Implementing a structural approach in preschool number activities. Principles of an intervention program reflected in learning

ABSTRACT

We report here on an intervention implementing a structural approach to arithmetic problem-solving in relation to learning outcomes among preschoolers. Using the fundamental principles of the variation theory of learning for developing the intervention and as an analytical framework, we discuss teaching and learning in commensurable terms. The research question is how teaching grounded on a structural approach and designed based on principles of variation theory is reflected in children’s learning of numbers. To answer this, three analyses were conducted, addressing: i) how the children’s ways of experiencing numbers changed after participating in the intervention, ii) how the theoretical ideas were afforded in the intervention program, and iii) synthesizing how the affordance was associated with the children’s arithmetic learning. One group of eight children participating in the intervention program was chosen for thorough analysis. Progression was observed in how the children changed their ways of experiencing numbers during the intervention that allowed them to enact more advanced arithmetic strategies, which was associated with the structural approach in teaching. The results also show how analysis focusing on aspects discerned in learning and aspects afforded in teaching provides a way of describing arithmetic learning with significant implications for teaching practices.

KEYWORDS:

• Arithmetic skills
• early childhood education
• intervention
• variation theory of learning

Background and aim

Based on earlier studies and theoretical conclusions that understanding numbers as part-whole relations is important for mathematical development in general and the flexible use of strategies in arithmetic problem solving in particular (Baroody & Purpura, 2017; Cheng, 2012; Fuson, 1992; Mulligan et al., 2013; Schmittau, 2003), we conducted an intervention study to thoroughly investigate enacted teaching with a structural approach to arithmetic problem solving among preschoolers. The structural approach in teaching basic arithmetic is in our program based on Davydov’s (1982) ideas of representing numbers as relations of parts and whole, by making connections and using bridging actions to illustrate such relations (Schmittau, 2004). This teaching approach further emphasizes composition and decomposition of numbers as the outset for solving addition and subtraction tasks, rather than operating on single units by counting up and down on the number line (see also Zhou & Peverly, 2005 for a Chinese perspective of a structural approach in teaching). As expected, the intervention was successful: the children learned to solve arithmetic tasks to a large extent compared with a control group (Kullberg et al., 2020), and did so by attending to the part-whole relations in the tasks and handle additive relations in more advanced ways (Kullberg & Björklund, 2019). The specific aim of this paper, however, is to understand how the children’s learning reflects the principles implemented in the intervention; thus, what was taught and what was learned.

Taking a variation theoretical perspective (Marton, 2015), we consider learning as a change in one’s way of experiencing the world around us, and furthermore that we act in accordance with how we experience this. Hence, powerful ways of acting stem from powerful ways of experiencing. A certain way of experiencing is a function of what aspects that come into the fore of attention thus, are discerned. Teaching have to, in this sense, bring fore and make such connections that the learner has not earlier been able to discern, in our case numbers’ part-whole relations. What we aimed for in the intervention was thereby that the children would change their ways of experiencing numbers in that they would be able to attend to the part-whole relations in arithmetic tasks rather than using an approach based on single-unit counting. Our research question is: How is the teaching grounded on a structural approach and designed based on principles of variation theory, reflected in the children’s learning of numbers? To answer this question, we conducted three analyses: firstly, of how the children’s ways of experiencing numbers changed after participating in the intervention; secondly, of how the theoretical ideas were afforded in the intervention program; and thirdly, synthesizing how the affordance was associated with the children’s learning of arithmetic skills.

Principles framing the intervention

The intervention program is grounded on i) the variation theory of learning (Marton, 2015) and ii) theoretical principles of number learning from a structural perspective (e.g., Davydov, 1982; Neuman, 1987; Schmittau, 2003, 2004). According to variation theory, how something is experienced depends on the aspects that come to the fore of our attention and are discerned. Hence, discerning and differentiating aspects of an object of learning is what constitutes a child’s way of experiencing his/her surrounding world. Consequently, seeing learning as a change in ways of experiencing is a fundamental principle in variation theory (Marton, 2015; Marton & Booth, 1997). Originating from the phenomenographic tradition (Marton, 1981), learning something means to discern aspects of what is to be learned (the object of learning) that have not previously been discerned, thereby perceiving the world in a more differentiated way (e.g., Gibson & Gibson, 1955). However, in order to discern the necessary aspects of an object of learning, the child has to experience a difference within or between these aspects. An aspect of a mathematical task can be opened up as a dimension of variation in which commensurable features may vary in the learners’ awareness. When an aspect not previously discerned is presented as a dimension of variation, his/her ways of experiencing this object of learning may be changed. These principles – discernment and variation – can be applied when designing tasks and activities to draw learners’ attention to essential relations and make them aware of, for instance, number relations. Subsequently, based on these principles it is possible to analyze what is afforded to learn as well as how aspects are brought to the foreground in a certain situation through different patterns of variation and invariance, and thus certain aspects are made possible to discern. Affordance does, however, presuppose that the learner is enabled to perceive the afforded as an affordance; that is, whether an artifact presents a mathematical aspect in a way that is possible for the learner to perceive. Here, the principles of how patterns of variation are enacted come to the fore, since the way a concept is presented also influences what the learner discerns and thus what the presentation affords.

In this way, learning and teaching are described in commensurable terms: as afforded dimensions of variation (the enacted object of learning), and as experienced dimensions of variation (lived object of learning). We suggest that these theoretical notions offer a coherent foundation that is possible to apply to data in order to relate what is taught and learned. Through a very close analysis of the enacted object of learning in terms of how the planned aspects were made possible to discern by means of variation, together with analyzing learners’ ways of experiencing (described in terms of discerned aspects), we can study how our intervention program is reflected in the development of children’s learning.

The other theoretical principle underlying the intervention comes from research on a structural approach to teaching arithmetic to young children that emphasizes children learning number relations as the outset for arithmetic problem solving rather than learning strategies based on counting single units. The skill to compose sets builds on children’s ability to coordinate the cardinality and ordinality meaning of numbers (e.g., Davydov, 1982; Fuson, 1988), that is seeing five as constituting a collection of five single ones, and knowing that adding, for instance, one object will be represented with the consecutive number word. This is considered essential for developing understanding of numbers and how to handle numbers in a powerful way in problem-solving. Sets can be composed by counting single units to make the whole but also by experiencing a pattern of composed units, for instance, by subitizing (Clements, 1999; Wynn, 1998). Consequently, experiencing numbers as composed sets opens up for comparing exact quantities and their relation. Number relations, considered critical for further advancement, go beyond counting skills and refer to the ability to represent and handle quantities in multiple, flexible ways, for example, thinking about the number 5 as 1 and 4 as well as 2 more than 3. The point of departure in our reported intervention is that learning about number relations in terms of their part-whole relationship extends one’s ability to work with numbers. This means that the child attends to the parts and the whole simultaneously, as composed sets, and is able to work with the triad of numbers found in any arithmetic task, for instance, as 3/5/8, which enables the child to solve tasks such as 3+_ = 8, but also see the connection to 3 + 5 = _, _+5 = 8, 8–3 = _ and 8-_ = 5 (Baroody & Purpura, 2017).

We presume that knowledge of number relations may develop through teaching with a structural approach. A structural approach in teaching is directed at developing children’s awareness of units and ability to reason about parts and the whole simultaneously (Davydov, 1982; Schmittau, 2003, 2004). This is done by making connections between and within examples possible to discern for the child through bridging actions and illustrative representations. Representations are seen as essential as they support the learner in discerning relations, for instance, with finger patterns (Brissiaud, 1992; Neuman, 1987) or artifacts that visualize units constituting composite sets.

Teaching in accordance with a structural approach aims at children developing an understanding of number relations and an ability to analyze and discern local relationships, which will eventually allow the child to identify more general mathematical relations and properties (Venkat et al., 2019). As opposed to such an approach is a counting-based way of teaching basic arithmetic, which emphasizes counting single units as primary strategy for solving arithmetic tasks. However, strategies such as counting single units do not support the child in experiencing the relation between and within numbers (Gray & Tall, 1994). Children who get stuck in this way of handling arithmetic tasks, as added ones, are found to be limited in their numerical development (Baroody & Gannon, 1984; Cheng, 2012; Neuman, 1987). Persisting single unit counting is found to be associated with learning difficulties among older learners and obstacles for solving multiplication and division problems efficiently (Ellemor-Collins & Wright, 2009; Schollar, 2015). Such learning problems have been traced back to pedagogy of concrete counting methods in early grades (Ensor et al., 2009). This is why we assert that it is desirable that children learn about number relations as the outset for arithmetic problem-solving in the early years.

Intervention programs attending to part-whole relations

Intervention studies in early childhood education have shown that part-whole relations are possible to teach to young children. For instance, Jung et al. (2013) showed that the part-whole relations of numbers can be taught to 4–5-year-olds through certain teaching activities. However, they conclude that some abilities related to number relations (subitizing and more–less comparison) seemed to be essential for the success. Others, take part-whole relations as the outset for developing arithmetic strategies, e.g., Fischer (1990), teaching the relation between small sets and subsets, and Cheng (2012), teaching decomposition strategies based in categorizing sets and subsets. Effective interventions that promote children’s knowledge of number relations thus seem to build on preexisting knowledge that allows the children to apply arithmetic strategies in successful ways.

Furthermore, we find support from Mulligan and her colleagues’ intervention studies (e.g., Mulligan & Mitchelmore, 2009; Papic et al., 2011) that teaching should address how the child is enabled to discern number relations. Results from their interventions of implementing attention to pattern and structure in preschool and primary grades indicate that children showing a higher awareness of relationships more easily learn about properties of numbers. A structural approach in teaching thus seems to help the child to detect patterns and relations within and between numbers, which is advantageous in arithmetic problem-solving.

Studying outcomes of interventions on a theoretical grounding

There is no simple cause–effect between teaching and learning. Still, there is a need to know how teachers’ actions affect student learning (Nuthall, 2004) and what elements of an intervention are reflected in students’ learning. Intervention programs in early childhood (in terms of implementing a certain pedagogical idea or way of teaching that is new to the participants) usually show growth in children’s knowledge and skills (Mononen et al., 2014). It is even shown that all children benefit from participating in a heightened attention to mathematics content, regardless of the curriculum of choice (Baroody et al., 2009). This might indicate that any heightened attention to mathematical content is beneficial to a certain extent, but it is rarely investigated which specific teaching principles that make the difference.

The majority of interventions in preschool use a battery of activities that are to be conducted systematically and frequently for a certain period of time by the preschool teachers (e.g., Clements & Sarama, 2011). Some programs have a basis in theoretical conjectures as to how learning arithmetic would be facilitated, e.g., Cheng (2012) drawing on Piagetian classification abilities as fundamental, and Mulligan et al. (2013) building on a generative structural awareness. Mostly, intervention outcomes are assessed through standardized tests that include general mathematical abilities and arithmetic skills in particular. Most often, these show children managing to solve more advanced arithmetic tasks after the intervention. However, qualitative analyses of how children understand some of the objects of interest in the interventions are also used, which broadens the picture of what an intervention actually develops among the children. Mulligan et al. (2013) is one example in which the outcomes are tested based on children’s ways of expressing structural awareness in drawings, rather than the frequency of correct answers alone. We also suggest an alternative approach to measuring outcomes from an intervention. Drawing on the variation theory of learning (Marton, 2015), we take an experiential approach to learning and to how powerful teaching is to be conducted in terms of patterns of variation. The theory states that some ways of experiencing an object (e.g., a task or a situation) include discerning more and necessary aspects. This indicates a qualitative analysis of different ways of experiencing this object (Marton, 2015) as a contrast to the frequency of producing correct answers.

The importance of monitoring the implementation of a new activity or program, and of observing children’s learning outcomes in relation to the implementation (Durlak, 2010), is well established in the literature. Still, there is a limited number of such studies to date that direct attention toward both the plan itself and its theoretical foundation, as well as the way the plan is carried out. The lack of such studies may be due to the complex task of studying a relation between what is taught and what is learned. Variation theory proposes that this is an epistemological question, and that looking at children’s learning as an experiential process or rather as a function of what is possible for a child to discern in a given situation, could open for thoroughly studying what in an intervention theoretically and empirically influences learning outcomes. We adhere to the significance of studying not only whether an intervention works but why it works. This means that we do not merely analyze the outcome(s) of the intervention but also monitor and analyze the affordance of the implementation itself, as suggested by Durlak (2010).

The intervention

The intervention program aimed at developing children’s ways of experiencing numbers, in that they will be able to attend to numbers’ part-whole relations in arithmetic situations. The program focused on a limited content for learning (the relations among the first ten natural numbers) and took its departure point in how children experience numbers rather than their strategies or skills for solving arithmetic tasks. Individual task-based assessments were conducted before, immediately after, and one year after the intervention (see Appendix A) to assess both the children’s pre-knowledge when designing the intervention program activities and their learning outcomes. We chose a very specific object of learning (experiencing numbers’ part-whole relations) and adhered to the children’s individual experiences. Our conjecture is that, in a problem-solving situation, a child acts in accordance with how s/he experiences the task at hand. The intervention thereby focused on extending the child’s way of experiencing numbers. The child who participates in activities in which certain aspects of numbers are made possible to discern will presumably change his/her way of seeing numbers, and thus also what is possible to do with numbers in arithmetic tasks.

The preschool context

The intervention was implemented by nine preschool teachers at five Swedish preschools in a suburban area outside a larger Swedish city. All teachers and children were Swedish speaking, some with another home language. A total of 65 five-year-olds participated in the intervention.

Swedish preschool is a voluntary pedagogical practice with no knowledge goals to achieve, but follow a national curriculum that is emphasizing children’s interests, play, and care. The attendance rate is 95% of all 4–5-yearolds (National Agency for Education, 2019). There are traditionally no scheduled lessons, instead teaching is to be conducted based on play and children’s own initiated activities and thematic work (National Agency for Education, 2018). Given this, the implementation of theoretically driven designed activities was a new way of conducting teaching for the participating preschool teachers. Only four main activities were used, but these were thoroughly evaluated and further developed during the intervention period to enact the theoretical principles and the individual children’s need to discern critical aspects. The process of implementing the program lasted eight months, during the children’s last preschool year (as 5-year-olds). One important feature of the program was that each activity could be assessed and developed in close collaboration between the researchers and the teachers, in relation to the children’s responses to the activities. The program was therefore conducted as an iterative process of teacher-researcher collaboration and development of the four designed tasks. Meetings were held every two weeks (in total 12 meetings) to evaluate the enacted activities and further develop and plan forthcoming ones, based on the teachers’ video documentation of the conducted activities. The documentation from the teachers’ enacted activities with the children was an important asset for developing and fine-tuning the program.

Intervention activities

The activities implemented in the program are designed to facilitate children’s discernment of number relations. These are conducted in an order that will allow the children to discern basic aspects before introducing aspects that induce more advanced ways of experiencing (and thus making use of) numbers. The main aspects foregrounded in the intervention activities were: representations of numbers, finger patterns representing number sets, numbers as composite sets, part-whole relations of numbers, and the commutative principle.

From the pre-assessment, we identified children not experiencing numbers as composite sets but rather as static objects, usually illustrated by a fixed finger pattern (Björklund & Reis, 2020). We thereby found it necessary to start the intervention focusing a basic aspect that would enable the children to decompose numbers into parts and to represent these with their fingers. We here brought fore the aspect representations of numbers by offering children to discern that numbers can be represented in different modes, such as dots on a die or with fingers. Other ways of representing numbers were later offered also by using bead-strings in combination with fingers. Finger patterns as representations of number sets was an aspect introduced to facilitate instant recognition of a composition of unfolded fingers representing a number. Finger patterns were used throughout the intervention program in all of the activities. The aspect numbers as composite sets was to be discerned, as children were encouraged to identify and create sets without counting single units, for instance, when emphasizing patterns such as a whole hand representing five or through the composition of similar-colored beads on a string in groups of five. To enable the children to experience the aspect part-whole relations of numbers, we further used finger patterns to direct attention toward how one set can be composed by and still constitute two smaller sets. By working with numbers’ part-whole relations represented as finger patterns, the aspect commutativity was also brought fore in that the parts and the whole were simultaneously presented in the tasks. All of these aspects were brought for in the teaching through enacted patterns of variation and invariance (see Table 1 for an overview of the activities and aspects to be discerned).

Table 1. Descriptions of the four main activities in the intervention program¹ and aspects intended to be experienced through patterns of variation and invariance

Activity 1	The Statement game^2 The children decide what number they think the die will show before throwing it. When they have agreed on a number, they show this number with their fingers using both hands. The teacher encourage them to show the number in different ways. Aspects to be made explicit in the enacted teaching:  – representations of numbers  – finger patterns representing number sets  – numbers as composite sets
Activity 2	The Snake Game A string with ten beads grouped as five in one color and five in another color. The teachers encourage the children to represent the whole number with their fingers. Then, some beads are hidden and the children show with finger patterns the number of beads they see on the string. By looking at their unfolded/folded fingers, they are to “see” the missing part (number of hidden beads). Aspects to be made explicit in the enacted teaching:  – representations of numbers  – finger patterns representing number sets  – numbers as composite sets  – part-whole relations of numbers  – the commutative principle
Activity 3	Finger Patterns The children identify finger patterns shown by the teacher with a number word. The “undivided five” is emphasized as only numbers >5 are chosen. The teacher asks how many fingers to add/take down to make a different finger pattern (>5), and then follows the children’s suggestions. The teacher reverses the task and asks how many are to be added/taken down to give the first number again. Aspects to be made explicit in the enacted teaching:  – representations of numbers  – finger patterns representing number sets  – numbers as composite sets  – part-whole relations of numbers
Activity 4	Context Problems Short number stories of different types within the number range 1–10 are presented. The children model the problems on their fingers. A systematic order of the presented problems emphasizes the number relations. Aspects to be made explicit in the enacted teaching:  – finger patterns representing number sets  – numbers as composite sets  – part-whole relations of number  – the commutative principle

¹No graphical numerals are used in the activities.

²The activity is inspired by the French research group Sensevy et al. (2015).

As can be seen in Table 1, the same aspects were intended to be experienced within the enactment of different activities. For instance, finger patterns representing number sets and numbers as composite sets were foregrounded in all the activities, but in the Statement game, the emphasis was foremost on partitioning numbers into two composite sets and representing them as a finger pattern, while in the Snake game composite sets in combination with fingers representing numbers were a prerequisite for experiencing the part-whole relations of numbers, whereas the commutative principle, for instance, was afforded only in the Snake game and the Context problems.

Sample

For the purpose of this paper; to study how children’s learning is reflecting principles grounding the intervention, we selected one group (hereafter called the target group), who had started from a low mean score, or in other words, the majority answered none or only one arithmetic task (in the number range 1–10) correctly. In the post-assessment (immediately after the intervention) the target group showed a significant change in the number of produced correct answers (mean score 6.37, SD 1.60, where 1 point was scored for each correct answer of the 8 tasks). The third follow-up assessment (one year later) confirmed these changes and showed a further increase in developed arithmetic skills.¹ Since our interest is in drawing attention to qualitative differences in ways of experiencing numbers and how such potential changes are afforded by the intervention, we selected this target group’s experienced teaching and learning outcome (N = 8) for more thorough investigation.

Data and analysis

To analyze the changes in ways of experiencing numbers on a group level, we included the number of observations of all eight children and eight arithmetic tasks/items from the pre- and post-assessments (total number of possible observations from each assessment = 64). All assessments were conducted individually and were video-recorded with the children’s legal guardians’ informed consent. The tasks were given orally by one of the researchers. No manipulatives other than seven marbles (Tasks A and B, see Appendix A) and children’s fingers were used. The primary focus in the assessment was on the children’s ways of encountering the tasks, to allow us to interpret how their actions reflected their ways of experiencing the numbers given in the tasks. Follow-up questions supported the children in reasoning, explaining, and using their fingers to express how they figured out their way to solve the tasks.

Data from the intervention’s teaching activities consist of teachers’ video documentation of their conducting the designed activities with the children. In the target group, 23 activities were video-recorded (total 210 minutes).

To answer our research question – How is teaching grounded on a structural approach and designed by principles of variation theory reflected in children’s learning of numbers? – we conducted the following three analyses:

1. how the children’s ways of experiencing numbers changed after participating in the intervention,

2. what dimensions of variation the teaching in the intervention afforded the children to discern, and

3. how the discerned aspects were connected to the afforded aspects.

Analysis (i) changes in children’s ways of experiencing numbers

Focusing on the children’s learning progression in terms of changes in their ways of experiencing numbers, data from the individual assessments before and after the intervention were coded based on expressions in words and actions observed when the children were trying to solve the arithmetic tasks. Every observation (N = 64) was coded by three researchers, individually and collectively, and disagreements were solved in collective analysis sessions with the whole research group. The coding was done on a micro-level, attending to both verbal answers and explanations and particularly the way the children used their fingers or made subtle gestures such as nodding their head or systematic eye-movements. Showing a finger pattern instantly or producing a pattern by raising one finger at a time, were then interpreted as expressions of different discerned aspects of numbers.

When children were observed differentiating several necessary aspects (of which, for instance, the coordination of cardinal and ordinal aspects was essential along with the part-whole relation aspect), this induced a more advanced way of experiencing numbers, while the ways of experiencing numbers that did not allow a child to operate on an arithmetic task in any successful manner were characterized by undiscerned aspects or uncoordinated aspects. Fall-out was coded when a child did not give an answer or was not given the task (due to two consecutive wrong or missing answers). The observations were categorized according to six different ways of experiencing numbers, also found in the project’s larger data set² (see also Björklund & Runesson Kempe, 2019, for more details about the categorization framework):

Experiencing numbers as words

This category includes instances when a child repeats one of the number words appearing in the task but without explanations of any relational meaning or says random numbers either solely as an answer to a task or a sequence of words (not in an orderly fashion). If the task is to find out how seven beads have been divided and hidden in two hands, the child might respond: ”Five, eight, nine [taps her finger at the right hand of the interviewer, once for each number word], ten [points at both hands], six [points at the right hand] and seven [points at the left hand]”. Thus, numbers are words used in certain contexts but there is nothing you can do with them to solve the task (e.g., Wynn, 1992).

Experiencing numbers as names

In earlier studies, experiencing numbers as names has been observed by Neuman (1987) and described as enacting a procedure of “word tagging” by Brissiaud (1992). The child regards number words as names given to a counted item, and the last uttered word then denotes only that single object. The task “you have ten candies and eat six of them, how many are left?” may then result in the child showing all the fingers on the left hand and the thumb on the right and saying: “Ate six. This is six [jiggles her right thumb]. And then I took the thumb away. So, it’s five.” When experiencing numbers as named objects, these cannot be added to or subtracted from anything, since they do not have a cardinal meaning.

Experiencing numbers as extent

This means having some sense of numbers’ cardinal value. The child gives plausible approximations, close to the correct answer. When figuring out how seven marbles are divided in two hidden hands, the child may reason: “One in that one [points on the closed right hand] five in that one [points at the closed left hand]. Or eight in that one [points at the left hand] and two in that one [points at the right hand].” Numbers are in this case not seen in an ordinal sense, which means that the child makes no attempt to count in order to determine the quantity. This also makes it impossible to add or subtract in other ways than giving an approximate estimation of more or less than the given numbers in the task.

Experiencing numbers as countables

This is strongly influenced by the ordinal aspect of numbers, as the child counts to determine a number, often using his/her fingers to count on and to keep track of counted units (Fuson, 1992). If the task is “You have two shells and your friend five. How many do you have together?” the child might respond: ”One, two [points at his little and ring finger] wait, I have to start over, one, two, three, four, five [pointing at fingers starting from long finger, index, thumb and thumb, index on the other hand. Then starts again from the little finger] one, two, three, four, five. Five!” Numbers are thus seen as added units of ones, which induces difficulties to see how numbers relate to other numbers or included in a composite set, not least when solving arithmetic tasks such as missing addends or missing start number (see also Steffe et al., 1982).

Experiencing numbers as structure

The parts and the whole in an arithmetic task are experienced as related to each other, and are thus possible to handle as a triad (see Baroody, 2016). The difference, compared to experiencing numbers as countables, is the way the child attends to the parts as composite sets that relate to each other and to the whole. The same task of adding two shells and five may then be answered by the child putting her left hand with all fingers unfolded on the table, then the right hand with thumb and index finger instantly unfolded, looking at her hand shortly, then saying “Seven!”. This opens up for a different way of handling arithmetic tasks, since the aspect of part-whole relations is foregrounded (see Venkat et al., 2019). However, the number relations are experienced as local relationships, since the children do not know them as automatized facts, but have to use some structuring act (by fingers or mentally) to experience the relation and then solve the task.

Experiencing numbers as known facts

This category reflects an advanced understanding of number relations (Askew & Brown, 2003). Either the number relations are known to the child and s/he does not need to calculate, or s/he has an advanced way of reasoning and using retrieved facts to find the answer. Experiencing numbers as known facts differs from the category of experiencing numbers as structures, in that the child does not have to compose the sets, for instance, on his/her fingers, to recognize the relations.

These categories are descriptions of differences in (un)differentiated and (un)coordinated necessary aspects of numbers. Different ways of experiencing numbers open up various possibilities to solve arithmetic tasks of different complexity (e.g., straightforward addition tasks can be solved by counting single units, while missing addend tasks require an awareness of part-whole relations), which entails a more powerful use of arithmetic strategies. The analysis regards the different ways of experiencing numbers found in this particular group of children at two occasions, and how these ways of experiencing numbers change from one category to another during the intervention.

Analysis ii) dimensions of variation afforded in the intervention

The four activities were presented to the target group at 44 occasions (see Table 2), with varying numbers of children participating each time (2–8 children). Each occasion lasted 2–16 minutes (mean duration 8.5 min).

Table 2. Summary of times the activities were carried out in the target group

Activities	Statement game	Snake game	Finger patterns	Context problems	Total
N	12	12	3	17	44

The summary of conducted activities shows that the Finger patterns activity was only implemented at three occasions, whereas the other three were enacted more often. The mean value of the eight children’s participation in the activities was 21 times (interval 15–24, median 23).

The video-documented observations comprise the data for analysis. This analysis focused on the enacted object of learning by describing how the structural approach through theoretical principles was implemented in the activities. This means that we scrutinized each observation concerning the enacted object of learning (how the teaching activities were conducted by the teachers). The enacted objects of learning were analyzed in terms of what dimensions of variation (DoV) were opened up, for example, by identifying which contrasting features of a specific aspect came through in the interaction and thus afforded the children to discern certain critical aspects of numbers. For instance, the Snake game was primarily designed to emphasize part-whole relations of numbers, but what was actually afforded through the acts and interactions of the teachers and children could vary, and thus became our object of analysis. For instance, the teacher might show the snake to the children and hide three beads in her hand. She then notices that one child shows a correct finger pattern but has counted the beads as single units. The teacher then acts in the following way:

Teacher:

“Remember there are five here [making a circle with her hand around the five red beads of the ten-snake] … you don’t have to count those … six and seven [points to the two white beads] just like your fingers …

Our way of analyzing DoV then implies that through the teacher offering the child an alternative to counting single beads, highlighting five as a benchmark and composite set, the child was afforded to discern the two ways of determining the number of the set and, in particular, how seeing numbers as composite sets provides a more efficient way of handling the task. Some activities were more likely to induce a certain DoV to be opened, while others were more restricted; nor did the teacher open up every possible DoV at every occasion. By identifying aspects opened as dimensions of variation, what the children were actually afforded to discern in the enactment of the activities were revealed.

Analysis iii) connecting discerned aspects to afforded aspects

In the third and synthesizing analysis, we looked for correspondence between which aspects were discerned and which were afforded. Our departure point was the identified changes in ways of experiencing numbers in the pre-assessment and post-assessment (Analysis i). The afforded aspects in the intervention activities (Analysis ii) were then considered to be related to the changes in the children’s ways of experiencing numbers. In accordance with variation theory, a change in the way of experiencing is constituted by aspects discerned, which is observable in the way children attend to numbers in solving arithmetic tasks. Specific interest is then directed toward those aspects that the children did not discern before the intervention but did discern after the intervention, and in what ways these aspects were afforded in the intervention. In this way, we were able to draw conclusions as to why the intervention worked and what in the intervention facilitated the development in arithmetic skills.

Results

Changes in children’s ways of experiencing numbers

In the first analysis, we investigated the learning progression on a group level. The conclusion is that the children did learn how to solve arithmetic tasks and produce correct answers within the number range 1–10; but more interestingly, the qualitative analysis showed substantially changed patterns in the children’s ways of experiencing numbers that consequently led them to enact powerful strategies. We categorized all observations of the target group from the pre- and post-assessments based on how they experienced numbers in eight arithmetic tasks. The frequency of observations in each category is shown in Table 3.

Table 3. Frequency of observations for each category of way of experiencing numbers

	Words	Names	Extent	Countables	Structure	Facts	Fallout	Total
Pre	0	8	36	7	6	1	6	64
Post	0	0	8	0	42	12	2	64

Table 3 shows that during the intervention period the number of observations changed from the majority being in the category experiencing numbers as extent (pre-assessment) to the majority being in the category experiencing numbers as structure (post-assessment). This means that in the later assessment the children were able to discern necessary aspects they had not discerned before. Thus, after the intervention numbers have a more differentiated and complex meaning, which afforded the children to attend to numbers in a different way, enacting other and more powerful strategies in solving the arithmetic tasks. In particular, the change from the extent to structure implies that most of the children have learned to discern the part-whole relations of numbers.

Seeing numbers as names or as extent are prominent ways of experiencing numbers in the pre-assessment, whereas no observations of numbers as words were found. When experiencing numbers as names, the child discerns the ordinality of numbers as their primary feature, which is often expressed through the child referring to the number sequence, suggesting some number that relates to a number given in the task but lacks relation to the given quantities (undiscerned aspect of cardinality and part-whole relations; see Figure 1). In the pre-assessment, we also found quite a great deal of observations of numbers as extent (36 of 64 = 56%). These are expressions of plausible answers that reflect a notion of quantity and the cardinal value of numbers, but the children have not found the answer by attending to number relations or retrieving the answer as a known fact since the answers are often incorrect even though the children might be very confident of their answer (see Figure 2). These ways of experiencing numbers, as names and as extent, do not enable the child to make use of any strategies for solving arithmetic tasks, since the child perceives numbers as either cardinal values or ordinal, but not simultaneously. The number of observations in these categories decreases significantly in the post-assessment, indicating that the children learn to discern these critical aspects as well as the part-whole relations of numbers simultaneously.

Figure 1. Excerpts showing the same child’s change from experiencing numbers as names in the pre-assessment to experiencing numbers as structure in the post-assessment

Figure 2. Excerpts showing the same child’s change from experiencing numbers as extent in the pre-assessment to experiencing numbers as structure in the post-assessment

We made seven observations of numbers as countables in the pre-assessment, clustered to two children with three observations each and a single instance by a third child. To them, numbers are single units that can be counted (countables), but since they are not seen as constituting composite sets, the child is unable to discern any part-whole relation as well. There were no observations of children experiencing numbers as countables after the intervention. This means that after the intervention, the children did not in the post-assessment attend to numbers as single units, for instance, in ”counting all” strategies to solve the tasks, either on their fingers or by counting number words.

In the pre-assessment, only six observations were made of the children expressing seeing numbers as structure and thus attending to their part-whole relations, in addition to one observation of numbers as known facts. A substantial change was observed in the post-assessment, in which 53 observations were made in the categories numbers as structure and numbers as known facts. A typical example of how children attend to the part-whole relations in a task is by representing the given numbers on their fingers and without counting, recognizing the number relations as finger patterns (see Figures 1 and 2).

Our main finding in this analysis is the changes in the children’s ways of experiencing numbers during the eight-month interval. An example of the changes found is illustrated in Figure 1. The task is the same in both the pre-assessment and the post-assessment: “You have ten candies and eat six of them. How many are left?” (10–6 = _).

The change from ways of experiencing numbers that would not afford the child to solve arithmetic tasks to a powerful way of handling numbers in arithmetic tasks is also illustrated in Figure 2, in which the same child solves the oral task “You have three glasses, but are going to set the table for eight people. How many more do you need?” (3+_ = 8). Even though her finger patterns seem similar in the two observations, her verbal expressions and explanations of how she finds the answer to the task show very different ways of experiencing numbers in the pre- and post-assessments.

Figure 2 illustrates how before the intervention the child seems to discern the cardinal aspect of numbers, creating a finger pattern that is related to a number given in the task but she makes no effort to figure out the relation between the given numbers (3 and 8) and the missing part. Our interpretation is that she does not attend to the given number 3 (the existing number of glasses); if eight people are supposed to have one glass each, then eight glasses are needed. In the post-assessment, however, she also discerns the part-whole relation between the sets she has created as finger patterns, clearly seeing the number relations, as she is able to discern how 3 and 5 are included in the whole 8. The number relations are not yet known facts since she has to create a structure with her fingers.

Of particular interest is the fact that since the number of observations of experiencing numbers as structure or known fact cover 53 of 62 (85%), the awareness of number relations (thus allowing the child to deal with part-whole relations in the task) is not task-related but rather applicable to a variety of arithmetic tasks. In the pre-assessment, only one observation was made of a child expressing numbers as known number facts. The post-assessment shows 12 observations of known number facts, which are expressions of children being able to deal with numbers as composite sets and in accordance with basic arithmetic principles.

We see a straightforward trend in the change in ways of experiencing numbers among the children, with a strong increase in the number of observations of advanced ways of experiencing numbers (structure and known facts) that enable the child to solve a variety of arithmetic tasks. No observations were made of children counting single units (experiencing numbers as countables) in the post-assessment, which means that they had developed ways of seeing numbers as composite sets and numbers’ relation to other numbers and parts within numbers. That is, the number of observations of numbers experienced as structure are in the majority in the post-assessment, at the same time as numbers being seen as known facts increases. This is partly an expected progression, as retrieving from known number facts is more proficient than operating on the numbers in a task by composing sets and relating them to the whole in a creating act, usually by structuring on one’s fingers.

Dimensions of variation afforded in the intervention

In the following, we present the enacted object of learning in terms of what dimensions of variation (DoV) were opened in the teachers’ enactment – in other words, what the children were afforded to experience regarding numbers through the enactments.

Representations of numbers

Given the importance of using different representations in learning numbers in early childhood mathematics, the intervention offered concrete representations in the enactment. In the Statement game, the children were afforded to experience that numbers can be represented by dots and fingers. This DoV was opened when the same number was to be represented through different representations: on the die (number of dots) and with their fingers. Similarly, in the Snake game, representations of numbers were foregrounded in the enactment and made it possible to discern that the same number and number relations could be represented by beads on a string and on one’s fingers. In the intervention, finger patterns were primarily promoted as means for representing number relations, which made this aspect – numbers can be represented – critical for how the children perceived the affordances in the other tasks.

Finger patterns representing numbers

All the activities afforded the children the idea that fingers are useful representations, not just for keeping track of counted numbers but as a powerful means of experiencing numbers as relations of parts and the whole (structured finger patterns). However, for modeling the part-whole relations on their fingers and seeing them as composite sets of units (rather than as single units alone), the children needed to discern that a specific number can be represented on one’s fingers as different finger patterns. Initially, many children expressed a static image of a finger pattern as being a number; thus it became critical for them to experience that fingers represent number sets and that these sets can be represented in different patterns, still constituting the same number. In Observation 1, the children are afforded to experience this critical aspect that numbers can be represented as different finger patterns. By pointing out several ways of representing the same number and thus opening up a DoV constituting the different finger patterns shown in the game, the meaning of finger patterns as representations of numbers became possible to discern. This idea was executed when the game directed the children to agree on which number they thought the die would show and represent this number with fingers on both their hands.

Table

Observation 1
1	Four children show 6 in different ways.
2	Teacher	How many do you have, Helen?
3	Helen	Six (shows 4/2: index and middle fingers on one hand and index, middle, ring, and little fingers on the other).
4	Teacher	And you, Harry?
5	Harry	Six (shows 3/3: thumbs, index and middle fingers on both hands).
6	Teacher	And you, Kate (shows 5/1: whole hand and middle finger) and you Annie (shows 4/2)? Did you show it in the same way?
7	Annie	Yes, Helen and I …
8	Helen	… and Kate
9	Teacher	Now … how many do you have? (points to Helen’s hand with four unfolded fingers) And how many does Kate have there (points to Kate’s hand with five fingers unfolded on one hand and middle finger on the other hand)?
10	Teacher	How many different ways do we have?
11	Annie	Two and four, like Helen and me. Harry had three plus three … and Kate had five and one (points to the friends’ finger patterns).
12	Teacher	How many ways?
13	Annie	Three.
14	Teacher	Three different ways of showing 6.

In Observation 1, all children represented 6 with their fingers. The activity enabled them to see that there are different ways of representing the same number on the fingers of two hands. The teacher compared the children’s ways of showing 6, asking: “same way?”, “different ways?” and “how many ways?” (lines 6, 10, 12). The teacher thus contrasted the children’s different finger patterns, and when they disagreed she compared the number of unfolded fingers (line 9). The different representations of numbers (yet in the same mode) simultaneously shown by the children participating in the game were highlighted by the teacher’s simultaneous focus on the number asked for. Thus the Statement game, when enacted in the way shown above, afforded the children to see that the same number can be represented with different finger patterns. Although fingers representing numbers were also present in the other activities, the Statement game was the first to open up this DoV as a primary goal.

Numbers as composite sets

To learn about number relations, one necessary aspect is to recognize numbers as composite sets. This was introduced in the Statement game when the attention was directed toward how the finger patterns were composed differently by the children but was further elaborated on in the following activities. This occurred when the teachers offered an alternative to counting in ones, to rather “see” the pattern of fingers or beads as composite sets, which can eventually be part of larger sets or partitioned into smaller sets (opening up the dimension of part-whole relations; see next section). By contrasting the procedure of counting single units with seeing collections of items as composite sets when the same number is to be determined, the children were afforded to discern two different features of how numbers are composed. In the Snake game, it was manifested how the children were constantly offered alternatives to counting single units. For instance, when they had identified the hidden part, 7, using finger patterns (five and two more fingers unfolded), they sometimes seemed unsure whether there were actually seven beads on the string, even if the beads were grouped together in five of each color. They started to count the beads one by one. The teacher then reacted: “Remember, there are five here (making a circle with her hand around the five red beads of the ten-snake) … you don’t have to count those … six and seven (points to the two white beads) just like your fingers (points to a child’s finger pattern)”. The teacher offered variation by directing the children’s attention to the undivided five, presenting an alternative to counting the beads as single units. Thereby, the DoV that enabled the children to discern numbers as composite sets was opened. The teacher’s enactment in Observation 2 will furthermore illustrate how this aspect was brought to the fore:

Table

Observation 2
1	Teacher	Ten elephants came to a cottage, where there were only eight beds. How many elephants didn’t get a bed to rest in?
2	Leon	(Hesitates, copies Annie’s incorrect finger pattern)
3	Teacher	How can you know? Have a look! If you have ten elephants (puts her ten fingers on the table) where are your fingers?
4	Leon	(Puts his ten fingers on the table)
5	Teacher	They’re going to sleep in the cottage, but there are only eight beds. How can you do it? Show me eight fingers!
6	Leon	(Shows eight unfolded fingers, a whole hand and three more fingers on the other hand)
7	Teacher	How do you know this is eight?
8	Leon	Mmm
9	Teacher	Five and three make eight (touches Leon’s whole hand and the other three fingers)
10	Leon	(Turns his right hand with the three unfolded fingers around, looks at his folded fingers) Two.
11	Teacher	Next time there were ten elephants again, but this time there were nine beds. How many didn’t get a bed?
12	Leon	(Counts nine fingers, one at a time, touching each finger) One.
13	Teacher	I saw you counted. How many fingers do you have on this hand? You counted, one, two, three, four, five. Do you have to count them (pointing to Leon’s fingers)? How many fingers on this hand? What’s the number after 5?
14	Leon	Six
15	Teacher	Yes, exactly. Then you can count, five (touching Leon’s whole hand), six, seven, eight, nine (touching each finger on the other hand, one at a time).

In Observation 2 the teacher noticed that Leon was unsure of how to represent the task on his fingers (lines 2–3). She emphasized the “eight beds” (the known part in the task) and asked him to show eight fingers (line 5). Having shown eight fingers (one hand and three fingers), she emphasized the “whole hand” constituting five, pointing at the finger pattern structure (5 + 3). Then, Leon counted his nine fingers one by one (line 13). The teacher noticed his single-unit counting and offered him an alternative, emphasizing the undivided five “this hand” (line 13) and then counting on the following numbers on the fingers of the other hand (line 15). Then, extending the set and asking him what comes after the 5, she offered him the possibility to use 5 as a benchmark for counting on from 6 up to 9. Furthermore, by directing attention to sets that the child can recognize (5 as the whole hand) and composing new sets, such as 5 and 3 making 8, a necessary DoV is opened up when the child sees how finger patterns constitute, and include, sets of different kinds. Altogether, these acts and actions afford the child to see numbers as composite sets – through the acts of determining the same sets of items in different ways; composing a number of two sets (8 of 5 and 3); or using one composite unit (5) and counting on single units (6, 7, 8, 9) – thus coordinating the ordinality and cardinality meaning of numbers.

In terms of DoV being opened up, the children were afforded to recognize numbers as composite sets without counting single units and to extend their subitizing range (conceptual subitizing) based on the undivided five. Extending the children’s conceptual subitizing range by recognizing finger patterns that include the undivided five also came through in the Finger pattern activity when the teacher showed a finger pattern including five (e.g., seven or eight), affording the children to discern the “five in the eight” and similar number relations. In the enactment, the five was kept invariant and the number of fingers added to five varied.

Part-whole relations of numbers

In the activities, the children are given several opportunities to discern that collections of items, constituting composite sets, can be seen as parts of a larger whole (see Observation 2, where five and three are determined to be the same as eight). However, the activities also afforded the children to discern how a number can be seen as constituting two parts, allowing for the decomposition of a whole into its parts and opening up for relating the sets as a part-whole relation. This, in turn, affords the child to see any given part and the whole as related, and the missing part can be identified as the other part within this relation. The finger patterns facilitated the children in discerning how composite sets formed a whole and how to “see” a missing part.

The Finger pattern activities thus served as a tool not only to open up for different ways of representing numbers, but also to structure number relations in a sensual way, which afforded the children to experience the relation between parts and the whole simultaneously. When finger patterns represent two sets of units (folded and unfolded fingers), the usefulness of structuring the part-whole relation on the children’s fingers emerged when they were asked to create one part and then to identify the whole and to “see” the hidden part. The Snake game was developed particularly to afford the children to “see” a hidden part (missing addend) by structuring the known numbers (one part and the whole) on their fingers. Some children then looked at their knuckles, while others turned their hand around to “see” the missing part (for more detailed descriptions, see Ekdahl, 2020). A proficient use of finger patterns thus enabled the children to create the structure in ways that allowed them to handle the tasks. Learning about number relations in these terms meant that the children experienced that parts within the same whole are related to each other and to that same whole. The same number can then be decomposed in different ways (as was intended in the Statement game) and, furthermore, the parts can be composed to that same whole again. When this part-whole relation is discerned, it affords the child to see that by knowing the whole and one part, the unknown part can also be identified. This was especially foregrounded in the Snake game and the Context problems.

In the enactment of the Snake game, however, one example of hiding one part of the whole snake is not enough to create the necessary contrasting features that would open up the DoV part-whole relations. Therefore, the teacher presented several examples of visible and hidden parts of the ten-snake (alternatively the five-snake), as shown in Observation 3.

Table

Observation 3
1	Teacher	How many (shows the whole snake) on both hands together?
2	Children	Ten (put two hands on the table).
3	Teacher	I’ve hidden some (covering six beads with her hand). How many do you see?
4	Annie	Five
5	Teacher	Look carefully!
6	Leon	Four (raises four fingers).
7	Teacher	Yes, show four fingers, all of you, and put the other hand beside them (puts her left hand on the table with four unfolded fingers). Tell me, how many are hidden?
8	Helen shows four folded and six unfolded fingers. The other children (Leon, Harry, Annie and Ivy) show four unfolded and six folded.
9	Annie	Six (looks at her whole left hand with fingers folded, and right thumb folded and index, middle, ring and little fingers unfolded).
10	Teacher	Do I have six beads under here? Are you sure?
11	Harry	Yes
12	Teacher	Let’s try another one (covers five white beads in her hand, and holds the bead string close to the children). Harry?
13	The children put one of their hands on the table with the fingers unfolded. Leon starts counting the visible beads.
14	Teacher	Both hands, how many beads are hidden? How many are under my hand?
15	Children	Five!
16	Teacher	How do you know?
17	Ivy	Because there are five blue and five white.
18	Teacher	So, you didn’t see any whites? (shows the whole bead string again)

In Observation 3, the teacher directs attention to the beads that are visible as composite sets, constituting parts of 4/6 and 5/5, after first determining that the whole is 10 (lines 1–2). She continues (not included in the excerpt) by offering other examples (3/7) of visible and hidden parts of the same whole number (10), thereby opening a DoV that affords the children to discern how parts are related to each other within the whole of ten. Thus, different number relations were explored in the Snake game activity, which made it possible for the children to discern that numbers constitute part-whole relations. The Snake game was limited to a fixed whole (five or ten). This enabled the children to specifically discern how the parts varied while still constituting the same whole, which was found to be essential, since experiencing the part-whole relations of numbers demands simultaneous attention to both the parts and the whole (lines 17–18). By only varying the parts, it was regarded that it was made possible to discern the aspect and the basic meaning of number relations.

However, to develop an understanding of the complex relations within numbers, it was found important to afford the children to handle any kind of part-whole relation rather than only a fixed five or ten. Therefore, in the Context problems, the whole could be any number within the number range 6–10. The focus on identifying the missing part in the problems formed the structure of part-whole relations of numbers. In the Context problem activity, the teacher presented a sequence of number stories in which, for instance, there were eight bears from the start and three ran away (8–3 = _), then they returned and the next time four ran away (8–4 = _), then five ran away (8–5 = _), and so on. In each number story, the total number of bears (whole) was the same while the number of bears running away differed. For each example, the children modeled the problem on their fingers. Through this variation – with several values of the number of bears running away (part) and the remaining number of bears (other part), and how these were related to the original number of bears (whole) – the DoV opened up afforded the children to discern different part-whole relations. Finger patterns afforded them a way to solve the Context problems by structuring the parts and the whole.

The commutativity principle

Operating on the relation between two parts in a specific part-whole relation can also open up for discerning a commutative pattern. This came through especially in the interaction concerning Context problems. In some stories, the structure of the story stimulated the learners to solve the problem by starting with the larger number (part) instead of following the story’s semantic structure. For instance, “Two bears were playing, then six more bears came. How many were there altogether? (2 + 6)” could be solved by starting with six and adding two (6 + 2). Sometimes the number stories were presented in a sequence, such as “Two bears were playing and six bears joined in”, followed by the next story, “Six were playing and two more joined in”. In Observation 4, the commutative principle was opened up as a DoV by presenting the same sum and same addends, but differing in the order of the addends in a sequence of two examples. Below, the teacher has just presented the story “Three elephants were out walking and five came along …”.

Table

Observation 4
1	Teacher	Three and five are eight together. If there are five elephants walking, and they meet three, how many are there altogether?
2	Children	Eight
3	Teacher	How did you know?
4	Marc	You did it in the same way.
5	Teacher	I did? Was it the same question? Again?
6	Kate	You didn’t ask the same question, but you used the same numerals.
7	Teacher	It was the same numerals, but not the same question. What happened with the numerals, then?
8	Kate	Now, you had five and three (shows five fingers on her left hand and three on her right hand).
9	Marc	Before you had three and five …
10	Kate	Then five and three … (shows a whole hand and three more unfolded fingers)
11	Teacher	Oh! (shows five unfolded fingers on her left hand and three unfolded fingers on her right hand). Is it the other way around? (Crossing her two hands, same finger pattern). Three and five are eight. Thus, five and three are also eight. And vice versa, with the numerals. Did you notice?

In the teacher-child dialogue in Observation 4, commutativity was opened up as a DoV when the children noticed the commutative pattern between the two number stories. They were encouraged to reason about which parts were the same and which differed (lines 5 and 7). By contrasting the order of the addends in the midst of invariance (same part-whole relation, 3/5/8), the number structure of the task was emphasized and thus afforded them to strengthen their knowledge of number relations by bringing to the fore the flexibility that commutativity provides in arithmetic operations.

The commutativity principle affords a flexible use of number relations but is also dependent on an awareness of the part-whole relations of numbers. The activities in the intervention program afforded both of these to the children: number relations and the ordering of the addends to solve the problem in a powerful way.

Connecting discerned aspects to afforded aspects

In the following, we present results from the synthesis of the aspects we found the children had discerned in the post-assessment, after the intervention, and the aspects they were afforded to discern through the intervention. The results help us answer how the intervention was reflected in the children’s learning of arithmetic skills.

Due to the abstract nature of numbers, representations are found to be necessary for learning about them. However, representations can be more or less powerful in making number relations visible. In this intervention program, the representation of numbers with finger patterns was chosen as one representation in parallel with, for example, bead strings. These became essential resources for opening up necessary aspects of numbers, for instance, that five fingers on one hand and three fingers on the other can be seen both as sets of three and five as well as one composite set of eight (in which three and five are easily recognizable parts if represented as said finger pattern or on the bead string). Thus, finger patterns as representations were necessary as they both allowed the children to discern how numbers can be represented in different modes and served as a resource for the structural approach, as the finger patterns could illustrate aspects that were difficult to discern without some manipulative and concrete means.

The children’s ways of experiencing numbers, as shown in the pre-assessment and during the intervention, clearly revealed that certain aspects were critical to discern for the participating children to become able to solve arithmetic tasks. Before the intervention, most of the children experienced numbers as names or extent, which means that they were unable to attend to numbers’ cardinal and ordinal meaning simultaneously, which is a prerequisite for the most elementary arithmetic strategies such as counting all and counting on (see Baroody, 1987; Fuson, 1988). Furthermore, these ways of experiencing numbers indicate that children have not discerned numbers’ part-whole relations, which is necessary, for instance, in solving missing addend and missing start number tasks. Since the majority of the children learned to discern these aspects during the intervention, here we connect these critical aspects to what they were afforded to discern in the enactments of the activities. A mere focus on a single aspect such as ordinality or cardinality may induce counting procedures as ways to determine sets, which was intentionally avoided in the intervention program. Opportunities to discern ordinality and cardinality, and particularly how numbers constitute both these aspects simultaneously, were afforded in activities in which numbers were attended to as composite sets and these sets were related to other sets, as contrasting examples of the same numbers but different within-number relations. The simultaneous discernment of numbers’ cardinal and ordinal meaning was strongly emphasized in the three first activities in the intervention; that is, the Statement game, the Snake game, and Finger patterns. In these, the teachers afforded the children to compare sets and particularly to determine quantities without counting single units but seeing, for instance, a whole hand as a composite five and adding one finger on the other hand to create a new composite set (6) which included the easily recognized five/whole hand. This facilitated the fundamental aspects of ordinality and cardinality to be discerned and experienced simultaneously.

Seeing numbers as composite sets was found to be important, not only for manifesting numbers’ dual meaning of cardinality and ordinality but particularly for discerning numbers’ part-whole relations. All the activities were designed to emphasize the part-whole relations of numbers, and thus afforded the children to identify number relations through different patterns of variation. That is, the intervention implemented a structural approach to teaching and learning arithmetic. We can see the teachers affording the children to relate composite sets to other sets and a whole, facilitating their discernment of the relation through different patterns of variation. The teachers explicitly afforded the number relations to be discerned by emphasizing what constituted the parts and what was the whole, in the Snake game by controlling for the whole 10 as the starting point and end point of the operations, reminding the children of the semi-decimal structure on the bead string and on their fingers.

Results from the post-assessment reveal that all but ten instances (of 64 observations) reflect the structural approach implemented in the intervention. The instances in which numbers were seen as extent in the assessment after the intervention are concentrated to the most difficult tasks (see Task C, Appendix A), where it can be observed that the children are unsure of what numbers to consider as parts and as the whole (Kullberg et al., 2020). Thus, we suggest that the children were afforded to discern important aspects of numbers, but there seems to be even more complexity to understanding part-whole relations of numbers that was not anticipated in the design and enactment of the activities that are brought to the fore in certain tasks but do not become critical in other tasks. Commutativity is largely such a task-related aspect that was afforded in the intervention activities (primarily the Snake game and Context problems). It is not a necessary aspect for developing basic arithmetic skills but is important for further advancement in arithmetic problem-solving. However, while discerning the part-whole relations of numbers, commutativity was also afforded as an extended DoV of the part-whole relations in our intervention.

Conclusion

Our study explicitly focused on the relation between what was taught and what was learned, through a micro-analysis of how aspects of numbers were afforded in an intervention and how these were reflected in children’s learning. Our intervention is an exception compared to many other intervention studies, in that the children have been encouraged to attend to numbers primarily as composite sets using finger patterns as an aid to visualize the part-whole relations. In other interventions described in the literature, children have most often counted to determine or compose sets (Fischer, 1990; Paliwal & Baroody, 2017).

One of our main results is that we identified a gap in the ways the children in the target group experience numbers, which influenced their abilities to enact powerful arithmetic strategies. Although it was a limited program in terms of duration and number of activities and learning occasions, through our intervention we developed ways to close this gap, with long-lasting results. Furthermore, of particular interest is that since the number of observations of experiencing numbers as structure and known fact amounts to 85%, the awareness of number relations that allows the child to deal with part-whole relations in the task is not task-related but is rather applicable to different arithmetic tasks.

What our study shows is that the children in our study make a necessary change toward experiencing numbers as structure and we would suggest, through the systematic work affording the discernment of necessary aspects of numbers, particularly relating to a semi-decimal structure (five is a whole hand), rather than determining a quantity by counting single units (cf. Fischer, 1990). This pedagogy has provided the children opportunities to discern aspects of numbers and number relations that liberate how numbers can be experienced, rather than teaching them strategies as the primary goal. This does not mean that counting is unimportant in this development, but that it may not necessarily be the primary way to teach children arithmetic skills since very few observations were made of the children using single-unit counting strategies to solve arithmetic tasks before the intervention, and none in the post-assessment. A structural approach with strong emphasis on number relations and representations illustrating this particular aspect of arithmetic tasks thereby offers an alternative to the commonly used counting-based approach (counting single units). More research is however needed to determine to what extent the structural approach is applicable in early childhood mathematics education.

As discussed by Baroody et al. (2009), most intervention programs have a learning goal driven from earlier research and what have been known to be successful strategies in arithmetic problem-solving. In our study, however, we conjecture that children can only make use of, for example, decomposing strategies if they already have an understanding of numbers’ part-whole relations, as suggested by Cheng (2012). Based on our thorough analysis, we suggest that it was the way of opening dimensions of variations of the part-whole relations that enabled the children to change their way of experiencing the numbers toward more powerful ways – ways that allowed them to make use of efficient strategies and features of the arithmetic tasks. Our contribution to the field of knowledge is thereby not only showing a progression that can be associated with a structural approach in teaching; we also show that learning analyzed in terms of aspects discerned, and teaching in terms of aspects afforded to discern, is an alternative way of describing arithmetic learning on a micro-level, compared to assessing success or failure through standardized tests. The methods we use are qualitative, guided by the theoretical framework, which is a strength as our way of analyzing allows us to discuss teaching and learning in commensurable terms with significant implications for teaching practices. Our contribution is thereby also an illustration of how to study the outcomes of an intervention program with young children on a highly detailed level. A detailed analysis like this is nevertheless time-consuming and demanding, and is not applicable to all available data, but does help us better understand arithmetic teaching and learning.

The development we observe in the target group can be confirmed by the earlier reported significant increase in arithmetic proficiency and large effect size in the whole intervention group (Kullberg et al., 2020). This particular study is small-scale, however, and does have limitations as to what conclusions we can draw from the analysis. We cannot say anything about the children’s gains in the ability to solve arithmetic problems in general or in new contexts, or their learning of formal numerical expressions such as 3+_ = 8, since the same tasks are used for pre- and post-assessment rather than standardized tests or similar. Our study is thereby limited in what kinds of generalizations we are able to claim. Nevertheless, the micro-analytical approach, on a theoretical grounding, allows us to draw conclusions about the ways in which a structural approach can be implemented in preschool activities that have an impact on children’s developing awareness of numbers and how to make use of this in arithmetic problem solving – a progression observed from the children experiencing numbers as names to being able to structure arithmetic tasks on their fingers or retrieving known number facts only eight months later. This particular study cannot reveal whether these children would have developed similar ways of experiencing numbers without participating in this intervention, or by participating in a different one where counting single units would have been promoted. However, our primary aim was to gain deeper knowledge of the relation between what was taught and what was learned, as a contribution to theory about children’s learning of basic arithmetic. More research, with control groups and a larger sample, is needed to establish the success of implementing this particular intervention program.

Additional information

Funding

This work was supported by the Swedish Research Council under Grant [721-2014-1791].

Notes on contributors

Camilla Björklund

Camilla Björklund is Professor in Education at the University of Gothenburg, Sweden. Björklund is involved in research projects within the field of mathematics learning and teaching in early childhood education, which are characterized by practice-oriented research questions and designs. Research areas of interest are mathematics learning and teaching in the early years of preschool and primary school, and teachers’ professional pedagogical development. She has frequently published scientific reports and books for teacher students and practicing teachers, particularly within the field of teaching about numbers and arithmetic in the early years.

Anna-Lena Ekdahl

Anna-Lena Ekdahl is Lecturer, PhD in Didactics, at Jönköping University, Sweden and visiting associate at WITS School of Education, University of the Witwatersrand, Johannesburg, South Africa. Ekdahl is particularly interested in teaching and learning numbers and additive relations in preschool and early primary years. She is involved in intervention projects in Sweden and in South Africa. Projects are characterized by practice-oriented research questions and close collaboration with teachers. Results from projects have been published in scientific papers.

Ulla Runesson Kempe

Ulla Runesson Kempe is Senior Professor in Education at Jönköping University, Sweden and visiting professor at WITS School of Education, University of the Witwatersrand, Johannesburg, South Africa. Her research interests are learning and teaching, particularly in mathematics and the teaching profession in general. She was involved in introducing Learning study – a theory-informed version of lesson study – in Sweden. Over the years she has been responsible for several research projects involving teachers as researchers, funded by the Swedish Research Council. Her PhD was the first one where variation theory of learning was used as an analytical framework and has been engaged with the development of the theory since then. She has been involved in several international research projects studying and comparing classrooms in different countries.

Notes

1. The third assessment is not included in this particular analysis since teaching activities between Assessment 2 and 3 are not recorded. The year after the intervention, the children were enrolled in preschool classes at different schools, with different peers and teachers.

2. Examples of children’s responses are from the target group data set.

Appendix A

Tasks given in the assessment

A. 7 = _+_ Seven stones are counted and then hidden in the interviewer’s closed hands.How many can there be in the one hand and how many in the other hand.Multiple options possible.

B. 7 = 4+_ The interviewer opens one of her hands, showing four marbles. If one knowsthere are seven altogether, how many are there in the closed hand?

C. 5 = 4+_ You and your friend collected five shells together, you collected four of them,how many did your friend collect?

D. 9–7 = _ If you have nine shells and your friend has seven shells, how many more shellsdo you have?

E. 2 + 5 = _ You have 2 sea shells and receive 5 more, how many do you have then?

F. 10–6 = _ If you have ten candies and eat six of them, how many are left?

G. 3+_ = 8 You have three glasses, but are going to set the table for eight people, how manymore glasses do you need?

H. _-3 = 6 On the morning of your birthday party, you blew up balloons. At the party, threeballoons broke, and there were only six balloons left. How many balloons didyou blew up that morning?