Sequential development of algebra knowledge: A cognitive analysis

Mathematics Education Research Journal

1998, Vol. 10, No.2, 87-102

Sequential Development of Algebra Knowledge: ACognitive Analysis 1

Hitendra Pillay, Lynn Wilss, and Gillian Boulton-Lewis Queensland University of Technology Learning to operate algebraically is a complex process that is dependent upon extending arithmetic knowledge to the more complex concepts of algebra. Current research has shown a gap between arithmetic and algebraic knowledge and suggests a pre-algebraic level as a step between the two knowledge types. This paper examines arithmetic and algebraic knowledge from a cognitive perspective in an effort to determine what constitutes a pre-algebraic level of understanding. Results of a longitudinal study designed to investigate students' readiness for algebra are presented. Thirty-three students in Grades 7, 8, and 9 participated. A model for the transition from arithmetic to pre-algebra to algebra is proposed and students' understanding of relevant knowledge is discussed.

The transition from concrete and numerical foundations of arithmetic to the symbolic and abstract reasoning of algebra has been conceptualised as a didactic cut by Filloy and Rojano (1989) and as a cognitive gap by Herscovics and Linchevski (1994). A consequence of this cut/gap has been recognition of the need for an operational level of pre-algebraic knowledge between arithmetic and algebra. This study proposes a model that depicts the transition from arithmetic to algebra and includes an operational level of pre-algebra. However, instead of explaining this transition as a cut/gap, it is explained in terms of a sequence of development from arithmetic to algebraic knowledge.

Development ofMathematical Knowledge Research in cognitive science is increasingly challenged by educational studies into how students learn subject matter and as a result has revealed the importance of knowing the cognitive manoeuvres engaged in during learning. Studies in cognition suggest that knowledge is a hierarchical network of concepts and attributes, connected by relational propositions which are organised from simple to complex concepts (Ericsson & Polson, 1988). It suggests a cumulative and gradual development with quantitative and qualitative changes occurring in both what is learnt and how learning occurs (Gott, Kane, & Lesgold, 1994; Halford & BoultonLewis, 1992). In light of this, it is plausible to suggest that mathematical knowledge is no different; it is incremental, sequential, and developed through a cumulative process. What is not clear is how the cumulative process occurs during the transition of mathematical knowledge from arithmetic to algebra. On the other hand, what is clear is that a relationship between algebra and arithmetic exists. This is evidenced in statements such as "algebra is not separate from arithmetic; indeed The research reported in this paper was funded by a large grant from the Australian Research Council (No. A79531738) to G. M. Boulton-Lewis, T. J. Cooper, and B. Atweh.

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it is in many respects 'generalised arithmetic'" (Booth, 1988, p.29) and also in Kieran's (1981) proposed sequence for teaching algebraic equations that is "anchored" (p. 322) in arithmetic. Janvier (1996) also argued that algebra involves the use of arithmetic operations on an unknown. However, secondary students often seem unable to apply basic algebraic concepts and skills and to understand manY'of the underlying structures. This is particularly' evident when a distinction is drawn between performance and understanding as outcomes of instruction (Rosnick & Clements, 1980). The following section discusses the development of algebraic knowledge by examining what constitutes algebra, the requisite arithmetic knowledge needed to operate algebraically, and the role of unknowns and variables.

Understanding Algebra Arithmetic and Algebraic Knowledge What is meant by algebraic knowledge has been well researched. Kieran (1990) argued that such knowledge begins when students learn to manipulate the symbolic language of algebra in addition to functioning with operations in an equation. It is also seen as the ability to accept lack of closure (Collis, 1974) where the focus becomes relationships rather than a computed answer. Scandura (1971) ,argued that it is the relationships between elements and transformational aspects that seem to underpin algebraic knowledge. Transforming and defining relationships algebraically requires understanding, for example, that the equals sign denotes an equivalence relationship rather than an instruction to find an answer or perform an operation (Filloy & Rojano, 1989; Kieran, 1981). Linchevski (1995) viewed this transformation from a psychological perspective, arguing that operating algebraically required students to move from a unidirectional mode of reading an equation to multi-directional processing of information. Kieran (1992), on the basis of the psychological model proposed by S£ard (1991), distinguished between procedural (such as solving 2x + 5 = 11) and structural (such as simplifying 3x + y + 8x) conceptions of algebra. She implies that these conceptions are hierarchical, with the procedural conception being at the lower level. Thus students need to form increasingly abstract views of arithmetic; for example, they need to view addition in algebra as an object (structUre) in Sfard's sense (see Linchevski, 1995). A sound arithmetic knowledge base has been recognised as essential to operating within an algebraic framework. For example, Booth (1989) stressed the importance of students' understanding various structural notions in arithmetic. Herscovics and Linchevski (1994) exemplified the importance of a good grounding in arithmetic by analysing the knowledge required to solve 4 + n - 2 + 5 = 11 + 3 - 5. They stated that students need to be able to use commutativity to obtain (n + 4) - 2 + 5 and then associativity to perform (4 - 2) + 5. Booth (1988) reported that students hold an inadequate conception of commutativity believing that division, like addition, is commutative. Furthermore, Linchevski and Herscovics (1994) conducted a study that found Grade six students over-generalised order of operations, failed to perceive cancellation of terms in an equation as they operated

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sequentially from left to right, and displayed a static view of the use of brackets. Understanding of the distributive law is also essential for algebraic functioning (Demana & Leitzel, 1988). Linchevski and Herscovics (1996) consider that appropriate preparation in arithmetic in upper primary school can help to overcome obstacles and foster the development of new pre-algebraic skills.

Unknowns and Variables The move from arithmetic to algebra also requires students' conceptions of operations performed on numbers to change so that the concept of operating on variables may develop (Filloy & Rojano, 1989). Cortes, Vergnaud, and Kavafian (1990) view such a move as constituting an epistemological jump requiring a shift from arithmetical knowledge to assimilating new notions and procedures. Linchevski and Herscovics (1996) suggested a possible source of difficulty in this "shift" in that students' assimilation of literal symbols must include letters to represent unknowns, generalised number, and functional variables. They stress that each of these notions of variable (literal symbol) implies a different level of difficulty. Other conceptual obstacles in interpreting letters in algebra have included a lack of understanding of concatenation. Herscovics and Linchevski (1994) found that when 14 year old students were asked to add 4 onto 3n, 64% gave an incorrect answer. Ursini and Trigueros (1997) proposed that understanding of variable as unknown implies recognising and identifying in a problem situation the presence of something unknown that can be determined by considering the restrictions of the problem; the ability to substitute for the variable, the value or values that make the equation true; and determining the unknown by performing the required arithmetic and/ or algebraic operations.

Pre-algebraic Knowledge The above discussion suggests that the move from arithmetic to algebra is complex; as a consequence many students may develop inadequate algebraic knowledge. As stated earlier, difficulties encountered by students operating on algebraic equations have been described as a cognitive gap (Herscovics & Linchevski, 1994) or didactic cut (Filloy & Rojano, 1989). Linchevski and Herscovics (1996) proposed that students could not operate spontaneously on or with the unknown and that grouping algebraic terms is not a simple problem. They also argued that students viewed algebraic expressions intuitively as computational processes (d. Sfard & Linchevski, 1994) and suggested that in teaching, instead of moving from variable to expression to equation, arithmetical solution of linear equations might be more suitable initially for learning to operate on or with the unknown. This means that students would need a sound understanding of symbols, operations, and laws of arithmetic (Boulton-Lewis et al., 1995) which can then be generalised to algebraic conceptions of variable, equation, expression, and equality. It is important to note the complexity of each one of these aspects of algebraic knowledge (Sfard, 199) and to reiterate that if these aspects are not well understood, subsequent learning can become difficult. Thus, when consideration is given to all the complexities of learning to function algebraically the central

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question is: How should students be introduced to specific algebraic principles and to operating within this framework? The following presents a possible solution.

Analysis ofa Sequence for Learning Algebra Biggs and Collis (1982), on the basis of Collis's (1975) research, described development of algebraic concepts in terms of the SOLO (Structure of Observed Learning Outcomes) Taxonomy. According to this taxonomy responses range, on the basis of the structural organisation, from incompetence (prestructural) to expertise (extended abstract) in hierarchical order. As a person learns more, levels of structural organisation of knowledge recur in a cyclical fashion, for increasingly more formal modes of learning. Biggs and Collis (1989) proposed this as a move from sensorimotor, through ikonic and concrete-symbolic (typical of most secondary school learning) to formal-1 and then formal-2 modes of knowing (evident in tertiary students). Hence we would expect students learning algebra in schools to develop responses in the concrete-symbolic mode, that is to relate their knowledge of operations to the symbols that represent them. Biggs and Collis (1982) found responses for number, operations, and closure were sequential, as follows: • •

•

•

Unistructural: success with arithmetic operations where one closure is required (for example, if q = 8 + 4 then q equals ?) Multistructural: success both with large numbers involving single operations and with a series of operations in sequence with small numbers (for example, if n = (6 x 8) + 4 then n = ?); closure is made in sequence with a series of small numbers and is not necessary with single operations on large numbers Relational:" generalised" elements, that is large numbers and x standing for particular numbers, are used; the idea of operations is generalised sufficiently so that there was no longer the need to close each operation immediately Extended Abstract: a new level of functioning; closure is not required, and problems with operations on variables are solved (for example, if (a * 3) * 4 = 8 then * = ? and a = ?)

We have proposed a model (see Figure 1) that depicts the move from arithmetic to algebra as a sequence of development of knowledge that incorporates pre-algebraic knowledge. Intrinsic to the model are cognitive implications from Biggs and Collis's (1982) sequential development of algebraic knowledge, in that knowledge is cumulative and each stage is prerequisite to subsequent stages. We exemplify this by analysing the knowledge needed to solve an equation with two variables, such as x + 3 = 2x + 1. Firstly, a student must recognise symbols that represent an equation including variables as unknowns. Secondly, the student has to determine whether to use an inverse or balance approach in handling the equals sign and to identify the sequence of operations. Thirdly, operations must be carried out correctly to determine the value of the variable. The difficulty for students is they have to integrate knowledge of (a) symbols, numbers, and variables; (b) basic computations; (c) arithmetic laws for individual operations and sequences of

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operations; (d) meaning of equals; and (e) operations on variables. We suggest that solution is dependent upon a hierarchical sequence of mathematical concepts, as depicted iIi. Figure I, and explained below.

2+3=5 35 -;- 7 + 8 = 13 . Operational laws, numerical answers and equals as meaning each side of equals is the same value.

Lowest level solution: numericalprocedures

x+ 7 = 16 3(x+ 7) = 24 3x Recognition of unknown then variable in equations then expressions, concatenation,

x+3=2.x-l

equals as meaning each

x + 3y + 4x - 2y = 15

side of the equation is the the same value.

LowestlevclsofutiOn:mvene procedures

"-

~

More than one unknown or variable, operating on or with the unknown/variable, equals as equivalence.

Lowest level solutiOn: balance procedures

Figure 1. A model of sequential development of knowledge for understanding algebraic concepts. Initially students must possess arithmetic knowledge including operational laws such as inverse, commutative, and distributive; be able to operate numerically; and be able to apply inverse arithmetic equation solution processes. In addition, we propose that students be taught the concept of equals as indicating that each side of the "=" sign is the same value. While we acknowledge that instructional practices include treating equals as denoting the answer, we believe that this on its own constitutes an inappropriate conception of equals. We maintain that students must learn that equals means "equality" or indicates an "equal relationship" early in arithmetic instruction, to form a basis for understanding more complex arithmetic and algebraic concepts. Following this we propose that pre-algebraic concepts be established. These include concatenation; unknown and variable; a focus on understanding equals to mean each side of an equation is the same value; using inverse procedures to solve a linear equation with an unknown; and discussion of expressions. Finally, knowledge of algebraic equation solution

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methods, incorporating operating on and with the unknown using balance procedures and accompanied by an understanding of equals as an equivalence relationship, are essential. It should be noted that the meaning of equals does not change when moving from arithmetic to pre-algebraic to algebraic understanding. In each case the "=" sign denotes that each side of the equation is equivalent. However we propose that instruction of equals incorporate terminology that is specific to each stage. Therefore arithmetic instruction of equals should focus on the equals sign and the fact that each side of this sign is the same value; pre-algebraic instruction should move to a focus on the equation and that each side of the equation is the same value; and algebraic instruction should incorporate equals as indicating equality of sides or an equivalence relationship. The purpose of this study was to explore students' arithmetic knowledge and their early understandings of algebraic concepts. This was done in order to determine factors that may constitute a pre-algebraic level of understanding in an effort to validate the model. These factors are described at the conclusion of this paper.

Method

Sample In Grade 7, 51 students from four .Brisbane state primary schools participated. They were tracked through to the first year of high school, Grade 8 (40 students), and finally Grade 9 (33 students). Analyses were conducted on data from the 33 students who remained throughout the duration of the study. Four schools were in a middle socio-economic area; one primary school was in a lower middle socioeconomic area.

Tasks and Procedure Interviews were conducted with Grade 7 students before any formal algebraic instruction occurred; in Grade 8 after instruction in operational laws, use of brackets, and solution of arithmetic word and number problems; and in Grade 9 after instruction in finding an "unknown" in an equation and solving an equation using balance procedures. Questions investigated were the commutative (+, -, X, + in 35? 76=76 ? 35) and distributive (6 x 13 = 60 + 18) laws; inverse operations (5 x 71 = 355, 355 ? 5 = 71; 64 - 29 = 35, 35 ? 29 = 64); order of operations (32 + (12 x 8) + 3); meaning of equals in an incomplete (28 + 7 + 20 = ?) and complete (28 + 7 + 20 = 60 - 36) equation; meaning of unknown (0 + 5 = 9; x + 7 = 16) and variable (0 + 5; 3x); and solution of equations using numerical and algebraic processes (3(x + 7) = 24; x + 3 = 2x - 1). Individual interviews were videotaped. Students were encouraged to complete each task; if they could not, the interviewer proceeded to the next task.

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Analysis Interviews were transcribed and analysed to identify key categories. The computer program NUD.IST (Richards & Richards, 1994) was used to classify responses under these categories. Responses for laws, inverse operations, and order of operations were categorised as satisfactory or unsatisfactory as a basis for learning algebra. Responses for the other tasks were categorised as follows:

-inappropriate: indicating lack of knowledge required for the task; equals as the answer;

-

-

-

arithmetic: recognition and use of arithmetical operations, laws, and numerical answers (Sfard & Linchevski, 1994); equals as each side of the equals sign being the same value; pre-algebraic: recognition and use of unknown, variable, concatenation, and use of inverse procedures fa find an unknown in an equation (Herscovics & Linchevski, 1994); equals as each side of the equation being the same value; and algebraic: recognition and use of relationships expressed in simplified form (Booth, 1988) and employing algebraic processes, such as a balance approach, to solve an equation by operating on or with the unknown; equals as equivalence.

Results Commutative and Distributive Laws, Inverse Operations, and Order of Operations Data on responses for the commutative and distributive laws are shown in Table 1. In Grades 7 and 8, the majority of students could not explain commutativity satisfactorily. Inappropriate responses reflected a lack of knowledge of this law. Additionally, many students focussed on the equals sign indicating the answer, many responding that 76 was the answer; this resulted in the perception that no sign could be used to complete the equation correctly. By Grade 9, most students gave a satisfactory explanation for commutativity. The majority of students in Grades 7 and 8 could not give a satisfactory explanation for the distributive law; most exhibited no knowledge of this law. Some recognised that 60 may have resulted from 6 times 10 but did not relate this 10 to the 13. While a substantial number of students still could not explain this law in Grade 9, more than half were able to provide a satisfactory explanation. Inverse operations were explained satisfactorily by most students in each Grade (26,30,33; 22, 29, 33 for each task, respectively). By Grades 8 and 9 most students explained order of operations satisfactorily (26 and 23 respectively) compared with only 9 satisfactory explanations)n Grade 7.

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Table 1

Examples and Response Frequencies for Commutative and Distributive Laws Example responses Satisfactory

Gd.7

Unsatisfactory Commutative law (35 ? 76

Divide in first side, minus in the second side because 35 plus 35 equals 76.

Number satisfactory Gd.8

Gd.9

16

25

= 76 ? 45)

Add and times. These two symbols don't matter about order.

14

Distributive law (6 x 13 = 60 + 18) 6 times 10 and 6 times 3 is 18. You have to make the 13 into 10 and 3.

If you multiply 6 by 13

12

16

19

then take 13 away you might get 60. I don't know, if you multiply 6 by 13 you get 18.

Meaning of Equals Table 2 summarises the responses for meaning of equals in an incomplete and a complete equation. For each grade, the majority of students explained" =" in 28 + 7 + 20 = ? as the answer, which was categorised as inappropriate. Only one response in Grade 8 and three responses in Grade 9 revealed knowledge that "=" denoted that both sides had to be equal; this was categorised as arithmetic. For" =" in 28 + 7 + 20 = 60 - 36, the majority of responses moved from inappropriate in Grade 7, as students stated equals meant the answer; to inappropriate or algebraic in Grade 8, as students explained equals as either the answer or denoting equivalence; to algebraic in Grade 9 with most students explaining equals as equivalence or showing a balanced equation..

Meaning of Unknown and Variable The majority of students in each grade (16, 22, and 21 students, respectively) indicated that the 0 in 0 + 5 = 9 represented an unknown number; similarly for the x in x + 7 = 16. However, most Grade 7 students did not know the meaning of 3x = 12 and gave an inappropriate explanation. In Grade 8, most students explained the concatenated x either arithmetically as a times sign or prealgebraically as an unknown number. In Grade 9 most explanations were prealgebraic. Similarly, for the x in the expression 3x, most students in Grades 7 and 8 responded arithmetically that it was a times sign. However, by Grade 9 the majority of students stated pre-algebraically that it was an unknown number or that it represented any number. For 0 + 5, most Grade 7 students (26) stated prealgebraically that 0 represented an unknown number or any number, five students

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Table 2

Examples and Response Frequencies for Meaning of Equals Examples

Task/ Category

Frequency Gd. 7

Gd. 8

Gd. 9

28 + 7 + 20 =?

Inappropriate

It's asking for the answer which would go beside the equals sign.

33

32

30

Arithmetic

It wants you to make everything on this side equal to the other side. You could write that on the other side it has to be equal, both sides.

o

1

3

28 + 7 + 20 = 60 - 36

Inappropriate

It means that that is the answer to that. The 60 - 36.

21

13

5

Arithmetic

There's 2 sides to the equation so if you divide 28 by 7 and add 20 you get the same as 60 take 36.

9

8

9

Algebraic

It's sort of like a balanced equation so 28 divided by 7 plus 20 is the same as 60 minus 36.

3

12

19

gave inappropriate responses, and two stated it was the answer. In Grades 8 and 9, the majority of students stated pre-algebraically that 0 was an unknown number (29 and 32 respectively) or any number. In Grade 9, only one student responded arithmetically that 0 was the answer. Examples and response frequencies for x + 7 = 16, 3x = 12, and 3x are shown in Table 3.

Solution of a Linear Equation For 3(x + 7) = 24, the majority of students in Grade 7 did not know how to solve the equation. Some of these responses indicated a lack of understanding of brackets; students initially added 3 and 7 to get 10 and subtracted this from 24 to get 14 for x. Some students used a pre-algebraic inverse procedure. By Grade 8, the majority of students still did not know how to solve the equation; however, there was a substantial number of students who did use a pre-algebraic inverse procedure. Arithmetic processes were evidenced by a small number of students who either used trial and error or an inverse method to find what they referred to as "the space after the x." These students regarded the x as a times sign and therefore were looking for a number to immediately follow it. By Grade 9, most students used prealgebraic inverse processes. There was still a small number of students who did not know how to solve the equation and gave inappropriate responses. Only three students were able to use a complete balance procedure, which was categorised as

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algebraic. In Grade 9 only, students were asked to solve an equation which required knowledge of balance procedures and simplification, namely x + 3 = 2x - 1. Most students used a balance approach, some students did not know how to solve the equation, and a small number of students used an arithmetic approach of trial and error. Examples and response frequencies are shown in Table 4. Table 3

Examples and Response Frequencies for Meaning of Unknown and Variable Task/

Frequency

Examples

Category

Cd. 7

Gd. 8

Gd. 9

x + 7 = 16

x

Inappropriate

a number.

any

3

1

1

Arithmetic

There'd be one number there and one number there (either side of the x) cause it would be like 2 times 5 plus 7 equals 16.

12

8

6

Pre-algebraic

A number that is missing, add to 7 equals 16. The number missing is thex.

18

24

26

Inappropriate

It's 3x. [Interviewer: Do you know what x is?] No. It's just a normal x and there's 3 of it. [Interviewer: Do you mean like when writing?] Yeah sort of.

18

9

4

Arithmetic

3x equals 12 so 3 times something

9

12

4

It is 3 times an unknown number and it equals 12.

6

12

25

3x means I don't know. x is an

9

4

4

19

15

2

5

14

27

IS

certain

number,

3x = 12

has to equal 12. It's 4. Pre-algebraic

3x Inappropriate

abbreviation-I'm not sure about it. Arithmetic

Three times. The x is the times.

Pre-algebraic

3 times whatever amount x IS whatever value x IS. x IS a variable.

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Table 4 Examples and Response Frequencies for Solution ofan Equation Task/ Category

3(x + 7) = 24 Inappropriate

Examples

Frequency Gd. 7 Gd. 8 Gd. 9

Add 3 to 7. [Interviewer: Do you know what you are trying to find out?] No.

27

15

6

Arithmetic

Do the ( ) first. x could be a number. You have to do x plus 7 then 3 times that equals 24. 24 minus 7 is that right? I'd use trial and error-try 2, 7 plus 2 is 9 then 3 times 9, no that doesn't work out to 24 so go 1 plus 7 is 8 and 3 times 8 is 26 isn't it? It's 24 so I'd use trial and error. x is 1.

o

6

1

Pre-algebraic

Work out the value of x. 3 times 7 is 21 then 24 take 21 is 3 so divide 3 by 3, x is 1. You have to divide into the first 3.

6

12

23

Algebraic

Written: 3(x + 7) = 24. 3x + 21 = 24, 3x + 21 -21 = 24 - 21, 3x = 3, 3x/3 = 3/3, x = 1

o

o

3

x + 3 = 2x-l

Inappropriate

You can't add the x and the 3 because they are not like terms. You can't take 1 away from 2x because they are not like terms. I don't know. We haven't these.

10

Arithmetic

Work out the x's, they will be the same thing. I do lots of guess and checks. I'd start with 2 times 2 but that wouldn't work because you have to minus 1 and that (LHS) would equalS. 23's is 6 and that one is 5 no that doesn't work. It would work with 4.

5

Algebraic

Find x and it will be the same number on both sides. Wrote: x + 3 = 2x . , I, x + 3 - 3 = 2x -1 - 3, x = 2x - 4, x + 4 = 2x - 4 + 4, x + 4 = 2x, x - x + 4 = 2x - x, 4 = x.

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Discussion

!

Resultsof this study indicate that most students in Grades 7 and 8 did not have a satisfactory understanding of commutative and distributive laws to use as a basis for algebra. Results also showed that most students believed equals, in the incomplete equation, .denoted the answer. This perception resulted in inappropriate responses for the commutative law for some students who focused on the equals sign and stated that none of the signs would fit. Thus students not only failed to see the full structure of the equation, they also failed to see the relationship between elements of the problem; and as Scandura (1971) argued, algebra is based on relationships. These results also correspond with the beliefs of MacGregor (1996) and Demana and Leitzel (1988) in that students have inadequate conceptions of these arithmetic principles. It was not until Grade 9 that most students had sufficient understanding of commutative and distributive laws to apply these to linear equations. Such inadequacies point to the need for explicit instruction in these arithmetic principles if cognitive difficulties for students beginning algebra are to be reduced. For equals in the complete equation, understanding moved from inappropriate in Grade 7, to inappropriate or algebraic in Grade 8, to algebraic (denoting an equal or balanced relationship) in Grade 9. Kieran (1981) noted that students require an equivalence understanding of equals to operate algebraically. In each grade almost one third of the students interpreted "=" arithmetically, that is each side of the equals sign as the same value. This suggests that while students' knowledge of "=" had developed, there was still a substantial number of students who did not understand "=" in an alg'ebraic sense. Therefore these students would not be able to carry out transformations or, as Linchevski (1995) suggested, understand multidirectional relationships. We believe that providing explicit instruction of equals initially at an arithmetic level then subsequently at a pre-algebraic level (that is, that each side of equals then each side of the equation is the same therefore operating can occur from either side) will provide the foundation to facilitate movement from an arithmetic to algebraic understanding of equals. Most students, over the three years, knew that D in the expression and equation represented an unknown number. This could be interpreted as understanding that was based on prior arithmetic knowledge as D is often used to denote a missing number in early arithmetic (Herscovics & Linchevski, .1994; Kieran, 1981). As understanding emerged in Grade 9 some said that it was a variable. These results indicate that understanding D as an unknown number appears to be a suitable foundation from which to introduce the concept of any number or variable. We consider this constitutes, in part, pre-algebraic understanding. Understanding of x in 3x was more cognitively demanding. Herscovics and Linchevski (1994) found that students have difficulty in interpreting concatenated letters. Some students provided explanations that were grounded in their arithmetic knowledge su~h as stating that x was a multiplication sign. Booth (1988) and MacGregor and Stacey (1993) suggested delaying the omission of the "x" sign, \

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thus allowing students more time to assimilate the concatenated letter. We feel this would be most suitably placed as part of pre-algebraic instruction. Results for solving the linear equations indicated that some students exhibited a lack of understanding of brackets, a finding that was also made by Linchevski and Herscovics (1994). This, in addition to a lack of understanding of concatenated x meant that most students in Grades 7 and 8 could not solve the linear equation. Herscovics and Linchevski (1994) and Filloy and" Rojano (1989) noted that students have difficulty in operating on letters in equations and defined this as a gap between arithmetic and algebra. However the results of the present study indicate that, from Grade 7, students have a good understanding of inverse operations. In attempting to solve the linear equation, students often disregarded what they did not understand, that is the x or brackets, and subsequently applied a sequential inverse process-thereby basing their solution procedures, albeit incorrectly, on what they knew from arithmetic. We propose that sequential inverse procedures constitute a sound basis for learning the simultaneous procedures of operating on equations with unknowns and would be appropriately placed at a pre-algebraic level of functioning. By Grade 9 just over half the students could use balance procedures to solve an equation with an unknown on each side of the equals sign. It is interesting to note, however, that the remaining half either did not know how to solve the equation or used arithmetic solution methods even though they had just received instruction in balance procedures. This finding serves to reinforce the need for instruction that will facilitate understanding of balance procedures for students; we suggest pre-algebraic instruction as a means of achieving this.

Conclusion It is our contention that for students to understand algebraic concepts they must have a firm understanding of arithmetic laws and operations followed by pre-algebraic principles; we have conceptualised this as a sequential development of knowledge as depicted in the model in Figure 1. This study showed that students' understanding of commutative and distributive laws and order of operations developed sequentially, and that by Grade 9 students had achieved sufficient understanding of these principles to enable them to operate algebraically. Similarly, students' understanding of equals in a complete equation developed sequentially over the years from inappropriate in Grade 7 to mostly inappropriate and algebraic in Grade 8, with some students also explaining equals arithmetically, and then to algebraic in Grade 9. These results fit, in part, the sequence for understanding equals that is proposed in the model. Additionally, we propose that a pre-algebraic understanding of equals be taught, that is, both sides of the equation are the same value. In doing so, understanding of equals will move sequentially, facilitating understanding of an equivalence relationship that is necessary to operate algebraically. Most students either could not explain the concatenated x in Grade 7 or they explained it arithmetically as a times sign. However, as understanding developed, some students were able to explain it pre-algebraically as an unknown number in Grade 8 and then as a variable in Grade 9. This finding also supports the model in that it is proposed that unknowns then variables should form a part of pre-

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algebraic instruction. Equation solution procedures also progressed from mostly inappropriate in Grades 7 and 8 to pre-algebraic in Grade 9; in addition, most students were able to apply the algebraic balance procedures that were taught in Grade 9. Thus, understanding of numerical procedures then inverse procedures to find an unknown provided a sound basis from which to learn algebraic balance procedures. Clearly, for each task, understanding had progressed in a sequential manner from arithmetic to pre-algebraic and in some instances to algebraic over the three years. These findings support the sequence of instruction as proposed in the model. Results of this study indicate that there is a need for instruction that will facilitate, for students, the ability to move from operating arithmetically to operating algebraically. Thus the more complex concepts of algebra can be developed sequentially from the lower level arithmetic concepts by allowing students time to assimilate quantitative and qualitative changes in their knowledge structure. This means that initially students will need to understand arithmetic principles such as those proposed in the first stage of the model depicted in Figure L Once these principles are understood, students will have a sound foundation from which to develop more abstract concepts through undertaking pre-algebraic activities. For example, students' understanding of an unknown can be extended to variable, solution processes can include solving to find an unknown in an equation, and understanding of equals can be introduced as each side of an equals sign being the same or equal. It is feasible to argue that students who undertake a sequence of mathematical instruction that incorporates a pre-algebraic level will be better prepared to move to algebraic concepts. Each increasingly more complex level of sequential development, as proposed in the model, will make an equal demand on processing load until concepts become well known and are subsumed into a concept at the higher level. Overall, this study highlights the need for an operational level of pre-algebra to address inadequacies in students' prerequisite knowledge and to prepare students for the symbolism and operations of algebra. This is suggested on the basis that results for the three years of the study support the sequence of instruction as proposed in the model. Specifically, we suggest that pre-algebraic instruction should include use of letters as unknowns in equations; then variables in expressions, concatenation, and use of inverse procedures to find an unknown in a linear equation; and extending the concept of equals to each side of an equation being the same value. Additionally we suggest that, prior to initiating pre-algebraic instruction, students' knowledge of operational laws and use of brackets be reviewed and if necessary reinforced. We believe that providing instruction at a pre-algebraic level, as outlined in the model, will serve to extend arithmetic knowledge and prepare students for the complexities of algebra. Future research should investigate teaching strategies in primary school mathematics that may avoid causing cognitive obstacles to understanding operational laws, meaning of equals, brackets, and use of symbols as place-holders.

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Authors Hitendra Pillay, Faculty of Education, Queensland University of Technology, Locked Bag No.2, Red Hills QLD 4059. Email: . Lynn A Wilss, Faculty of Education, Queensland University of Technology, Locked Bag No.2, Red Hills QLD 4059. Email: . Gillian M. Boulton-Lewis, Faculty of Education, Queensland University of Technology, Locked BagNo 2, Red Hills QLD 4059. Email: .