy in order to spend less than $50 in total?

(Heinemann Mathematics 10, 1994, p. 190)

Students working through these and similar examples may not see why an algebraic representation is useful. We agree with textbook authors that it is good to use simple, well-understood situations to demonstrate new ideas. However, we think that the usefulness of algebraic methods is, by and large, not getting across, and therefore more complex problems demonstrating the efficiency and power of algebra should be frequently used.

Curriculum emphasis not oriented to the future

The questions in Figure 1 may be too easy to demonstrate algebra as a powerful tool, but they are intended to give students experience of formulating equations from real situations. The advent of new technology for mathematics makes this one of the skills upon which an algebra curriculum should now focus. Within a decade or less, access to computer algebra will mean that the ability to solve equations `by hand' will not be a major goal of algebra learning. Already, for about $300, students can purchase `calculators' which can do all the routine algebraic procedures taught at school. The price of these calculators is expected to fall during the next few years, so that all students might soon be expected to own one. In this circumstance, carrying out routine algebraic techniques will not be a valuable skill, whereas understanding what they do and a facility with reading and writing the notation will be critical. The crucial part played by the human in the path from a problem to its solution will be the formulation stage, where information is represented algebraically. In this regard, it is disturbing to see that students now are given very limited experience in formulating equations from real situations. The typical textbook has a larger number of exercises requiring routine manipulation of symbols, followed by a much smaller number of problems. The data presented in Table 1 quantify this for one type of problem. The average number of routine exercises per book is 14, whereas the average number of problems per book is 2.6. Moreover, when the problems appear only at the end of sections and chapters (as is usually the case), the implication is that they are of secondary importance. Formulating equations and describing situations algebraically, the step that turns algebra from a symbol-pushing routine into something meaningful and useful, appear not to be valued.

Table 1 Total number of examples(a) in Year 10 textbooks(b) that are presented as routine manipulations or as problem situations

               Number of equations   Number of problem
Textbook       for routine solving      situations

Textbook 10A           34                   1
Textbook 10B            0                   0
Textbook 10C           13                   9
Textbook 10D            7                   1
Textbook 10E           10                   0
Textbook 10F           16                   7
Textbook 10G           18                   0
Average                14                   2.6

(a) These are questions containing an equation with the unknown appearing on both sides or a problem situation leading to such an equation.

(b) These are the most recent editions of the Year 10 textbook series for Victorian schools held in University of Melbourne library.

Subtle reductions of goals

The equations students learn to solve when beginning formal algebra are linear equations, for example, 4x + 5 = 19 where the unknown x is on only one side, and 4x + 5 = 6x - 2 where x is on both sides. Although experts see these two types of equations as being very similar, it is well established that for beginners they present very different cognitive challenges (Bednarz & Janvier, 1996; Kieran, 1992). Most people can easily solve a problem that is modeled by an equation of the first type in their heads, without any algebra. The music cassette problem in Figure 1 is one of these. (More precisely, it is modeled by an inequation. This does not affect the substance of the argument.) This problem is easy to solve without algebra, either by trial and error or by logical reasoning and arithmetic. It is also easy to solve with algebra. If we let c be the maximum number of cassettes, then the equation is 3c + 22 = 50. This equation is easily solved because the only quantities that need to be manipulated during the solving process are numbers: 50 - 22 = 28; 28 / 3 = 9.3. The ease of solving this problem is due to its mathematical structure -- indicated by the form of the algebraic equation -- as well as to the actual numbers involved.

Problem situations that result in equations of the second type (i.e. with the unknown on both sides) are usually much more difficult for people to solve without algebra. The Budrose Beach question of Figure 1 is one of these. It is easy only because the solution is a small whole number, so that trying a couple of likely numbers is a viable strategy. More generally, however, for this second type of problem, algebra is genuinely useful, turning intellectual challenges into routine procedures. In the light of the difference between the two types of problems, it is disturbing to see that current textbooks have few, if any, of the harder type. In some new versions of textbooks, and hence probably in some schools, there are none, even in Year 10 (see Table 1). Moreover equation-solving methods that only work on the first, easy type of problem are now being promoted by some textbooks and teachers as the preferred methods. To enable students to keep using these methods, some of the textbooks listed in Table 1 have significantly reduced the scope of the algebra taught. In this respect at least, there has been a `dumbing down' of the algebra curriculum.

One reason why the harder types of linear equation problems should be included is to give students some early payoff from learning algebra -- they can now solve problems that are otherwise hard. However there is another reason, related to the development of algebraic thinking. To solve the harder type of equation, operations have to be carried out with or on the unknown quantity. The recognition that unknowns can be used as if they were knowns is one of the key conceptual changes in the transition from arithmetic to algebraic problem-solving methods (Herscovics & Linchevski, 1994). The shift from arithmetic (calculating from what is known) to algebra (reasoning with symbols that represent unknowns) is recognised by the research community as a difficult step, essential for progress. Eliminating the harder type of equations from the curriculum essentially means that students apparently learn algebra but, in reality, they do not have to make the transition from thinking about quantities in problems in an arithmetic way to thinking about them in an algebraic way. In a sense, algebra is being taught without algebraic thinking.

Issues about teaching and learning algebra

We have shown how attempts to make algebra relevant to students' lives succeed at one level but fail at a deeper level. The contexts used to introduce ideas are apparently useful, but the content taught is less useful than before. In this section, some other issues about the organisation of the algebra curriculum and the methods chosen to introduce topics are discussed.

Identifying the foundations for algebra

In the last decade, there has been widespread change in how algebra has been introduced in the secondary school. Many textbooks now present algebra as a way of writing functional relationships from numerical or spatial patterns. They also give students practice in using letters to represent unknown numbers and write expressions, with a focus on manipulating symbols rather than expressing and extending number knowledge. The treatment of these two aspects of algebra (viz., describing functions and manipulating symbols) by many teachers reflects a currently popular view that experience with pattern is the principal primary school precursor of algebra (National statement, p.192). An emphasis on pattern has taken the spotlight from what we see as an equally important precursor: understanding the properties of numbers.

School algebra begins as generalised arithmetic -- the letters stand for numbers (despite the message of some misleading teaching materials -- see below) and they combine together in accordance with the rules that govern numbers. For fluency in algebra, students need to know properties of numbers almost as second nature. For example, if you take a number and divide it by a second number, then divide the result by a third number, you get the same answer as dividing the first number by the product of the second and the third: expressed in symbols (a/b)/c = a/(bc). Students need to know this not as a rule to be remembered but as describing a familiar property of which they are certain. Practice in simplifying and expanding algebraic expressions would not achieve this. Achieving the thorough understanding of the properties of number operations that supports algebra learning is a very substantial task for the mathematics curriculum, with no quick fixes. Some revitalisation of this area is needed now, to accompany and balance the revitalisation of curriculum materials that has flowed from the emphasis on pattern.

Students' numerical experience also impacts on the teaching of algebra through the type of numbers that are used in algebra work. One reason why algebra problems are often so easy to solve is that only the simplest numbers are used. Recognising students' difficulties in coping with decimals, fractions and negative quantities, teachers and textbooks use only small whole numbers. Teachers find that interesting problems needing algebraic solutions are too hard for many students, because solving them may require a secure understanding of fractions, decimals or negative numbers as well as arithmetic concepts such as equivalence, ratio, percentage, rate, or exponential growth. Algebra learning should support the learning of these arithmetic concepts instead of being taught as a separate topic in which only the simplest arithmetic ideas appear (Stacey & MacGregor, 1997a).

Flawed teaching methods

Our analysis of error patterns showed that some errors clustered in particular classes or schools and some were distributed widely. We hypothesise that widely distributed errors arise from the way in which human thinking and common prior knowledge interact with cognitive features of the new material to be learned. The famous `reversal error' is one example (MacGregor & Stacey, 1993b). Other widespread errors, especially among beginning students, arise because students simply guess the meanings of letters. For example, they may assign numerical values to letters, such as evaluating 2n as 28 because n is the 14th letter of the alphabet. Others interpret letters as initials, for example thinking that 3c may stand for `three cats'.

With more advanced students, both these errors cluster in classes. Assigning numerical values to letters based on alphabetical position is aggravated by teachers using code-breaking puzzles in other parts of mathematics. If given attention, it is easily fixed (MacGregor & Stacey, 1997). Interpreting algebraic letters as initials for objects, however, is not easily fixed. For many years, it has been known that there are underlying cognitive and linguistic reasons for this tendency (see Kieran, 1992). Yet teaching approaches that reinforce the misconception (the `fruit salad' approach) are still prevalent in textbooks, and these textbooks are still selected by schools. See, for example,Jacaranda Mathematics 9, (1994, p.52) `Let x = apples and y = oranges'; Mathematics today Year 9 (1994, p.67) `Use G for goats and C for cows'; New Maths 10 (1990, p.22) `Two shrubs and three shrubs' represented as 2s + 3s. Signs of this error and its serious effects on understanding were evident in students several years after they had been introduced to a fruit salad interpretation of letters in Year 7 (MacGregor & Stacey, 1997). This misinterpretation and the alphabetical interpretation of letters referred to above are simple instances demonstrating that the quality of instruction affects student outcomes.

Quantity, intensity and integration of algebra instruction

In our study of students' understanding, we found that success, at the school level and the class level, was very variable. In some classes, almost all students could solve problems using standard techniques of algebra correctly in a routine way. In other classes, students had few automated skills and generally avoided an algebraic approach. Instead they tried untutored intuitive ways to solve problems. Some of the differences between classes and schools reflect differing curricula. Some reflect the commitment to teaching algebra -- to the quantity, the frequency and the intensity of the experience.

As one part of our research, we looked at middle-secondary school students' progress in algebra over ten months in one school (Stacey & MacGregor, 1996). A few students improved, the majority stayed the same, and some completely gave up using algebra. Schools need to question whether they are actually providing students with a sufficient amount of algebra in a sufficiently intense experience, and with sufficient practice, for the instruction to have any lasting effect.

In view of the need for frequent exposure to algebra, it is disturbing to find that now, more than ever before, the mathematics curriculum is being taught in a fragmented way with no links between topics. One mathematics co-ordinator responded to our request for participation in our research by writing, in the third month of the school year, `I am sorry that we cannot participate but we have already done algebra for this year'.

Our analysis of textbooks showed that ideas developed in one chapter are often not reviewed anywhere else in the year's program. For example, outside the algebra sections of their textbooks, students rarely see algebraic letters used except in formulas or as labels indicating the quantity to be found in diagrams or formulas. Their exercises almost always have numerical (rather than algebraic) answers. Figure 2 shows typical examples of the exercises with the greatest use of algebra in the geometry chapters of currently used textbooks for Year 10. These examples require no knowledge of algebra at all beyond the solving of easy equations (e.g. 2x + 3x = 120 and 3x = 150, substantially below Year 10 level) . Mostly the algebraic letters are merely used to identify which quantity is to be calculated arithmetically from known geometric properties. This contrasts markedly with earlier practice. We give one example below.

[Figure 2 ILLUSTRATION OMITTED]

In a textbook widely used in the 1970s (Mathematics for Form 4, first published 1969, reprinted each year to 1976), students use algebraic methods in work on number properties, measurement, ratio and geometry (see Figure 3). For example, in the last question in Figure 3, the quadratic equation (x + 5)/4 = 6/x has to be formulated and solved, providing practice of recently developed skills. In that book, algebra is also used for formal and informal proofs such as questions 1 and 2 in Figure 3. We could find no similar use of algebra in current textbooks. In the 1990s, students are commonly led to believe that a valid generalisation can be assumed from a few examples without any need for proof, as in the following example:

Find answers for:

(a) [1.sup.3]

(b) [1.sup.3] + [2.sup.3]

(c) [1.sup.3] + [2.sup.3] + [3.sup.3]

Predict the answer for [1.sup.3] + [2.sup.3] + [3.sup.3] + [4.sup.3] + [5.sup.3] + [6.sup.3] + [7.sup.3]

(Mathematics today Year 10, 1990, p.8)

[Figure 3 ILLUSTRATION OMITTED]

It seems to us important for students to know that a prediction based on a few instances may not be correct.

We would argue that question 1 in Figure 3 is unmotivated and would promote better learning if students first discovered that

[15.sup.2] = 100 x 2 + 25   [25.sup.2] = 200 x 3 + 25
[35.sup.2] = 300 x 4 + 25   [45.sup.2] = 400 x 4 + 25

so that the generalisation and its proof could gain concrete meaning. However it does at least propose algebra to explain the general reasons why a result is true, something which is lost in current curricula.

The mathematics education community continues to hope that students will appreciate the unified structure of mathematics. However, at the present time, skills and concepts developed in one topic are rarely used in another. Consequently much teaching time in all areas of mathematics is spent re-teaching ideas that students have forgotten. Teachers regard `forgetting' between topics as a normal process. There seems little expectation that students will remember content from one year to the next, and no evidence that active steps are being taken in curriculum organisation to maximise the chance of it happening. Continuous nurturing of ideas has been replaced by annual repetition.

Conclusion

It is clear from our data that there are some successes and some failures in the current teaching of algebra. There are some very nice new materials and important new emphases, some new materials that seem to have few advantages, and some misleading methods that continue to be used despite years of condemnation. There is a clear commitment to using real situations to illustrate abstract ideas, so that the acquisition and application of ideas proceed hand in hand. However a close analysis reveals that the goals of algebra have been reduced, without debate in Victoria at least. The usefulness of the contexts used to illustrate the ideas is not matched by the actual usefulness of the material being taught. In a subtle way, some programs now teach `algebra' without really confronting students with algebraic thinking.

Some schools in our sample did very well. In other schools, students had had very little or no experience of using algebra to solve problems. Even at Year 10, some students rely only on logical reasoning, either because they have not been taught powerful mathematical methods or perhaps because they have not had sufficient experience to make progress with them. This may well have come about because of the goal of `algebra for all'. Students who are going to use algebra in higher mathematics need to develop a fluency and symbol sense that will let them operate with algebra as a language and as a set of problem-solving methods. This ability requires time (probably several years) and practice to develop. These potentially professional users of algebra seem not to be well served by the minimal exposure that students in some schools are being offered. Most of them have relatively strong arithmetic and logical skills. By the end of primary school, for example, they could have solved most of the word problems that some of today's texts are giving in the Year 10 algebra sections. These students need to be given a stronger algebra diet and encouraged to make the conceptual jumps to algebraic thinking. Other students have weak arithmetic concepts. Strengthening their understanding of number and their ability to solve problems by logical reasoning, and even by systematically guessing answers, should be a high priority. For these students (assuming that schools are prepared to identify them), we need to debate what is the goal of their algebra experience and how it is best taught.

Wheeler (1996) comments that in thinking about the mathematics curriculum, and algebra in particular, not enough distinction is made between the knowledge that every member of society should possess and the knowledge which society as a whole (but not every individual) should preserve and pass on. The watered-down version of algebra given out now in many schools seems a poor compromise to meeting the diverse needs of students. Algebra is hard to teach and hard to learn. However the difference in achievement between different schools is a sign of hope, suggesting that with commitment it is possible to teach a large proportion of the school population.

Our findings underline the importance of teachers having deep knowledge of mathematical content as well as insights into students' thinking. Teachers must identify the fundamental ideas that need to be taught, and understand the difficulties and misunderstandings that are likely to occur, so that an effective and coherent approach to algebra in Years 7 to 10 can be established. Schools hoping to improve their outcomes in algebra learning can look to:

* commitment to algebra in the curriculum, in various ways;

* the detailed content of the curriculum, including the type of problems set, the teaching approaches, and integration across topics;

* the adequacy of the foundation in arithmetic for supporting algebraic ideas.

During the past decade, teachers have experimented, very often successfully, with ways to help students give concrete meaning to variables, equations and functions. At the same time, there seems to be an ambivalence about the goals of algebra learning and a lack of recognition of the contrast between arithmetic and algebraic ways of solving problems. Teachers need to find a balance between strengthening students' arithmetic thinking and developing the powerful new patterns of reasoning that algebra provides. The algebra curriculum currently being taught in many Victorian schools is significantly out of balance.

Keywords

algebra
curriculum policy
mathematics education
problem solving
secondary school mathematics
textbooks

Acknowledgements

This article draws on research funded by three grants to Kaye Stacey from the Australian Research Council. We wish to express our appreciation to the many teachers and students who co-operated with the projects.

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Professor Kaye Stacey is Foundation Professor of Mathematics Education and Dr Mollie MacGregor is a Research Fellow in the Department of Science and Mathematics Education, University of Melbourne, Parkville, Victoria 3052.