Making Associativity Operational

To bring about such an understanding, Fujii and Stephens (2001) propose the use of quasi-variables, by which they mean numbers that vary within “a number sentence or group of number sentences that indicate an underlying mathematical relationship which remain true whatever the numbers used are” (p. 259). For this approach to work, one needs to be or to become a “proceptual thinker” (Gray & Tall, 1994) who can resist immediate direct calculations and an operational conception of the equal sign (Kieran, 1981) and concurrently realize which particular features should be stressed and ignored. Only then would it be possible to see a specific equality like 78 − 49 + 49 = 78 as a generic equality and “the carrier of the general” (Mason & Pimm, 1984) in which both 49 and 78 can be understood as a quasi-variable indicating that the sentence belongs to a type of number sentence. Put simply, one needs a structural conception of the equal sign. In this paper, we propose a different approach that does not rely on such a conception.

The equal sign is a symbol to indicate the notion of equality, and equality tells us whether two mathematical objects, possibly presented to us in wildly different ways, are to be considered equal (Mazur, 2008). Strictly speaking, the equal sign represents equivalence. The number expression 8 + 2 + 4 is equivalent to (or “is the same as”) (8 + 2) + 4 or 8 + (2 + 4). Students can use some structured materials, say Cuisenaire rods, to model the number expressions and see either way they would end up with the same number of blocks, that is, (8 + 2) + 4 = 8+ (2 + 4).We call this focus on the equality as equivalence, the structural conception in which we see the both sides of the equal sign intact (Carpenter, Levi, Franke & Zeringue, 2005; Stephens, 2006; Molina & Ambrose, 2008; Stephens, Knuth, Blanton, Isler, Gardiner & Marum, 2013).

However, when students transform 8 + 6 into 10 + 4 (i.e. 8 + (2 + 4) into (8 + 2) + 4) and then to 14, the focus is on the transformations that have taken place while intermediary structures might “have only transient existence” (Caspi & Sfard, 2012, p. 50) in the process of calculations. We call this focus on the equality as transformation, the operational conception in which we replace one number expression with another through “equality-preserving transformations” (Kieran, 2014) without necessarily paying attention to the expressions for their own sake. Both conceptions of equality can play a productive role in “thinking relationally about equality” that is one of the main themes of “algebraic thinking and the routes by which its growth might be encouraged” (Kieran, 2014, p. 1) in arithmetic. However, it is not an easy route.

Based on historical and empirical evidence, Caspi and Sfard (2012) argue that the first level in the development of algebra is processual, where the focus is on numerical calculations in the order of their execution on constant (or as in Sfard & Linchevski (1994), fixed) values, either known or unknown. With letters, it would be challenging to break away this default linear structure and with numbers even more challenging. Arithmetic is where the so-called left-to-right approach commonly used in early years of school (Booth, 1989; Kieran, 1989) goes hand in hand with the students’ “tendency to think about expressions involving binary operations in terms of a sequential procedure” (Larsen, 2010, p. 42). Accordingly, both expressions 8 + 2 + 4 and 7 + 9 + 1 have the same chance of being treated from left to right, as (8 + 2) + 4 and (7 + 9) + 1 regardless of the addends involved. Arithmetic is where the operational conception of the equal sign is most frequently realized as instructions to do something and as an obstacle for the structural conception, that is, equality as equivalence (Kieran, 1981). Accordingly, the response to the open number sentence 8 + 6 = ◻ + 4 would be 14 (Stephens et al., 2013).

Yet, arithmetic is where we wish to cultivate students’ relational thinking about equality. One way to do so is to bring out the relational meaning of the equal sign whenever it is used. However, it is doubtful that simply telling students what the equal sign means effectively develops understanding (Carpenter, Franke & Levi, 2003). Instead, we might use a structural program in which students see structural expressions and equations in their entirety. For example, we might develop a set of true/false (e.g. (4 + 7) + 9 = 4 + (7 + 9)) or open number sentences (e.g. 56 + 75 + 25 = ◻) that might encourage our students to experience and to make explicit the associative property (Carpenter et al., 2003; Carpenter et al., 2005). In this way, students may find a correct operational sense of when and how it is appropriate to use the associative property to transform one mathematical expression into the other. However, when working with prealphanumeric students and in the absence of writing symbols to record number sentences, a structural program would be hardly accessible. Yet, we might encourage an operational conception of equality. This paper proposes an approach to do so, using the associative property as an example.

It is common to think of (a + b) + c = a + (b + c), but not (a + b) − c = a + (b − c), as the associative property. However, using the concept of additive inverse, we can see (a + b) − c as (a + b) + (− c), and then as \( a+\left(b+\left(-c\right)\right) \), which is the same as a + (b − c). The same argument applies to (a − b) + c = a + (c − b) and (a − b) − c = a − (b + c). Of course, this way of looking at associativity is too advanced for a preschooler or a primary schooler. However, children can use these equalities when operating with natural numbers and without thinking of the concept of additive inverse. For example, while transforming 13 − 5 to 13 − 3 − 2 by decomposing 5 = 3 + 2 (with respect to 10), one implicitly replaces 13 − (3 + 2) with (13 − 3) − 2. The equality 13 − (3 + 2) = (13 − 3) − 2 provides two different ways of structuring and restructuring 13 − 3 − 2. In general, the same thing can be said about (a + b) + c = a + (b + c), (a + b) − c = a + (b − c), (a − b) + c = a + (c − b), and (a − b) − c = a − (b + c) in relation to a + b + c, a + b − c, a − b + c, and a − b − c, respectively.

Moreover, the associative property can also be translated in the language of transformations. Consider that \( \pm b\pm c \) “can be perceived as a composition of two transformations that yield a third transformation” (Peled & Carraher, 2008, p. 320). Thus, one might move from 4 − 2 + 1 to 4 + (− 2 + 1) (where “− 2” is thought of as a change), and then, to 4 − 1 as an attempt to figure out the total change that has been applied to the stating number (here, 4). In this paper, the focus is on a + b − c and a − b + c, interpreted as adding-removing and removing-adding, respectively.

The purpose of this study was to investigate the possibility of developing an algebraic understanding of the associative property in the absence of alphanumeric symbols. Our first step was to interpret associativity operationally, in terms of transformations. In this respect, the aim was to help children to avoid the temptation of calculating from left to right and instead encourage them to restructure the given expression. For example, we intended that children be able to interpret \( 3+4-2 \) as \( 3+\left(4-2\right) \), rather than automatically as \( \left(3+4\right)-2 \). Learning to restructure an arithmetic expression is an important step towards relational understanding of equality in algebra. However, such restructuring only involves known specific numbers in a result-oriented calculation. In contrast, one essential component of algebraic thinking is the ability of dealing with the indeterminate quantities that are unknowns or variables (Radford, 2014).

Hence, we had to address two problems. To address the first problem (i.e. total change), we adopted the didactical strategy of dissuading the learner from touching (i.e. calculating with) the starting number (Asghari , 2012), hoping that it also addresses the second problem (i.e. indeterminacy). For example, working with the expression \( 3+4-2 \). If we prevent the actualization of the initial and subsequent operations (3 + 4 and \( 7-2 \) respectively) and instead invite children to operate on the starting number (here, 3) while leaving it intact, then, they might transform \( 3+4-2 \) to \( 3+\left(4-2\right) \), and then, to 3 + 2. In doing so, the specific starting number (here, 3) might turn into what is called a specular number that is specific for the learner, but it is treated as a non-specific number (Asghari, 2012). As such, one of the operands (here, 3) is treated as an indeterminate value, while the problem is restructured into a new one through equality-preserving transformations.

The idea of specularity seemed promising since its focus is on “the process acted on the object” (Asghari, 2012, p. 37) rather than the structural features of the problem. Whether it works in the absence of writing symbols was the subject of a one-on-one design experiment (in the sense of Cobb, Confrey, diSessa, Lehrer & Schauble, 2003) of which the research question was as follows.

In preschool children ages 5–6 years and in the absence of alphanumeric symbols, can the idea of specularity promote an operational algebraic understanding of associativity?

We conducted a series of sessions with a small number of preschoolers. The sessions were test-beds for practical variations of the idea of specularity as a means to promote operational algebraic understanding (of associativity). Preschool children who participated in the study were from the same preschool in the north of Tehran. In Iran, educational programs at the preschool level vary from one preschool to another. In this particular preschool, the focus of their program was mostly on hands-on activities. As far as mathematics is concerned, their focus was on counting activities and the well-known activities of trying to find a one-to-one correspondence between sets and hence to show whether the sets have the same cardinality or not. Since numeral recognition and numeral writing is part of the centered national mathematics program for the first year of school, the preschool program was not obliged to reach any particular target in that direction; at the best, their point was to create a sense of familiarity with numerals. We also did not use numerals in the study.

The study had two main cycles in which two different groups of students participated. The first group consisted of nine children. The first cycle was the developmental stage of the study in which different tasks were designed and tried out. A year later, we worked with a different group consisted of five children. Now, the study was in a more advanced stage and only a carefully chosen number of tasks were used to get the desired outcome. More importantly, as a result of our growing understanding of the situation, a fundamentally new task was added to the previous tasks.

A number of individual task-based interviews were used in both cycles. Each interview that lasted between 5 and 15 min (once a week) was videotaped and subsequently transcribed for further analysis. The maximum number of interviews with each child (done by Leyla, known to children as auntie Leyla and one of the authors of the present article) was 20 in the first cycle and 7 in the second cycle. Generally, in both cycles, the first few tasks were numeric. The tasks (not reported here because of space limitations) included counting a number of objects (toys, counters, etc.) and one-step addition and subtraction problems with or without hands-on material. The aim of the numeric tasks was to decide the range of numbers that would be meaningful to each child. In the main stage of the study, we used five types of tasks, all being variations of the same theme: structuring/restructuring a to-be-specular number sentence with two consecutive operations on a number intended as a specular number (specific, but untouchable). In task 1 (hereafter, “task x” is representative of its type), the starting number was known and “small”, in task 2, known and “big”, in task 3, unknown and specific, and in task 4, known and small again; finally, in task 5 (added to the study in the second cycle), we used two tasks simultaneously with two unknown starting numbers. The details and reasons behind all these choices will be explained below. The following data come from both cycles of the study.

The didactic strategy used in the current study was making the starting number untouchable during certain known operations. Accepting an untouchable number would not be as easy as it looks in our reports above. In fact, there were many occasions, before and after getting used to the idea, where children felt obliged to make the number touchable to be convinced about the correctness of the answer by counting the objects inside each box. Thus, when the starting number was untouchable but knowable and representable on the fingers (task 1), the child’s approach to the task was hardly predictable. Overall, it seems that if applying the imposed order of the problem were feasible, the child would choose the left-to-right approach. Sarah and Parsa successfully solved several of such problems by direct calculations at the start of the interview process. Thus, the change of their strategy (task 4) could be due to their exposure to the previous tasks (say, task 3). However, such tasks could make the child aware of the possibility of the change of structure, but not necessarily of the suitability of such a change for the problem at hand. True, task 3 forces the child to restructure the problem whether it is [6] + 4 − 1 or [7] + 4 − 3. However, restructuring 6 + 4 − 1 as 6 + (4 − 1) does not necessarily make the calculations easier for the child (if it does not make it harder). In arithmetic, the proper use of associativity is a decision that is taken based on each and every three operands and the numeric relations they have together. It is based on such relations that a person decides to restructure one expression but not the other. To provide such awareness, there needs to be a different strategy allowing children to reflect on different ways of structuring the same number expression. This is where a structural approach (e.g. Carpenter et al., 2003) might come handy. In a way, our children learned to restructure without necessarily seeing the structures.

The Second Cycle

The first cycle of the present design experiment clarified the notion of operational algebra as equality-preserving restructurings. However, as far as algebra is concerned, what is being done is being done on a quantity that is subject to change; this involves a simultaneous awareness of what is being done and what is being changed. Though the problem of assessing such awareness is notoriously difficult (Carraher et al., 2000; Mason, Stephens & Watson, 2009), the main distinguishing feature of the second stage of the study is to offer a solution. Task 5 was given after all the first four tasks had been completed.

Task 5: the Starting Number Is Specular.

There are two toys; each one has two boxes in two different colors (Fig. 4). The boxes of one of them are blue and red; both boxes contain the same unknown number of foods, say [x]. The boxes of the other toy are green and yellow; both boxes contain the same unknown number of foods, say [y]. These two unknown numbers, [x] and [y], are not necessarily equal to each other. Leyla chooses the red box (of the first toy) and the green box (of the second toy), adds four foods to each one of them, and then removes two foods from each one of them. Each toy only has access to its own boxes, and each toy chooses the box with more food. The task is to determine which box each toy chooses:

Helen (a 6-year-old girl): They choose Red and Green.

Leyla: How do you know?

Helen: Because you added four to them and removed two from them.

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Using each one of the tasks alone (like task 3), we would have had another indication of the child’s ability of relational thinking. Could we interpret this as an indication of the child’s awareness that in the transition from ([a] + 4) − 2 to [a] + 2, the starting number can vary? By the definition of specularity, the answer could be “yes.” However, our students do not learn because of our definitions! There is a possibility that in every single task, the specular starting number is mentally fixed, hence, at the best, an indication of relational arithmetic. Task 5 was set up to try and rule out this possibility.

Task 5 asks the child to compare ([x] + 4) − 2 with [x] and ([y] + 4) − 2 with [y]. Both [x] and [y] are specific numbers (though kept unknown). Moreover, [x] and [y] are not necessarily equal to each other. In the transition of ([x] + 4) − 2 to [x] + 2 and ([y] + 4) − 2 to [y] + 2, there is no resort to specificity of [x] and [y], respectively. This is experiencing both transitions simultaneously as specific cases of the same structural change. It seems that there is a sense of generality involved in articulating, “Because you added four to them and removed two from them.” This is to see that the total change would be valid for any starting number, though the range of “any” is determined by one’s (here, Helen’s) conception of the numbers involved.

We presented and examined an approach to algebra that was based on arithmetic. In essence, like Carpenter et al. (2005) we advocated thinking relationally about equality. They have tried to foster it structurally. We attempted it operationally and extended previous research on early algebra to verbal mathematics. We interpreted the operational conception as the ability of transforming one structure into the other. Using a series of carefully designed tasks based on the idea of specularity (Asghari , 2012) we prompted 5–6-year-old children to think operationally in the absence of alphanumeric symbols. The focus was on the associative property.

We showed the possibility of developing an operational algebraic understanding of the associative property at the preschool level. In arithmetic, this appears as the ability of restructuring a given arithmetic expression to an equivalent one. In algebra, this is the ability of transforming an operation between an unknown and a known ([x] + 4) − 2 into an operation between two knowns: [x] + (4 − 2). Considering that these are what students need in mental and written methods of calculations (McCallum et al., 2011) and in a good part of algebra (Wasserman, 2014), our work highlights the surprisingly high level of what can be experienced by preschool children. More importantly, it is what the result of this study suggests for “the development of algebra”:

In contrast to what Sfard and Linchevski (1994) and Caspi and Sfard (2012) say, our findings suggest that verbal and operational are not necessarily processual and bound to numeric computations. They have the potential to foster thinking relationally about equality with both known and unknown quantities. However, as always, there are some words of caution.

Due to the purely verbal nature of our study, we could not directly encourage “different ways of looking at the same number expression” that is necessary for the flexible use of the associative property in arithmetic. In a way, in our tasks, one structure collapses into the other without any trace. As soon as the numerals come into play, we can (must) encourage students to see operational transformations as different ways of structuring the same expression. Such awareness is the sine qua non of the structural conception of algebra and should be meaningfully linked to the operational conception via symbols (for some suggestions to link from the operational to the structural, see Khosroshahi & Asghari, 2016; for the other way round, see Carpenter et al., 2003).

However, with great potential of symbolic representations comes great challenge. With the advent of written base-ten numerals comes the so-called left-to-right approach commonly used in early grades. With the equal sign comes its computational drive. With the letters for unknown quantities comes “the addictive power of algebraic manipulations” (Kieran & Sfard, 1999, p. 15). Simply speaking, there is a long journey from operationally and verbally transforming ([x] + 4) − 2 to [x] + 2 (where [x] is a specular number) to structurally and symbolically understanding the equality x + 4 − 2 = x + 2 (where x represents a variable). As one example among many, consider how in the verbal situation of our design, the operations were automatically attached to the operands (e.g. add four, remove two), all of which should be kept in memory (though not reported above, there were many times that we needed to remind children of the numbers used in the task at hand). In contrast, in a symbolic situation, the operations and operands are recorded, but children need to learn how to correctly interpret each and every one of them (that is not as easy as it might look; see Herscovics & Linchevski, 1994).

What we did was as an exploration into where an algebraic journey of children might start. Surprisingly, this start can include an experience of working with unknown quantities through equality-preserving transformations. This also includes an affirmative answer to our research question that whether the idea of specularity can promote an operational algebraic understanding of associativity in the absence of alphanumeric symbols. However, we cannot claim that the participants of this study would be able to use what they experienced later on as primary school students. In fact, even in task 4 and in the middle of their exposure to many similar tasks, we could not unequivocally predict whether a child would restructure the given problem or not. Believing that providing an algebraic experience of arithmetic for students needs targeted tasks and, more importantly, an algebraic mentality, all we wish to claim is that we have offered both. There is still much to do before the tasks we used in a one-to-one situation can be adapted for the use in the classroom (for an attempt, see Khosroshahi & Asghari, 2016), where an algebraic journey would not exist if an algebraic habit of mind is not infused into school curriculum (Blanton & Kaput, 2003), whether we advocate an operational or structural approach.