In our analysis of children’s ways of structuring, we found two main categories: Instantly showing finger patterns as sets, and Instantly showing finger pattern sets and counting single units, the latter further including two sub-categories of ways to enact the structural approach (see Table 1). These are in the following discussion described and problematized in terms of how the structuring is enacted and what constitutes the respective ways of structuring when solving the task ‘You have three glasses, but are going to set the table for eight people; how many more glasses do you need?’ (3 + _ = 8).
Instantly showing finger pattern sets without counting single units
In our sample we found seven instances of children structuring numbers on their fingers without counting single units—they created a structure of composite sets to solve the task. These children created composite sets of numbers as finger patterns and all of them were fluent in this act, producing a structure on their fingers from which they instantly ‘saw’ the answer (see Fig. 1). The excerpt below shows how Hannah, who by simply looking at her fingers could see the given part (3) and the whole (8) and thereby determine the unknown part (5) (see Fig. 1):
Hannah: Three glasses and eight are going to eat, how many more. Three (shows thumb, index, middle finger on her left hand) and eight (puts her whole right hand on the table; looks at her hands) Five more.
Shows simultaneously 3 and 8. Sees 5 in the 8.|
What characterizes this way of structuring is that the child instantly creates the first heard addend (3) and then the whole (8) by including the first addend in the whole, simultaneously. The three fingers constituting the first addend are thereby seen as a part of the whole eight. This way of seeing ‘the 3 in the 8’ also makes her see the unknown part, ‘the 5 in the 8’. Thus, numbers are perceived as composite sets, and are related in a part-part-whole fashion that helps her see the answer without having to count. The task at hand does help her to see the answer easily, since the second addend (5) can be recognized as a whole hand. Nevertheless, structuring in this way demands attention to how parts and the whole are related, and most importantly that parts are included in the number that constitutes the whole.
As discussed above, we have drawn the conclusion that experiencing how parts are included in a whole in a part-part-whole relationship is an important aspect that is critical for structuring a task of composite sets. The following excerpt shows one child (Sofia) who did not solve the task even though she enacted a structural approach, because she lost track of the whole and treated the task as if the whole were 10 instead of 8:
Sofia: (shows index, middle and ring fingers on her right hand and keeps all fingers on her left hand folded, looks a moment at her hands) I need to bring seven.|
Interviewer: How did you do that, show me, first you showed three there (imitates the child’s finger pattern), how many do I need to bring in order to make eight? How did you do it then?
Sofia: It’s seven (looks at the interviewer’s hands), all these (touches the interviewer’s five folded fingers on the left hand and the folded thumb and little finger on the left hand).
Shows three and simultaneously relates it to the whole ten composed of the rest of the folded fingers. Sees the unknown part as 7 on her hands.|
Incorrect answer—uses 10 as the whole.
Our conclusion is that it is crucial that children not only experience numbers as parts of larger numbers, but also perceive the whole as specific to the task within which the two addends are to be found. When children fail to perceive the correct whole simultaneously with the parts they may very well structure part-part-whole relationships, but end up with wrong answers (see Sofia, above). The enactment, then, has one number (shown as a finger pattern of 10 in the example above) foregrounded, but not the two numbers presented in the task (3 and 8).
Our sample contained one instance of structuring that did end up with a correct answer even though the child did this by structuring in a distinctly different way, as shown in the excerpt below. Benjamin uses another way of composing and re-grouping between the parts, adding one unit (finger) to one set and then another unit (finger) to the other set, then making use of a structure composed by the doubles “3 and 3” and “4 and 4”.
Benjamin: Three (shows the thumb, index, middle fingers on his left hand) then it’s eight (unfolds little, ring, middle fingers on his right hand, holding the hands close together) three more. Or I got three so it’s going to be (pauses and looks straight ahead) four more. Three and it’ll be, or no it’ll be five more because then this is four here (unfolds ring finger on his left hand) and four here (unfolds all but the thumb on his right hand)
3 and 8. Makes use of doubles (4 + 4). Moves units between the parts.
Benjamin seems to recognize 4 on one hand and 1 on the other as one composite set (4 + 1), which leaves the first addend (3) as one set. The whole (8) is emphasized only in the beginning, when he creates a pattern of 3 on his right hand as he starts grouping the parts within the whole on his fingers. He creates a structure composed of two parts (using doubles), 4 and 4, to which he can relate the numbers in the task.
Instantly showing finger pattern sets and counting single units
Most of the children (n = 21) who enacted a structural approach did so by both counting to create a set (represented as a finger pattern) and instantly showing numbers represented as composite sets on their fingers without counting (finger patterns), in the same act. Most of them managed to solve the task. In general, we can see that all but one of the children in this category started with the first given addend (3). This is not surprising since the semantic structure of the task is that you first have a set of three, presented orally, and then you are given the information that there are eight people coming. And lastly, the question is posed as to how many more glasses (than the three) you need to get. In order to solve the task, it is important to establish of what the whole is constituted, and what the part is, thus creating a structure of parts and the whole. Two children in our sample did not see the whole and the given part simultaneously, and thus lost track of the missing part, for example Peter in the excerpt below:
Peter: One, two (unfolds left thumb and index finger) and then I need to get…|
Interviewer: Three at first, three glasses.
Peter: Three glasses (unfolds the left middle finger as well) and then I’ll get one, two (unfolds the ring and little finger) three, four, five, six (unfolds one finger at a time on his right hand) maybe six more? (shows the interviewer nine unfolded fingers).
Creates the known part (3) by unfolding one finger at a time.|
Creates the missing part by counting on one finger at a time from the first part.
In the excerpt above, Peter is solving the problem by ‘counting on’. What we can conclude is that he and the other child who ‘counted on’ in a similar way failed to find the answer because they lost track of how many units they needed to add to get the whole. Peter’s verbal expressions reveal a rather empirical interpretation of the task, as he first creates the given part (3) and then says “then I need to get…” followed by a single counting act to add units. It does not seem as if he is experiencing numbers as related in a part-part-whole fashion, which hinders him in figuring out when to stop counting and adding on units to the given part.
While most of the children did perceive the parts and the whole simultaneously, how the structure was created differed among them. Discerning what the parts and the whole are simultaneously seems to be a critical aspect of any kind of structuring, but the observations of different ways of structuring also bring to the fore that other aspects are critical. In the following we take a closer look at two distinctly different ways of combining instant finger patterns with counting.
Counting on the unknown part on separate hand
Nine children were found to create the parts (3 + 5) on separate hands by both instantly creating one set on one hand and adding units on the other hand, until the whole is represented. A typical example of this can be found below. Oliver starts by creating the set of 3 and then continues counting on, on his other hand: “four, five, six, seven, eight”. Since his whole hand is included in the counting on procedure, he gives a rapid answer: “Five” (see illustration in Fig. 2):
Oliver: Three (unfolds the little finger, index and thumb on his left hand). One, two, three (points with his right index finger when counting the fingers, then unfolds all fingers on the right hand and points with his left thumb while counting, still holding his original pattern of three on the left hand) four, five, six, seven, eight. Five.
Creates the given part (3). Counts on until reaching the whole (8). Sees the 5 in the 8. The unknown part is created on the other hand.|
Seven of the nine children who enacted structuring in this way produced correct answers, which indicates that they experienced the first addend (3) as a composed set and kept track of the counted numbers until reaching the whole (8). Once having counted on to 8, the children seemed to experience their finger pattern as a structure of the parts and the whole related to each other, and instantly recognized the unknown part (5).
Counting on by partitioning the unknown part on separate hands
Composing sets can be done by counting and by instantly seeing a finger pattern, but the structuring can be enacted in yet another way than the one described above. Like most of the children, 12 of them started by creating the first given addend (3). However, like Mike in the excerpt below, they continued composing the whole (8) by counting their full hand (3 + 2) and then continuing on the other hand (+ 3). Some children then counted the missing part (one, two, three, four, five) or recognized the part instantly as 5 (see Fig. 3):
Mike: (Shows the thumb, index and middle finger on his right hand, looks at his left hand (folded), then unfolds the ring and little finger on his right hand one at a time, then the thumb, index and middle finger on his left hand. He then folds the right hand’s ring and little fingers and unfolds them again, then points with the right thumb the unfolded fingers on his left hand) Five more glasses.
Creates the given part (3)|
Counts on until the whole is created (8).
Counts the unknown part (2 + 3).
Sees 3 + (2 + 3) in the 8.
This way of solving the task seems to be a quite demanding act since the child has to experience, first of all, that the given part (3) is included in a part-part-whole relation with the 8, and that the added fingers on his first hand (2) have to be added to the 3 on his other hand, leading to a structure in which the unknown part is partitioned into 2 and 3, yet constitutes a composite set of five. The children who started enumerating the missing part starting with ‘one’ (usually on the ring finger) succeeded and managed to see the 5 in the 8 (like Mike in the excerpt above). Some who started enumerating the whole (continuing from ‘four’) seemed to lose track of the parts, both given and unknown, as shown in the excerpt with Leah, below:
Leah: One, two, three (unfolds three fingers, one at a time, thumb, index and middle finger, short pause) four, five (unfolds ring and little finger) six, seven, eight (thumb, index and middle finger on her other hand). I got two? (takes her hands away)|
Interviewer: Hm. How many glasses was it that you needed to bring?
Leah: Two glasses and I have two children.
Creates the given part (3). Counts single units to create the whole. The unknown part is not differentiated from the whole (8).|
Discerns the part and the whole, but loses the parts of the whole.
Leah starts with the known part (3), and continuing to count on to make eight. However, she does not see the unknown part as differentiated from the whole. Instead she answers two, which could be interpreted as the remaining fingers of ten. Another plausible explanation could of course be that she does not fully understand the semantics of the task.
Structuring numbers in the task by partitioning the unknown part on separate hands illustrates to a larger extent the comprehensiveness of the part-part-whole relation, whereby the parts are included as a composite set of the whole. This differs from the earlier category (4.2.1), in which the whole was created and illustrated as two separate parts. Different ways of enacting a structural approach thus imply that different aspects are critical.
When comparing the second and the third interview (conducted with a one year interval) we found a pattern in the ways the children’s enactment had changed (see Table 2). The sample is too small to take statistical measures and more tasks should be analyzed to make general statements about the findings, but there are tendencies in the children’s enactment that are of educational interest.
Among the children who were counting on the unknown part as a set composed on the other hand (4.2.1), there were diverse results in the third interview in how they solved the same task. Two of the nine children in this category developed their way of seeing numbers as being composite sets which do not need to be counted (4.1), while three acted in similar structuring ways, creating sets using finger patterns and counting single units as ‘counting on’. However, we also saw that three of these children enacted the more primitive strategy of ‘counting all’ or simply guessing in the later interview, which was a surprise. Solving the task by counting emphasized that the children experienced numbers as single units rather than composite sets structured in a part-part-whole fashion.
The third interview reveals that some ways of structuring seem to foster a continuation of using a structural approach while other ways of structuring seem not to facilitate the discernment of part-part-whole relations to the same extent, in this group of children. One explanation is that the structuring acts in interview II where children create parts and the whole on separate hands by counting on (4.2.1), may be an indication of the child experiencing local relationships; however this action does not seem to induce a generalization of their structuring. In contrast, when the children create the unknown part as partitioned on two hands but still constituting a composite set of five (4.2.2) they seem to experience the number relations differently, as they in the later interview enact in the same way (which by all means may be due to their discerning of the local relationship only, and having mastered this particular triad of numbers), or they have learnt the number relation as a fact and do not need to calculate or operate with the numbers to find the answer.