The First Cycle
Structure of Tasks, Illustrated on Task 1.
There are two small toy animals, a frog and a ladybird, and two empty boxes (Fig. 1). Each toy is attached to one of the boxes. Leyla counts three candies and places them in the frog’s box. She does the same for the ladybird’s box. Then, Leyla counts four new candies and adds them to the frog’s box, and finally, she visibly removes two candies from the same box. Although the child is allowed to see the whole process and to know the beginning number of candies in each box and the number of candies added and removed, it is not allowed to see or touch (say, for counting) the content of the two boxes, unless it is asked for by the child. The task for the child is to determine which toy has more candies.
Various variations of the task above were designed and used with different background stories, hands-on materials, and numbers and yet similar underlying idea. Generally, each task started with two “number holders” initially holding the same number of objects. The starting number could be known and “small” (task 1 and task 4), known and “big” (task 2), or unknown (task 3 and task 5). The starting number was intended to be a specular number, simply by hiding the (number of) objects, say, inside a box. Hereafter, square brackets around the starting number indicate its intended role (e.g. [3] in task 1). Then two operations in which what being added or subtracted was in the range of 1 to 10, one after the other, were applied on one of the holders (e.g. the frog’s box in task 1). The problem was to compare the number of objects of the holders with each other (e.g. [3] + 4 − 2 and [3] in task 1). In this paper, the pairs of operations are adding-removing and removing-adding. In relation to the design of the tasks, there are two points that should be highlighted.
First, in the absence of writing symbols to record the problem, it is not easy to track which number is compared with which number. For example, in the task above, if we had used only one box, it would not have been clear for the child whether the final amount ([3] + 4 − 2) should be compared with the intermediate amount ([3] + 4) or the initial amount ([3]). The use of two different boxes reduces such ambiguities. Moreover, the presence of two number holders could keep the starting number specific (though, untouchable) during the whole process and also allow the child to check his or her answer (if necessary).
Second, the tasks ask for the comparison of the initial and final states, not for the calculation of the final result. Asking for the final result would be a direct invitation to the left-to-right calculation when starting with a known number (task 1) and meaningless when starting with an unknown number (task 3).
It is now time to see the tasks through children’s eyes.
Task 1: the Starting Number Is Known and Small.
Task 1 is one of more than hundred tasks with a known and “small” (yet, untouchable) starting number that were used in both cycles of the study. This version of the tasks allowed for various solutions as follows (notice that though the term left-to-right is reminiscent of written calculations, for simplicity, we sometimes use it to refer to the order imposed by the presentation of the tasks).
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1.
Following the left-to-right order that includes moving from ([3] + 4) − 2 to 7 − 2 and then to 5 which is subsequently compared with the initial number (3). Here, the focus is on the result of each operation one after the other.
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2.
Changing the structure of the task from ([3] + 4) − 2 to [3] + (4 − 2) and then following one of these three options: (i) calculating the total change (4 − 2 = 2) and then applying it to the initial number (3 + 2 = 5), which is subsequently compared with the initial number (3); (ii) calculating the total change (4 − 2 = 2) and realizing that it is a “positive” change: [3] + 2 is more than [3]; (iii) realizing the direction of change (whether it is “positive” or “negative”) without calculating the amount of change: the number added (4) is more than the number removed (2), thus the final is more than the initial.
In many cases, children could not explain how they have solved the task at hand. For example, Melina (a 5-year-old girl) said: “this ([2] + 2 − 1) has more; it has three, but that one has two.” When Leyla asked: “how do you know?” she answered: “I just know.” When the calculations were a bit more cumbersome, there was more chance to observe the strategy used. The following excerpt illustrates how Sarah (a 6-year-old girl) used the left-to-right strategy to solve task 1:
Sarah: May I use my fingers?
Leyla: Yes, of course.
Sarah: Three plus four is seven [she held up three fingers on her left hand and four fingers on her right hand. Then she counted all the fingers held up]. Then you removed two [she closed two fingers out of four held-up fingers on her right hand and counted all the fingers that remained up on both hands]. It’s five.
Leyla: So which toy has more candies?
Sarah: Frog.
There were also cases that the child could explain how the task has been solved. For example, Inez (a 6-year-old girl) used her fingers in a left-to-right calculation to compare 5 − 2 + 3 (= 6) and 5. But, when faced with [6] + 3 − 2, she explained her strategy:
Inez: This ([6] + 3 − 2) has more; it’s seven.
Leyla: How do you know that it’s seven?
Inez: Because, you added three, and took away two.
She stresses on number 2 meaning that it is less than the number added. Then, she proceeds by applying the total change (3 − 2 = 1) to the initial number 6 + 1 = 7.
Inez’s change of strategy from one task to the other was typical of this stage of study in which children worked with known starting numbers. The same child could solve each task by either of the two strategies, whether the task was adding-removing or removing-adding. However, the available evidence from more than a hundred tasks suggested that the size of the numbers influences the child’s strategy. For example, Inez could easily find the result of [5] − 2 + 3 from left to right by calculating the result of 5 − 2 (= 3) and then the result of 3 + 3 (= 6). The left-to-right approach to [6] + 3 − 2 would lead to 6 + 3 (= 9) and 9 − 2 (= 7) which both were beyond the calculating ability she showed earlier when answering the numeric tasks.
It is important to notice that “number size” is a relative concept. Thus, while the magnitude of the left-to-right calculations in [6] + 3 − 2 was big enough to lead Inez to restructuring the problem, even something like [4] + 6 − 5 could not hinder Parsa (a 6-year-old boy) from using the imposed structure of the task. True, the presentation of the tasks did not allow children to use physical objects to represent and solve the problem, but it did not stop them to model the problem on their fingers or mentally carry the calculations when the numbers involved were “small” enough for them. In general, the more a child was able to operate on the initial number, the more he or she was inclined to apply the imposed structure of the task. Thus, for more able children, the known starting number (though untouchable) was more specific and less specular. For these children (like Parsa), the tasks were just an arithmetic addition and subtraction task. For those (like Inez) who were at times impelled to restructure the problem (say, from ([6] + 3) − 2 to [6] + (3 − 2)), restructuring was mainly directed towards finding the final result of the numeric sentence. In a way, the restructuring was attached to the starting number. The aim of the next two tasks was to bring the restructuring to the fore, receding (calculations with) the starting number into the background.
Task 2: the Starting Number Is Known and Big.
In task 2, the magnitude of the starting number and the middle and final results is “big” enough to block the imposed order of the calculations and encourage the use of restructuring strategy. Here, the candies of a bag of a hundred candies are emptied into a box. Leyla adds three candies to the box and then removes four candies from the box (Fig. 2).
Leyla: Are there more or less than hundred candies in the box?
Sarah: Oh, it’s very easy. It is less than a hundred.
Leyla: Why?
Sarah: You put three [candies] in place of that four [candies], and so, “hundred” went down one.
Leyla: What should I do if I want it to be a hundred again?
Sarah: You should add one.
In using a big number, Sarah’s inability to model the problem on her fingers turned into the ability of perceiving (100 + 3) − 4 as 100 + (− 4 + 3) and then as 100 − 1 (here, − 4 + 3 represents a composition of two transformations).
A big number would work providing that the child could apprehend it as a specific whole. Thus, while 20 (and for some children, even 10) was big enough to encourage restructuring, a number like 100 could be too big. There were times that children comprehended a “big” number as “too many” (rather than as a fixed whole), where a bit more or less than “too many” was still “too many”! In the following excerpt, Fatima (who was competent with numbers in the range of 1 to 10) compares 100 + 3 − 2 with 100:
Fatima: It’s more than hundred.
Leyla: Why?
Fatima: Aha! It’s less than hundred!
Leyla: Do you remember what I did?
Fatima: Yes, you added three [candies], and then, removed two [candies].
Leyla: How many candies were inside the box?
Fatima: Hundred.
Leyla: Now, how many candies are inside the box?
Fatima: Twenty!
A “big” number, if meaningful to the child, could encourage the change of structure while leaving the starting number intact. However, again, the transformations are bound to a specific number (say, 100), at least for us as observers. The next task is to address the indeterminacy of the starting number.
Task 3: the Starting Number Is Unknown and Specific.
Here, the starting number is big enough that cannot be subitized and small enough that can be counted and used in calculations (if allowed or wanted). Here, there are two equal height towers (with the same number of cubes), one red and the other blue. Leyla removes one cube from the red tower (puts it on the table) and adds three cubes to the same tower. Sarah does not see the towers, but she sees that one red cube comes out from behind a pad and then three red cubes go behind the pad (Fig. 3).
Leyla: Which tower is taller now?
Sarah: Red has two more.
Leyla: How do you know?
Sarah: Because one of those three [cubes] replaces the one you removed; and the other two [cubes] make the red tower taller than the blue tower.
With an unknown starting number, there is no other way to solve the task but by restructuring it. Sarah restructured the initial arithmetic structure of the task from ([a] − 1) + 3 to ([a] − 1) + (1 + 2) and then to [a] + 2. This is decomposing 3 = 1 + 2 with respect to the number taken away and then using the associative law twice: first in restructuring ([a] − 1) + (1 + 2) to (([a] − 1) + 1) + 2, and second, in transforming ([a] − 1) + 1 to [a]. The second transformation can be also thought of as making use of “the principle of inversion” which is known to be used by children even younger than Sarah (Geary, 2006).
Sarah spontaneously answered two questions: “which one has more?” and “how many more?” Most children (including Sarah), most of the times, just answered the former question, mentioning the direction of the total change, without calculating the amount of the total change. Thus, a common answer could be something like, “Red is more because you removed one, but added three.” In fact, for most children, the difference questions (“how many more?”) were hard even for two known numbers, let alone two unknown numbers (say, [a] and [a] − 1 + 3 in the task above). For two known numbers, the typical answer was to say the bigger number (see also Hudson, 1983). For two unknown numbers (task 3), if they were asked the question, the typical answer was, “I don’t know because we didn’t see inside it.” However, if the question was, “How many must we add to this box ([a]) so that it will have the same number as this box ([a] – 1 + 3)?” then it was more likely to get a correct answer. Phrasing the difference question in terms of transformations made it like an equalizing problem in arithmetic where “the question is directly about how much to add to one set to make it equal to the other set” (Nunes & Bryant, 1996, p. 130). Unlike arithmetic problems, the situation here is not start-known and result-known. However, it is not quite start-unknown and result-unknown either. The starting number and accordingly the final number, both are intended as specular numbers. They are unknown but could be disclosed by the child whenever wanted. As such, children could think of “this number” (the number of objects in the box), though it was unknown (Asghari, 2012; Radford, 2010). More importantly, they could operate on one possible value for the number holders.
Parnian (a 6-year-old girl): [Skillfully modeling 4 − 3 on his fingers while comparing blue tower ([a]) and red tower ([a] + 4 − 3)] Red is one [cube] taller.
Leyla: Can you tell me how many are the reds?
Parnian: [After pausing a few seconds] No, I can’t. I didn’t see how you built the towers.
Leyla: Okay. I give you a clue. The number of blues is ten.
Parnian: [Immediately] Eleven.
Even sometimes, they could operate on more than one possible value.
Sarah: If this ([a] − 3 + 2) is nine, this ([a]) is ten.
Leyla: And if this ([a] − 3 + 2) was five?
Sarah: Six.
Whether thinking of possible values (one at a time) is an indication of the generality perceived by the child is the subject of the discussion below and the second cycle of the study. For now, it is important to say that task 3 makes the child aware of the possibility of the change of structure by forcing the change. Whether the child applies such restructuring in the absence of such a force is the subject of task 4.
Task 4: the Starting Number Is Known and Small.
The starting number is small and known. The whole expression is within the calculating ability of the child. However, now, unlike task 1, the child has experienced several restructuring tasks. Here, the task was to compare [4] + 4 − 1 (red cubes) with [4] (blue cubes).
Leyla: Which group has more cubes?
Sarah: The red one. It has three more cubes [than the blue group].
Leyla: How many cubes does it [the red group] have now?
Sara: [while using her fingers to add 3 to 4] it has seven.
Without restructuring, the order of approaching the task is to calculate the final answer according to the imposed order of the task and then compare it with the starting number (see Sarah, task 1). With restructuring, the total change can be calculated first, and then, the total number (if needed). This is what Sarah did when the initial arithmetic structure of the task was changed from ([4] + 4) − 1 to [4] + (4 − 1) and then to 4 + 3. This is also what Parsa did when he calculated [6] + 3 − 2 at this stage of his work, while earlier, he had solved all the start-known problems with “small” numbers by direct calculations from left to right (see task 1 above). What they did is an indication of awareness of the possibility of the change of structure. However, it is not an indication of a free choice of strategy. It is more likely that the similarity of the presentation of the task with the previous ones (in particular, the unknown-number version) somehow triggered what they did. We will discuss this point further in the next section.