1 Introduction

For some time now, mathematics educators have come to realize the importance of engaging childrenFootnote 1 with mathematical activities as early as possible (Baroody & Lai, 2022; Clements et al., 2020; Ginsburg et al., 2008). An essential content area for this age is, of course, number, and within that content area is being able to compare the number of objects in different sets, as well as dividing a set of objects into equal groups. While several studies have explored young children’s abilities to compare the numerosity of sets (Baroody, 2004; Mix, 1999), few have investigated the ways in which children divide a set of objects into equal groups. Yet, dividing objects into equal groups is a natural activity for children, part of their everyday lives that can be linked to fair sharing. Furthermore, dividing a set of objects into equal groups may be considered as a prelude to skip counting and the basis for multiplication and division (Sarama & Clements, 2009).

Yet, content without processes is not enough. “The solution strategies children use are critical components of their learning, especially in mathematics” (Clements et al., 2020, p. 1). Educators (Linder et al., 2011) and curriculum guidelines (Common Core State Standards for Mathematics, 2010) suggest that early on children should be exposed to several strategies for solving a specific problem, with children eventually learning to use the various strategies flexibly and appropriately for various situations. Strategic flexibility is also thought to link procedural and conceptual knowledge (Baroody, 2003). By investigating children’s solutions and accompanying strategies when dividing a set into equal sets, we can gain insight into their conceptualization of sets, equal sets, and what it means to divide a set into equal sets, as well as the strategies they use to perform this activity. Such background knowledge can help early childhood educators plan ways in which to enrich children’s repertoire of strategies and their flexible use of those strategies.

In the current study, we investigate the ways in which kindergarten children divide a set of objects into equal groups, when the number of groups as well as the number of items to be placed in each group is unspecified. The study further investigates the relationship between the number of given objects in the set, n, and children’s solutions to this task. Specifically, we investigate the following four cases, n = 8, 9, 22, and 23.

Recognizing that there are several solutions to the task of dividing n objects into an unspecified number of equal groups, each child was asked to solve the same task (i.e., for the same n) more than once and to think about a different solution each time. This is in line with another aim of education, that is, to promote mathematical creativity (Leikin, 2018; Sriraman, 2017). Most mathematics educators who studied mathematical creativity did so within a school context, with school-age students. For example, Liljedahl and Sriraman (2006) described mathematical creativity in the classroom as when students view a familiar problem from a new angle and find unusual or insightful solutions. Thus, requesting a child to solve a task that he or she has already solved, but to solve that problem in a different way, or to find a different solution to that same problem can encourage one to seek new paths and viewpoints.

Several mathematics educators have implemented such tasks, often called open-ended tasks (Klein and Leikin, 2020), to assess school students’ mathematical creativity (Levenson, 2011; Molad et al., 2020). For an open task to promote creativity, Haylock (1987) recommended that the task include the possibility of using a wide range of ideas, have at least 20 possible appropriate responses among several of which are obvious to students, some responses which will likely be obtained by only a few students, and they should not be mathematically trivial. In previous studies of students’ mathematical creativity (Leikin, 2018; Levenson, 2011; Molad et al., 2020) and kindergarten children’s mathematical creativity (Tsamir et al., 2010b), participants’ solutions, as well as the strategies used to reach those solutions, were used to evaluate three measures of mathematical creativity: fluency, flexibility, and originality. Within mathematics education research, fluency and flexibility might have different connotations depending on the context of the study. For example, when the topic of the study is arithmetic, computational fluency refers to carrying out computations quickly and accurately, as well as appropriately (Baroody, 2003). Computational flexibility means being able to choose the most appropriate strategy for the situation. However, within the context of studying creativity and solving open tasks that have many solutions, fluency is reflected by the number of different solutions found for a task, while flexibility refers to “apparent shifts in approaches taken when generating responses to a prompt” (Silver, 1997. p. 76). Identifying those shifts is usually carried out by categorizing solutions or the strategies used to reach the different solutions (Levenson & Molad, 2022). Originality may be measured by the level of insight or conventionality with respect to the learning history of the students (Leikin, 2009). Silver (1997) argued that educators can view mathematical creativity as “an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population” (p. 79). Being that dispositions develop from an early age and considering that, with few exceptions (Tsamir et al., 2010b), studies of mathematical creativity were carried out among school-age students, it becomes important to promote and evaluate children’s mathematical creativity when they are young. Thus, another aim of this study is to investigate kindergarten children’s mathematical creativity within the context of creating equal groups.

2 Creating equal groups

Equivalent sets do not necessarily have the same elements, but they do have the same cardinality. Equal groups have the same elements, which of course implies that they have the same cardinality. In the current study, children were given n bottle caps all of which were the same size and color. Thus, we refer to the creation of equal groups.

There are several numerical competencies related to the ability of creating equal groups from n objects such as object counting, subitizing, number decomposition, and set comparison. We briefly discuss these competencies and then review studies related to children’s ability to create equal groups.

Object counting, also called enumeration, is carried out for the purpose of saying how many objects are in a set. Gelman and Gallistel (1978) outlined five principles of counting objects. The first, called the stable-order principle, is based on being able to count verbally, that is, to recite the numbers consistently in the conventional order. A common error might be to skip a number. The second principle is the one-to-one principle, which involves assigning one count word to each object. Common mistakes related to this principle can be skipping over an object or assigning more than one counting number to an object (Fuson, 1988). Next is the cardinality principle, knowing that the last number mentioned when counting objects represents the number of objects in that set. Without cardinality, a child may recount the objects which have just been counted or state a random number when asked how many objects are in a set (Fluck & Henderson, 1996). The fourth principle is the abstraction principle, meaning that any set of discrete objects can be counted. Finally, the order-irrelevance principle means knowing that one may count the objects in any order (e.g., from left to left, from right to left) and that enumerating objects in different ways results in the same cardinality.

Identifying the numerosity of a set may also involve subitizing. Subitizing is “the direct perceptual apprehension and identification of the numerosity of a small group of items” (Clements et al., 2019, p. 14). Researchers (Clements et al., 2019; Sarama & Clements, 2009) differentiated between two types of subitizing. Perceptual subitizing is recognizing and then naming the number of items perceived, without consciously involving other mental processes. Conceptual subitizing involves applying perceptual subitizing and then composing parts into a whole. For example, the six-dot pattern on a die may be recognized as six when the child perceptually subitizes the three and three and uses memorized facts to compose six. Thus, conceptual subitizing is related to number composition and decomposition and the notion that a set may be decomposed into subsets and then combined once more to compose a single set.

Despite the importance of being able to create equal groups, few studies have investigated the ways in which children create such groups. Instead, most studies have examined children’s abilities to compare sets. Early researchers (Bryant, 1974; Gelman, 1972; Piaget, 1952) found that children employ six different strategies when comparing sets: length (which row of items is longer?), number (cardinality), correspondence, heaped (high density), proximity (does one set have 25 items and the second 23 items or does the second set have 10 items?), and guessing. In those early studies, researchers used physical items laid out in rows (Piaget, 1952) or cards with illustrations of dots laid out in rows (Gullen, 1978). Further studies investigated children’s strategies when comparing sets of items bunched in a pile to those strategies used when items were placed in rows (Baroody & Gatzke, 1991). Those and other studies found that children often used estimation, rather than counting (Zhou, 2002), when deciding which set has more. Whereas subitizing can be used for a maximum of four or five items and counting is a relatively slow process, estimation, or more precisely, numerosity estimation, is a quick perceptual process often used to judge the numerosity of a large group of items (Luwel & Verschaffel, 2008). In a more recent study (Tsamir et al., 2010a), it was found that when children are requested to compare two piles of homogeneous items (9 and 12 bottle caps), some counted the objects in both sets to compare them, while others lined up the objects in two rows and used one-to-one correspondence to compare the number of objects in each row.

Linked to both one-to-one correspondence and creating equal groups is the activity of setting a table, where each place setting requires, for example, one cup. In addition to the one-to-one correspondence of setting a cup beside each place, there is also the fact that we are now creating equal groups of one. If the place setting requires two forks, then we are creating equal groups of two. In their study of young children setting a table, Tirosh et al. (2020) found that one challenge occurred when children kept on placing items on the table until they had used all of the items given to them. That is, when given a stack of cups (more than eight) and a table set for eight, the children kept on going around the table handing out cups, until no cups were left. The authors suggested that this might reflect a cultural norm that all the items given in an activity must be used up in that activity.

Dividing a group of objects into equal groups is also related to fair sharing. In an early study of young children’s understanding of sharing (Pepper & Hunting, 1998), 92% of the children were able to divide a set of 12 crackers equally between two dolls. However, when given 21 cookies to be shared equally between three dolls, only 60% of the children successfully completed the task. The context of the sharing task can be especially important for young children. In a recent study (Hamamouche et al., 2020), children who were asked to divide a set of toys between two stuffed animals so that each had the same number of toys performed significantly better than children asked to divide the same number of toys among a cardboard circle and triangle placed on the table. This difference between the social contexts was not found for other arithmetic operations such as addition.

In the current study, children engaged in an open task that had multiple possible outcomes, as well as multiple strategies for reaching those outcomes. Relative to both the creation of equal groups and open-ended tasks is the study by Tsamir et al. (2010b). In that study, children were presented with one set of three bottle caps and one set of five bottle caps and were requested to create from those two sets, two new sets with an equal number of caps. There are potentially five solutions to that task: two sets with four caps in each (4,4), two sets of three caps in each (3,3), two sets of two (2,2), two sets of one (1,1), and two empty sets (0,0). Although most children came up with more than one solution, less than 25% came up with the solution (0,0). In that study, children were explicitly requested to form two equal groups. In the current study, children were requested to form equal groups, without stating how many equal groups should be formed. Furthermore, in Tsamir et al.’s (2010b) study, children were presented with a relatively small even number of objects. In the current study, we were interested to know if there would be a difference between if children were given an even or an odd number of objects and if there would be a difference if they were given a small or large number of objects. This led to the following research questions.

When creating equal groups from n items:

  1. (1)

    Will the parity and magnitude of n affect the number and types of solutions given?

  2. (2)

    Will the parity and magnitude of n affect the strategies children use to create equal groups?

  3. (3)

    To what extent can this type of activity elicit creativity (as measured by fluency, flexibility, and originality) among kindergarten children?

3 Method

3.1 Context

In Israel, where this study takes place, there is a mandatory mathematics curriculum for preschool (Israel National Mathematics Preschool Curriculum (INMPC), 2010). Accordingly, by the end of kindergarten (age six), children should be able to count verbally until 30 and enumerate at least 20 objects, although it is recommended that the teacher encourage children to count more than 20 objects. For set comparison, it states that children should be encouraged to use two strategies: counting the objects in the sets and using one-to-one correspondence. There is a separate heading entitled “dividing a set of objects into equal groups.” Under that heading, it is suggested that children practice dividing concrete objects, first into two equal groups and then into more groups. The curriculum notes that some children will divide the objects into a certain number of groups by handing out one object at a time until all objects are given out. This relates to using a strategy of one-to-one correspondence. The curriculum also mentions that the number of objects need not always be divided equally into the targeted number of groups and that the teacher should discuss with children various solutions to such problems, such as when five apples need to be shared among two children. Relevant to this study, one solution mentioned for that apple task is giving each child two apples and putting the fifth on the side. Finally, children should learn that there are different ways to divide a set of objects into equal groups, such as dividing eight objects into eight groups with one object in each, four groups with two objects in each, and two groups of four. Interestingly, the curriculum also states, “in mathematics there is another solution, less accepted—one group with eight items.” The possibility of an empty set is not mentioned.

3.2 Participants and procedure

Participants in this study were 34 children (5–6 years old) learning in the same kindergarten with the same teacher. Ethical approval was received by the Chief Scientist of the Ministry of Education and the Institutional Review Board; all parents signed an informed consent form. The study was conducted toward the end of the school year; all children would be attending first grade in the coming year. The researcher (third author) sat with each child individually in a quiet corner of the kindergarten and placed n identical bottle caps on the table in front of the child. Four tasks were implemented with the children, all in one session. For the first task, 8 caps were placed on the table; then, 22 caps were presented, followed by 9 and then 23 caps. Each time, the child was asked to create equal groups of caps. After the child created the groups, the researcher gathered the caps back into one group and asked the child: is there another way to create equal groups? This was repeated until the child declared there were no other ways to create equal groups or when the child said “I am done.”

Sessions with each child lasted between 1 and 15 minutes, depending on the number of solutions and strategies the child used. Each session was recorded with two video cameras, filmed from different angles. The cameras recorded speech, as well as hand movements, body movements, and the table. Recordings were transcribed by the third author.

3.3 Data analysis

A first analysis differentiated between correct and incorrect solutions. An incorrect solution was when the child said that she/he had created equal groups, but the groups had an unequal number of bottle caps. A second analysis differentiated between equivalent solutions. That is, if a child created from eight bottle caps, for example, a solution consisting of four caps in one pile and four in another and then later created a solution of four caps in one row and four caps in another row, the solutions were only counted as one (4,4). For each child, a fluency score was then calculated based on the number of different correct solutions found per task. Note, the children were not told that all the bottle caps had to be used when creating equal groups. Also, they were told to create “equal groups,” groups being in the plural. Thus, a solution for the 9-cap task could be, for example, two groups of four caps denoted as (4,4) and one cap pushed to the side. Accordingly, an a priori analysis of the tasks indicated that the first task (eight bottle caps) had 13 possible solutions (consisting of at least two groups, including the possibility of creating two empty sets, although theoretically there could be an infinite number of empty sets); the second task (22 caps) had 53 optional solutions; the third task (nine caps) had 15 optional solutions; and the fourth task (23 caps) had 54 optional solutions.

A third analysis focused on flexibility; two types of flexibility were analyzed. First, solutions were categorized based on the number of equal groups created for a task. That is, when creating equal groups from, say, eight caps, did the child divide the caps once into two equal groups, once into three equal groups, and once into four equal groups or did the child only create solutions with two equal groups. This flexibility we will call “grouping flexibility.” Thus, given eight caps, one child could have a fluency score of four (1,1), (2,2), (3,3), (4,4), but a grouping flexibility score of one. Or a child could have a fluency score of four (2,2), (3,3), (2,2,2), (1,1,1,1) and a grouping flexibility score of three. In line with Levenson and Molad (2022), we also analyzed “strategic flexibility.” Children’s strategies for creating equal groups were first analyzed deductively based on those strategies mentioned in the preschool mathematics curriculum (Israel National Mathematics Preschool Curriculum (INMPC), 2010) for set comparison—using one-to-one correspondence and object counting. Inductive analysis was then used to categorize additional strategies. This resulted in a total of five categories (see Table 5 in the next section). Categorization was carried out first by the third author. The first author then viewed all of the videos together with the third author to validate the categorization. Full consensus was reached.

Two solutions were considered original, based on previous studies related to understanding sets. The first was the solution of creating empty sets. For example, Piaget and Inhelder (1969) considered the empty set challenging “because a class without any elements is incompatible with the logic of concrete operations, i.e., operations in which form is inseparably bound up with content” (p. 149). Similarly, Linchevski & Vinner (1988) called the empty set counterintuitive. In Tsamir et al.’s (2010b) study of kindergarten children, the solution which involved creating two empty sets was found significantly less than other possible solutions. The second original solution was to create equal sets with one item in each. According to Fischbein (1999), the collection model for sets renders a “single element collection” (p. 56) intuitively unacceptable.

4 Findings

The first part of this section describes the different solutions found per task and the number of equal groups children created for each task. The second part focuses on the children, how many solutions were found by the children (fluency), their grouping flexibility and strategy flexibility, and finally their originality.

4.1 Descriptions of solutions

Table 1 presents the frequencies of the various solutions according to the number of caps presented to the children. For example, in the 8-cap task, the solution of two equal groups of two caps each (2,2) was presented six times, while the solution of two equal groups with four caps in each (4,4) was presented 31 times. When nine caps were presented to the children, the solution of four equal groups consisting of two caps each (2,2,2,2) was presented six times. Note that in Table 1 (and in all the other tables), a line (–) represents a solution that is irrelevant for that task, while a 0 represents a relevant solution that none of the children suggested.

Table 1 Frequency (%) of final solutions per task (N represents the total number of solutions for that task)
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As seen in Table 1, more solutions (82) were presented for the 22-cap task, than for any other task. Interestingly, when the number of caps was even (both for the 8- and 22-cap tasks), the number of solutions found was higher than for the cases with an odd number of caps. Furthermore, in the tasks with an even number of caps, more solutions were given when a large number of caps (22) were presented than when a small number of caps (8) were presented. While this may seem trivial given that there are more solution options for the 22-cap task than for 8-cap task, note that the opposite occurred when an odd number of caps were placed on the table. In that case, less solutions were given when a large number of caps (23) were presented than when a small number of caps (9) were presented. Interestingly, the (4,4) solution was the most frequent solution when eight or nine caps were presented to the children, but was hardly present when a large number of caps were on the table. For all four tasks, nearly all possible solutions for creating two equal groups were found by the children, the exception being the (2,2) solution for the 23-cap task, which none of the children presented.

In the following analysis, the empty set solutions were excluded, as in theory there is an infinite number of such solutions. Hence, the 8-cap task has 12 possible solutions, 4 of which are two equal groups (1, 2, 3, or 4 caps in each group). Similarly, the 22-cap task has 11 two-equal-groups solutions out of 52, the 9-cap task, 4 of 14, and the 23-cap task has 11 two-equal-groups solutions out of 53. In sum, the two-equal-groups solutions make about a quarter of all possible solutions. Goodness of fit tests, performed for each task separately, showed that the two-equal-groups solutions were more common than all other solutions combined. Specific results are shown in Table 2.

Table 2 Frequency of the two-equal-groups solutions as compared to all other solutions. Numbers are N (%). All p < 0.001
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Looking closer at Table 1, when eight or nine caps were presented to the children, some children created four equal groups. However, when 22 or 23 caps were presented, none of the children created four equal groups. On the other hand, children created seven equal groups, a solution not present when eight or nine caps were presented. Interestingly, in three tasks (9, 22, and 23 caps), a solution of nine equal groups was an option. Yet, only when nine caps were placed before the children did two children present this solution. The creation of ten and eleven equal groups occurred only when 22 caps were presented to the children, but not when 23 caps were presented.

4.2 Fluency

Table 3 presents the frequencies (and relative frequencies) of fluency scores for each task, as well as the mean fluency score per task. For example, when 22 caps were presented to the children, six children had a fluency score of four, that is, they presented four solutions to the task of creating equal groups from 22 caps. Additionally, in the task with nine caps, 19 children suggested only one solution for the creation of equal groups (so, their fluency score was one.)

Table 3 Frequency (%) of children’s fluency scores per task (N = 34)
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We used a 2 (magnitude: small, large) by 2 (parity: even, odd) repeated measures ANOVA model to test for differences in fluency between different tasks and for a magnitude-by-parity interaction. In addition to a significant parity effect (F(1, 33) = 4.3, p = 0.046), we found a significant interaction between the magnitude and parity of the number of caps (F(1, 33) = 4.3, p = 0.045). Simple effects analysis, presented in Fig. 1, showed that for odd numbers, there was no difference between the small and large numbers (9 and 23), while for even numbers, fluency for the 22-cap task was significantly higher than that for the 8-cap task. Alternatively, it was found that for small numbers there was no difference between 8 and 9, while for large numbers fluency for 22 was significantly higher than that for 23.

Fig. 1
figure 1

Simple effects of the magnitude by parity interaction affecting fluency

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Taking a closer look at the results of the 22- and 23-cap tasks in Table 3, one child failed to create equal groups in the 22-cap task, and nine children (26%) failed to create equal groups from 23 caps (e.g., one child said, “it’s impossible,” and others had errors in what they deemed as a solution). This is noteworthy as all solutions found in the 22-cap task could have been utilized in the 23-cap task; yet they were not. To test whether the probability of a child offering no solution is larger in the 23-cap task than for all other tasks, we used Cochran’s Q statistic for comparing within-subjects categorical measures. Results showed a significant difference between the tasks (Q = 22.8, p < 0.001), with 9 of 34 children not producing any correct solution to the 23-cap task.

4.3 Grouping flexibility

Grouping flexibility measured the number of various groupings when creating equal groups (e.g., creating solutions with two equal groups and also creating solutions with three or four equal groups). Results of the grouping flexibility (see Table 4) showed that most children employed one type of grouping per task.

Table 4 Frequency (%) of children’s grouping flexibility scores per task (N = 34)
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A repeated measures ANOVA was conducted to test for the effect of magnitude and parity on grouping flexibility. We found a significant magnitude by parity interaction (F(1,33) = 5.6, p = 0.025). Simple effects analysis, presented in Fig. 2, showed a significant difference between small and large odd numbers (9 and 23), but not between small and large even numbers (8 and 22). When comparing odd to even numbers, a marginally significant difference (p = 0.055) was found between odd and even large numbers (22 and 23) but not between the small numbers (8 and 9).

Fig. 2
figure 2

Simple effects of the magnitude by parity interaction affecting grouping flexibility

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Looking at Table 4 (grouping flexibility) while also considering Table 3 (fluency), we see that most children created one type of grouping, perhaps not surprising considering that most children found only one solution (see Table 3). Yet, we also see that when children found more than one solution, they were still likely to use the same type of grouping. For example, for the 8-cap task, 18 children found more than one solution; however, only six children created more than one type of grouping. Likewise, for the 22-cap task, 19 children found more than one solution; however, only five children created more than one type of grouping.

4.4 Strategic flexibility

The second type of flexibility we called “strategy flexibility.” To analyze this, we categorized the strategies used to create the equal groups (see Table 5).

Table 5 Strategies for creating equal groups
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As seen in Table 6, different tasks elicited different strategy use. The counts in the cells of this table reflect the number of children that used this strategy at least once in solving each task. Note that a child could be counted more than once for each task if the child used more than one strategy.

The association between task and preferred strategy was tested using a chi-square test on the frequencies presented in Table 6. A significant association was found (x2(12) = 33.5, p < 0.001). Specifically, the counting strategy was dominant for grouping large numbers while subitizing was the preferred strategy for grouping small numbers. The last category was called “random” as no visible strategy was discerned. However, given that we are considering only correct solutions, it is likely that children estimated the number of caps placed in each group. This strategy was used only when the number of caps was large. One-to-one correspondence was used infrequently and will be discussed later.

Table 6 Frequencies (%) of strategy use per task
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We then calculated for each child a strategic flexibility score by seeing how many different strategies were employed by that child. As can be seen in Table 7 (N represents the number of children that offered at least one solution for that task), most children employed either one or two strategies, regardless of the number of caps presented in the task. Comparing the findings in this table to those of Table 3 (fluency scores), we note that a few children (between four and five children) used a combination of two or more strategies when creating a single solution, such as beginning with a one-to-one correspondence strategy but then continuing by forming shapes.

Table 7 Frequencies (%) of strategic flexibility scores per task
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Although up until now we have considered each task separately, we now consider the number of different strategies children employed across all four tasks, where a maximum score of five could be achieved (see Table 8). Comparing the results of Table 7 to Table 8, it may be said that while children employed between one and two strategies when working on a single task, when considering all four tasks, they generally employed between two and three strategies. In other words, they did not necessarily stick to the same strategy when solving all four tasks.

Table 8 Frequencies (%) of number of strategies used across all four tasks
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4.5 Originality

Lastly, we describe originality. Recall that two solutions were considered original—creating equal empty groups and creating equal groups with one cap in each. Eight children came up with at least one original solution for one of the tasks. Interestingly, the same child (C2) gave the empty set solution for the first three tasks, but a different child gave that solution for the fourth task. For the 8-cap task, C2 first divided the 8 caps into two groups of four and then two groups of three, two groups of two, and two groups of one. After the researcher placed the 8 caps once again in front of C2 and asked C2 if he could divide the caps in a different way into equal groups, C2 moved one cap at a time to a different part of the table until all had been moved and said “zero, zero.” In other words, C2 only divided the caps into two equal groups (thus having a grouping flexibility of one.) Similarly, the second child that found the empty set solution for 23 caps also had a grouping flexibility score of 1 for each of the tasks, consistently dividing the caps into two groups only. Regarding the solution of creating equal groups of 1, none of the children thought to create eight equal groups of one for the 8-cap task. This solution, using all the caps to create n equal groups of 1, only came to mind beginning with the 22-cap task, and then, only three different children presented that solution.

5 Discussion

The first aim of this study was to investigate the number and types of solutions preschool children (5–6 years old) find when asked to solve an open-ended task that involves the creation of equal groups. Specifically, it investigated differences between creating equal groups from an even number and an odd number of objects and between creating equal groups from small and large amounts of objects.

The first important finding is that despite the option of creating a various number of equal groups, the most frequent solution by far was the creation of two equal groups. Certainly, the inclination to create two equal groups is understandable for even numbers. In that case, children could use all the caps and create two equal groups. Recall that, in Tirosh et al.’s (2020) study, children continued distributing items around the table, even though it was unwarranted, in order to use up all the items. Indeed, for the 8-cap task, over 90% of the children reached the solution (4,4) which simultaneously uses all the caps and creates two equal groups. Following this finding, it is interesting to see what happened in the 9-cap task. If a child was strongly inclined to divide the caps into two equal groups, then one cap would have to be left out. However, if the child was more inclined to use all the caps, then he/she could create three equal groups of three (3,3,3). In fact, less than 15% of the children reached the solution (3,3,3), and more than 76% of the children gave the solution (4,4). Furthermore, for the 22-cap task, creating two equal groups was the most frequent solution, but it was not always (11,11). That is, children created two equal groups and pushed extra caps to the side. We tentatively conclude that when dividing objects into equal groups, children’s intuition is to divide the objects into two groups. According to Fischbein (1999), one characteristic of intuition is that it is coercive. In this case, when there was a so-called choice between using all the caps or creating two equal groups, children tended to create two equal groups. This may have implications later on, when decomposing numbers or breaking wholes into parts. Might children readily accept decomposing a number into two parts, but not into more parts? On the other hand, knowing that children intuitively seek out two equal groups might relate to why teaching the addition strategy of “near doubles” (i.e., when adding 4 + 5, add instead (4 + 4) + 1) is encouraged, especially for children at risk (Baroody et al., 2012).

For the tasks with a small number of caps, there were no significant differences in fluency or in grouping flexibility. However, when the number of caps was large, there were near significant differences in both measures. This is curious. Theoretically, the same solutions for the 22-cap task could be found in the 23-cap task, yet this did not happen. A major factor impacting these results was that for the 23-cap task, approximately a quarter of the children failed to solve the task, while for the 22-cap task, only one child failed. A simple explanation for this could be that the 23-cap task was the last of four tasks, children were tired and thus made more mistakes. Another explanation could be related to the properties of the numbers—23 is not just odd, it is also prime. Not only can it not be divided into two equal groups using all the caps, but even if a child thought to divide the caps into more than two groups, there would always be leftover caps. Consequently, children who wanted to use all the caps had only two options, i.e., create 23 groups consisting of one cap each or create empty groups. Both options are counterintuitive (Tsamir et al., 2010b), as most children (and many adults) believe a set must contain more than one element.

Although this study focused on creating equal groups, it is related to the more general topic of number sense, a general aim of preschool mathematics (Andrews & Sayers, 2015). Examining properties of numbers, such as their parity, can promote this development in the sense that children can see number patterns and learn about number representations (Andrews & Sayers, 2015). The activities in the current study support this. Children divided the eight caps into four groups of two and saw that all the caps were used, but when dividing nine caps into four groups of two, one was leftover. Exposing children to sets with different numerosities is one way to offer them experiences with how numbers behave. Such activities can lead to further exploration regarding divisibility, equivalence, and sets. What might happen if children are given 30 items to divide into equal groups? Will children “discover” all the factors of 30? What might happen if children are given a small prime number, such as seven, to divide into equal groups? These are questions for further research.

The second aim of this study was to investigate the strategies children use to create equal groups. As pointed out previously, few studies investigated children’s strategies for creating equal groups and instead focused on set comparison (e.g., Zhou, 2002). In the current study, nearly all of the strategies children used for creating equal groups were found in previous studies related to comparing sets (e.g., Bryant, 1974; Gelman, 1972). This link between the two activities implies that both set comparison activities and activities that require the creation of equal groups offer children opportunities to practice several important strategies. On the other hand, the link with equal sharing activities may be only partial. In Pepper and Hunting’s (1998) study of preschool children’s ability to equally share cookies among dolls, the most often used strategy was one-to-one correspondence or two- or three-to-one correspondence. However, many children were not systematic and began the act of sharing with a different doll each time, ending up with an unequal sharing of the cookies. In the current study, one-to-one correspondence was hardly used. A possible reason for the difference in strategies might be the more abstract context of the current study. While an unnatural context might make it more difficult for children to correctly distribute items equally (Hamamouche et al., 2020), it might also encourage children to use a variety of strategies, in turn, encouraging strategic flexibility. In fact, as seen in the findings, for each number of caps, there was a preferred strategy and not necessarily the same one for each. Specifically, different preferred strategies were found for the small n tasks and the large n tasks, hinting that challenging children to engage with a large number of items may encourage them to use different strategies. Furthermore, the random or estimation strategy was only used when a large number of caps were presented. Researchers have recognized estimation as part of children’s everyday lives and have found that it may be related to additional arithmetic skills (Butterworth, 2010; Siegler & Booth, 2005). Yet, it is not always given the attention it deserves (Andrews, et al., 2022). Thus, although a preschool mathematics curriculum might only suggest that children create equal groups from 10 items (e.g., in Israel (Israel National Mathematics Preschool Curriculum (INMPC), 2010)), having children work with a larger number of items might promote the use of estimation, in addition to other strategies.

One strategy found in the current study, but not mentioned in previous studies of creating equal groups or fair sharing, was the shape forming strategy. In a way, this strategy is similar to perceptual subitizing, where seeing shapes and patterns plays an implicit, but important role in comparing sets (Clements et al., 2019). However, unlike in set comparison activities where the sets are given and the child must judge their comparative numerosity, in the current study, children proactively constructed shapes and patterns to create equal groups. Considering that several mathematics educators recognize the importance of visualization for mathematical understanding (Presmeg, 2014), an activity that can encourage young children to create their own visual tool is to be encouraged. In general, the results of this study, which explored situations that may not occur in children’s daily lives and may not be recommended by curriculum guidelines, show how research can expand our view of children’s strategies and abilities.

The third aim of this study was to examine the creativity (as measured by fluency, flexibility, and originality) elicited by the task of creating equal groups. Although the tasks had potential to occasion mathematical creativity, this potential was not met. Most children gave only one or two solutions for each task and were fixated on the two-equal-groups solutions, regardless of the number of caps presented. Few children offered original solutions. One might ask what we gained by using a creativity lens in our study. We have several responses to this question. We looked at children’s originality, not only to examine what solutions are novel but to seek out areas that need attention. For example, within set theory, it is counterintuitive to accept that sets can contain one element in each (Fischbein, 1999). Yet, children often share their cookies by giving out one cookie to each friend. It is curious then why children only thought to use all of the caps and create equal groups of one when they were challenged with a large number of caps. As educators, we might consider how this relates to decomposition, for example, when decomposing the number five into five ones. The lens of creativity also allows us to focus on children’s fixations, considered the opposite of flexibility by creativity researchers (Haylock, 1987). We not only note the frequency of strategies, but how often children change their pattern of thinking. In the current study, we say that children were fixated on the two-groups solution. It was difficult for them to break away from this way of creating groups. Educators need to challenge these fixations, lest they become rigid and difficult to undo when they get older (Fischbein, 1987). Finally, we also point out that viewing these results from a creativity lens has brought to light a possible relationship between the notions of fluency and flexibility as used by early childhood educators and how these notions are used by creativity promoters. If we wish to enhance computational fluency and flexibility (Baroody, 2003), then we might begin by having children seek out different solutions and strategies when solving a problem. Creativity promoting tasks do just that. Although we do not know if the children in this study were exposed previously to open tasks, Silver (1997) stated and Tsamir et al. (2010b) concurred, searching for more than one outcome is a habit of mind that can be developed and promoted at an early age. Future research might explore this relationship further.