Playing Gebeta in Preschool: Informal Pathways to Early Numeracy Through Directionality and Bundling

Abstract

Preschool children develop essential mathematical concepts through play, yet little is known about how traditional board games can support this process. This small-scale microgenetic case study investigates how children unfamiliar with the Ethiopian game Gebeta (a mancala-type game) learn to play the game and what mathematical competencies emerge. Video observations were conducted with 5-year-olds in Norway as they engaged in repeated play sessions. Event logs and transcripts were analysed with a focus on children’s developing strategies and difficulties. The results show that children quickly understood the basic rules but faced challenges with directionality (i.e., maintaining the correct direction of play) and differentiating between different types of game moves. Over time, they demonstrated evident progress in following the rules. They began to treat groups of counters as units, moving from one-by-one counting to bundling and unitising. These developments align with Bishop’s fundamental activities locating and counting. We conclude that Gebeta provides a playful context in which children can practice directionality and bundling, both of which are crucial for avoiding common errors in later school mathematics, such as off-by-one counting errors and misunderstandings of the number system. The study suggests that Gebeta can be introduced in early childhood settings without simplifying the rules, providing an engaging resource for early mathematics education.

1. Introduction

Play is a universal activity, taken seriously in all cultures, even though play and seriousness are often seen as literal opposites (Huizinga, 1949; Stelzer, 2023). It is a social activity, unlike most others, where participants become players and must agree to follow the rules to participate. This aspect links play (and especially games) closely to mathematics, as rules form the essence of both activities. In games, rules create a predictable environment, allowing players to understand which rules others are following (Bishop, 1991, p. 45). This must not be confused with an instrumental understanding that mathematics is about following rules (Skemp, 1976). Rules are meaningful when learners grasp both how to act and why the rules work as they do, combining procedural and conceptual understanding (Kieran, 2013).

Games can be used as an educational tool to teach mathematical concepts and procedures. Recently, play-based and game-based mathematical learning has gained importance as an educational method (Maffia & Silva, 2025; Stohlmann, 2022; Vlassis et al., 2023; Vogt et al., 2018). In their systematic literature review, Sousa et al. (2023) found that only two of the 45 analysed studies had mathematics as a learning outcome. Maffia et al. (2025) found thirteen papers, of which five focused on numbers and operations and two on plane geometry, targeting preschool to lower secondary students. Nevertheless, the search for games that stimulate and support mathematical learning poses challenges for early childhood practitioners. While some games are created explicitly for mathematics instruction (e.g., Vogt & Rechsteiner, 2016), when learning objectives take precedence over enjoyment, they become more classroom activities than genuine games (Martinez et al., 2022; Vogt et al., 2018). Our approach is therefore to investigate the mathematical potential of traditional games. There are already studies on traditional outdoor games (Öngören & Gündoğdu, 2021) and European board games such as “Ludo” (Gasteiger & Moeller, 2021) and “Chutes and Ladders” (Siegler & Ramani, 2009), but these are limited to physical games and games of chance. Inspired by an Ethiopian colleague, Solomon Abedom Tesfamicael, we chose the Ethiopian strategy game “Gebeta” (Tesfamicael & Farsani, 2024) and introduced it to Norwegian children unfamiliar with the game. The game is shown in Figure 1.

Tesfamicael and Farsani (2024) have shown which mathematical concepts, operations, and procedures are embedded in the Gebeta game. Afterwards, we investigated the affordances of Gebeta in early childhood mathematics education (Thiel et al., 2024). The follow-up study presented here aims to answer the following research questions:

• How do preschool children unfamiliar with Gebeta learn and play the game? What do they master, and where are they challenged?
• Which mathematical skills do the children learn by playing Gebeta?

1.1. Play in Early Mathematics Education

Vygotsky (1978) highlights the importance of social interactions and cultural tools in learning, emphasising play as a setting for children to acquire mathematical concepts. Play is a vital part of how children learn, offering a meaningful and engaging environment for exploring mathematical ideas while promoting cognitive, social, and emotional growth (K. R. Ginsburg, 2007; Vygotsky, 1967). H. P. Ginsburg (2006) found that children use mathematical thinking during play, and Lundvin and Palmér (2025) demonstrated that a play-responsive approach to early mathematics enhances children’s experiences with mathematical concepts. Still, early childhood practitioners often do not participate or support this development, fearing they may interfere with children’s play or dominate their agendas (Pyle & Danniels, 2017). Rather than intervening in children’s play, H. P. Ginsburg (2006) suggested playful teaching, where lessons are integrated into play in a creative and enjoyable way. Similarly, Walsh et al. (2017) note that modern versions of children’s play often reduce mathematics to meet curriculum objectives, positioning it as an add-on rather than a natural part of play. Bishop (1988), on the other hand, considers playing itself a mathematical activity because players must follow more or less formalised rules. Importantly, this engagement is not limited to applying procedures; it also involves exploring strategies, relations, and meanings that emerge from the rules. In this sense, play connects both to procedural dimensions of mathematics (e.g., carrying out strategies or predictions) and to relational (Skemp, 1976) and conceptual (Kieran, 2013) dimensions (e.g., reflecting on why certain moves are possible and how rules interact).

Papandreou and Tsiouli (2022) highlight that children draw on their existing mathematical “funds of knowledge” (Moll et al., 1992) during play to explore their current interests or solve problems. This underscores the importance for early childhood practitioners to observe and recognise children’s mathematical activities in play, enabling them to develop and extend their mathematical themes. However, H. P. Ginsburg et al. (2008) point out that although young children engage in mathematical play, it often does not lead to mathematization, i.e., interpreting experiences mathematically or understanding relationships between mathematics and the real world. This aligns with Gifford’s (2005) conclusion that children’s spontaneous play may not necessarily involve engaging in mathematics in a meaningful way, as children may lack interest in less engaging mathematical tasks. Therefore, there is a need for play scenarios and games that are rich in mathematical affordances and simultaneously enjoyable and exciting for children.

1.2. The Gebeta Game and Mathematics

Gebeta is a version of “Count and Capture—The world’s oldest game” (Rohrbough, 1955, title page). The different versions of the board game are called “mancala, wari, oware, soro, omweso, ayo, adi and hundreds of other names” (Bishop, 1991, p. 46). It is best known by the generic name “Mancala” and some game experts consider it one of the best mathematical games worldwide (Zaslavsky, 1994, p. 4). In Norway, it is marketed as “Kalaha”. The Ethiopian versions are called “ገበጣ” in Amharic, which is usually transcribed “Gabata” (Pankhurst, 1971). For our version, we use the spelling “Gebeta” (Tesfamicael & Farsani, 2024), which works in English and Norwegian. The pronunciation (expressed in the International Phonetic Alphabet, IPA) is [gəˈbä.t’a] with stress on the second syllable and a “popping” [t’] sound.

1.2.1. The Gebeta Game Rules

Gebeta is a two-player, two-row mancala-style game with six holes in each row (see Figure 2). The holes are called “homes”, i.e., “ቤት” [bä.t] in Amharic (Tesfamicael & Farsani, 2024, p. 11). Each player owns one row of homes. Some boards have two additional, larger pits (one on each side) that can serve as storage compartments. Forty-eight counters (traditionally seeds, beads, or pebble stones) are evenly distributed into the twelve homes, with four counters in each home. In the original game, a group of four counters is a “complete home” (Tesfamicael & Farsani, 2024, p. 12), but I call it a “family” (Thiel et al., 2024) to distinguish between holes and groups of counters. Players pick up the counters from any of their homes and sow them in an anticlockwise direction (directionality rule). In this context, “sowing” means distributing one counter at a time to adjacent homes. If the last counter lands in a home that is occupied, the player picks up the contents of this home and continues to sow. When the last counter lands in an empty home, the turn ends. When the last counter lands in a home that has three counters, it creates a family. The player captures the family by removing it and setting it aside. The turn ends.

If, during sowing, a family is formed elsewhere, the owner of that home captures it. The goal is to capture most families. If a family is not captured when it should have been, and a subsequent sowing causes this hole to contain five counters, “it becomes a Jen” (Piette & Stephenson, 2022, line 210). Players can no longer capture from a Jen. Captures occur when the last sown counter lands on it. In that case, the sowing player captures the final counter along with one counter from the Jen. I did not introduce this complex Jen-rule. Instead, I encouraged the children to help each other identify the families they created.

When Gebeta was introduced to me, I learnt the rule that the match ends when a player must pass three consecutive times. This requires endgame strategies (Thiel, 2025). Players count their pieces by placing four in each home. The player with more than the original 24 counters takes ownership of an opponent’s home for every additional family. The game continues with a new match. The game finally ends when one player owns all homes. These rules are almost identical to the Gabata version from Ghinda (Pankhurst, 1971; Piette & Stephenson, 2022).

1.2.2. Mathematical Concepts Embedded in the Gebeta Game

Tesfamicael and Farsani (2024) demonstrate that the game Gebeta embeds a rich array of mathematical concepts, operations, and procedures that can meaningfully support early mathematical learning. Drawing on the theoretical lenses of “funds of knowledge” (Moll et al., 1992) and “cultural commognition” (Sfard, 2015), they identified two core areas of mathematical thinking that the game fosters: early numerical thinking and algorithmic thinking. When sowing the counters, the players must keep a one-to-one correspondence (Piaget & Szeminska, 1941) between counters and homes. When picking up all counters from a hole, an empty home is formed, representing the concept of zero (Hartmann et al., 2022; Krajcsi et al., 2021). Players must know how many counters there are in every home to play strategically. They use subitising (Clements, 1999) or counting (Gelman & Gallistel, 1978) to figure out the numbers. If they pick up three counters (a cardinal number), they can reach the third home (an ordinal number) counted anticlockwise from their current position. These are fundamental concepts in early numerical thinking. Algorithmic thinking is evident in the need to follow step-by-step procedures to execute moves, applying strategic and conditional reasoning, and developing and refining efficient strategies based on changing game configurations.

Tesfamicael and Farsani (2024) argue that in Ethiopia, Gebeta is a culturally familiar and readily accessible activity that constitutes a powerful cultural artifact for mathematics education. It supports the development of mathematical thinking and communication from an early age, offering a valuable opportunity to bridge children’s everyday experiences with formal mathematics. Additionally, Thiel et al. (2024) found that even children who are not culturally familiar with the game, engage in mathematical activities and early numerical thinking when playing Gebeta. They used early number sense (Whitacre et al., 2020) and Bishop’s six fundamental mathematical activities, counting, measuring, locating, designing, explaining, and playing (Bishop, 1988), as a theoretical framework (see next section). The children participated in counting and locating activities, with notable examples of one-to-one correspondence, subitising, and bundling, incl. counting sets as units. Furthermore, the children used the game materials during their free play to create patterns and to develop their own rules. In this previous study, we observed only one play session per child, focusing only on mathematical activities, not learning. In the new study, I observed children’s development over time to analyse their learning.

2. Theoretical Framework

There are different ways to structure the academic content of mathematics. European early childhood mathematics education usually uses Bishop’s (1988) six fundamental mathematical activities: counting, measuring, locating, designing, explaining, and playing (Perry et al., 2020). According to Bishop (1991), mathematics is a cultural product. Mathematical situations arise in our lives, even outside of formal learning situations. Different cultures around the world have developed mathematics to meet similar needs. Even though the cultural expressions and mathematical tools can be different, mathematical activities are universal. Bishop has identified six such universal or fundamental activities and developed the “Mathematical enculturation curriculum” that builds on these activities (Bishop, 1991, p. 93). The Norwegian curriculum for the learning area “Quantities, spaces and shapes” in the “Framework plan for kindergarten content and tasks” (Norwegian Directorate for Education and Training, 2017, p. 53) is also based on these activities, even though this is not explicitly mentioned.

2.1. Counting

Counting is “a systematic way to compare and order discrete phenomena. It may involve tallying, or using objects or string to record, or special number words or names” (Bishop, 1988, p. 182). Kindergartens shall enable children to “play and experiment with numbers, quantity and counting and gain experience with different ways of expressing these” (Norwegian Directorate for Education and Training, 2017, p. 53). Numbers are essential both in mathematics and in everyday life. Although Bishop (1988) refers to it as counting, the activity encompasses almost every aspect of numbers. This includes both approximate, early and mature number sense (Whitacre et al., 2020). The related topics in the Learning Trajectories approach (Clements & Sarama, 2021) are (a) Quantity, number, and subitising; (b) Verbal and object counting; (c) Comparing, ordering, and estimating numbers; (d) Early addition and subtraction and counting strategies; and (e) Composition of number and place value. In the context of the game Gebeta, children predominantly experience specific early number sense skills: one-to-one correspondence, conceptual subitising, systematic counting, basic arithmetic, bunding, and the concept of zero (Thiel et al., 2024):

• One-to-one correspondence is needed to compare the cardinality of two sets. Two sets have the same cardinality, i.e., the same number of elements, if you can allocate each element of one set to one and only one element of the other set and vice versa (Piaget & Szeminska, 1941). When playing Gebeta, the players sow counters by placing one counter in each of the homes. Thus, there is a one-to-one correspondence between the sown counters and the homes that received a counter.
• Subitising is the ability to “see” how many items a set contains. This is easy for up to three elements. For larger sets, we subitise by structuring the set. For example, we can see that a set contains five elements when we subitise a group of two and a group of three. This skill is called conceptual subitising (Clements, 1999). When playing Gebeta, the players must identify “families” of four counters.
• Systematic counting is the ability to determine the cardinality of a set by enumerating. It requires an understanding of the five counting principles (Gelman & Gallistel, 1978):
  • The one-to-one principle demands that children maintain one-to-one correspondence between objects and number words when counting.
  • The stable-order principle demands that children can recite the number words in the correct order “one, two, three, four, five, …”
  • The cardinal principle says that the final number word used in the counting process is the total number of counted objects.
  • The abstraction principle is about what can be counted, i.e., both tangible and abstract objects. In the context of Gebeta, these can include counters, homes, steps, families, players, matches, and wins.
  • The order-irrelevance principle highlights that the same cardinal number results regardless of the enumeration order.
  When playing Gebeta, players count, for example, how many counters or families they have captured.
• Basic arithmetic is the ability to add or subtract small numbers, eventually by using tangible objects (Barth et al., 2005). Gebeta players must understand that a set of three counters will become a family of four if one is added.
• Bundling is the ability to understand sets of objects as units (Wege et al., 2023). In the Gebeta game, sets of four counters are treated as units, known as families. Before the game starts, the families are distributed into the homes. During the game, the players capture families. At the end of the game, players count the number of families they have captured. Bundling plays a crucial role in the number system. We bundle ten ones into a group of ten, ten groups of ten into a group of one hundred and so on.
• Children’s concept of zero develops over time. Nieder (2016) distinguishes four stages of zero-like concepts: (1) the sensory representation of zero as the absence of stimulation; (2) the categorial representation of zero as “nothing” in contrast to “something”; (3) the quantitative representation of zero as the cardinality of the empty set; and (4) the mathematical representation of zero as an abstract number. In the game Gebeta, empty homes have a special meaning. This can be used to introduce zero as the cardinality of the empty set, i.e., the number of counters that remain in a pit after all counters have been removed.

2.2. Measuring

Measuring is “Quantifying qualities for the purposes of comparison and ordering, using objects or tokens as measuring devices with associated units or ‘measure-words’” (Bishop, 1988, p. 182). Kindergartens shall enable children to “gain experience of quantities in their surroundings and compare them” (Norwegian Directorate for Education and Training, 2017, p. 53). These can be quantities related to length, area, volume, weight, time, and others. The Learning Trajectories approach focuses on geometric measurement: length, area, volume, and angle (Clements & Sarama, 2021). The Gebeta game does not require measuring (Tesfamicael & Farsani, 2024).

2.3. Locating

Locating is much more than just determining the location of something. Bishop (1988, p. 182) defines it as exploring “one’s spatial environment and conceptualising and symbolising that environment, with models, diagrams, drawings, words or other means.” Kindergartens shall enable children to “use their bodies and senses to develop spatial awareness” (Norwegian Directorate for Education and Training, 2017, p. 53). Clements and Sarama (2021) refer to this as spatial thinking, which encompasses spatial orientation and spatial visualisation. Spatial orientation is needed for navigation. Navigating requires locating my position and the position of the goal, as well as identifying the direction I must move to reach the goal. Spatial visualisation is the ability to create mental images of spatial relationships and manipulate those images. This is, for example, needed to construct new objects. Navigating without a physical map requires visualising a mental map of the landscape.

When playing Gebeta, children must locate their position on the board and move in the correct direction (to the right or to the left). Directionality refers to understanding the importance of spatial order (right, left, forward, backwards, up, down) and plays a vital role in the development of numeracy. On the one hand, the brain encodes numbers in a mental image of a spatially directed number line (Dehaene et al., 1993; Göbel et al., 2018; Shaki & Fischer, 2008). On the other hand, our numerical and arithmetic symbols have directionality; for example, 9 and 6 represent different numbers, even though the graphical symbol is just rotated by 180°. Similarly, 12 and 21 represent different numbers because it matters whether we read from left to right or from right to left. Additionally, being aware of the importance of direction helps prevent mistakes like 72 − 35 = 43 (The student calculated 5 − 2 instead of 12 − 5.). Another type of mistake commonly found in third-grade classrooms is the off-by-one counting error, which occurs when starting with the wrong number, for example, counting 9, 10, 11, 12, 13, 14 to solve 9 + 6 = 14 (Nelson & Powell, 2018).

2.4. Designing

Designing is “Creating a shape or design for an object or for any part of one’s spatial environment. It may involve making the object, as a ‘mental template’, or symbolising it in some conventionalised way” (Bishop, 1988, p. 183). Kindergartens shall enable children to “investigate and recognise the characteristics of different shapes and sort them in a variety of ways” (Norwegian Directorate for Education and Training, 2017, p. 53). The Learning Trajectories approach distinguishes between identifying shapes (and their properties) and composition and decomposition of shapes (Clements & Sarama, 2021). This activity is not facilitated in the Gebeta rules.

2.5. Playing

Playing is (according to Bishop, 1988, p. 183) “Devising, and engaging in, games and pastimes, with more or less formalised rules that all players must abide by.” It contributes to mathematical ideas such as puzzles, paradoxes, models, rules, procedures, strategies, prediction, guessing, chance, hypothetical reasoning and game theory. Playing has a central place in the Norwegian kindergarten. In the section about the learning area “Quantities, spaces and shapes”, play is mentioned three times: The learning area “covers play and investigation”, “kindergartens shall enable children to … play and experiment with numbers” and staff shall “create opportunities for mathematical experiences by enriching the children’s play and day-to-day lives with mathematical ideas” (Norwegian Directorate for Education and Training, 2017, pp. 53–54). Games are mentioned, too. Here, playing and games are instructional practices and pedagogical tools rather than mathematical content areas. This is also the case with the Learning Trajectories approach (Clements & Sarama, 2021). However, for Bishop (1991, p. 102), the “‘Mathematics is a game’ metaphor is a very powerful idea.” He explicitly mentions Mancala as a board game that can “be traced back to some modelling of reality. That modelling is critical in mathematical development” (Bishop, 1991, p. 46). Gebeta requires players to coordinate multiple interdependent rules: select a starting home, collect counters, start sowing at the next pit, sow in an anticlockwise direction, switch rows appropriately, apply the capture condition, and decide whether to continue or end their turn. Thus, playing Gebeta requires algorithmic thinking (Tesfamicael & Farsani, 2024), which can be defined as “as a cognitive process that involves the ability to decompose complex problems into manageable components, recognize patterns, generalize solutions, and design systematic steps or algorithms to solve problems effectively” (Yusuf & Noor, 2024, p. 4).

2.6. Explaining

Explaining is the most general of the fundamental activities. It “lifts human cognition above the level of that associated with merely experiencing the environment. It focusses on the actual abstractions and formalisations themselves which derive from the other activities” (Bishop, 1991, p. 48). Kindergartens shall enable children to “discover and wonder about mathematical relationships” (Norwegian Directorate for Education and Training, 2017, p. 53). Being able to explain how and why things are connected is essential in all mathematical understanding. Encouraging children to explain and argue develops their mental skills in logical reasoning, as well as the use of mathematical language. The most fundamental explanatory activity is classifying, i.e., arranging objects into collections based on similarities and differences (Piaget & Inhelder, 1991). Explaining plays a role in all content domains of the Learning Trajectories approach, but is most strongly related to the domains “Patterns, Structure, and Algebraic Thinking” (Clements & Sarama, 2021, p. 279). In the context of playing Gebeta, explaining helps the children understand the rules as well as the possible consequences of specific moves. However, this is already covered by the category playing. Bishop (1991, p. 103) admits that playing and explaining seem to overlap. Thus, explaining requires that the children directly engage “in the way Mathematics explains, in sort of ‘answers’ one can obtain to Mathematical questions, in the kinds of questions themselves and in the power (and limitations) of Mathematical explaining.”

3. Materials and Methods

3.1. Study Design

Due to limited resources, this is just a small-scale case study using an opportunity sample. Since the aim is to investigate what young children learn when they play Gebeta, a microgenetic design is a suitable method. The microgenetic method is an approach that can yield data about cognitive developmental change mechanisms (Siegler & Crowley, 1991). Even though the cost in time and effort of microgenetic studies is often high, there are several examples of such studies that accomplish their goals with limited resources. For example, Rengifo-Herrera (2025) observed 12 young children on three days. In the meta-study by Brock and Taber (2017), 12 out of 38 studies had four sessions or fewer. According to Lavelli et al. (2005, pp. 42–43), microgenetic designs are characterised by four key features:

• The focus is on individuals observed throughout a period of developmental change, making the child the primary unit of analysis. In this study, the focus was placed on five children in a Norwegian kindergarten (see Section 3.2).
• Observations occur before, during, and after a phase of change within a specific domain, not just before and after. Thus, I observed the children during four play sessions over a one-month period. This is a realistic timeframe to become familiar with a new game.
• There is a high frequency of observations during the transition period, with intervals considerably shorter than the time needed for the developmental change. Here, the transition is from not knowing the game to playing the game without help. Thus, in the first session, I introduced the game and helped the children learn the rules. In the following sessions, support was gradually reduced, allowing the children to play more independently.
• Behaviours are analysed in depth. Microgenetic studies usually combine qualitative and quantitative methods (Siegler & Crowley, 1991). However, due to the small sample size, only a qualitative method was employed: Interaction Analysis (see Section 3.3).

3.2. Data Collection

Over four days, I collected video footage of five children, aged five to six years, playing Gebeta at a Norwegian kindergarten, which is a daycare centre for children from birth to six years of age. The kindergarten was chosen due to personal contact with the head teacher. Thus, this is an opportunity sample, not a random sample. The kindergarten is a small private daycare centre with only two groups. The group “Mini” has 12 children aged 1 to 3 years, and the group “Maxi” has 17 children aged 3 to 5 years. The participants were the oldest children in the “Maxi” group. They wanted to participate and had permission from their parents. The parents were informed by letter and gave written consent. The practitioners and the researcher told the children about the project, and they expressed their willingness orally. They could withdraw their participation at any time. To protect children’s privacy, pseudonyms are used. Some of the participants had parents with migration backgrounds, but all children were born and raised in Norway. The participants were two girls, Ada and Britt, and three boys, Carl, Dan, and Emil. Ada, Britt, Carl, and Dan were unfamiliar with the game before the first session. Emil was not present on the first day. However, this was no problem because he already knew the game from playing with his parents. The observations were conducted between 9:30 and 10:30 a.m. in a separate room in the kindergarten. In each session, only the following people were present in the room: the four or five participating children, a practitioner, and the researcher. The practitioners varied from session to session. They did not participate in the game but gave the children a feeling of security. The camera was mounted on a tripod and remained stationary during the recording.

Figure 3. Data collection: Session dates and length and played matches. Dark circles are playing children; light circles are watching children. A—Ada, B—Britt, C—Carl, D—Dan, E—Emil, R—researcher. (The last game, Emil vs. Dan, was not finished).

Figure 3. Data collection: Session dates and length and played matches. Dark circles are playing children; light circles are watching children. A—Ada, B—Britt, C—Carl, D—Dan, E—Emil, R—researcher. (The last game, Emil vs. Dan, was not finished).

The first session lasted 36 min (Figure 3). After we (Ada, Britt, Carl, Dan, a practitioner, and I) had moved to the assigned room, I introduced the game and explained the rules for about four minutes. Then, the children started playing against each other. I observed the children and further explained the details of the rules when the players encountered relevant situations. The girls’ match ended before the boys’ match. Thus, they rearranged the counters to their starting position and began another game. The second session took place the following week. When Emil’s match was finished, he started a new game against Dan. When Britt’s match was finished, she and Carl watched for a while and then left the room. When Emil’s second game was finished, he still wanted to continue, but Dan did not. Dan left the room, and Emil played against me. The third session was held one week later, and the final session took place about one month after the first. Britt arrived later. Emil, who had previously watched the game between Dan and Ada, decided to play against Britt. After both matches were finished, Emil started a match against Dan, but they got distracted and abandoned the game before it was finished. The total recorded time is 115 min.

3.3. Interaction Analysis

During the game sessions, the children interacted with each other, the researcher, the practitioner and the material. Therefore, Interaction Analysis (Jordan & Henderson, 1995) is the appropriate method of data analysis. However, due to the small scale of the project, not all steps described by Jordan and Henderson (1995) are applicable. The following steps were used:

• The first step is a content log. The log entries are indexed by videotape number and timestamp and contain summary listings of events as they occur on the tape (Jordan & Henderson, 1995, p. 43). The log entries are broad descriptions of what is happening, not detailed transcripts.
• Instead of Group Work, I coded the data alone. First, a deductive classification was conducted using the predefined coding scheme, i.e., Bishop’s fundamental mathematical activities as presented in Section 2. Afterwards, the data was coded inductively by identifying patterns of repeated actions in the children’s interactions with the material and each other. Through repeated replays, increasingly detailed aspects of the participants’ social skills and their abilities to collaboratively construct meaning became clearer (Jordan & Henderson, 1995, p. 43).
• As specific tape segments became significant, content logs were expanded into transcriptions (Jordan & Henderson, 1995, p. 47). Focusing solely on relevant segments enables me to transcribe those parts more thoroughly.

4. Results

The results are reported in two sections. Section 4.1 presents which of Bishop’s fundamental mathematical activities the children were engaged in during the play sessions. Section 4.2 describes the patterns of repeated actions while playing Gebeta.

4.1. Bishop’s Fundamental Mathematical Activities

This section reports how the 108 entries of the content log relate to Bishop’s fundamental mathematical activities (see

Table 1. Percentage of Bishop’s fundamental mathematical activities in the content log.

Activity	Percentage
counting	69%
measuring	0%
locating	11%
designing	6%
playing	13%
explaining	1%

).

4.1.1. Counting

The majority of the content log entries were coded as counting, which is the prevalent mathematical activity during the game. Most children distributed the counters correctly, one by one, into the homes by placing one counter in each consecutive hole. Only in four instances did Britt skip a home. I coded 27 content log entries as related to subitising. This includes not only instances when the children identified groups of four counters (families) that they could capture. Nine times (six times on day 1 and three times on day 2), I observed that the children overlooked families that they could have captured. This did not happen on days 3 and 4. Many times, the children counted the number of counters they had captured. Approximately one-third of the content log entries are related to bundling, a theme that I will discuss in Section 4.2.2.

4.1.2. Measuring

As predicted by the theory, the children were not engaged in measuring while playing Gebeta.

4.1.3. Locating

More than ten per cent of the log entries are related to locating. However, these are all instances when the children did not distribute the counters correctly, since only those caught my attention. The most common mistake was that the children sowed the counters in the wrong direction. I will discuss the children’s challenges regarding directionality in Section 4.2.1.

4.1.4. Designing

In 6% of the content log entries, the children were engaged in Bishop’s mathematical activity designing. These incidents occurred during the play but were not part of the established rules. All but one child used the captured counters to create patterns or pictures during the opponent’s turn. Examples are shown in Figure 4.

4.1.5. Playing

More than one-eighth of the log entries were related to playing. For one, these are instances when the children misapply, explain, or argue about the rules. The third theme that I will discuss in detail in Section 4.2.3 is algorithmic thinking, specifically how children understand and interpret rules.

Another topic that falls under Bishop’s category of playing is strategy. I have observed several instances when the children made strategic moves. Here are some examples:

• On day 2 (timestamp 44:23), during Emil’s match against me, Emil had one counter in C1 and D1, three in E1 and two in F1. Instead of choosing the home with the most counters, he moved the counter from D1 to E1 and captured a family (cf.,

  Table A2. Transcript of the final moves of Emil’s match against me on the second day.

  Line	Timestamp	Activity *	Comments
  		Emil and I have each captured 16 counters. There are still 16 counters remaining on the board. One counter is in C1, D1, and C2; two in F1 and F2; and three in E1, E2, and D2.
  1261	44:05	R: I have four groups of four.	I have grouped my counters.
  1262	44:07	Emil (groups his counters): It is easier to count like this.
  1263	44:09	Emil (counts the counters): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16.	He counts the individual counters even though they are grouped.
  1264	44:17	R (looks at his 4 groups of 4): I have 16, too.
  1265	44:19	Emil: Yes.	He seems to know that 4 times 4 is 16.
  1266	44:21	R: So, we have the same amount. It is your turn.
  1267	44:23	Emil moves a counter from D1 to E1 and creates a family.	This is a strategic move.
  1268	44:25	R: Yes, you win this one.
  1269	44:27	Emil (moves the family from E1 to his side): Do you want to continue?
  1270	44:29	R: Yes, it is my turn.
  1271	44:31	R (picks up the counters from F2): I take these.
  1272	44:33	R (sows into E2 and D2): Then, I win those.
  1273	44:35	R moves the two captured families to my side.
  1274	44:37	Emil: Do you have more than I now?	This is a reasonable conclusion.
  1275	44:39	R: Yes, I believe I have a little more.
  1276	44:41	R: But if you win all remaining ones…
  1277	44:43	Emil moves a counter from C1 to D1.	This is a strategic move. He avoids moving the counters from F1 to my side.
  1278	44:45	R (moves the last counter in my row from C2 to B2): I must take this one.	This is a move without agency.
  1279	44:47	Emil moves the counter from D1 to E1.	Emil proceeds with his strategy.
  1280	44:49	R moves the counter from B2 to A2.	The last phase is like a race. Unfortunately, Emil lost this time. We would have succeeded if his single counter had started at B1 instead of C1 or if my last counter had started at B2 instead of C2.
  1281	44:51	Emil moves the counter from E1 to F1.
  1282	44:53	R: Unfortunately, you must continue with this one.
  1283	44:55	Emil picks up the counters from F1.
  1284	44:57	Emil sows the counters into F2, E2, and D2.
  1285	44:59	R: Now, it is finished.
  1286	45:01	Emil: It is your turn.
  1287	45:03	R: Yes, but you have nothing in your row.
  1288	45:05	Emil: So what?
  1289	45:07	R: The game is finished.
  1290	45:09	R: I believe I have a little more.
  1291	45:11	Emil silently counts both his and my families by pointing at each bundle from a distance.	Emil counts the bundles.
  1292	45:13	Emil (pushes his counters together): Yes, you win!	Emil mentally compares the numbers.

  * Text in italics describes actions. Standard text quotes utterances. Homes are labelled like on a chessboard (see Figure 2). Emil owns A1 to F1, and I own F2 to A2.

  , line 1267).
• In his next move (timestamp 44:43), he avoided moving the counters from F1 to my side. He instead moved the counter from C1 to D1 (cf., Table A2, line 1277).
• On day 3 (timestamp 07:22), during Emil’s match against Dan, Dan had one counter in E2 and C2, and three in B2. He took three counters from B2 and started to sow them. Then, he realised that there was only one counter in C2. Thus, he put the three counters back into B2, took the counter from C2, and moved it to B2. And captured the family in B2.
• On day 4 (timestamp 12:02), during Emil’s match against Britt, Britt had one counter in B2 and D2, and three in E2. Emil showed Britt how she could capture a family by moving the counter from D2 to E2 (cf.,

  Table A3. Transcript of essential parts of the matches on the fourth day.

  Line	Timestamp	Game: Emil vs. Britt *	Game: Dan vs. Ada *	Comments
  166	05:33	Emil and Britt restart their game.	Dan and Ada have already played some turns. Dan has captured three families, and Ada has one. It is Dan’s move.
  167	05:34	Emil (takes the counters from F1): Well! I play.	R (points at D2 and talk to Ada): Those are yours.
  168	05:35		Ada collects the family from D2.
  169	05:36		R (points at B2 and talk to Dan): You must continue here.
  170	05:37	Emil sows into E1, D1, C1, and B1.		Emil sows clockwise.
  171	05:38		Ada attempts to take the counters from B2.
  172	05:40		R (points at Dan): No, he shall do it.
  173	05:41		Ada shrugs her shoulders.
  174	05:42		Dan: But that is on Ada’s side.
  175	05:43	Emil (takes the counters from B1): Right?
  176	05:44	Practitioner: Yes.
  177	05:45		Dan attempts to take counters from B1. R (points at B2): Here.
  178	05:47	Emil (sows into A1): One.		Emil counts while he sows.
  179	05:48	Emil (sows into A2): Two.	Dan takes all counters from B2.
  180	05:49	Emil (sows into B2): Three.
  181	05:50	Emil (sows into C2): Four.
  182	05:51	Emil (sows into D2): Five.
  183	05:53	Emil (takes all counters from D2): Well.
  184	05:54		One of Dan’s counters falls into A2. R (points at A2): You can start with this one.
  185	05:56	Emil continues sowing into E2 and F2.	Dan sows into A1.
  186	05:57	Emil sows into F1.	Dan sows into B1. Ada: This one is full.
  187	05:59	Emil sows into E1.	Carl: Yes.
  188	06:00	Emil (sows into D1): Five.		Emil has counted silently.
  189	06:01	Britt: Yes.	Dan sows into C1.
  190	06:02	Emil takes all counters from D1.
  191	06:03		Dan sows into D1.
  192	06:04		Ada (reaches to Dan’s side): Oh, you have forgotten this one.
  193	06:05	Emil sows into C1, B1, and A1.	Ada takes a counter from Dan’s side and sows it into E1.
  194	06:07		Dan sows into F1.
  195	06:08	Emil sows into A2, B2, and C2.	Dan sows into F2.
  196	06:09		Dan sows into E2 and picks up the four counters from E2.
  197	06:10		Dan moves the counters to his side. R: But–yes, now it is your turn (points at Ada).
  198	06:12	Emil takes all counters from C2.	R (points at D1): Then you can take out these, too.
  199	06:14		Dan: These?
  200	06:15	One of Emil’s counters falls into B2.	R: Yes.
  201	06:16	Practitioner: Oh. Emil tries to pick up the extra counter from B2.	Dan takes the counters from D1 and moves them to his side. R (to Ada): And now, it is your turn.
  202	06:17	The practitioner picks up the counter and moves it to D2.	Ada (attempts to take counters from B1): Okay, I take this one.
  203	06:19		R: No, you must take from your side.
  204	06:20	Practitioner (points at E2): Now, you must continue here.	Ada picks up the counters from C2.
  205	06:21		Ada sows into A2.
  206	06:22	Emil sows in E2 and F2.	Ada sows into A1.
  207	06:24		Ada sows the last counter into B1.
  208	06:25	Emil continues sowing in F1 and E1	R: Now, you can take these.
  209	06:26	Emil (puts the last counter in the empty home D1): Britt, it is your turn now.	Ada picks up all the counters from B1.
  210	06:28	Practitioner (points at D2): Yes, then you take this one.
  211	06:29	Practitioner moves the counter from D2 to Emil’s side.	Ada sows into B1.	Ada starts at the same place where the counters came from.
  212	06:30	Practitioner (points at D1): Because the last one was here, you win this one.		The practitioner knows the Kalaha rules and believes that we are playing this version.
  213	06:33		Ada sows into C1.
  214	06:34	Practitioner (looks at me): Is it like this? Britt takes all counters from F2.
  215	06:35	R: No.	Ada sows into D1.
  216	06:36	Emil shakes his head and moves the counter back to D2. R comes over to look at the game. Britt starts sowing at F2.	Ada sows into E1.
  217	06:38	Britt sows into E2 and D2. Practitioner: Don’t you do it like this?	Ada sows into F1.	Britt starts at the same place where the counters came from, and sows left to right, anticlockwise.
  218	06:40	Britt sows into C2. Emil: No, wait! That’s the wrong way, wrong way, wrong way.	Ada sows into F2.	Emil started the game clockwise. Therefore, anticlockwise is the wrong direction.
  219	06:43	Britt sows into B2. Emil: Wrong way.	Ada sows into E2.
  220	06:44	Emil (picks up a counter from C2): That’s the wrong way!	Ada sows into D2.
  221	06:45	Emil (picks up the counter from B2): That’s the wrong way!	Ada sows into C2.
  222	06:47	Emil: No, not this way! You do it the wrong way!
  223	06:49		Ada sows the last counter into B2.
  224	06:50	Emil: That’s the wrong way!
  225	06:52		Ada (points at B2): I did this one at the end.
  226	06:54		Ada: I have finished it.
  227	06:57	Emil picks up the extra counter from B2.
  228	06:58	Practitioner: I think I see how you do it.	Dan groups his families.	Dan groups on his own.
  229	07:00	R: He has started the other way.
  230	07:02	R: Therefore, you must continue playing the other way.
  231	07:04	Britt points at E2.	Dan looks at Britt’s game.
  232	07:06	Britt (points at Emil’s row and moves her finger from F1 to A1): I must distribute here? Emil sows Britt’s counters into F1 and E1.	Dan and Ada watch what Emil is doing.	Britt is confused but starts to understand.
  233	07:08		Dan: It is not like this.	Does he mean the direction?
  234	07:09	Emil continues sowing into D1 and C1.
  235	07:10		Carl attempts to take the counters that Ada has captured.
  236	07:11	Emil puts the last counter into B1 and picks up the two counters from B1.	Ada pushes Carl’s hand away.
  237	07:13	Emil sows the counters into A1 and A2.	Dan (to Ada): Are you ready?
  238	07:14		Ada: Yes!
  239	07:15	Emil hesitates.
  240	07:17	R (points at A2): Just take these here.	Ada (points at B2): This was the last one. Carl takes a counter from B1 and pretends to eat it.
  241	07:19	Britt picks up all counters from A2. Emil (points at B2): And then you give this one, (at C2) this one, (at D2) this one, (at E2) this one.	Dan points at B1, and Carl puts the counter back.
  242	07:23	Britt sows the counters into B2, C2, and D2.	Dan takes all counters from B1.	Britt sows correctly.
  243	07:25	Emil: Right.	Dan sows into C1.	Dan sows correctly.
  244	07:27	Britt attempts to put the next counter into A2.	Dan sows into D1.	Britt loses track when she gets disturbed.
  245	07:29	Practitioner (stops her): No, we continue here, Britt (points at E2).	Dan sows into E1.
  246	07:31		Dan picks up all counters from E1.
  247	07:33	Britt puts a counter in D2 and E2.	Dan sows into F1.
  248	07:35	Emil moves a counter from D2 to F2.	Dan sows into F2.
  249	07:37	R (points at E2): You must take these and continue.	Dan sows into E2.
  250	07:39		Dan: This one.
  251	07:41	Emil puts a counter in F1.	Dan: It is yours.	The children help each other.
  252	07:43	R (points at F1): Then you win those here now.	Ada: What? (takes the four counters from F2.)
  253	07:45	Emil takes all counters from F1.
  254	07:46	Emil moves the counters to Britt’s side.	Ada (looks at the counters in her hand): Okay.
  255	07:47		Carl (points at A2): Whose are those?
  256	07:48		Ada: Do I get four?
  257	07:49		Carl: Whose are those?
  258	07:50		Ada: I get four.
  259	07:51		R: Yes.
  260	07:52	Practitioner: If she finishes on your side.
  261	07:57	Emil: If I finish with four, they belong to me.	Ada: I have got four, and you have one more.	Ada now has three families, and Dan has four.
  262	08:03		Dan puts his hand into C1. R (points at C1): Then you win these.
  263	08:05	Emil takes all counters from E1.	Dan picks up the four counters from C1 and moves them to his side.	Dan now has five families.
  264	08:06		R: Now, it is Ada’s turn.
  265	08:07	Emil sows counters into D1.	R points at Ada’s row, which starts with A2.
  266	08:09	Emil sows counter into C1.	Ada takes all counters from E2.
  267	08:10		Dan groups his counters.
  268	08:11	Emil sows counter into B1.	Dan: I will group them.
  269	08:12		Ada attempts to sow from A2. R (points at D2): No.
  270	08:13		Ada sows into D2.
  271	08:14		Ada sows into C2 and picks up all counters from C2.
  272	08:17	Emil sows into A1. Britt (shows me the counters in her hands): Four.	Ada sows into B2.
  273	08:18		Ada sows into A2. R: And now, you get these.
  274	08:19	Emil sows a counter into A2.	Ada: Yes! (picks up the four counters from A2.)
  275	08:20	Emil sows counters into B2 and C2.	Ada: I have got four.
  276	08:21	Emil takes all counters from C2.	Ada shows Dan the four counters in her hand.
  277	08:22		Ada: I have got four.
  278	08:23	Emil sows into D2.	Ada (shows Britt the counters): I have got four, Britt.
  279	08:24	Emil sows into E2. Britt raises her hand, holding the four counters.	Ada: I have got more than you. Dan picks up counters from F1.	Ada has four families, and Britt has one.
  280	08:25	Emil sows into F2.	Ada: But that’s fine, because you can get more later.
  281	08:28	Emil picks up the four counters from F2.	Dan sows into E1 and D1.	Dan sows clockwise (the wrong direction).
  282	08:29	Emil moves the counters to his side.	Dan sows into C1.
  283	08:31		Dan: I have the most of all.
  284	08:32		Ada (points at her counters): Look at me!
  285	08:33	Emil looks at Ada’s counters.
  286	08:34	Emil (points at Britt): Okay!
  287	08:35		Ada (points at A1): I have started from there.
  288	08:36		Ada (points at A2): Not here. Dan (counts his families): I have one, two.
  289	08:38	Emil (points at the homes on his side): You cannot take from this, this, and this one.
  290	08:39	R (points at Britt’s homes): You must well take from your side.
  291	08:40		Dan (still counts his families): Three, four.
  292	08:41	Britt moves her hand to E2.
  293	08:42	Britt: Can I take these?
  294	08:43	R: Yes.
  295	08:45	Britt picks up all counters from E2.
  296	08:49		Ada (starts to group her counters): Here are just four.
  297	08:51	R (moves his finger over F2, F1, and E1): And then it goes this way.
  298	08:53	Britt sows into F2.
  299	08:55	Britt attempts to sow into E2. R (points at F1): Here.	Ada: I believe I have six.	Britt still wants to sow left-to-right in her row.
  300	08:57	Britt sows into F1.	Dan: Six? That are more.	Dan estimates the amount.
  301	08:59	Britt sows into E1.	Dan: I think so.
  302	09:01	Britt sows into D1.	Ada makes a group of five counters.
  303	09:03	Emil picks up the four counters in D1 and moves them to his side. Britt sows into C1.
  304	09:07	Britt sows into B1.	Ada (removes the fifth counter from her group of five and starts a new group): I do it just as I want to.
  305	09:11	Emil moves a counter from C1 to A1. R (asks him): What do you do?
  306	09:13	Emil: She had two here (points at C1).
  307	09:15	R: Ah, yes.	Ada makes a new group of four.	Ada groups correctly.
  308	09:17	R (asks Britt): Do you still have one?	Dan: No, you must do it right.
  309	09:19	Emil points at B2.
  310	09:21	Britt sows into B2.	Dan: That’s not good. Ada: Let me think.
  311	09:23		Ada (has four groups of four): Just here.
  312	09:25	Britt sows into C2.
  313	09:27	Emil picks up all counters from C1.	Ada (counts the counters in the first group): 1, 2, 3, 4.	Ada does not count the groups.
  314	09:29		Ada: I do it just as I want to.
  315	09:31	Emil sows into B1 and A1.	Dan: But you must only count the four counters.
  316	09:33	Emil sows into A2.
  317	09:35		Ada: No, I just count like this.
  318	09:37	Emil sows into B2.	Dan: Okay, I do the same as you.
  319	09:39	Emil sows into C2.	Dan starts counting the counters in each group.
  320	09:41	Emil sows into D2.	Dan: 1, 2, 3, 4.
  321	09:43	Emil sows into E2 and F2.	Dan: 5, 6, 7, 8.
  322	09:45	Emil sows into F1.	Dan: 9, 10, 11, 12.
  323	09:47	Emil takes the counters from F1 and sows them to E1 and D1.
  324	09:49	Emil (to Britt): It is your turn.	Dan and Ada continue counting silently.
  325	09:51		Ada (to Emil): I have 14.	Ada has 16 counters (4 times 4).
  326	09:53	Emil (counts Ada’s families): 1, 2, 3, 4.
  327	09:55		Ada: No, I have 14. Dan: I have 15.	Dan has 20 (5 times 4). Errors of perseveration (Radatz, 1979)?
  328	09:57		Dan: I have ... (starts counting again) 1, 2, 3, 4,...	He is unsure if 15 was correct.
  329	09:59	Emil (counts Ada’s families again): 1, 2, 3, 4.
  330	10:01		Ada: No, I have 14.
  331	10:03	Emil (counts Dan’s families): 1, 2, 3, 4, 5.
  332	10:05	Emil: It is like this.
  333	10:07		Ada: No, I have 14.
  334	10:09		Dan (starts counting once again): 1, 2, 3, 4,...
  335	10:11	Emil (points at one of Dan’s families): No, but you see, this one is here like this.	Dan (continues counting with the next group): 5, 6, 7, 8,...
  336	10:14	Emil: Look here. This is a group of four. Practitioner (to Britt): It is your turn.	Dan (continues counting with the next group): 9, 10, 11, 12,...
  337	10:16		Ada rearranges her counters in a row.
  338	10:18	Practitioner (asks me): Is it she who shall start?
  339	10:19		Dan (continues counting with the next group): 13, 14, 15, 16,... Ada: We can do it as we want to.
  340	10:21	Britt looks at Ada.	Dan counts the last group silently.
  341	10:22	Practitioner: Britt?	Dan: I have 20.	He is correct now.
  		The children continue playing.
  394	11:59	Britt has one counter in B2, one in D2, and three in E2.	Dan has one counter each in A1, B1, C1, and E1, and three in D1.
  395	12:02	Emil shows Britt how she can capture a family by moving the counter from D2 to E2.	Dan moves the counter from C1 to D1 and captures the family.	The children make strategic moves.
  396	12:05	Britt moves the counter from D2 to E2.	Ada: Don’t take from me!
  397	12:06	Emil picks up the family from E2 and moves it to Britt’s side.	R: Now, he is finished. It was his turn. Now it is your turn.
  398	12:09	Emil picks up the counters from F1.	Ada moves a counter from B2 to A2.
  399	12:11	Emil sows into E1 and D1.	R: Your turn is finished.
  400	12:14	Emil captures the family from D1.
  401	12:18	Britt (looks at her counters): Look here, look how many I have.
  402	12:21	Emil: Britt, it is your turn.
  403	12:25	Emil (to me): Can you look at us?	R moves to Emil’s side.
  404	12:27	Emil (points at the board): There are only four left.
  405	12:29	Ada (talks to Britt): I have done it very well. Dan looks at Britt’s counters.
  406	12:32	Ada pushes her counters together. Dan: Britt has very much.
  407	12:34	Ada: Britt and I did a lot yesterday.
  408	12:37	Ada: It is summer.
  409	12:39		Dan (counts the counters on the board): 1, 2, 3, 4.
  410	12:43		Carl (points at E1): And this one.
  411	12:45	R: Now, at the end, we can say that it is finished. Emil groups his counters.	Ada and Dan point at E1.	The number of counters on the board must always be a multiple of four.
  412	12:50	R: Then you can count how many you have.
  413	12:52	Emil (points at Britt’s counters): You must count these here.
  414	12:54	Britt: I can count.
  415	12:55		Ada: I have 17. I have 17.
  416	12:59		Dan (counts groups of four even though his counters are arranged in a row): One.
  417	13:02		Dan: Two.
  418	13:03	Emil: You must just count like this.
  419	13:04	Emil (shows Britt how he counts his families): 1, 2, 3, 4.
  420	13:05		Dan: Three.
  421	13:06	Britt starts grouping her counters.		Britt can group.
  422	13:07		Dan: Four.
  423	13:08	R: Since you capture groups of four, you can just count how many groups of four you have.
  424	13:12	The practitioner helps Britt group her counters.	Dan (repeats his count of the fourth group): Four.
  425	13:14	Britt pushes together the groups that the practitioner had created. Practitioner: I have just made groups of four.		Britt wants to group on her own.
  426	13:17	Practitioner: What do you do now?
  427	13:19	R: But Britt can do it on her own.
  428	13:21		Dan (repeats his count of the fourth group): Four.
  429	13:22	Britt continues grouping her counters.	Dan stops his attempt to count groups of counters that are not grouped.
  468	14:33	Britt finished her grouping.	Dan and Ada distribute the counters back on the board into the starting arrangement.
  469	14:34	R: Now, you can count how many groups of four you have.
  470	14:37	Britt (counts the counters in one group): 1, 2, 3, 4.
  471	14:39	R: How many groups?
  472	14:41	Britt (starts counting the counters in the next group): 1, 2. Emil (shows Britt how to count groups): No, like this 1, 2, 3, 4.		Britt does not count how many counters she has nor how many groups. She checks if there are four counters in each group.
  473	14:45	Britt: No. I don’t do it like this. Emil (continues counting Britt’s groups): 5, 6.
  474	14:47	Britt counts the counters in the first group again.
  475	14:49	Britt: 1, 2.
  476	14:51	Emil: It is not like this.
  477	14:53	Britt: 3, 4.
  478	14:55	Britt (counts the counters in the second group): 1.
  479	14:57	Britt: 2.
  480	14:58	Britt (starts over again in the second group): 1.
  481	14:59	Britt: 2
  482	15:00	Britt: 3.
  483	15:01	Britt: 4.
  484	15:02	Practitioner: Let her count.
  485	15:03	Britt (counts the counters in the third group): 1.
  486	15:04	Britt: 2, 3, 4.	Ada: Can we change, Britt?	Ada wants to start a new match with Britt.
  487	15:06	Britt (counts the counters in the fourth group): 1, 2, 3, 4.	Ada: Britt, you and I can play against each other.
  488	15:12	Britt (counts the counters in the fifth group): 1, 2, 3, 4.
  489	15:13	Britt (counts the counters in the sixth group): 1, 2, 3, 4.
  490	15:15	Britt (counts the counters in the seventh group): 1, 2, 3, 4.
  491	15:18	Britt (counts the counters in the third group again): 1, 2, 3, 4.
  492	15:20		Dan leaves the table. Ada (collects Dan’s counters): I can take yours.
  493	15:21	Britt: Four.
  494	15:22	Britt: I have four.		Does she mean ‘groups of four’?
  495	15:24	R (points at Britt’s families): Britt, can you count how many groups of four you have?
  496	15:27		Ada distributes Dan’s counters into the homes on the board.
  497	15:29	R (points at Britt’s families): 1, 2, 3, 4,...
  498	15:31	Britt (counts the families): 1, 2, 3, 4.
  499	15:33	Britt (continues counting the families correctly): 5, 6, 7.		Britt can count the groups.
  500	15:38	Emil distributes his counters into the homes on the board.
  501	15:40	Practitioner: So, Britt, how many groups do you have?
  502	15:44	Britt: 7. Emil starts picking up Britt’s counters.		Britt is a CP-knower.

  * Text in italics describes actions. Standard text quotes utterances. Homes are labelled like on a chessboard (see Figure 2). Britt and Ada own F2 to A2, and Emil and Dan own A1 to F1.

  , line 395).

In some cases, the children expressed their attitudes towards the game. On day 1 (timestamp 05:17), Dan said that the game is difficult. Ada disagreed, but later (timestamp 32:15) said it was boring, while Britt replied that it was fun. Nevertheless, on day 1, Ada played a second match against Britt after their first match finished (see Figure 3). On day 4 (timestamp 15:04), she also wanted to play against Britt after her game with Dan (cf., Table A3, line 486–487). All children decided to participate voluntarily on all four days and informed me that they also played Gebeta in their free time. This is a strong indicator that they liked the game after they understood the rules.

4.1.6. Explaining

As already covered in Section 4.1.5, from the second day onward, I observed several instances in which the children explained the rules to one another. An example is given in Table 5 in Section 4.2.3. Additionally, there was one action that belonged to Bishop’s activity explaining, but not to playing. The children classified the counters by colour. They did this when setting up the game’s start configuration and when tidying up after the game was finished.

4.2. Patterns of Repeated Actions

Through the inductive interaction analysis, three patterns of repeated actions were identified. Often, children struggled with the game’s directionality (see Section 4.2.1). Furthermore, they were challenged by the game’s demand to bundle counters into families (see Section 4.2.2). Although it was not always easy to recall and follow the game’s rules, the children demonstrated algorithmic thinking in several instances (see Section 4.2.3).

4.2.1. Directionality

Most of the children sowed correctly anticlockwise most of the time. Only four times on day 1 (timestamps 04:24, 06:05, 12:54, and 27:05) and two times on day 2 (timestamps 08:05 and 12:41), Ada sowed clockwise. This happened when she started at the right-most home, A2. Instead of sowing right-to-left in the opponent’s row, A1, B1, etc., she started in her row at B2. I corrected her, and it did not happen again on days 3 and 4. On day 1, Britt received a lot of help from me and Ada. On day 2, she attempted to make her first move independently. She demonstrated moments of correct sowing, maintaining the one-to-one correspondence between counters and homes in the correct direction (timestamp 11:30). However, she frequently restarted sowing from the home she had just emptied (timestamps 10:38, 10:58, 11:56, and 18:05). Other children occasionally did the same (e.g., Ada on day 4, timestamp 06:29), but not as often as Britt. This procedural gap was particularly evident at row transitions, where Britt sometimes attempted to continue in the same row rather than switching (day 2, timestamp 11:08) or reversed direction from anticlockwise to clockwise (day 2, timestamps 11:10, 12:56, 17:12, and 18:15). A possible explanation is that Britt thinks about a different sowing pattern: left-to-right (see Figure 5), instead of anticlockwise (see Figure 2).

4.2.2. Bundling

The observations show that all children were able to distribute the counters into the homes. When it comes to bundling the captured counters, the children showed different paths of development during the observation period.

• Ada: On day 1 (timestamp 17:30), after their first match, I showed Ada and Britt how to bundle the counters, and they did it. Then (timestamp 18:38), I asked them to count the families, but they counted the counters separately. However, after I had shown them how to count the families (timestamp 19:57), both girls were able to do it. Already after the second match on day 1 (timestamp 32:54), Ada grouped her counters on her own initiative, but she counted the individual counters instead of the families as units. This remained unchanged during all four days. On day 4 (see Table 2), Ada grouped, but preferred to count the counters: “I do it just as I want to” (line 314), even though Dan insisted that she must count the groups (line 315):

Table 2. Transcript of timestamps 08:49–09:31 of Dan’s match against Ada on day 4.

Line	Timestamp	Activity *	Comments
296	08:49	Ada (starts to group her counters): Here are just four.
299	08:55	Ada: I believe I have six.	Britt still wants to sow left-to-right in her row.
300	08:57	Dan: Six? That are more.	Dan estimates the amount.
301	08:59	Dan: I think so.
302	09:01	Ada makes a group of five counters.
304	09:07	Ada (removes the fifth counter from her group of five and starts a new group): I do it just as I want to.
307	09:15	Ada makes a new group of four.	Ada groups correctly.
308	09:17	Dan: No, you must do it right.
310	09:21	Dan: That’s not good. Ada: Let me think.
311	09:23	Ada (has four groups of four): Just here.
313	09:27	Ada (counts the counters in the first group): 1, 2, 3, 4.	Ada does not count the groups.
314	09:29	Ada: I do it just as I want to.
315	09:31	Dan: But you must only count the four counters.

* Text in italics describes actions. Standard text quotes utterances.

Table 2. Transcript of timestamps 08:49–09:31 of Dan’s match against Ada on day 4.

Line	Timestamp	Activity *	Comments
296	08:49	Ada (starts to group her counters): Here are just four.
299	08:55	Ada: I believe I have six.	Britt still wants to sow left-to-right in her row.
300	08:57	Dan: Six? That are more.	Dan estimates the amount.
301	08:59	Dan: I think so.
302	09:01	Ada makes a group of five counters.
304	09:07	Ada (removes the fifth counter from her group of five and starts a new group): I do it just as I want to.
307	09:15	Ada makes a new group of four.	Ada groups correctly.
308	09:17	Dan: No, you must do it right.
310	09:21	Dan: That’s not good. Ada: Let me think.
311	09:23	Ada (has four groups of four): Just here.
313	09:27	Ada (counts the counters in the first group): 1, 2, 3, 4.	Ada does not count the groups.
314	09:29	Ada: I do it just as I want to.
315	09:31	Dan: But you must only count the four counters.

* Text in italics describes actions. Standard text quotes utterances.

• Britt: On day 4 (see Table 3), she bundled the counters on demand. She even insisted on doing it by herself without help (line 425).

Table 3. Transcript of timestamps 13:06–13:22 of Emil’s match against Britt on day 4.

Line	Timestamp	Activity *	Comments
421	13:06	Britt starts grouping her counters.	Britt can group.
423	13:08	R: Since you capture groups of four, you can just count how many groups of four you have.
424	13:12	The practitioner helps Britt group her counters.
425	13:14	Britt pushes together the groups that the practitioner had created. Practitioner: I have just made groups of four.	Britt wants to group on her own.
426	13:17	Practitioner: What do you do now?
427	13:19	R: But Britt can do it on her own.
429	13:22	Britt continues grouping her counters.

* Text in italics describes actions. Standard text quotes utterances.

Table 3. Transcript of timestamps 13:06–13:22 of Emil’s match against Britt on day 4.

Line	Timestamp	Activity *	Comments
421	13:06	Britt starts grouping her counters.	Britt can group.
423	13:08	R: Since you capture groups of four, you can just count how many groups of four you have.
424	13:12	The practitioner helps Britt group her counters.
425	13:14	Britt pushes together the groups that the practitioner had created. Practitioner: I have just made groups of four.	Britt wants to group on her own.
426	13:17	Practitioner: What do you do now?
427	13:19	R: But Britt can do it on her own.
429	13:22	Britt continues grouping her counters.

* Text in italics describes actions. Standard text quotes utterances.

When asked to count the groups (see Table 4), she counted that there are four counters in each group (lines 470–491). However, she could count the groups after I had shown her once again how to do it (lines 495–499). When the practitioner asked, “So, Britt, how many groups do you have?” she answered correctly without having to count again. This indicates that Britt has understood the cardinal principle.

Table 4. Transcript of timestamps 14:33–15:44 of Emil’s match against Britt on day 4.

Line	Timestamp	Activity *	Comments
468	14:33	Britt finished her grouping.
469	14:34	R: Now, you can count how many groups of four you have.
470	14:37	Britt (counts the counters in one group): 1, 2, 3, 4.
471	14:39	R: How many groups?
472	14:41	Britt (starts counting the counters in the next group): 1, 2. Emil (shows Britt how to count groups): No, like this 1, 2, 3, 4.	Britt does not count how many counters she has nor how many groups. She checks if there are four counters in each group.
473	14:45	Britt: No. I don’t do it like this. Emil (continues counting Britt’s groups): 5, 6.
474	14:47	Britt counts the counters in the first group again.
475	14:49	Britt: 1, 2.
476	14:51	Emil: It is not like this.
477	14:53	Britt: 3, 4.
478	14:55	Britt (counts the counters in the second group): 1.
479	14:57	Britt: 2.
480	14:58	Britt (starts over again in the second group): 1.
481	14:59	Britt: 2
482	15:00	Britt: 3.
483	15:01	Britt: 4.
484	15:02	Practitioner: Let her count.
485	15:03	Britt (counts the counters in the third group): 1.
486	15:04	Britt: 2, 3, 4.
487	15:06	Britt (counts the counters in the fourth group): 1, 2, 3, 4.
488	15:12	Britt (counts the counters in the fifth group): 1, 2, 3, 4.
489	15:13	Britt (counts the counters in the sixth group): 1, 2, 3, 4.
490	15:15	Britt (counts the counters in the seventh group): 1, 2, 3, 4.
491	15:18	Britt (counts the counters in the third group again): 1, 2, 3, 4.
493	15:21	Britt: Four.
494	15:22	Britt: I have four.	Does she mean ‘groups of four’?
495	15:24	R (points at Britt’s families): Britt, can you count how many groups of four you have?
497	15:29	R (points at Britt’s families): 1, 2, 3, 4, …
498	15:31	Britt (counts the families): 1, 2, 3, 4.
499	15:33	Britt (continues counting the families correctly): 5, 6, 7.	Britt can count the groups.
500	15:38	Emil distributes his counters into the homes on the board.
501	15:40	Practitioner: So, Britt, how many groups do you have?
502	15:44	Britt: 7. Emil starts picking up Britt’s counters.	Britt is a CP-knower.

* Text in italics describes actions. Standard text quotes utterances.

Table 4. Transcript of timestamps 14:33–15:44 of Emil’s match against Britt on day 4.

Line	Timestamp	Activity *	Comments
468	14:33	Britt finished her grouping.
469	14:34	R: Now, you can count how many groups of four you have.
470	14:37	Britt (counts the counters in one group): 1, 2, 3, 4.
471	14:39	R: How many groups?
472	14:41	Britt (starts counting the counters in the next group): 1, 2. Emil (shows Britt how to count groups): No, like this 1, 2, 3, 4.	Britt does not count how many counters she has nor how many groups. She checks if there are four counters in each group.
473	14:45	Britt: No. I don’t do it like this. Emil (continues counting Britt’s groups): 5, 6.
474	14:47	Britt counts the counters in the first group again.
475	14:49	Britt: 1, 2.
476	14:51	Emil: It is not like this.
477	14:53	Britt: 3, 4.
478	14:55	Britt (counts the counters in the second group): 1.
479	14:57	Britt: 2.
480	14:58	Britt (starts over again in the second group): 1.
481	14:59	Britt: 2
482	15:00	Britt: 3.
483	15:01	Britt: 4.
484	15:02	Practitioner: Let her count.
485	15:03	Britt (counts the counters in the third group): 1.
486	15:04	Britt: 2, 3, 4.
487	15:06	Britt (counts the counters in the fourth group): 1, 2, 3, 4.
488	15:12	Britt (counts the counters in the fifth group): 1, 2, 3, 4.
489	15:13	Britt (counts the counters in the sixth group): 1, 2, 3, 4.
490	15:15	Britt (counts the counters in the seventh group): 1, 2, 3, 4.
491	15:18	Britt (counts the counters in the third group again): 1, 2, 3, 4.
493	15:21	Britt: Four.
494	15:22	Britt: I have four.	Does she mean ‘groups of four’?
495	15:24	R (points at Britt’s families): Britt, can you count how many groups of four you have?
497	15:29	R (points at Britt’s families): 1, 2, 3, 4, …
498	15:31	Britt (counts the families): 1, 2, 3, 4.
499	15:33	Britt (continues counting the families correctly): 5, 6, 7.	Britt can count the groups.
500	15:38	Emil distributes his counters into the homes on the board.
501	15:40	Practitioner: So, Britt, how many groups do you have?
502	15:44	Britt: 7. Emil starts picking up Britt’s counters.	Britt is a CP-knower.

* Text in italics describes actions. Standard text quotes utterances.

• Carl: In all his matches, Carl bundled the counters and counted the families on his own initiative. He must have learnt this skill before the project.
• Dan: On day 1 (timestamp 19:26), he bundled his counters but counted them individually. On subsequent days, he bundled them himself and insisted that the other children count the bundles as well (e.g., Table 2, line 315). Only when Ada refused to count the families did Dan do the same (timestamp 09:37). At the end of day 4 (timestamps 12:59–13:22), he even tried to count the bundles without grouping the counters, but this was too difficult.
• Emil was familiar with the game already before the project started. He bundled and counted families all by himself (e.g., day 2, timestamp 45:11). On several occasions, he showed the other children how to do it. You can find examples in Appendix A (Table A3, lines 326, 329, 331, and 419). Nevertheless, he sometimes preferred to count the individual counters. On day 2 (timestamp 44:07), after he had counted his 16 counters, I stated that I had 16, too. Emil acknowledged. How did he know? Most likely, he subitised that my four groups of four contain the same amount as his four groups. However, when he later compared his five families to my six families, he counted the groups (see Figure 6).

4.2.3. Algorithmic Thinking

It was challenging for the children to recall and apply all the steps. I have observed several instances where the children sowed in the wrong direction, started sowing in the wrong home, failed to notice that they had created a family, or broke the rules in other ways. However, most of the time, the children followed the rules correctly. Sometimes, the children explained the rules to one another and helped each other. When they explained the rules, they demonstrated the step-by-step procedures with physical demonstrations of what the players should do. However, there was a clear distinction between explaining actions and actual game moves. For example, when Ada explained the sowing at the beginning of day 2 (see

Table 5. Transcript of timestamps 00:30–00:46 of Emil’s match against Ada on day 2.

Line	Timestamp	Activity *
16	00:30	R: Do you remember how to start?
17	00:32	Ada (moves her hand to Britt’s A1): Well, first you must do it …
18	00:34	Ada (picks up the counters): …like…
19	00:36	Ada (shows the counters in her hand): …this.
20	00:38	Ada (closes her hand and moves it to Bitt’s A2): Then, we take one…
21	00:40	Ada (points with her thumb down at Britt’s A2): …here.
22	00:42	Ada (points with her thumb down at Britt’s A3): And here.
23	00:44	Ada (points with her thumb down at Britt’s A4): Here.
24	00:46	Ada (points with her thumb at Britt’s A5): And there.
25	00:48	Ada: And so on and on and on.

* Text in italics describes actions. Standard text quotes utterances. Homes are labelled like on a chessboard (see Figure 2). Ada and Carl own F2 to A2, and Emil and Britt own A1 to F1.

), she picked up the counters, but she did not sow them. She just pointed one-by-one at the homes where the counters should be placed. Even though she used the word “take” when demonstrating the sowing, her gesture made it clear that she meant putting down or dropping the counter.

The observations showed that Gebeta is robust against rule violations. Not every mistake that the children made was immediately corrected. I only noticed many mistakes afterwards during the data analysis, because during the gameplay, I could not focus on both parallel matches simultaneously. (An example is shown in Appendix A,

Table A1. Transcript of Britt’s first turn in her match against Carl on the second day.

Line	Timestamp	Activity *	Comments
314	10:34	Britt takes all the counters from A1.
315	10:38	Britt puts the first counter back to A1. R: Here is the first one. (points to B1)	She starts at the same place where the counters came from.
316	10:40	R: That was well the one you had taken. R takes the counter from A1.
317	10:42	R: This is the first one. I put the counter in B1.
318	10:44	R: The second. Britt puts a counter in C1.
319	10:46	R: The third. Britt puts a counter in D1 and E1. R: And the fourth.
320	10:48	R (points to E1): And now you take all here.
321	10:50	Britt takes the counters from E1.
…
325	10:58	R (is focused on the other match): Yes. Britt starts sowing at E1.	She starts at the same place where the counters came from.
327	11:02	Britts puts a counter in F1.
328	11:04	Britt (looks at the three counters in her hand): There are only these left here.
329	11:06	R (points at the homes in the second row): Yes, you must continue.
330	11:08	Britt puts a counter in A1. R (points at F2): Here, you must continue here.	She attempts to continue in the same row, not the other one.
331	11:10	Britt moves the counter from A1 to A2. R (points again at F2): Here!	She attempts to sow clockwise.
332	11:12	R (points at A2 and moves finger to F2): Here, here, here. Britt takes the counter from A2.
333	11:14	R (points at F2): Here. Britt puts the counter in F2.
334	11:16	R (points at E2): And there. Britt puts a counter in E2 and D2.
335	11:18	R (points at D2): And now, you must take all these.
336	11:20	Britt takes all counters from D2. R (points at C2): And continue.
337	11:22	Britt puts the counters on the table.	She thinks she has won the counters.
338	11:24	Britt picks the counters up. R (points at C2): You must continue from here.
…
341	11:30	Britt sows the counters to C2, B2, and A2.	She sows correctly.
…
344	11:38	R (points at A1): Here. Britt puts a counter in A1.	She needs help at the transition from one row to the other.
345	11:40	R (points at B1): And here. Britt puts a counter in B1.
346	11:42	R (points at B1): And now, you take all these. Britt takes all the counters from B1.
347	11:44	R (points at C1): And distribute.
348	11:46	Britt looks at the counters and asks something unintelligible.
349	11:48	Britt puts the counters on the table. R: You must distribute them.	She thinks she has won the counters.
…
353	11:56	Britt starts sowing at B1. R (points at C1): Here. It starts here.	She starts at the same place where the counters came from.
354	11:58	R (takes the counter from B1): Not here.
355	12:00	R (puts the counter in C1): It starts here.
356	12:02	R (points at D1): And then here. Britt puts counters in D1, E1, and F1.
…
362	12:14	Britt (looks at the two remaining counters in her hands): I have only two.
363	12:16	R (points at C2): Yes.	Because I thought the two counters came from the empty home D2.
364	12:18	R (points at C2): Here. Britt hesitates. R: Where do they come from?
365	12:20	R (points at D2): They have been here?
366	12:22	Britt puts a counter in D2. R (takes the counter from D2): But you must distribute...	She starts at the same place where the counters supposedly came from.
367	12:24	R (puts the counter in C2): ...here and here. (points at B2)
368	12:26	Britt puts the counter in B2 and waits.	She does not know how to continue.
373	12:36	Britt still waits. R (points at B2): Was the last one here, now?
374	12:38	Britt: Yes. R: Then you must take all these here.
…
382	12:54	R (points at A2): Then, you continue here.
383	12:56	Britt puts a counter in A2 and the next one in B2.	She sows in a clockwise direction.
384	12:58	R (moves the counter from B2 to A1): No.
Britt’s first turn continues for a while in the same manner.
498	17:12	Britt distributes counters from A2...	She sows in a clockwise direction.
…
505	17:25	... to F2.
507	17:31	Britt continues distributing from A1…	In her row, she sows in an anticlockwise direction.
508	17:36	... to E1.
510	17:40	Britt (attempts to take the counters from C1): Can I take these because there are so many?	She does not remember that she must continue from E1.
511	17:42	R: Hm? Britt: Can I take these because there are so many?
513	17:46	R: Was the last one there?
514	17:48	Britt: Yes.
515	17:49	R: Really?
516	17:50	R: Yes, then you can take these.
…
522	18:03	Britt (looks at the counters in her hand): There are so many.
523	18:05	Britt starts sowing at C1 and continues to D1 and E1.	She starts at the same place where the counters came from.
525	18:09	R points at F1. Britt puts a counter in F1.
527	18:11	R points at F2. Britt puts a counter in F2.
529	18:15	Britt sows counters in D2, E2, and F2.	She sows in a clockwise direction.
…
532	18:20	R (moves the counter from F2 to C2): The last one is here.
533	18:23	R takes the counters from C2 and distributes them to B2 and A2.	R finishes Britt’s turn.
534	18:26	R takes the counters from A2 and distributes them to A1, B1, and C1.
535	18:27	R (takes the counters from C1 and distributes them to D1 and E1): Yes. Now, here are four and here are four.
536	18:30	R (takes the four counters from B1 and E1 and moves them to Britt’s site): And now, you win these.

* Text in italics describes actions. Standard text quotes utterances. Homes are labelled like on a chessboard (see Figure 2). Britt owns A1 to F1, and Carl owns F2 to A2.

: I corrected Britt in line 315, but not in line 325.) Despite the children’s mistakes, the game’s flow remained intact. This suggests that not every error requires immediate correction. However, during the learning stage, it is essential to address the mistakes to help children understand and internalise the proper rules.

During the final stage of a match, when there were only a few counters left, I often observed that the children ran out of patience. The last moves can be boring when there is not much you can do. Thus, sometimes I allowed ending the game even though there were some counters left on each side. However, in most cases, the children played until one player had to pass once (instead of three times). Then the player who did not pass captured the remaining counters. Even though this rule is easier than the one described in Section 1.2.1, there were nevertheless opportunities for endgame strategies: Players were actively avoiding moving counters over to the opponent’s side. (You can find an example in Table A2, line 1277). Finally, I skipped this rule that a game consists of several matches when I realised that the children lacked the perseverance to keep playing match after match. They were eager to play one match at a time, perhaps a second match against a different opponent, but did not want to play several matches against the same opponent.

5. Discussion

The findings show that children playing Gebeta spontaneously engage in these activities: they had to count one-by-one and count captured collections, and they had to locate the next homes and maintain a consistent directional pattern. In fact, counting was the predominant mathematical action during play. The children subitised small groups (families of four) and carefully tracked how many counters they had captured. Correcting their directionality also required them to focus on spatial language, reinforcing concepts of forwards/backwards and clockwise/anticlockwise that are crucial in mathematics. Additionally, most children used the Gebeta materials to engage in designing activities, even though designing is not necessary when playing Gebeta. This indicates that the material is appealing to children and provides opportunities to explore more mathematics than initially expected.

The observations reveal that young children quickly engage with Gebeta through guided play, demonstrations, and trial-and-error learning. Initially unfamiliar with the game, the 5-year-olds learnt basic play patterns by imitation (watching the instructor and peers) and by having the rules modelled for them. For example, once shown how to pick up all counters from a home and sow them one by one, most children could follow this procedure, demonstrating one-to-one correspondence between counters and homes, which is a key early counting skill (Gelman & Gallistel, 1978). They readily counted out moves and kept track of counters, often using pointing and verbal counting. By the end of the intervention, children consistently practised placing one counter per home in sequence, even if some still needed prompting about where to start or which way to turn. In this way, they quickly mastered many basic aspects of the game: understanding that all counters in a chosen hole must be distributed, that turns alternate, and that completing a “family” of four counters earns a capture. However, they continued to struggle with correct procedural direction and group counting.

5.1. Directionality

Specifically, directionality (the rule that counters must be sown anticlockwise to the next home) was a central challenge. Some children often restarted from the same home they had just emptied or continued in the same row rather than switching. For example, one child repeatedly attempted to continue sowing in the row she started in instead of moving into the opponent’s row, and at times she reversed the intended direction (sowing clockwise) until prompted otherwise. This indicates that the concept of moving to the next adjacent home in the correct sequence was not yet intuitive. Such difficulties highlight the emerging nature of locating skills (understanding spatial direction and order) in early learners (MacDonald, 2025). In Gebeta, children had to locate the correct next home repeatedly. Early errors suggest that they were still building that sense. Games with explicit directional moves are known to support this skill (Ramani & Siegler, 2008). Siegler and Ramani (2009) have shown that linear games involving numbers facilitate the development of numerical skills, while circular games do not. Gebeta is an interesting case because the board is linear, but the playing direction is circular, and the board does not display numerals, yet the game requires counting and moving in the correct direction. In practice, working out the correct anticlockwise path in Gebeta encouraged the children to develop spatial sequencing habits. Over successive turns, they became more accurate at beginning to sow in the next home (rather than the emptied one), avoiding off-by-one errors in counting. This practice of attention to sequential order has clear links to early school arithmetic and may prevent typical mistakes as described by Nelson and Powell (2018).

5.2. Bundling

The children also showed developing facility with bundling. In Gebeta, this emerged as treating each assembled group of four counters (a “family”) as a single unit. By the end of the intervention, most children had begun to count captured families as units, rather than as separate counters. Initially, however, they tended to count all counters individually. With minimal instruction, they learnt that a bundle of four counters could be treated as a single unit. For example, once shown how to bundle her captured counters into piles of four, one child immediately recognised that four groups of four made 16, effectively applying the cardinal principle of counting and hinting at multiplicative reasoning. The children were moving beyond counting single items to recognising larger, structured quantities, a key step toward understanding place value (Wege et al., 2023). In fact, bundling in Gebeta can be seen as an informal base-4 grouping: when they counted collected counters as families, they practised unitising (treating four as one unit), much like grouping tens in our number system. This supports number-system understanding, since children who recognise “3 families” as meaning 3 times 4 counters are effectively engaging in early multiplication.

5.3. The Adults’ Role

When it comes to the question of how to help children learn mathematics, the balance between guided instruction and play-based learning is a crucial topic (E. A. Wood, 2022). In my observations, the presence of the researcher was an essential element of the learning process. In line with Vygotskian perspectives on the zone of proximal development (Vygotsky, 1978), the adult provided scaffolding by modelling moves, clarifying rules, and prompting children when they encountered difficulties (D. J. Wood et al., 1976). Without such guidance, children might have taken considerably longer to grasp central features of the game, such as directionality or bundling. At the same time, the practitioners’ non-intrusive presence gave the children a sense of security and encouraged sustained engagement. This highlights that while Gebeta has rich mathematical affordances on its own, these affordances become educationally meaningful only when an adult deliberately draws attention to them, asks productive questions, and gradually withdraws support. Thus, the role of the adult is not to dominate the play but to sensitively mediate it, ensuring that mathematical ideas embedded in the game surface and are appropriated by the children.

5.4. Summary

In summary, the game of Gebeta naturally highlights two key mathematical themes: procedural directionality and bundling. Challenges related to directionality (such as starting the sequence in the correct hole and moving anticlockwise) engage spatial locating skills. These mirror classroom exercises, like reading number lines correctly or understanding the order of digits in place-value problems. Grouping counters into families develops counting and number sense, helping children view multi-quantity collections as single units, which is a foundational step toward understanding place value and multiplication. Both themes correspond with Bishop’s core activities: locating (movement in space and direction) and counting (numerical quantification).

These learning patterns confirm that Gebeta is rich in mathematical affordances. Previous research on Mancala-type games notes that such games involve counting, planning and problem solving (Tesfamicael & Farsani, 2024; Usta & Cagan, 2022). This is evident in Gebeta: players must constantly update their count of counters in each home and anticipate where the last seed will fall, which can change ownership of a family. In my observations, after each sow, the children often calculated “How many do I have in that home now?” or “If I move here, will I get a family?” This kind of procedural thinking, chaining one counting move to the next, is precisely the kind of activity that early mathematics education aims to support. Alehegn (2022, p. 1) explains that Gebeta teaches children how to count: “During Gebeta play children learn number sequence, forecast future landing spots, and engage in procedural thinking.” In other words, children naturally exercised early number sense, counting and basic prediction in a fun game context.

These observations align with recent research on play-based mathematics learning. Systematic reviews (Maffia & Silva, 2025; Sousa et al., 2023) emphasise that only a small portion of play studies explicitly document mathematics learning outcomes, highlighting the importance of examining a traditional game like Gebeta that naturally promotes mathematical engagement. Recent empirical work demonstrates that board games can support number sense and spatial reasoning (Gasteiger & Moeller, 2021; Vlassis et al., 2023; Vogt et al., 2018). The present findings expand this literature in two ways. First, while many studies have focused on games of chance (such as “Chutes and Ladders”) or linear board games that reinforce number lines (Ramani & Siegler, 2008; Siegler & Ramani, 2009), Gebeta is a strategy game with circular movement and no printed numerals. Yet, it elicited consistent counting, attention to sequential order, and the development of spatial locating skills. Second, research on play-based mathematics often emphasises early arithmetic or geometry (Maffia et al., 2025), whereas this study demonstrates how a culturally grounded game can foster less commonly studied but equally fundamental competencies: directionality and bundling. In this way, the study offers a new perspective to the field by showing that traditional strategy games can cultivate mathematical ideas that go beyond those typically addressed in classroom-oriented play activities.

5.5. Limitations

Despite these promising observations, the study has limitations. The sample was small, not randomly selected, and observed over only a few sessions. Therefore, I cannot claim generalizability. I did the coding on my own. This might limit reliability. However, in the previous study (Thiel et al., 2024), three researchers coded similar observations, and the inter-rater reliability was sufficient. All children were from one educational context (a Norwegian kindergarten) and had similar exposure. Children from other backgrounds might respond differently. I also guided play and sometimes simplified explanations (for instance, emphasising anticlockwise movement), so a pure free-play situation might yield slower mastery of rules. Future research could address these issues by studying a larger, more diverse group of children over a longer period. It would be valuable to compare Gebeta with other board games, such as Chutes and Ladders, to see if the directionality and bundling benefits are unique to mancala-type games. Formal assessment of learning outcomes (e.g., pre-post tests of number skills) would clarify how much playing Gebeta improves specific mathematical competencies. Finally, it would be interesting to investigate how long the learning effects last and whether children transfer skills from Gebeta to tasks like counting in everyday play or solving arithmetic problems.

6. Conclusions

This study asked how preschool children unfamiliar with Gebeta learn and play the game. The observations showed that the children quickly appropriated the basic rules through modelling, guided play, and peer interaction. They mastered core procedures such as distributing counters one-by-one, recognising when a turn ended, and capturing families. At the same time, they were challenged by the procedural complexity of the game, especially maintaining the correct anticlockwise direction and treating groups of counters as units rather than as individual items.

The second research question concerned the mathematical skills that children develop through Gebeta. The findings indicate that the game fosters early number sense, including one-to-one correspondence, systematic counting, and subitising. Moreover, Gebeta provided rich opportunities to practise bundling, i.e., treating groups of four counters as single units, which is foundational for understanding place value. Directionality emerged as another key competency: children repeatedly engaged with spatial orientation and sequencing, which are critical for avoiding common counting and arithmetic errors in school mathematics.

Taken together, the study demonstrates that Gebeta offers a culturally rich and engaging pathway into early mathematics. Its rules invite children to practise both spatial and numerical reasoning in ways that complement other play-based approaches, highlighting the potential of traditional strategy games in early childhood education.