Keywords

Introduction

This paper is about the representations 6-year-old students use when working on a problem-solving task on combinatorics. Most studies on young students and representations have focused on numbers and quantitative thinking (Sarama & Clements, 2009). In a longitudinal study, a connection was found between the representations the students used and the extent to which they managed to solve a combinatorics problem-solving task (Palmér & van Bommel, 2017, 2018). However, when working on this problem-solving task, iconic representations did not generate more complete solutions than pictographic representations. Quite the opposite; pictographic representations seemed to imply more systematization and less duplication (van Bommel & Palmér, 2021). This is somewhat surprising as iconic representations are considered to be on a higher level of abstraction than pictographic representations (Heddens, 1986). For example, low achieving students have shown to more often use pictorial and iconic representations that are also poorly organised whereas high achieving student more often use well-structured abstract representations (Mulligan, 2002). Thus, the connections found between representations and how students solved the combinatorial problem did not apply to results from previous studies of young students’ use of representations (see e.g., Hughes, 1986; Mulligan & Mitchelmore, 2009; Piaget & Inhelder, 1969).

In the longitudinal study, some hypotheses for these results were posed (Palmér & van Bommel, 2018; van Bommel & Palmér, 2021). For example, it was suggested that the time issue and the connection between pictographic representation and the context of the task may explain why students who use pictographic representations make few duplications. Also, it takes longer to draw a pictographic representation than to draw an iconic representation, giving students who drew a pictographic representation more time to reflect on the task. Iconic representations are easier to draw, which makes the process faster and which may be why the solutions with iconic representations quite often contain several duplications (Palmér & van Bommel, 2017). To elaborate further on the previous somewhat contradictory results and on the hypotheses described above, after students solved the combinatorial problem-solving task they were interviewed about their choice of representation. The question focused on in this paper is: What rationales do young students express for their choice of representation?

The paper begins with a background on combinatorics and representations. After that, it focuses on the methodological aspects of the study, followed by results and analysis. The paper ends with a discussion in which the limitations of the study are also addressed.

The Combinatorics Task and Representations

The problem-solving task on combinatorics is one of several tasks used in a Swedish longitudinal intervention study investigating the potential of using problem solving as the start for the mathematics education of 6-year-olds. In Sweden, 6-year-olds have not yet begun formal schooling but attend what is called preschool class, a year of schooling intended to provide a smooth transition between preschool and school (Swedish National Agency for Education, 2014). Problem solving is part of the Swedish preschool class curriculum, but combinatorics is not. However, research has proven that within a proper and meaningful context young students can indeed work with combinatorial tasks finding permutations (English, 2005). Combinatorics can also be connected to pattern (a predictable regularity) and structure (how elements are organized and related) where Mulligan and Mitchelmore (2009) emphasise that awareness of pattern and structure is critical to mathematical learning.

The combinatorial task the students worked on entailed determining how many different ways three toy bears could be arranged in a row on a sofa. Thus, it was an enumerative combinatorial task involving counting permutations, in this case for n = 3. To make the task meaningful for the students, it was presented as a conflict between the toy bears, where they could not agree on who should sit at which place on the sofa. One toy bear then suggested that they could change places every day. The students’ task was to find out how many days they could sit in different ways on the sofa. As an introduction to the task, the students were shown three plastic bears in three different colours.

In studies on representations, one focus is on linkages and development between informal and formal representations (e.g., Hughes, 1986; Heddens, 1986; Carruthers & Worthington, 2006). Heddens (1986) focused on the connection between objects (concrete representation) and signs (abstract representation). He introduced representations of two levels of abstraction between these two representations, pictures and tally marks. He referred to pictures of objects as semi-concrete, and to tally marks (where the symbols or pictures do not look like the objects they represent) as semi-abstract. When documenting permutations in this study, all students used pictographic or iconic representations. The pictographic representations were drawings of the plastic bears in the three colours (example Fig. 1, uppermost), and the iconic representations were lines or dots in the three colours (example Fig. 1, bottommost).

Fig. 1
An illustration of the pictographic and iconic representations of arrangements of 3 toy bears on a sofa. The top layer consists of 3 bear diagrams in 3 different colors. The bottom layer consists of 3 abstract and colored designs arranged in a row.

Example of pictographic (uppermost) and iconic (bottommost) representations

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Connecting these two representations to abstraction, the use of pictographic representations, as in drawing bears, implies a semi-concrete level, while using iconic representations such as lines or dots implies a semi-abstract level, which is then considered more abstract than the semi-concrete level (Heddens, 1986).

Also older students often start by copying the picture of the items to be combined when working with combinatorics. These representations gradualy become more systematic and refined throughout the solving process (Rønning, 2022). Listing items systematically is one difficulty for young students when working on combinatorial tasks (English, 2005). English (1991) found three approaches used by young students when working with combinatorics: the random stage, the transitional stage and the odometer stage. The random stage entails trial and error, where checking becomes important to avoid duplicates. At the transitional stage, students start to adopt patterns in their documentations, but the pattern is not kept all through the task. Instead, the students revert to the trial-and-error approach. At the odometer stage, the students use an organized structure for selecting combinations throughout the whole solving process. One item is held constant while the others are varied systematically.

In the study focused on here, these stages were combined in the analysis with the degree to which students produced duplicates. In earlier interventions with this task, the students showed a preference for iconic over pictographic representations. Only a few students (4 out of 114) managed to find all permutations (Palmér & van Bommel, 2018). The pictographic representations seem to imply more systematization and fewer duplications than the iconic representations (van Bommel & Palmér, 2021). A new consideration was formulated regarding students’ rationales for choosing specific representations.

To summarize, based on above the theoretical framing in this study consists of the two dimensions of representations and systematisations. When analysing representations, Hughes’ (1986) notions pictographic and iconic representation are used (Fig. 1). As some children use both pictographic and iconic representations this gives thee possible outcomes. When analyzing systematization, English’s (1991) notions random stage, transitional stage and odometer stage are used. These stages are in turn divided into two outcomes based on whether or not the students produce duplications (random and transitional stage) or to what degree all solutions were found (for odometer). Finally, the three outcomes of representations are connected with six outcomes of systematization (see Table 1).

Table 1 Categorization of documentations based on representation and systematization
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Method

As mentioned, the problem-solving task on combinatorics is from a longitudinal intervention investigating the potential in using problem solving as the start for the mathematics education of 6-year-olds. This intervention is conducted through educational design research (Anderson & Shattuck, 2012). The intervention has been ongoing for several years, involving more than 40 Swedish preschool classes in different design cycles with different foci. The empirical material in this paper is from one small pilot design cycle within this intervention focusing on the students’ rationales for the representations they use when working on the combinatorial tasks.

Selection of Preschool Classes

Five classes were selected for this design cycle based on their teachers’ interest in participating. The teachers working in these preschool classes are educated as preschool teachers, which implies that they have completed a 3-year university course in preschool teacher education. The teachers have participated in several of the previous design cycles and hence were familiar with the aim of the study and the problem-solving task on combinatorics. As preschool class is only 1 year of education, there are new students in the classes each year, so the same lessons can be re-used each year. Thus, the teachers had implemented the problem-solving task on combinatorics several times before with other students. The students were familiar with problem solving but not with combinatorics. In line with the ethical rules described by the Swedish Research Council (2017), the students’ guardians were given written information about the study and approved their children’s participation. Altogether, 48 students from these five preschool classes got approval and thus participated in this design cycle.

The Problem-Solving Lesson

When introducing the problem-solving task, the students were shown three small plastic bears in three different colours. After the introduction, the students worked individually. They were given white paper and pencils in different colours but no instructions regarding what or how to document on the paper. After working individually, the students were divided into pairs to compare and discuss their documentations. However, they were not allowed to change anything on their documentations. Finally, all students were gathered for a joint discussion based on their documentations.

Interviews and Analysis

To explore the students’ choice of representation, a short interview was conducted after the problem-solving lesson. The interview guide was developed by the researchers and communicated to the teachers. The teachers have been taking an active role in the larger education design research study and, as in previous design cycles, it was they who conducted the interviews (Palmér & van Bommel, 2021). The researchers instructed the teachers, in writing and verbally, on how to carry out the interviews and how to take notes. In the interviews, the teachers showed the students their documentation from working on the task and asked them, Why did you choose to draw bears/dots/lines/.. when solving the task? The teachers then documented the students’ answers.

The researchers analysed the students’ answers to the interview questions together with their documentation from the lesson. Based on the limited selection of students, this analysis was qualitative, explorative and mainly at a group level (see Cohen et al., 2018). First, the students’ documentation was categorized deductively, identifying whether a pictographic and/or an iconic representation was used, and also identifying the stage of systematization used and the number of solutions: random with or without duplicates, transition with or without duplicates, and odometer with all solutions or not (see English, 1991, 1996). After that, the students’ answers to the question on their choice of representation were categorized inductively, based on the representation used. Thus, answers from students using pictographic and iconic representations respectively were analysed collectively, and during the analysis we looked for patterns.

Results

When working on the task, all students used pictographic and/or iconic representations (Table 1).

When working on the combinatorial task, the students need to pay attention to each object as well as to the relation between the objects; therefore both pictographic and iconic representations are well suited. The students showed a preference for pictographic representations. As shown in Table 1, 27 students used a pictographic representation (Fig. 2 uppermost), 14 students used an iconic representation (Fig. 2 bottommost), and seven students started with a pictographic representation but switched to an iconic representation while solving the task (Fig. 2 middle).

Fig. 2
An illustration of the representation of plastic toys using pictographic and iconographic methods. A majority of the students prefer the pictographic method, followed by iconographic representation by another group of students. Some students begin with pictographic technique and switch to the iconographic technique. Some use dots and line to represent the bears.

Examples of pictographic and iconic representations in students’ documentations

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As in previous cycles (Palmér & van Bommel, 2017, 2018; van Bommel & Palmér, 2021), few of the 27 students using a pictographic representation made duplications. Of the 27 students using a pictographic representation, 17 used trial and error when working on the task, but only one produced duplicates. The students using an iconic representation were fewer (14) and more distributed in terms of systematization stage when documenting their solutions. The use of iconic representations implies a semi-abstract level, more abstract than the use of pictographic representations (Heddens, 1986). Also as in previous studies (Palmér & van Bommel, 2017, 2018; van Bommel & Palmér, 2021), a larger proportion of the students who used iconic representations made duplications. Further, three students using iconic representations found all permutations. We do want to point out that this study includes few participants, however, these results are well in line with previous studies (Palmér & van Bommel, 2017, 2018; van Bommel & Palmér, 2021).

In the interview, the students who used a pictographic representation gave very homogeneous answers to the question of choice of representation. Their answers indicated that the rationale for drawing bears was that the task was about bears. These students answered, for example, Because it is bears and You showed us bears. Thus, for these students the choice of representation seems to have been quite obvious: you document what the task is about, capturing the context of the task.

For students who used iconic representations, the rationales for their choice of representation can be divided into two themes. Some of these students answered that they drew lines or dots because they were not able to draw bears; it was too difficult. Thus they put forward a technical reason to choose an iconic representation: for example, I chose to do dots as it was so hard to draw bears.

Other students answered that they drew lines or dots simply because it was easier or faster than drawing bears. Here we see a rationale indicating an understanding that icons (dots, lines) can represent bears. The students expressed that to solve the task, the bears per se are not important; for example, You can draw circles instead of bears or It works just as well with squares.

The seven students that used both pictographic and iconic representation showed various degrees of systematization, and there were no clear patterns between their solutions and their answers to the interview question. However, when we focused on why they changed representation, a pattern occurred. Most often, these students explained their choice of representation as changing to an easier or faster representation, for example, I started with the bears but it was faster to draw lines. Compared to the themes above, the technical issue of not being able to draw bears obviously does not apply as they started by drawing bears. Here they express the insight that icons can replace the pictographic representation, although they did not articulate it as specifically as the students who used only iconic representations did. The rationale was more a practical one: It was faster to draw lines.

Discussion

The starting point for this small-scale pilot study was previous results of a problem-solving task on combinatorics where iconic representations did not generate a higher level of correct solutions than pictographic representations, which may be considered somewhat surprising (Palmér & van Bommel, 2017, 2018; van Bommel & Palmér, 2021). The sample in this study is too small to allow for generalizations, but based on the students’ answers in the interview, we will make some refined hypotheses to be further investigated.

The division of systematization stages within the different forms of representation in this study is similar to that in previous cycles (Palmér & van Bommel, 2017, 2018; van Bommel & Palmér, 2021). In the interview, the students who used a pictographic representation (drawing bears) expressed that the context of the task led their choice and answered that they drew bears because the task was about bears. These answers indicate that the students chose a representation on a level of abstraction that is suitable for them, in this case a representation on the semi-concrete level, bridging the gap between the concrete and the abstract level (Heddens, 1986). When documenting their solutions with a representation at this level of abstraction, these students make very few duplications regardless of stage of systematization (i.e., random, transition, odometer; see English, 1991, 1996).

The 14 students who used iconic representation were more distributed in terms of systematization when documenting their solutions. In the interviews, some of these students expressed an understanding that the bears per se were not important to solving the task. Others gave answers indicating a technical reason for choosing this type of representation – they wanted to draw bears but resorted to drawing lines or dots because drawing bears was not possible for them. This latter group of students may have been induced to work with a representation at a level of abstraction that was not suitable for them. Thus, the context of the task, which was intended to make the task meaningful for the students, may have instead hindered some of them from using a representation at an appropriate level of abstraction. This may explain why students who used iconic representation are more distributed in terms of systematization when documenting their solutions.

As English (2005) suggested, combinatorics can be used with very young students. English points out that a proper context will make it possible to work with combinatorics and finding permutations. Our research points out that it is the context that influences the choice of representation, which in turn can both hinder and support students in completing the problem-solving task. Adjusting the context with regard to complexity to draw this context will give us a better insight into whether or not students’ choice of representation reflects their level of abstraction.

Some of the students expressed that they chose a pictographic representation, drawing bears, because that is what the task was about. This might indicate a socio-mathematical norm that the students are consciously or unconsciously aware of. In our study, it is the students’ own teachers who conducted the interview, and these socio-mathematical norms might be difficult to challenge or contest. Our collaboration with the teachers has been ongoing (Palmér & van Bommel, 2021), and the teachers are well aware of what we want to capture and what they are expected to do. It is important to realize that we cannot be sure that students’ responses to the interview questions would be the same if the interviews had been conducted by the researchers instead of the teachers.

We have mentioned that the small number of participants in this study does not allow us to make generalizations. However, it is important to consider that this small-scale pilot study was not about reaching final conclusions but about seeing in what direction we could further develop our interventions and give directions for further design cycles (see Anderson & Shattuck, 2012). If we focus on the different levels of abstraction, we could work on the concrete level and let students work with actual toy bears, or we could omit any representation and ask students to merely colour (e.g., In how many ways can a predesigned flag be painted using three different colours?). Both options would give us an opportunity to focus on whether or not the students are able to find all permutations and whether or not duplicates are created. However, our results show that the students’ level of abstraction and their representations might not be compatible, and thus it would be interesting to investigate further. With a focus on representations, we could develop the intervention by letting students work on a similar problem-solving task on combinatorics but where the object is easier to draw, such as pens or buttons (e.g., In how many ways can three pens/buttons of different colours be arranged in a row?). At an earlier stage of the intervention, we already designed an application where the students could work with pictographic representations (van Bommel & Palmér, 2021), prolonging the semi-concrete phase and giving the students an opportunity to internalize the problem before starting to document it. A decline in duplicates in students’ documentation was observable, but the pictographic and iconic representations still showed differences in stages of systematization. Thus, a focus on objects that are simpler to draw could be an option and would give us a better insight into whether or not the form of representation chosen by the students is in line with and supports their own level of abstraction.