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ZDM

, Volume 40, Issue 1, pp 39–53 | Cite as

Algebraic thinking with and without algebraic representation: a three-year longitudinal study

  • Murray S. Britt
  • Kathryn C. Irwin
Original article

Abstract

An Algebraic Thinking Test was given to 116 students aged 12–14, at the end of each of three years. This age span crosses two levels of school in New Zealand. This test assessed their ability to represent compensation in the four arithmetic operations both numerically and with letters for variables. The analyses of these results, together with the results from separate interviews designed to report individual progress of students in the New Zealand Numeracy Project, showed that students who had developed advanced mental strategies for dealing with additive, multiplicative and proportional operations, were the students who were capable of making full use of the alphanumeric symbols of algebra. These results, taken together with earlier studies by the authors, led to a proposal for a “pathway for algebraic thinking” accessible to all students.

Keywords

Operational Strategy Professional Development Program Lead Teacher Algebraic Thinking Decimal Fraction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

In a recent plenary address to the North American Chapter of the International Group for the Psychology of Mathematics Education, Radford (2006) presented a compelling argument in support of a generalization approach to algebra in which he noted that, “the algebraic generalization of a pattern rests on the noticing of a local commonality that is then generalized to all terms of the sequence and serves as a warrant to build expressions of elements of the sequence that remains beyond the perceptual field” (p. 5). As well, Radford adds a third element, using the commonality to provide a direct expression or rule to specify any term of the sequence. But while it is customary to require that learners use the symbols of algebra to express such rules, Radford acknowledges that rule-making proceeds through various layers of awareness articulated through different semiotic systems; words, gestures, pictures, graphs and symbols. Irwin & Britt (2005a) illustrate such a layer of awareness in a study that contributed to an evaluation of The New Zealand Numeracy Project, a national project in which students throughout New Zealand are encouraged to devise and experiment with a range of mental operational strategies in arithmetic. In that study, we argued that students who could apply an operational strategy to sensibly solve different numerical problems were disclosing an awareness of the relationships of the numbers involved as well as the underlying structure of the strategy. We claimed that successful application of such operational strategies demanded an awareness of the generality of the operational strategy thus illustrating algebraic thinking. Students’ explanation of their thinking revealed that they were treating the numbers as if they were variables. Fujii & Stephens (2001) refer to numbers used in this way as quasi-variables (see also Warren & Cooper, 2002). The results of our study led to a view of algebra, particularly as it pertains to the pedagogy of introductory algebra, in which we do not see algebra as following arithmetic so that arithmetic has an ending that coincides with the beginning of algebra. Instead our view is consonant with those of Hewitt (1998, p. 20), who argues that algebra enables arithmetic to be carried out, and of Steffe (2001, p. 563), who argues that children’s knowledge of number together with numerical operational knowledge that is effective and reliable is essentially algebraic in nature.

Much has been written about the difficulties encountered during the transition from arithmetic to algebra (see for example, Filloy and Rojano, 1989; Herscovics & Linchevski, 1994; Kieran, 2001). But as Carraher, Schliemann, Brizuela and Earnest argue, acceptance of such a transition arises from an impoverished view of elementary mathematics in which mathematical generalization is postponed until the onset of algebra instruction (2006, p. 89). We argue similarly. The notion of algebra in arithmetic, in which generalization provides the basis for successful numerical operational thinking, dismisses the claim for transition and offers algebra for all through algebraic thinking with and without the symbols of algebra. Also, whereas Carraher et al (2006) argue for the early inclusion of algebraic symbols as a valuable tool for early algebraic thinking, our view is that young children need to work with several layers of awareness of generality prior to their introduction to the semiotics of algebra.

In this paper we begin our discussion of algebraic generality by illustrating how the development of algebraic thinking can evolve, without recourse to the symbols of algebra, from students developing an ongoing awareness of the underlying structure of operational strategies in arithmetic. We first draw on the New Zealand Numeracy Project (Ministry of Education, 2007), to illustrate the development of operational strategies in arithmetic and thus algebraic thinking for students at different ages, and then focus on a recent longitudinal study in which we attempt to ascertain what level of strategy development is likely to be necessary for students to extend their expressions of generality from using numbers themselves as quasi-variables to a semiotic layer of awareness that embraces the literal symbols of algebra.

2 Algebra within arithmetic: algebraic thinking

2.1 The New Zealand Numeracy Project and algebraic thinking

In 1999, the New Zealand Ministry of Education introduced a professional development program in mathematics known as the Numeracy Project, motivated by the need to improve the number sense and operation sense of students by introducing a flexible approach to solving problems in numerical situations (see for example, McIntosh, Reys & Reys, 1992; Slavit 1999; Wright 1994). A primary purpose of the professional development project was to get teachers to focus on the strategies and knowledge that each student brought to a problem. At first the project was intended for year 1–3 students (aged 5–7 years). In 2001, the project was extended to year 4–6 students (aged 8–10 years). In 2002, following a pilot study involving over 4,000 year 7–10 students (aged 11–14), the project was expanded into the Secondary Numeracy Project (Ministry of Education, 2007), and offered to some 13,600 year 7–10 students. By 2007, nearly all elementary schools were using at least some aspects of the Numeracy Project and an increasing number of secondary schools were becoming involved.

In order to provide a focus for this professional development program, a Number Framework that mostly embodied the achievement aims and objectives in Levels 1–4 (years 1–8) of the New Zealand Curriculum (Ministry of Education, 1992) was developed. In the Framework, a distinction is made between strategy and knowledge (Ministry of Education, 2006a). The strategy section describes the thinking students use to estimate answers and solve operational problems with numbers. The knowledge section describes the key items of number knowledge that students need to learn and without which they will be unable to broaden and advance their repertoire of strategies. The two are linked in that strategies create new knowledge that in turn provides a foundation for the development of new strategies. Tables 1 and 2 show the progression of strategies that we argue in Irwin & Britt (2005a), form a strong basis for ongoing opportunities for students to develop an awareness of generality and hence of algebraic thinking.
Table 1

The Number Framework for Stage 1–4 operational strategies that involve counting

Operational domains

Global stage

Addition and subtraction

Multiplication and division

Proportions and ratio

Counting

Stage 0

Emergent

Emergent

Unable to count a given or form a set of up to ten objects

Stage 1

One-to-one counting

One-to-one counting

Unequal sharing

One to one counting

Able to count a set of objects

Able to count a set of objects

Unable to divide a region or set into two or four equal parts

Unable to form a set of objects to solve simple addition and subtraction problems

Unable to form a set of objects to solve simple multiplication and division problems

 

Stage 2

Counting from one

Counting from one

Equal sharing

Counting from one on materials

Counts objects to solve simple addition and subtraction problems

Solves simple multiplication and division problems by counting one-to-one with the aid of materials

Able to divide a region or set into a given number of equal parts using materials

Needs to use materials such as counters or fingers

  

Stage 3

Counting from one

Counting from one

Equal sharing

Counting from one by imaging

Counts objects by visualizing or imaging

Counts all the objects in simple multiplication and division problems by imaging the objects

Able to share a region or set into a given number of equal parts using materials or by imaging

Unaware of 10 as a counting unit

Uses materials to solve multiplication and division problems with larger numbers

 

Solves multi-digit problems by counting all the objects

  

Stage 4

Counting on

Skip counting

 

Advanced counting

Counts on or back to solve simple addition and subtraction problems

Skip counts to solve simple multiplication and division problems using materials or imaging

 
Table 2

The Number Framework for Stage 5–8 operational strategies that involve part-whole thinking

Operational domains

Global stage

Addition and subtraction

Multiplication and division

Proportions and ratio

Part-whole

Stage 5

Early addition and subtraction

Multiplication by repeated addition

Fraction of a number by addition

Early additive

Uses a limited range of mental strategies to solve addition and subtraction problems. E.g., 8 + 7 is 8 + 8−1 (doubles) and 39 + 26 = 40 + 25 = 65 (partitioning and compensation)

Uses a combination of known multiplication facts and repeated addition. E.g., 4 × 6 is (6 + 6) + (6 + 6) = 12 + 12 = 24

Uses addition facts to find the fraction of a number. E.g., \( \tfrac{{\text{1}}} {{\text{3}}} \) of 12 is 4 since 4 + 4 + 4 = 12

 

Uses known multiplication facts with repeated addition, to anticipate the result of division. E.g., 20 ÷ 4 = 5 since 5 + 5 = 10 and 10 + 10 = 20

Solves division problems mentally using halving or deriving from known addition facts. E.g., when seven pies are shared among four children each gets one pie plus \( \tfrac{{\text{1}}} {{\text{2}}} \) of a pie plus \( \tfrac{{\text{1}}} {{\text{4}}} \) of a pie

Stage 6

Advanced addition and subtraction of whole numbers

Derived multiplication

Fraction of a number by addition and multiplication

Advanced additive–early multiplicative

Can estimate answers and solve mentally addition and subtraction problems that involve whole numbers by choosing appropriately from a broad range of advanced mental strategies. E.g., 324 − 86 = 324−100 + 14 and 1242−986 = 1242 + 14 − (986 + 14)

Uses a combination of known multiplication facts and mental strategies to derive answers to multiplication and division problems. E.g., 4 × 8 = 2 × 16 = 32 (doubling and halving) and 9 × 6 is (10 × 6) − 6 = 54

Uses repeated halving or known multiplication and division facts to solve problems that involve finding fractions of a set or region, and division with remainders. E.g., \( \tfrac{1} {3} \) of 36 = 12 since 3 × 10 = 30, 6 ÷ 3 = 2, and 10 + 2 = 12.

Stage 7

Addition and subtraction of decimals and integers

Advanced multiplication and division

Fractions, ratios, and proportions by multiplication

Advanced multiplicative–early proportional

Can estimate answers and solve mentally addition and subtraction problems that involve decimals, integers and related fractions by choosing appropriately from a broad range of advanced mental strategies. E.g., 3.2 + 1.95 = 3.2 + 2 − 0.05 = 5.2 − 0.05 = 5.15

Chooses appropriately from a broad range of mental strategies to estimate answers and solve multiplication and division problems. E.g., 24 x 6 is (20 × 6) + (4 × 6) or 25 × 6 − 6; 81 ÷ 9 = 9 so 81 ÷ 3 = 3 × 9; and 4 × 25 = 100, so 92÷4 = 25 − 2 = 23

Uses a range of multiplication and division strategies to estimate answers and solve problems with fractions, proportions, and ratios. E.g., \( 13 \div 5 = (10 \div 5) + (3 \div 5) = 2\frac{3} {5}; \) 3:5 is equivalent to 24:40 since 8 × 3 = 24 and 8 × 5 = 40

Stage 8

Addition and subtraction of fractions

Multiplication and division of decimals, multiplication of fractions

Fractions, ratios and proportions by re-unitising

Advanced proportional

Uses a broad range of mental partitioning strategies to estimate answers and solve problems that involve adding and subtracting fractions including decimals

Chooses appropriately from a broad range of mental strategies to estimate answers and solve problems that involve the multiplication and division of decimals and the multiplication of fractions. E.g., 4.2 ÷ 0.25 = (4.2 × 4) ÷ (0.25 × 4) = 16.8 ÷ 1

Chooses appropriately from a broad range of mental strategies to estimate answers and solve problems that involve fractions proportions and ratios. E.g., 6:9 is equivalent to 16:24 since \( 6 \times 1\tfrac{1} {2} = 9 \) and \( 16 \times 1\tfrac{1} {2} = 24 \) or \( 9 \times 2\tfrac{2} {3} = 24 \) and \( 6 \times 2\tfrac{2} {3} = 16 \)

Combines ratios and proportions. E.g., 20 counters in ratio of 2:3 combined with 60 counters in ratio of 8:7 gives a combined ratio of 1:1

  

The strategies illustrated in Table 2 involve part-whole thinking in which students recognize that numbers are abstract units that can be treated simultaneously or can be partitioned and recombined.

An essential part of the Numeracy Project is a diagnostic test given to students individually as an interview. The intention of the interview is to provide quality information about the knowledge the student has and mental strategies the student can apply successfully. That information is recorded in the form of judgments and accompanying notes related to the stages of progress reached according to the Stages in the separate Knowledge and Strategy sections of the Number Framework. Three strategy items from the test are as follows.
  • Sandra has 394 stamps. She gets another 79 stamps from her brother. How many stamps does she have then? One sensible strategy for this is to add 6 to 394 and subtract 6 from 79 so that the task becomes 400 + 73 or 473. Another strategy may be to split 79 into 6 + 73 and then combine 394 and 6 before adding 73 to the result, 400. Either strategy illustrates Stage 6, Advanced Additive part-whole thinking.

  • At the car factory, they need 4 wheels to make each car. How many cars could they make with 72 wheels? A sensible strategy for this task may be to see that 80 wheels are needed for 20 cars since 4 × 20 = 80, so that 20 − 2 = 18 cars need 72 wheels. Another strategy might be to note that 10 cars need 40 wheels since 10 × 40 = 40, and the remaining 32 wheels are required for a further 8 cars. So altogether 72 wheels are needed for 18 cars. Either strategy illustrates Stage 7 Advanced multiplicative part-whole thinking.

  • It takes 10 balls of wool to make 15 beanies (hats). How may balls of wool does it take to make 6 beanies? One sensible strategy for this task may be to se that two balls of wool will make three beanies so that four balls of wool will make six beanies. A second strategy may be to see that since 10 balls of wool make 15 beanies, 1 ball of wool will make \( 1\frac{1} {2} \) beanies so that 4 balls of wool will make 6 beanies. Either strategy illustrates Stage 8 Advanced Proportional thinking.

Students are grouped for teaching purposes according to the highest stage for their operational strategy thinking and a teaching program, drawn from the published array of activities designed to reflect the Number Framework and therefore the national curriculum, is devised to meet the needs of each group in both strategy development and knowledge acquisition (see http://www.nzmaths.co.nz/numeracy/project_material.aspx). Adjustments are made to the teaching program as a result of ongoing assessment of strategy and knowledge development.

Since 2000, when it began as a modest experiment, the Numeracy Project has been subjected to a number of evaluations (see http://www.nzmaths.co.nz/numeracy/References/Reports.aspx). These evaluations have provided ongoing national data for the project over a five-year period. Evidence, including measures of effect sizes related to the performance of students, including those from different ethnic groups, on the Number Framework and of issues related to the validity of the Framework and reliability of tests administered by classroom teachers (r = 0.81), have supported the continuing development of the project (see Irwin & Niederer, 2002; Irwin, 2004; Tagg & Thomas, 2006a, 2006b; Thomas, Tagg & Ward, 2006; Young-Loveridge, 2004, 2006), and have led to the establishment of standards for students’ learning in number for each of the first four levels of the national curriculum (for ages 5 through 12), which is currently in its final stage of redevelopment (see Ministry of Education, 2006b).

2.2 Algebraic thinking in a Numeracy Project primary classroom

In order to clarify the algebraic nature of operational strategy teaching and learning, we illustrate a typical teaching sequence from the Numeracy Project. We describe the actions and responses of Mary, a student aged 8, as she engages with a series of tasks designed to help her advance from Stage 4 (Advanced Counting) to Stage 5 (Early Additive) strategy thinking.

Mary’s initial task is to use tens-frames (see Fig. 1) to help devise a sensible non-counting strategy to work out 9 + 4. Prior to engaging with this task, Mary had shown she could count to 99 and interpret 2-digit whole numbers according to the place-value system for tens and ones, she could instantly recognize the numbers of counters displayed on tens-frames and she knew pairs of numbers that added to 10.
Fig. 1

Using tens-frames to develop early understanding of additive compensation

Mary’s teacher began by asking her to put counters on a tens-frame to show nine and then a further four counters on another tens-frame to show four (Fig. 1a). Figure 1b shows the outcome of Mary’s actions that transform 9 + 4 into 10 + 3 so leading to her recognition of 13 as the answer.

Mary then went on to use the tens-frames to figure out the solution to several similar tasks, 8 + 5, 7 + 6, and 9 + 7 before she was challenged by her teacher to see if she could work out 19 + 4, 27 + 6, and 38 + 7 without recourse to the tens-frames. She could revert to using the tens-frames if she was unsure what to do. She was asked by her teacher to explain her thinking after each response. For 19 + 4, she said she took one from the 4 and put it on the 19. So in her mind she could see 20 and 3 making a total of 23. She similarly transformed 27 + 6 into 30 + 3 = 33 and 38 + 7 into 40 + 5 = 45 each time describing an additive compensation strategy for which the underlying structure or generalization may be represented algebraically as a + b = (a + c) + (b − c). Such descriptions of her thinking represent a layer of awareness of generality expressed in words.

There are several aspects associated with Mary’s thinking that warrant further analysis. Firstly, the role assigned to Mary’s use of the tens-frames is one of “Image-Making”, an early level of understanding drawn from the Pirie–Kieren Model for the Growth of Mathematical Understanding (Pirie & Kieren, 1989, 1994), that forms the basis of a teaching model recommended for use with the Numeracy Project materials (Ministry of Education, 2007). In the Numeracy Project Teaching Model, students begin with a “Using Materials” phase designed to build concrete images that reflect her thinking and in so doing reflect the Image-Making level proposed by Pirie and Kieren. As she manipulates the counters on the tens-frames, she notices the space/s to be filled on one frame to make it complete and also where she can get the counter/s to do the filling of the space/s. She begins to develop an awareness of the consequence of this compensation action (Pirie–Kieren Image Having level), and so begins to visualize the structure of the transformation that she uses subsequently to solve the more challenging tasks, 19 + 4, 27 + 6, and 38 + 7. In the Teaching Model these tasks are included within a “Using Imaging” phase and are designed to challenge Mary’s thinking towards an awareness of the transformations involved in the tasks but in the absence of the tens-frames. And as this happens, she is likely to have begun to isolate the features that are common to each task. This Pirie–Kieren “Property Noticing” level of understanding—“Using Number Properties” phase in the Project’s Teaching Model—in which she is able to explain the roles of the various elements in the transformation, leads to the conclusion that Mary has generalized the transformation, that is, she has engaged in algebraic thinking. As with the Pirie–Kieren model, the Project’s Teaching Model incorporates opportunities for folding-back. For example, if Mary had not succeeded with the tasks at the Using Imaging phase she would fold back to work with more tasks in the Using Materials phase. She would then advance again to the Using Imaging phase when she had demonstrated success with these tasks.

2.3 Algebraic thinking in the numeracy project: a research study with students aged 12

In 2002, we carried out an evaluation of one aspect of the Numeracy Project (Irwin & Britt, 2005a). The goal of the evaluation was to gauge the extent to which students aged 12, who participated in the Numeracy Project, used operational strategies, deemed algebraic in nature, more successfully than students of a similar age who did not participate in the project. We wanted to test our conjecture that project students would show a greater awareness of the algebraic structure of problems in arithmetic. The 21-item test comprised six sections: compensation in addition, x + = (x + a) + (y − a); compensation in subtraction, x − y = (x + a) + (y + a); compensation in the distributive law of multiplication over addition, xy = x (y + a) − xa; equivalence with sums and differences in which one of four structures is, “If x + a = b, then x = b − a”; compensation in multiplication, \( xy = (ax){\left( {\frac{y} {a}} \right)}; \) and equivalence with fractional values, again in which one of four structures is, “If \( \frac{a} {b} = \frac{{an}} {x}, \) then x = b × n”.

Altogether the study involved 899 students aged 12, from four schools of which 431 participated in the Numeracy Project in 2002 and 468 did not. Students of this age were chosen for the study because they had not been taught formal algebra prior to the study. The test was subjected to a Rasch analysis with reliability 0.88 as estimated by Kuder-Richardson’s formula 20. An analysis of the scores of the students who had participated in the Numeracy Project was significantly higher than that attained by those who had not participated in the project F(1,895) = 47.44, < 0.01. The results suggested strongly that students’ involvement in the Numeracy Project was likely to lead, not only to improved outcomes for the arithmetic involved, but with respect to current and future algebraic activity, improved algebraic thinking skills. These results are in accord with the views of a number of researchers such as MacGregor and Stacey (1999) and Orton and Orton (1994) who have argued that the understanding of and skill in using arithmetical relations are a necessary pre-requisite for learning algebra. While we do not disagree with this we feel that we are able to explain why this is likely to be the case. In our view, it is not merely that students involved in such arithmetic activity develop greater skill in this area but rather it is because they have already begun their algebraic development as a consequence of their participation in the Numeracy Project where a considerable amount of their time had been given to working with the strategies as described in the Project Framework (see Tables 1, 2). That work, as we have previously argued, demands the development of awareness of generality representing algebraic thinking.

2.4 Algebraic Thinking in the Numeracy Project: a teaching study with students aged 13

As previously indicated, the Numeracy Project was expanded in 2002 to include some 13,600 students in secondary school. The Project teaching program for these students focused not only on using a range of operational strategies in number that we describe as algebraic thinking but also provided increasing opportunities for students to express generality with the symbols of algebra. In the following, we show aspects of a lesson conducted in a Secondary Numeracy Project girls-only secondary school by a student teacher with 30 students aged about 13, working mostly together in pairs.

Students were shown what was described as a drinks coaster, a square-shaped 5-by-5 mat that had 5 circles (counters) on each side giving a total of 16 circles on its edge (Fig. 2).
Fig. 2

5-by-5 mat (drinks’ coaster)

The students, all of whom had been adjudged as being at least Stage 6 (Advanced Additive—Early Multiplicative) in the Number Framework (see Table 2), were asked to devise with the aid of colored counters, different representations of the 5-by-5 coaster and to illustrate each representation with an operational strategy for counting the total number of circles on the edge of a coaster with different numbers of circles, for example, 100, 47, and 139 on a side. They were then to attempt to write algebraic expressions, derived from their representations, for the total number of circles on the edge of a coaster with any number, n, of circles on a side.

Each pair of students produced at least one strategy that they represented numerically, where numbers are used as quasi-variables, as well as algebraically for n circles on a side. The algebraic expressions of generality included: 2 × n + 2 × (n − 2) for the pattern in Fig. 3a, 4 + 4 × (n − 2) for the pattern in Fig. 3b, and 2 × (n − 1) + 2 × (n − 1) or in some cases 4 × (n − 1) for the pattern in Fig. 3c. Several students already knew the conventions for writing algebraic expressions and were, for example, able to represent 2 × n + 2 × (n − 2) as 2n + 2(n − 2).
Fig. 3

A series of 5-by-5 coasters illustrating numerical operational strategies for counting circles leading to algebraic generalizations for n-by-n coasters

A small number of students also produced the rule, 4 × n − 4, which they were able to explain as, “Each side has n circles so that makes four groups of n circles from the four sides. But each corner circle overlaps another corner circle, so four corner circles must be subtracted leaving a total of 4 × n − 4 circles”. Interestingly, none of these students represented this particular rule, arguably the simplest rule for the task, numerically prior to writing the rule algebraically. They said that they just saw the rule.

These students who were able to see generality within the figural representations and were able to express the generalization for a coaster with n circles on each side in a valid algebraic form may accordingly be classified as predominantly “Figural Generalizers” working at the “Representational Level of Generalizing”: which Rivera (2006, p. 4) notes, “is characterized by an ability to use symbols in expressing and communicating a generalization”. Comments made by a number of the students suggest that an important factor leading to successful rules or strategies for counting circles on the edge of coasters related to their use of colored counters to devise patterns of circles.

2.5 A research study in which generality is expressed by young children

Thus far we have focused our discussion of generality to students in primary or elementary schools and to beginning secondary classrooms. But it would be remiss if we were to ignore discussion of the algebraic thinking that children engage in prior to and during the beginning of their formal education. The earlier focus on additive compensation carried out by Mary (see Sect. 2.2) brings to the fore a study by Irwin (1996) who demonstrated that the concepts of compensation and covariation of the size of the whole with the size of the parts, without reference to numbers, was understood by children of 4 years of age. In her study, children aged 4 were asked if the amount of lollies (sweets) in two containers would be the same or different if some were moved from one container to another, if one was removed or added to one container, or if one was removed from one container and another added to the other container. Children of this age were certain that the total quantity would stay the same if an item was moved and would increase or decrease if one of the parts was altered. At age 5 and 6 they could explain these relationships, sometimes with a principle in their own language that showed they understood this as a generalization. For example, one child said the total number in a compensatory move would be the same, “The same, except (doll’s name) put one of the lollies from here to here”. Expressed algebraically, they understood that if x + y = W, then (x + a) + (y − a) = W, where x and y are the generalized unknown (uncounted) number of lollies in the separate containers, a stands for the number of lollies transferred from one container to the other and W stands for the number of lollies (uncounted) altogether.

However, if they were given a similar task with numbers only using doubles facts that they knew such as 5 + 5, and asked whether or not it would be the same as 4 + 6, they were unsuccessful until about age 7. As one child phrased it using a visual image for the equality of 10 + 10 and 11 + 9, “…because if you put one of the group of 11 over to the 9 group they would both be 10 and that means 20.” Since young children can understand this compensation concept when no numbers are attached, it may be that the complexity of learning to understand numbers distracts students from the knowledge that they had in a proto-quantitative form (Resnick, 1992) before going to school.

2.6 The growth of algebraic thinking: from numbers to symbols. The 2004–2006 study

Following the study (Irwin & Britt, 2005a) in which we assessed the difference between students engaged in the Numeracy Project and those not in the Numeracy Project, we wanted to explore the growth in algebraic thinking across the important move for students from intermediate school (students aged 11 through 12), to secondary school. Our hope was that the algebraic thinking skills gained in intermediate school would not be lost when students proceeded to secondary school, where traditionally algebra has meant “doing arithmetic with letters”, which Mason with Graham and Johnston-Wilder (2005, p. 309) contend, “has proved fruitless for countless generations”.

We reasoned that students, who had developed through their participation in the Numeracy Project, an awareness of the generality in a range of numerical operational strategies, would be able to extend their algebraic thinking to include the standard alphanumeric symbols of algebra. Students who could generalize in this way are at the “Representational Level of Generalizing” (Rivera, 2006). We argued that it was likely they would be able to capitalize on these generalizing skills as they progressed through secondary school.

This study was carried out as one of the regular evaluations of the Numeracy Project that is now used in almost all schools in New Zealand. We sampled students from eight schools in two cities, four from each. Specifically we focused on two questions: (1) What level of strategy development represented in the Numeracy Project Framework is likely to be necessary for students who have participated in the Numeracy Project to be able to extend their expressions of generality to include the literal symbols of algebra to represent compensation strategies in arithmetic that involve addition, subtraction, multiplication and division? (2) What are some of the critical professional characteristics displayed by teachers whose students are able to successfully use algebraic expressions to represent compensation strategies in arithmetic? For separate reports for each of the three years of this study see Irwin & Britt, (2005b, 2006, 2007).

3 Method

3.1 Participants

Four intermediate schools and four secondary schools agreed to participate in this study. Each secondary school was paired with the intermediate school that contributed students to it. Two pairs of schools were on the outskirts of each of two major New Zealand cities. New Zealand classifies the average socio-economic status of schools using a decile ranking in which Decile 1 designates students from poorer homes and Decile 10 designates students from upper socio-economic homes. Table 3 shows that students at the intermediate schools tended to be from lower and middle socio-economic families while the students in the secondary schools, which draw students from wider districts, had students from middle and upper socio-economic families.
Table 3

Characteristics of secondary schools that participated in the assessment of algebraic thinking in 2006

School pair number

Decile ranking of intermediate schoola

Number of students in the intermediate schoola

Decile ranking of secondary schoola

Number of students in secondary schoola

Number of Students for whom three results are available

1

2

222

4

726

13

2

3

280

4

1201

14

3

5

621

6

1583

61

4

7

530

8

1314

28

Total

116

aDecile ranking and number of students as given on Ministry of Education website (http://www.tki.org.nz/e/schools/)

Altogether, 116 students took the test on three occasions. The number of students in the different pairs of schools varied, as shown in Table 3. This reflects mobility of students, absences on one of the three occasions, and related factors. There is no reason to suggest that the sample was atypical.

The intermediate schools were all participants in the Numeracy Project. The secondary schools were chosen by finding out which secondary school the majority of students from the intermediate school moved on to. We asked schools to give the test to all students in year 8 (age 12) at intermediate school in 2004, year 9 at secondary school in 2005, and year 10 in 2006. In addition, each secondary school was asked to give the test to all of their year 9 and 10 students in each of the three years. Our main focus was on the students who took the test for each of the three years, thus giving us three data points for each of the students.

3.2 The assessment instrument

For this study, we devised a new algebraic thinking test that was subjected to multiple trials in four intermediate and secondary schools not included in the main study. A cycle of trialling, item difficulty analysis followed by item modification before continuing the cycle beginning with further trialling led to a test of algebraic thinking with reliability of 0.91 (Kuder-Richardson’s Formula 20). The test comprised four sections, one for each arithmetical operation given in each of the three years of the study. In each section, a solution was first presented showing how a hypothetical student had solved two problems using a compensation strategy for the operation. The first item for each operation involved generalizing the demonstrated method with 3-digit whole numbers, chosen to discourage students’ use of computational approaches while at the same time encouraging the use of the illustrated compensation approach. The second item required them to demonstrate the operation with decimal fractions. The inclusion of decimal fraction items arose partly out of a previous study (Irwin & Britt, 2004), in which we noted that decimal fraction questions increased the level of complexity of the task. We also wanted to provide further opportunity to gauge the effect of extending numerical generalizing beyond whole numbers as is required when letters are used as generalized numbers. The third item asked students to show how the operation would work when one of the numbers in the particular item was represented by a letter as a generalized unknown, the fourth item asked them the same question when the item involved letters and a decimal fraction, and the final item asked them to identify missing letters in an algebraic identity representing the compensation operation. Figure 4 shows the additive compensation tasks for Section A of the Algebraic Thinking Test.
Fig. 4

Section A: additive compensation in the Algebraic Thinking Test

The full test is available as Appendix K in Irwin & Britt (2007).

3.3 Procedure

Teachers in all schools gave the Algebraic Thinking Test to their students during the final term of the year in question. The tests were then returned to the authors for marking.

3.4 Method of analysis

All tests were marked either by the first author, or under the supervision of the authors, by pre-service secondary student teachers familiar with the Numeracy Project. Markers were told to mark items as correct if they followed a method reliant on understanding of the underlying compensation principle. Scores for each participant were then entered on a spreadsheet that enabled us to see which items each student was successful on. Spreadsheets were compared across years to identify students tested on all three occasions. We then gained totals for each subject and each item. A random coefficients’ analysis of the data was undertaken by means of SPSS’s mixed linear model (Kreft & de Leeuw, 1998; Singer & Willett, 2003).

4 Results

Three analyses were carried out to evaluate the data gathered over 3 years from the tests of algebraic thinking. The first analysis determined the correlation of individual students’ scores on the Algebraic Thinking Test with their three strategy scores obtained from the Numeracy Project Diagnostic Interview at the end of year 9 (age 13). The second analysis looked at the Algebraic Thinking Test scores for students who took the test in years 8, 9 and 10 in 2004, 2005 and 2006 respectively. The third analysis took a closer look at what was happening in the one pair of schools where the mean scores on the Algebraic Thinking Test were significantly superior to the other schools.

4.1 Relationship between scores on the Algebraic Thinking Test and the Numeracy Project Diagnostic Interview

In 2005, the scores that year 9 students gained on the Algebraic Thinking Test and on the Numeracy Project Diagnostic Interview were correlated for the three secondary schools in the Secondary Numeracy Project. These students had been given the diagnostic interview described in Sect. 2.1, “The New Zealand Numeracy Project and Algebraic Thinking”. Scores were calculated by awarding a score of 6 for a strategy judgment of Stage 6, a score of 7 for a strategy judgment of Stage 7 and so. While scores for each of these operational domains are reported individually for diagnostic purposes, we added them together, reasoning that students needed to have a good ability in all skills to be a competent mathematics student in this age group.

For 557 students for whom both scores were available, the correlation between scores for these strategy stages on the interview and scores on the Algebraic Thinking Test was 0.47 with significance <0.01. Thus we were assured that individual interviews demonstrated that students who were flexible in their thinking about numerical problems as fostered by the Numeracy Project were the same ones who could transfer from using numbers flexibly to using letters to express the generalizations.

A further analysis of the diagnostic test strategy scores shows that students with a total score above the median score of 19 across the three operational domains on the Numeracy Project Diagnostic Test were those who demonstrated an awareness of the underlying algebraic structure of the operation and could express the generalization algebraically on our Algebraic Thinking test. This means that students who were operating at or above the advanced additive stage on all three operational domains were the students most likely to transfer this flexible numerical thinking to algebraic thinking in which the generalization is expressed algebraically.

4.2 Development of algebraic thinking across three years

This analysis was of the scores of the 116 students who took the Algebraic Thinking Test on three occasions.

The students varied both in their initial attainment at the end of year 8 and in their pattern of development over the three-year period. An indication of this variability is shown in Fig. 5, which presents the results of a random selection of eight students. The figure shows that some students improved over the three years, some did not change, and some declined in performance. In order to take into account this variability among students over each year, a random coefficient analysis was undertaken. The analysis also accommodated the correlation between responses that arises when the same person is measured on several occasions. These are shown in Table 4.
Fig. 5

Each panel represents a randomly selected student (ID at top of panel). Waves 1, 2, and 3 refer to years 8, 9, and 10 respectively. The data points show a student’s score at each wave, and the straight lines are least-squares fits

Table 4

Correlation coefficients among the students’ scores over three years

  

Year 9

Year 10

Year 8

Pearson correlation

0.640**

0.639**

Year 9

Pearson correlation

 

0.714**

** Significant at the 0.01 level (2-tailed)

A statistical analysis of the data was undertaken by means of SPSS’s mixed linear model. Two models were fitted. The simpler model estimated, without differentiating among the schools, the average score at year 8 and the average rate of improvement over each subsequent year. The model’s estimated score at year 8 was 5.7 and the estimated improvement per year was 1.3 points. The solid line labeled “all” in Fig. 6 shows this model.
Fig. 6

Two best-fitting models: a the solid line, labeled “all”, shows the estimated score at year 8 and the estimated improvement over three years of the average student, without differentiating among the schools; and b a model of the estimated score at year 8 and the estimated improvement of the average student in each of the pairs of schools

The other model took into consideration the effect of attending different pairs of schools. This model provided a marginally better fit to the data than the simpler one. The goodness-of-fit quantity, −2 log likelihood, was 2028.3 for the simpler model and 2021.1 for the more complex one, a difference equivalent to chi-square of 7.2 with 3 df, and P = 0.066. If this is taken to show a worthwhile improvement in the model’s fit, then the resulting estimated initial score and improvement rate for students from each school can be shown in Fig. 6 by the lines labeled “1”, “2”, “3”, and “4”. For both models, the rate of change (1.3 points per year) represented a significant improvement.

Scores from intermediate school 3 differed significantly from schools 2 and 4 (< 0.05) but not from intermediate school 1 (P = 0.081) because of the large variance in the small number of student scores at that school, (t test). The difference between schools when all three years were taken into consideration approached significance F(3,112) = 2.620, (= 0.054).

A still more complex model, which allowed for different average rates of improvement for students from different schools, did not improve the fit; it therefore is not discussed further.

4.3 What was happening in one pair of schools

The secondary school in the schools designated as Pair 3 tested all students in years 9 and 10 over three years. This was not true for the other schools. Analysis of the data for secondary school 3 showed that the algebraic thinking of the students in year 9 and 10 at the school improved across the three years of the study (see Fig. 7). This could be the result of changes in teaching in the secondary school, of cohorts of intermediate students coming through with a better understanding of algebraic thinking, or a combination of both. It is likely that this improvement may be attributed, at least in part, to the influence of the teachers at the intermediate and secondary school.
Fig. 7

Mean scores of for the Algebraic Thinking Test in Secondary School 3 over three consecutive years

Teachers in all schools were interviewed but we chose to focus on the interviews from the two Pair 3 schools in order that we might identify factors that contributed to the success of their students. We interviewed the assistant principal who was the lead teacher for mathematics at the contributing intermediate school in 2004 and in 2006 and the teachers and head of the mathematics department at the secondary school in 2005 and in 2006. The important characteristics of the numeracy lead teacher in the intermediate school appeared to include a firm knowledge of relevant mathematics, of the Numeracy Project, and of the connections between number and algebra. Other teachers had enabled her to recognize when firmer supervision of her teachers was needed. She had the full support of her principal, who was informed about the Numeracy Project. He told us, “all credit went to the woman in charge of mathematics”. This lead teacher gave credit to the man who had led their school through the professional development aspect of the project. She wrote in an email to K. Irwin, (1/5/2006), “During the 2 years we were on the contract, our teaching was very focused on number and I would say it took about 80% of the (mathematics) teaching time. Also the facilitator may have contributed to our brighter students being encouraged to move from the strategies into algebraic thinking. He gave me a lot of good ideas for doing this”. In describing her own teaching, she said that in 2004 she had already taken her top group into algebra. When we asked if this was traditional algebra, she appeared almost shocked. Of course not, she took the students from their use of strategies for numerical calculation into the generalities that could be expressed algebraically. This was a natural growth for her. In the email quoted above, she reported that the school had chosen algebra as one of their target areas for year 7 for 2006. “A curriculum combining algebra and number can only be a help. Very sensible.” She thought that most of the teachers were managing to teach the Numeracy Project well but indicated that there were two groups of teachers that needed her help: the less confident teachers and those who did not want to change their practice.

She described the ways in which she worked with these less confident teachers as well as those who did not want to change their teaching practices. We formed the impression that she had a warm but professional relationship with the teachers that added to her effectiveness as a mathematics leader. Her overall view was that students were enjoying mathematics more now. Their attitude was positive, and they were not afraid of using letters in place of numbers. Their understanding of the generality of the numerical operational strategies and measurement formulae for perimeter and area deepened as they explored the use of different numbers in response to “Will it work all the time?”

The staff of this school had regular contact with the secondary schools that their students attended. They passed on results of students’ progress in the Numeracy strategies.

The head of the mathematics department at the secondary school in Pair 3 did not participate in the senior version of the Numeracy Project but knew about the project in some depth. The mathematics department had one teacher who taught only numeracy to those students who needed it. Others integrated the concepts into their traditional curriculum where they thought appropriate. In 2005, the teachers told us that they had put more emphasis on algebra than before. In 2006, the teachers in the department had taught number and algebra over terms 1 and 2 of the four-term year and then integrated algebra into the other topics that they taught during the rest of the year. At the interview in 2006, we told the mathematics staff that their students outperformed the three other secondary schools and asked why they thought that might be. The teachers’ response was that they did not think their students were very good. As the discussion developed, we learned that the head of the mathematics department rewrote the departmental scheme for mathematics every year. For 2006, he had incorporated some of the ideas from the Algebraic Thinking Test. In 2006, he had judged that students from intermediate schools who had had the Numeracy Project were noticeably different from those who had not participated in the project. The teachers rejected the suggestion that their students had done well on this test as the result of any of their own efforts, saying that all credit must go to the intermediate school. Senior teachers at the school said that algebra was the basis of all high school mathematics and had to be brought in whenever possible. “We concentrate on algebra because they are doing badly.” “Number underpins everything.” They accepted different methods if the students could justify them: “Cannot have one size fits all.”

5 Discussion

5.1 Algebra within arithmetic

In an earlier study (Irwin & Britt, 2005a), we showed that students who had participated in the New Zealand Numeracy Project were more able than students who had not participated in the Numeracy Project to demonstrate an awareness of the underlying algebraic structure of the operational strategies they used to solve problems in arithmetic. We noted that this awareness of structure amounted to an awareness of generality and argued that such students were therefore engaging in algebraic thinking. We also claimed that, rather than using the literal symbols of algebra, these students who had no prior experience with such symbols, were thinking of the numbers themselves as variables. We referred to these as quasi-variables (Fujii & Stephens, 2001) and subsequently argued that the arithmetic students used to solve problems was rooted in algebra. That is, without algebra there could be no arithmetic (Hewitt, 1998; Mason with Graham & Johnston-Wilder, 2005). In this study, we have extended this relationship between algebra and arithmetic by showing that students, who had participated successfully in the Numeracy Project at the Intermediate School for years 7 and 8, were capable of algebraic thinking that involved representing numerical generalities with the special symbols of algebra. We also showed that these algebraic thinking skills for such students continually improved as they progressed through the first 2 years of secondary school.

This claim arises from the analyses of the data collected over the three-year period of the study. We noted that despite individual differences in performance, average scores increased significantly over the 3 years from year 8 to year 10. Also, the correlation between the scores on the Numeracy Project Diagnostic test and the Algebraic Thinking Test, showed a significant relationship. These analyses taken together suggest strong support for a positive relationship between success in the Numeracy Project and subsequent algebraic thinking in which students use the literal symbols of algebra to express algebraic generalizations.

5.2 Teachers who effect improvement in algebraic thinking

We interviewed teachers in all of the schools and drew some conclusions from the interviews of School Pair 3. In particular, the roles of the lead teachers as key in providing support for the development of algebraic thinking warrant attention. The students in this pair of schools, designated as School 3 in the analyses, performed creditably on the algebraic thinking test in each of the three years of the study (see Fig. 6). In addition, Fig. 7 shows that in the secondary school of the School 3 pairing, students in year 10 performed better than the year 9 students in each of the three years they were tested with the Algebraic Thinking Test. School 3 also consistently out-performed the other pairs of schools (see Fig. 6). Much of the difference in performance in these schools was likely to have been attributable to the teachers directly responsible for the classroom mathematics teaching and overseeing that teaching.

There are several factors that seem to have been important contributors to the successful leadership roles of the assistant principal in the intermediate school who was in charge of teaching mathematics and of the head of the mathematics department at the secondary school.

The teacher responsible for leading mathematics in the intermediate school had a well-developed mathematical knowledge for teaching (Ball, 2002). She had taken university courses in mathematics designed specifically for teachers of intermediate and early secondary school students, she had acquired a deep understanding of the intentions of, and activities recommended in, the Numeracy Project, and she had grasped the notion of “algebra within arithmetic” as a result of the support she had received from the Numeracy Project’s School Facilitator who had been attached to work with her and her colleagues in their classrooms. In her leadership role, she had learnt how to successfully manage and encourage colleagues who struggled with change in their classroom environment. In addition she had leadership qualities that enabled her to recognize teachers who still worked from an impoverished view of arithmetic and found ways of helping them.

The secondary school teacher, who seemed to have made an important impact at his school, had been head of the school’s mathematics department for a number of years. He knew the mathematics he was expected to teach at all levels in the secondary school and seemed to be open to experimenting with alternative classroom teaching approaches. Although he had no direct classroom experience with the Numeracy Project, he knew of it and supported its overall goals and classroom intentions. Interestingly, although the students at his school had performed creditably on the Algebraic Thinking Test, he, together with some of his colleagues, felt that the students could do better. And in order to effect improvement, he had in late 2005, rewritten the school’s mathematics teaching scheme for 2006 to incorporate some of the ideas from the Algebraic Thinking Test. As part of this school-based curriculum development process he also seems to have convinced his colleagues of the efficacy of his teaching scheme modifications. Both of these teachers stood out as quality professionals who were respected by their colleagues and their respective school Principals.

5.3 Teaching algebraic thinking

When the analyses discussed above are taken together we believe they contribute to a very strong case to be made for the inclusion in mathematics curricula of activity that is based on the notion of “algebra within arithmetic”. We contend that, where the focus for arithmetic in classroom curricula is on developing an awareness of the underlying structure of the operations rather than on merely getting answers through procedures and algorithms, it is likely there will be important positive consequences for the development of algebraic thinking. Thus, students who reach a level where they can readily access mental operational strategies, such as those identified in the Numeracy Project, to sensibly and flexibly solve numerical problems, are likely to be able to grasp and work with algebra where full use is made of algebraic symbols to express algebraic generality and which is central to most secondary school mathematics curricula.

We argue that the students in this study who can express numerical relationships algebraically are working at the Representational Level of Generalizing as proposed by Rivera (2006) in his Theory of Generalizing Types. In our earlier study (Irwin & Britt, 2005a), which also focused on students generalizing, we claimed that students who were able to use quasi-variables (numbers for letters) to generalize numerical operational strategies were engaged in algebraic thinking. Furthermore, the study by Irwin (1996) revealed an awareness of generality among children aged 4 as they proposed additive compensation principles without needing to establish cardinality. These children had acquired this knowledge in a proto-quantitative form (Resnick 1992) prior to their going to school. However, because teachers at the beginning school level often get caught up in helping their children get answers to simple numerical calculations, this early ability to generalize goes unnoticed.

In order to consider the educational implications arising from this study, we have combined its finding with those from the studies by Irwin & Britt (2005a) and Irwin (1996). As a result of having this wider perspective, we propose a pathway for algebraic thinking that we believe is accessible to all students motivated to develop a relational understanding of number and operations with number. The pathway firstly involves young children engaging in activities of a proto-quantitative nature where they explore and describe in their own words, generalities arising from simple operational strategies that do not require extensive counting knowledge. It then involves primary school students deepening their understanding of a range of operational strategies in arithmetic and finally it involves middle school and beginning secondary school students expressing algebraically, generalizations drawn from numerical situations and figural representations. In proposing this pathway, we note that it incorporates many of the characteristics of both proto-representational and representational generalizing that form the theoretical framework in the Theory of Generalizing Types proposed by Rivera (2006).

New Zealand teachers and their students have had a growing involvement in the Numeracy Project since 1999. The experimental nature of the project together with its strong evaluative component and accompanying teacher professional development, has led to improvements in skill level and confidence in teaching numeracy. Increasingly, teachers are developing awareness that their work entails considerably more than helping students get answers to problems involving numerical situations. They are supported by the Numeracy Project resource material and a curriculum that encourages the concept of algebra within arithmetic. This national “experiment” has led to professional development for most primary school teachers and an increasing number of secondary school teachers. The challenge now is to enable all teachers to see the benefit of an approach that includes seeing algebra within arithmetic.

Notes

Acknowledgments

We are grateful to the Ministry of Education for funding this research project. We also express our thanks to R. J. Irwin for some of the statistical analysis.

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Copyright information

© FIZ Karlsruhe 2007

Authors and Affiliations

  1. 1.Faculty of EducationThe University of AucklandAucklandNew Zealand

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