The construction of number in concrete and electronic games: the Kalah game case

A construção do número em jogos concretos e eletrônicos: o caso do jogo Kalah

Abstract

Games have sparked interest as an object of study in multiple areas of knowledge. In this research, we will discuss the impacts of the new formats and configurations of games in current times, based on the Piagetian interactionist theory’s contributions. The main objective of the study was to analyze the effects of an intervention with the Kalah game, in its concrete version and in its electronic version, for the construction of the notion of number. Thirty-eight children took part in the investigation, their ages ranged from seven to ten years old, from both genders, enrolled at two county Elementary Schools, located in a small town in the state of Minas Gerais, Brazil, subdivided into 3 groups: EG¹ - Experimental Group, subjected to the intervention with Kalah in its electronic version; EG² – Experimental Group, subjected to the intervention with Kalah in its concrete version and CG-Control Group. As instruments of pretest and posttest, the operatory tests/proofs of conservation, classification, and serialization were utilized, to verify if there were qualitative and/or quantitative changes to the thought’s structures. The results indicated the most expressive differences in the assessed structures amongst the subjects from the Experimental Groups when compared to the Control Group, just as there was a distinction in the performances of the subjects from the experimental groups, after the intervention. During the sessions with the Kalah electronic game, it was found that the fundamental principles of the Piagetian theory could not be assured, possibly being one of the pedagogic interventions’ limitations.

Keywords
Electronic Games; Concrete Games; Genetic Epistemology; Number notion

Resumo

Desde muito tempo, os jogos despertam interesse como objeto de estudo em múltiplas áreas do conhecimento. Especificamente nesta pesquisa, discutiremos os impactos dos novos formatos e configurações dos jogos nos tempos atuais, a partir das contribuições da teoria interacionista piagetiana. O objetivo principal do estudo foi analisar os efeitos de uma intervenção com o jogo Kalah, na sua versão concreta e na sua versão eletrônica, para a construção da noção de número. Participaram da investigação 38 crianças, com idades entre sete e dez anos, de ambos os sexos, matriculadas em duas escolas municipais de ensino fundamental, localizadas em uma cidade de pequeno porte do estado de Minas Gerais, Brasil, subdivididas em 3 grupos: GE1- Grupo Experimental, submetido à intervenção com Kalah em sua versão eletrônica; GE2 – Grupo Experimental, submetido à intervenção com Kalah em sua versão concreta e GC-Grupo Controle. Como instrumentos de pré-teste e pós-teste foram utilizados os testes operatórios/provas de conservação, classificação e serialização, para verificar se houve alterações qualitativas e/ou quantitativas nas estruturas do pensamento. Os resultados indicaram diferenças mais expressivas nas estruturas avaliadas entre os sujeitos dos Grupos Experimentais quando comparados ao Grupo Controle, assim como houve distinção nos desempenhos dos sujeitos dos grupos experimentais, após o período de intervenção. Durante as sessões com o jogo eletrônico Kalah, constatou-se que os princípios fundamentais da teoria piagetiana não puderam ser assegurados, sendo possivelmente uma das limitações das intervenções pedagógicas nesta modalidade de jogo.

Palabras-clave
Jogos eletrônicos; Jogos concretos; Epstemologia genética; Noções de número

1 Introduction

Since long ago, games have been present in several cultures and contexts, highlighting the fact that men will play regardless of their time and place, and that ludic activities are as old as mankind. An example of this is the panel “Children’s Games” (1560), by Pieter Bruegel (1525-1567), in which the artist depicts approximately a hundred traditional children’s games, that were a part of the sixteenth century’s ludic activities and, even in different times and places, we’re capable of recognizing countless of them: hobbyhorses, kites, hopscotch, leapfrog, dolls, tag, stone skimming, hula-hoops, amongst others.

Classical theorists that have studied games through the sociological perspective, such as Huizinga (1872-1945) and Caillois (1913-1978), have highlighted their role in a society’s cultural dynamics. One of the main aspects defended by Caillois in his work “Man, Play and Games”, originally, Les Jeux et lês hommes, from 1958, is that a society’s culture can be comprehended through games. For him “what’s expressed through games is no different than what’s expressed through culture.” (Caillois, 2017, p. 112), and there is a close relationship between these connivance and compensation.

Gilles Brougère, philosopher and anthropologist, brings forth a contemporary interpretation of this phenomenon, in which he argues that there is a profound connection between games and culture, them (the games) being both a cultural process and product: “they truly consist of a social act that generates a specific culture (a set of different meanings) and are, at the same time, generated by a culture.” (Brougère, 1998, p. 29).

From this perspective, games and culture are developed amidst the ludic, with the ludic being defined in this study as freedom of expression, going against Buytendijk’s (1977) views. In this regard, Brougère (1998, p. 30) says that “games are, above all else, a ludic culture’s source of development (or creation, but this word is, sometimes, dangerous!)”

The author comprehends that, ludic activities change as the individuals and social groups change, due to time and place, climate conditions, habits, amongst other things. In this context, the child is seen as a social-historical individual, the maker of childhood’s ludic culture, developing distinct games according to their interests, fashions and other current influences (Brougère, 1998). Therefore, if the ludic culture is diversified according to social settings, age, sex, amidst others, “ergo, games are also transformed and/or adapted according to the demands, through which the individuals are conditioned” (Scaglia; Fabiani; Godoy, 2020, p. 192-193).

Certainly, were a child to look at the same Bruegel panel today, they would identify some of the children’s games depicted by the artist. However, they would notice the lack of others that are a part of current times, such as videogames. It’s notorious that the growing use of digital technologies is a twenty-first century staple, holding a prominent place in (and not only) children’s day-to-day life. Studies show that children from western countries, aging between 2 and 8 years old, have a recreational digital consumption of approximately 3 hours a day, resulting in over 1000 hours (1,4 month) per year of screen exposure.

A study done by TIC Kids Online Brazil 2018 (CGI.br, - Brazil’s Internet Management Committee, 2019), on children’s (9-10 years old) internet usage, showed an increase in the popularity of online games connected to other players, which went from 38%, in 2017, to 55%, in 2018. Therefore, playing on the internet is amongst this public’s main reported activities, whether it be multi-player, more prevalent amongst the boys (71%) than the girls (39%); or even by themselves – a common practice for 65% of the boys and 56% of the girls. It’s worth noting that the audience of games that are not multi-player is younger, children between the ages of 9 and 10 years old.

Although the TIC Kids Online’s studies are done with children aged nine or older, studies done by Brazil’s Pediatric Society (SBP) maintain that the access to gadgets with digital screens is being granted earlier and earlier, “Always with the goal of getting the child to “settle down’” (SBP, 2019, p. 3). Because of this, the presence of electronic games in children’s day-to-day life is inevitable, which can be explained by (Brougère, 1998, p. 26) in these words: “a new technique creates new ludic experiences that transform many children’s ludic culture. All of this shows the importance of the object in the development of the current ludic culture.”

In this context, it must be considered that this new element, when incorporated to children’s daily lives, more and more abruptly as time goes by, brings forth a new branch to be studied: what would be the impact of this transformation of the current ludic culture to child development? “To sum up: there’s a concern regarding this phenomenon when it comes to the development of the newer generations, due to its negative and standardizing aspects” (Arruda, 2011, p. 26).

Organs and institutions from several parts of the world, (World Health Organization - WHO); American Academy of Pediatrics- AAP; Brazil’s Pediatric Society - SBP), concerned about health, during early childhood and adolescence, formulated documents and manuals alerting society to the importance of parents, educators, pediatricians, among others, to be mindful of screen usage and aid children and teenagers to do it conscientiously and responsibly.

According to SBP, not only is being exposed to screens for longs periods of time harmful, but also, the passive distraction influenced by the consumption of “apps and videos on the screen” and the entertainment industries’ publicity, “[...] differ greatly from active play, a universal and timeless right of all children and teenagers, during the cerebral and mental development stage.” (SBP, 2019, p. 3).

About this, Desmurget (2021), neuroscientist, has recently published a book titled “The Digital Cretin Factory”, providing evidence of many studies throughout time, confirming the real dangers of screens, and alerting to the greater consequences, if we continue to use digital technologies without thinking critically. One of the aspects he pointed out that draws attention is that just thirty minutes per day of screen exposure is enough to start affecting a child’s intellectual development. It would affect all aspects, the somatic (obesity and cardiomyocyte maturation), the emotional (anxiety or aggressive behavior), besides the cognitive (for example, language, focus); and many others (Desmurget, 2021).

Other studies also corroborate the negative effects on several developmental aspects , which are mostly related to the type of content, excessive screen exposure, premature access, among others, such as: the development of a more aggressive and anti-social personality, restricted imaginative repertoire and academic interests (Singer; Singer, 2007); difficulty establishing boundaries between fantasy and reality (Zaia, 2008); poorer performance in school, less empathic behavior and larger acceptance of violence (Dongdong et al., 2012); varied health issues such as anxiety, depression, attention deficit and hyperactivity, sleep disorders, overweight and obesity, hearing loss, myopia, postural disorders etc. (SBP, 2018); premature thinning of the cerebral cortex in children exposed to more than seven hours per day – according to the study conducted by the National Institute of Health (NIH) (SBP, 2018).

Furthermore, concrete and virtual toys and games can be increasingly more functional and utilitarian; less active and creative children, speeding up the disappearance of a ludic and cultural patrimony that has lasted for centuries (Scaglia; Fabiani; Godoy, 2020).

Contrary to the previously mentioned authors, there are studies that present positive results and, in their majority, utilized games and digital apps for educational purposes, as for instance: the Literácia which aids the development of the written alphabetical system of students in the beginning stages of the reading/writing process (Kerscher, 2017); the rule-based electronic games Protocolos and Zona Trash 3, which were considered good instruments to evaluate the operative notions of deductive logic and spatial awareness, in children that have shown signs of ADHD (Rossetti et al, 2014).

On top of these, other games adapted to digital contexts such as Tower of Hanoi, Rush Time, The Peg Game, Sudoku, among others, have contributed to the improvement of the attention spam, maturity, higher problem-solving speed, autonomy and persistence (Ramos; Anastácio 2018); just as students have shown improvement solving math problems due to the electronic game, called The Room (Silva, 2008); Minecraft, Far Cry 4, Tomb Raider, Prince of Persia: The Forgotten Sands, Uncharted: Drake’s Fortune, Assassins’ Creed II and The Saboteur, have proven to be effective when it comes to acquiring knowledge of geographic concepts, such as space, scenery and orientation (Carneiro, 2019); the cognitive development of students that utilized Minecraft showed improvement, allowing for a greater comprehension of logical mathematical concepts in a more abstract manner (Schifter; Cipollone; Moffat, 2013); Many digital games have enabled the improvement of the individual’s writing level (Andrade, 2021).

Very well, being faced with the current scenario warrants the need of developing other studies that contribute to the conversation regarding the impact of screen exposure in children’s development and if this virtual environment qualifies as propitious for the child to play, learn, interact. From this point of view, the current investigation recognizes the new settings, resignifications, and current game formats and, because of this, suggests that this phenomenon be looked at through an educational and phycological perspective, being based on the assumptions of the interactionist and constructivist theory of the Swedish biologist and epistemologist Jean Piaget (1896-1980).

Thereby, it springs from the following questions: what would happen, when it comes to developing the basis of intelligence, when using a concrete game and the same game, but its electronic version? What implications would one type of playing or the other have on a child’s development? In which aspects? Can electronic games replace concrete games, seeing as many of them are already available in this new, easily accessible version? This new way of acting, that resulted from technological advancements, induces phycological changes in the individual and modifies the way they learn? Can it influence the development of basic logical foundations, comprehended through the lens of the Piagetian theory?

Based on these questions, the current study presents the result of a research that compared the effects of a pedagogical intervention, based on constructivist principles, utilizing the Kalah game in its concrete and electronic versions, for the development of basic logical foundations.

2 The construction of Number in Piaget

The epistemological stance, adopted by Piaget, is based on the interpretation that the acquisition of new knowledge is a gradual and continuous process, that begins at birth and continues to evolve until it reaches a stage where they are able to incorporate deductive reasoning, highlighting the fact that a child’s thought process differs from an adults’, because certain logical mathematical structures aren’t innate and, because of this, are not accessible during all ages. About this, (Piaget, 1964, p. 78) claims that:

An operation is an internalized action which becomes reversible and is coordinated with other operations into an integrated operatory grouping. However, since operations materialize only toward seven or eight years of age, there is a long “preoperational” period of development which corresponds to what I used to call the “prelogical” period. (Operations themselves are formed in two successive stages: the “concrete” stage between seven and eleven years of age, and the “formal” or propositional stage, which appears only at age eleven to twelve years.).

By dedicating himself to the study of the genesis of number, Piaget, and his contributors Inhelder e Szeminska, used different experiments with balls of play dough, chips, liquids, rods, flowers, and fruits etc. that made it possible to evaluate children’s thought processes and explain how the number is organized in tandem with the gradual evolution of the classifications and serializations. Those studies can be found in the works “A Gênese do Número na Criança” (1975) and “A Gênese das Estruturas Lógicas Elementares” (1975). Amongst the main results of these works, it stands out that the simple fact that a child can count verbally, doesn’t mean they have grasped the idea of number, because:

A five-year-old individual is very much capable of, for example, enumerating the elements of a row of five chips and think that, if you separate the five chips in two sub-groups of 2 and 3 elements, these sub-collections do not equate, when put together, to the whole initial collection. (Piaget; Szeminska, 1975, p.15).

For this reason, for Piaget, the number depends on a grouping operatory structure, that takes, in average, 7/8 years to be developed. To comprehend the levels of evolution and the processes used by the children whilst developing the logic of the numbers, and arrange the answers in stages, we resorted to the structures, considered achievements of intellectual development and that, in Piagetian terms, are not innate. We highlight the conservation, considered one of the first invariants identified in children, taking time to be elaborated and representing the closure, or almost, of the whole construction (Banks-Leite, 1997).

Another problem posed for the child to solve, when they’ve reached the concrete operations’ levels is that of the inclusion of class, in which it’s needed to compare or simultaneously consider the class and the whole, until a logical solution based on the reversibility is constructed, making it possible to hierarchize, combine, and detach classes “thinking of the same elements as simultaneously present in different classes, the part and the whole” (Carraher, 1983, p.96).

The assortment of element’s magnitudes, differentiated by their characteristics and criteria such as color, weight, height, volume and length is the achievement of the structure called serialization, indicating a logical need of order.

Considering that the construction of number by a child requires, not only the notions of class and sequences, but also the conservation of quantity, Zaia (2012) claims that, in order to acquire this notion, it’s essential to pick games that make the construction of a word for word correspondence possible, the comparison of the amounts of objects, establishing relations of equivalence and non-equivalence, the division of the quantities in equal parts and the quantification of objects.

Taking into account the educational possibilities achieved through playing, it is considered that a soliciting environment, through games, may favor the construction of the number from this study’s theoretical perspective, precisely because different types of games will elicit the need to classify, serialize and conserve in order to execute their plays. Piaget (2017, p. 140) believes that games are a powerful learning tool “that, any place where it’s possible to turn the reading, calculus or writing learning process into a game it has been found that the children fall in love with these, commonly thought of as boring, occupations.”

Through this perspective, there are several studies concerning games and their application in educational or psycho-pedagogical contexts, proving their innumerable benefits for intellectual development, corroborating it as an important instrument to evaluate, comprehend and intervene in children’s’ cognitive processes, from the Piagetian perspective (Brenelli, 1986, 1993; Zaia, 1996; Piantavini, 1999; Macedo; Petty; Passos, 2000; Macedo; Petty; Passos, 2005; Camargo, 2002; Oliveira, 2005; Dell’agli; Brenelli, 2010; Carvalho; Oliveira, 2014; Quinelato, 2015; Pessoti, 2015; Silva, 2018; Cesar, 2018). On top of these, other studies that focused on the construction of number through games are brought to attention (Nunes, 2019; Gonçalves, 2020; Carvalho, 2020).

Faced with this, one realizes that there are few studies done with electronic games, from the Piagetian theoretical perspective, and that prioritize planned and intentional actions, in order to make the development of cognitive structures possible. Thus, the innovation presented by this research is to attempt to comprehend what are the limits and range of a pedagogical intervention with electronic games, when compared to the same game in its physical form.

3 The Kalah Game

Kalah is an African game amongst the oldest of the world, probably originated in Egypt. The fundaments of this type of game consist in placing or capturing seeds that are distributed in holes lined in two parallel rows. The goal is to harvest the highest number of seeds for the oasis or Kalah (the biggest hole on the player’s right).

Studies with pedagogical or psycho-pedagogical goals using this game brought to attention positive results, especially concerning the development of the logical mathematical thought process (Macedo; Petty; Passos, 2000); (Missawa; Rossetti, 2008); (Dias, 2009); (Jaskulski et al., 2020); (Maurício et al., 2020). Currently, besides the classic physical board game version, there’s also the electronic version available for free on Google Play called Mancala. Images 1 and 2 show the game’s board for both versions: physical and electronic. The games consist of a board with 12 holes and 32 seeds, played in pairs (Figure 1).

Figure 1
Kalah’s physical version and Kalah’s electronic version Source: Authors, 2023

To begin the game, 32 seeds are distributed, placing 3 in each hole on the board, except in the middle ones, where there are 4 seeds; each Kalah begins the game empty. The players take turns transferring their seeds in an anti-clockwise motion, during each round it’s possible to move the seeds of only one hole at a time. So, if there are 5 seeds in the chosen hole, the player must distribute them, placing them one by one, in the ensuing holes, except if it lands on the opponent’s Kalah, which must be “skipped”. The game ends when a player runs out of seeds, the winner being the one that got the highest number of seeds for their Kalah.

On top of these basic rules, the game has two other special rules: getting the chance to play again and capturing the opponent’s seeds. The first happens when, during the distribution of the seeds, the last one is placed in the player’s own Kalah, the player will be allowed to go again. As for the second, when distributing the seeds, if the last one is placed in an empty hole, located on the player’s own side, it’s possible to capture the opponent’s seeds that are in the hole facing the empty one (meaning, on the opponent’s side) and place them in their own Kalah.

The problem situations triggered by this game require the players to come up with strategies that go through, necessarily, the establishing of word-for-word correspondence, the comparison of the amounts of seeds in virtue of establishing equality relations or not, the division of the amounts in equal parts and the quantification of the seeds that, among other things, also solicit constantly the development of the discreet or discontinuous quantities’ conservation structure.

In order to elevate the game to a more complex level, the players must anticipate both their own moves as well as their opponent’s, pondering what is the best hole to relocate the seeds to, performing mental calculations is indispensable. For example, how many seeds are needed in the first hole (from left to right) in order to get the chance to play again? Or even, if you move the seeds from a certain hole is there a chance of it (the seed) landing in an empty hole (belonging to the own player), making it possible to capture the seeds that are facing this hole (the opponent’s)?

Thereby, the concept of addition, for example, is quite conducive to be explored during the moving of the seeds, when comparing the total of captures carried out by the players, identifying who has more, less or the same amount, or even, the idea of division, because the children must equally distribute the seeds both among them (16 each) and among the holes before starting the game.

Other two important mathematical concepts, that can be explored in this game, are the notion of time, understood as a consequence of the displacements carried out and of place, which consists of the spatial configuration of the board, modified each play, which always requires the evaluation of the next moves that depend on a new arrangement of the seeds (Macedo; Petty; Passos, 2000).

4 Method

To develop the research, a quasi-experimental design was adopted, inspired by a model proposed by Campbell and Stanley (1979), renowned for their work on experimental and quasi-experimental methods. In a quasi-experimental design, researchers cannot control all variables, but they still seek to establish and analyze cause-and-effect relationships.

The study involved 38 children, aged between seven and ten years old, of both sexes, enrolled in two municipal elementary schools located in the state of Minas Gerais, Brazil, which serve populations with similar socioeconomic conditions.

For the experiment, three groups were formed: two experimental groups and a control group. Experimental Group 1 (EG1) was composed of 13 children, 4 girls and 9 boys, enrolled from the second to the fifth grade, with 4 students in the second year and 3 in each subsequent year. This group received intervention with the electronic version of the Kalah game. Experimental Group 2 (EG2) was composed of 13 children, 5 girls and 8 boys, enrolled from the second to the fifth grade, with 5 in the second year, 3 in the third and fourth years and 2 in the fifth year. This group received intervention with the concrete version of the Kalah game. The Control Group (CG) was composed of 12 children, 3 girls and 9 boys, enrolled from the second to the fifth grade, with 4 in the second and third years and 2 in the fourth and fifth years. This group did not receive any intervention.

The description given above can be seen in Table 1 below:

Table 1
Research Participants and Groups

Experimental Group 1 Participants – Electronic Games
2nd GRADE	3th GRADE	4th GRADE	5th GRADE
ALY (7)	LUI (7)	JOL (9)	VIT (10)
JUL (7)	GUH (8)	JON (9)	ELO (10)
MIC (7)	REN (8)	MAR (9)	JOE (10)
VIN (7)

Experimental Group 2 Participants – Concrete Games
2nd GRADE	3th GRADE	4th GRADE	5th GRADE
BRU (7)	ING (7)	CAR (9)	MAI (9)
KEL (7)	GUO (7)	CAU (10)	RAI (10)
JOS (7)	ADR (7)	PED (9)
TIA (7)
DAV (7)

Control Group Participants - No Intervention
2nd GRADE	3th GRADE	4th GRADE	5th GRADE
ROD (7)	GER (7)	RAF (9)	ISA (9)
NIC (7)	MAT (7)	JOP (9)	LOR (9)
JOV (7)	LUA (8)
ART (7)	VIO (8)

• Participants are identified with initials of fictitious names, followed by their age at the beginning of the investigation, in years
• Source: Research Data

To assess the effects of the interventions carried out in the experimental groups, all participants underwent pre- and post-tests, through the application of 3 operational tests: operational test of conservation of discrete quantities (tokens), operational test of inclusion of classes (flowers) and operational test of serialization (rods). These evaluations followed the protocols established in the works of Piaget and his collaborators (Piaget; Szeminska, 1975; Piaget; Inhelder, 1975), famous for their research on child cognitive development.

By applying these tests, it is possible to observe the level of construction at which the participants are in relation to the logical structures evaluated. From this analysis, three possibilities can be inferred: presence of structure still at a figurative level (NC-Non conservation, NI-Non-inclusion or NS-Non-Serialization), intermediate levels that indicate structural transitions (TR-Transition) and levels that indicate that the structure is at the concrete operational stage (OC-Operatory Conservation, OI-Operatory Inclusion, OS-Operatory Serialization). Specifically in the conservation situation, the transition level can be analyzed when the child uses three types of arguments: A) oscillation between responses (giving conservation responses but then giving non-conservation responses); B) resorting to empirical return (to explain the transformation by trying to perform a reverse action that nullifies the previous transformation, making everything the same again); and C) identity argument (admitting the same quantity with the argument that there was no change in the number of objects).

To compose the sample, the children were assessed regarding the presence or absence of operational logical structures. Subsequently, a random draw was made to define which students would participate in the experiment with the electronic game, which would participate in the experiment with the concrete game and which would make up the control group.

In order to secure the ethical aspects of this study, the procedures were approved by the Committee of Ethics in Local Research, through a Plataforma Brasil’s registration. The research data was analyzed and interpreted taking into account the qualitative and quantitative aspects. This way, the instruments used for data collection such as videos, records of the comments done by the researcher during the intervention sessions, as well as the transcriptions of the children’s answers, were compared in a qualitative manner, looking for identifying similarities and differences between the contexts of the concrete and electronic game. As for the instruments selected to diagnose the basic logic structures’ level, used during the pre and posttest, were analyzed according to the established criteria specific to each evaluated structure and followed the Piagetian protocols. As for the acquired data’s reliability, the study enlisted the help of two judges to evaluate the participant’s performance in the operatory tests. Judge A expressed 95% concordance concerning the basic logic structures’ tests (discreet quantities’ conservation, flower inclusion and serialization), whereas judge B expressed 100% concordance.

5 Results and Discussions

Many times, when proposing a study regarding games, the professional, due to a lack of knowledge, disregards the intervention’s role, attributing almost exclusively to the game the responsibility of acting in the individual’s learning process. The act of playing is not enough by itself, because it’s not about incidental learning but, instead, in which the pedagogical intervention is the required condition in the theoretical perspective that substantiate this paper.

This way, we believe that playing favors the development and learning process, insofar as the subject is provoked to reflect about their actions, being important, that the professional that oversees the matches, structures the pedagogical intervention in a way that it evokes challenges, makes gameplay analysis possible, instigates reflections, accomplish forethoughts, establish relations, etc., because, without this, the act of playing is restricted to common sense, and in Macedo, Petty and Passos (2000) view this already happens spontaneously with children.

Thereby, the pedagogical interventions were based on constructivist principles, brought to attention by Kamii and Devries (2009) and Mantovani de Assis (2002) – as for example, the development of physical and logical mathematical knowledge – as well as in the four stages that must be a part of studies with games, as illustrated by Macedo, Petty e Passos (2000): the exploration of the materials and the learning of the rules; the playthrough and the development of strategies; the resolution of problem situations, and analysis of the consequences of playing. The sessions with the Kalah game lasted for, on average, 1 hour and always took place in pairs, in a total of 4 sessions.

6 Pretest

Table 2, below, shows the results gotten in the pretest, upon the application of the discontinuous quantities’ conservation tests, inclusion of class and serialization, in the 3 evaluated groups. No subject in the pretest presented any sort of level III (operatory conservation) arguments. The answers varied between levels I and II, in the 3 research groups, the only difference being the arguments used by the subjects. As for class inclusion, the majority couldn’t compare the parts to the whole, noticing the possibility of including the subclasses in a broader class, meaning that, daisies and roses, when put together, make up a bigger class: flowers. To resolve the issue of class inclusion, the child must have reached the Reversibility stage, as it can be observed in Piaget and Inhelder’s explanation:

Table 2
Pretest Results – Basic Logic Structures

Discreet Quantities’ Conservation
Level	Experimental Group 1	Experimental Group 2	Control Group
NC	31%	38%	50%
TR – A	31%	16%	17%
TR – B	15%	23%	17%
TR - C	23%	23%	17%
OC	---	----	----
Class Inclusion
NI	54%	46%	75%
TR	46%	54%	25%
OI	---	----	----
Rod Serialization
NS	77%	62%	75%
TR	33%	38%	25%
OS	---	----	----
Subtitles

• NC - Non-Conservation /TR- Transition/ OC- Operatory Conservation
• NI- Non-Inclusion/ TR- Transition/ OI- Operatory Inclusion
• NS- Non-Serialization/ TR- Transition /OS- Operatory Serialization
• TR-A – Sometimes gives a conservation response, other times a non-conservation response; TR-B Resorts to empirical return; TR-C – Admits conservation but only presents the identity argument.
• Source: Reserarch Collection

In other words, the A < B relationship implies the reverse operation in the form of A= B − A’, in a way that B subsists as a whole, despite its parts A and A’ being separated in thought. Failing to preserve the whole B in such conditions, the subjects will, then, simply compare A and A’, and conclude, according to their appraisal, that there are more springs A than flowers [implied, other A’ flowers] (Piaget; Inhelder, 1975, p. 131).

In the same way, there were not, during the pretest, participants that were able to solve the problem presented by the serialization test, using a systematic method, which consists of identifying, firstly, the smallest (or biggest) element of all; then, the smallest out of the ones that are left and so on and so forth, characterizing the third stage (concrete operatory stage).

7 The Interventions

Figure 2
Kalah Game – PED (9,0) – Experimental Group 2

Below, Chart 1, shows the interventions structured with the Kalah exploring situations that asked the participants to establish relations of quantity, taking into consideration the studies of (Zaia, 1996) and (Macedo; Petty; Passos, 2000), both for the physical and electronic version.

Chart 1
Intervention with the Kalah Game

Section	Concrete	Electronic
1st	1) Free exploration of the board and game pieces. 2) Introduce the game’s rules, but first, ask whether they have any idea how to play. 3) Raise questions that make it possible for them to understand the game’s rules, such as: a) How many seeds can be put in each hole? b) How many seeds are there in total? c) In which direction must the seeds be distributed? d) Which holes can the player utilize during their round? e) During a game, when a player lands on their opponent’s kalah, what should they do? f) When does the game end? g) Who wins the game? 4) Play a few rounds, noticing whether they understand the game’s rules.	1) Free exploration of the board and game pieces on the tablet. 2) Introduce the game’s rules, but first, ask whether they have any idea how to play. 3) Raise questions that make it possible for them to understand the game’s rules. Such as: a) What happens when you click on this hole? How many seeds are placed in each hole? b) How many seeds are there in total? c) In which direction are the seed being distributed? How do you know this? d) Which holes can the player utilize during their turn? e) Why is it that you got the chance to play again? f) How did you manage to capture the seeds? g) When does the game end? h) Who wins the game? 4) Play a few rounds, noticing whether they understand the game’s rules.
2nd	1) Remind them of the game’s rules and play a few rounds, noticing the players’ behavior. 2) Problem situations based on the matches: a) Each player begins the game with the same number of seeds? How do you know this? b) If you choose this hole, where do you think you’ll put your last seed? c) What can you do to add more seeds to your Kalah? d) Insert a special rule for those that have already grasped how to play. e) Is it possible for a player to play more than once? f) How many seeds are needed for you to be able to play again?	1) Remind them of the game’s rules and play a few rounds, noticing the players’ behavior; 2) Problem situations based on the matches: a) Each player begins the game with the same number of seeds? How do you know this? b) If you choose this hole, where do you think you’ll put your last seed? c) What can you do to add more seeds to your Kalah? d) Is it possible for a player to play more than once? e) How many seeds are needed for you to be able to play again?
3rd	1) Play a few rounds, noticing the strategies the players use. 2) Problem situations based on the matches: a) Insert the second special rule: seed capture. b) When is it possible to capture seeds? c) During this round is there a possibility of one of the players capturing the other’s seed? d) Is it possible for a player to win during this round? e) Does it matter which hole you choose for your first move? Why? f) Now it’s your peer’s turn to play, if you were them, which move would you make? 3) Problem situations that should be recorded: a) Imagine that you’re player number 1 and you’ll move first. Which hole would you choose to start with? b) Look at the following game setup, if you were player number 2, which house would you move the seeds to? Why?	1) Play a few rounds, noticing the strategies the players use. 2) Problem situations based on the matches: a) When is it possible to capture seeds? b) During this round is there a possibility of one of the players capturing the other’s seeds? c) Is it possible for a player to win during this round? d) Does it matter which pit you choose for your first move? Why? f) Now it’s your peer’s turn to play, if you were them, which move would you make? 3) Problem situations that should be recorded: a) Imagine that you’re player number 1 and you’ll move first. Which hole would you choose to start with? b) Look at the following game setup, if you were player number 2, which house would you move the seeds to? Why?
4th	1) Write down the score of each round, noticing the following situations: a) Do you have more, less or the same number of seeds in your Kalah, when compared to your opponent? b) What do you need to do in order to have more seeds in your Kalah than your opponent? c) And less? And the same amount? d) During which round did you capture the most seeds for your Kalah? e) Observe round “X.” What would need to happen for the game to end in a tie? f) How many seeds, the player that had less by the end of the round would’ve needed to win the game? g) How was your experience playing this game? Register it in words, drawings.	1) Write down the score of each round, noticing the following situations: a) Do you have more, less or the same number of seeds in your Kalah, when compared to your opponent? b) What do you need to do in order to have more seeds in your Kalah than your opponent? c) And less? And the same amount? d) During which round did you capture the most seeds for your Kalah? e) Observe round “X.” What would need to happen for the game to end in a tie? f) How many seeds, the player that had less by the end of the round would’ve needed to win the game? g) How was your experience playing this game? Register it in words, drawings.

• Source: Reserarch Collection

The interventions that had as their main goal the exploration of the material and the learning of the rules, showed more possibilities with the concrete game, seeing as the children were able to discover color, shape, texture, consistency, favoring the empirical abstractions (Piaget, 1995), whereas in the virtual field, the exploration was limited to color and shape.

Another aspect to be highlighted is that in the digital version, the game board shows the number of seeds in each hole, whereas in the concrete version, the children needed to count them one by one in order to know the amount of seeds so that they could, only afterwards, distribute them in the holes, as per the game’s rules, taking advantage of the word for word equivalence, an important construction for the comprehension of the notion of number. As for the exploration of the game’s rules, with the electronic version, all the special rules are already included, and the child needs to figure out which action they performed in order to play again or to be able to capture seeds. With the physical version, the special rules were added after the 3^rd session, to the children that showed comprehension and coordination of the basic rules. This allowed the developments of a few relations seeing as the players needed to think about their moves ahead of time, planning to benefit from the special situations. Such a thing was not possible for the players of the electronic version, seeing as, since the 1^st session, they had to coordinate all the rules, even without understanding them.

The direction in which to move the seeds was a problem observed among the players of the concrete version, seeing that it’s the own subject that must develop said action, favoring the construction and motor coordination. As for the players of the electronic version, they only had to click one of the holes, that the seeds were placed one by one, automatically.

When it comes to the playing experience and the development of strategies, both with the electronic version, as well as with the concrete one, during the first rounds, the children didn’t show signs of planning when it came to picking holes to use; after a couple of interventions, this behavior was seen in older subjects (9/10 years of age).

With the digital version, the children presented a more random game, without showing signs of comprehending (or even planning) their plays. Thus, for example, when they were faced with the message to play again or to capture the opponent’s seeds, they weren’t able to explain the process that made the action possible. On top of that, playing electronically proved to be a more passive game, because the participants would wait for the tablet’s commands to indicate who would be the next to play or, even, who won the game. With the physical version, the children needed to count the seeds to know who won, coordinate the direction in which to move the seeds, identify if they could play again or not and “be mindful of the opponent’s plays.”

As previously mentioned, the special capture rule for the Kalah players, with the concrete game, was added only in the 3ª round. It was possible to observe that this created conflict, seeing as the smaller children weren’t capable of coordinating the basic and special rules simultaneously; something that was only accomplished by players from the age of 8. Two 7-year old’s players didn’t manage to play with the capture rule, because when it was added, they started to disregard the original rules.

With the electronic version, the players in this same age range demonstrated that they’d begun to comprehend this rule, but during the plays, few indications of a plan to capture seeds were observed, highlighting higher instances of automation and randomness, especially with the younger players.

To solve the problem situations involving the concepts that corresponded to the same amount, more, less, with the concrete version, the children had the possibility of coming back to the action in order to reconstruct it and manipulate the seeds to get to the result. With the electronic version, this was not possible.

In the last session, a few problem situations (with records) were proposed, elaborated, and contextualized, taking into account the situations experience by the children, during the sessions with the game.

One of the proposed records was taking note of the points obtained in each round, until the end of the game, to compare quantities later, establishing connections between who managed to capture more seeds, who got less, how many were still needed in order to win, etc. The players of the electronic version struggled to take note of the points, because they weren’t able to establish the number of seeds that were placed in the Kalah, seeing as the total was updated automatically by the game, making it harder to reconstruct the amount that was there before. Whereas with the concrete version, the players had an easier experience because they were able to, in each round, quantify their seeds, develop strategies to know how many they had in the previous round and how many they had obtained. It is known that these innumerable actions over the objects make classifications and serializations possible, which are needed to comprehend the notion of number.

Table 3 Posttest Results – Basic Logic Structures

Discreet Quantities’ Conservation
Level	Experimental Group 1	Experimental Group 2	Control Group
NC	17%	8%	42%
TR - A	8%	25%	25%
TR – B	8%	8%	8%
TR- C	59%	50%	25%
OC	8%	8%	----
Inclusion of Class
NI	33%	33%	58%
TR	17%	25%	42%
OI	50%	42%	----
Rod Serialization
NS	42%	62%	75%
TR	54%	38%	25%
OS	---	----	----

Source: Reserarch Collection. Prepared by the Authors

The general results, in the posttest, indicated progress of a more qualitative sort in relation to the structure of the notion of discreet quantities’ conservation in the three groups, however, it was found that the most significant changes occurred in the two experimental groups if compared to the control group. It’s worth noting that, in the pretest, there were no arguments that could be categorized as operatory conservation and, in the posttest, two children (one from EG1 and the other from EG2) presented answers in this level.

When it comes to the results gotten in the class inclusion test, it was discovered that there was significant improvement made by the subjects from experimental groups 1 and 2, who reached the operatory inclusion, which didn’t happen with the control group.

In stage 3 (operatory inclusion), the children needed to present arguments that took the notions of part and whole into account, and again only the subjects belonging to the experimental groups presented, said arguments, during the post-test. This way, out of all the subjects belonging to the experimental groups, 33% have yet to develop the basis of inclusion of class, being 4 in each group. As for the serialization structure, it was also possible to observe small improvements in the participants belonging to the experimental groups. Although few presented the conduct of operatory serialization, in the posttest (17%), this only happened among the subjects of the experimental group 2.

From the results obtained related to the logical structures of conservation, classification, and serialization, it’s possible to infer that the intervention with the electronic and concrete games favored the progress in the construction of these structures, seeing as the results in the posttest indicated that there were differences amongst the subjects belonging to the experimental groups, when compared to the control group. In the same manner, the differences found amongst the experimental groups proved to be more significant to the structure of serialization.

Favorable results after the intervention process were also observed in other studies that adopted a similar theoretical framework and used rule-based games (Bessa; Costa, 2019). Some studies highlight that games can be useful even for more complex content (Rostirola; Siple; Henning, 2022). Implications for teaching and understanding mathematics can also be observed from our data. As Silva and Luna (2019) pointed out, there is a correlation between logical reasoning and mathematical reasoning. This is an essential aspect defended by Piaget when addressing the teaching of mathematics, that is, possessing elementary and infralogical logical structures, at least at an operational level, is an important condition for engaging with mathematical problems, involving, for example, arithmetic and geometry. In the case of the present study, we observed that the interventions with the Kalah game were important for the construction of some of these essential structures, such as inclusion, seriation, and conservation. Without these structures, Piaget believed it would not be possible to understand the notion of a set (Piaget, 1998).

8 Similarities and Differences of the Electronic and Concrete Versions of the Kalah Game

The results showed significant differences regarding the exploration of the game’s rules and materials, because for the electronic version’s players the discovery of the physical properties of the game pieces and game board happened in a more perceptive manner, being possible to identify only color, shape, and infer sizes whereas, with the concrete one, such discoveries went on to cover textures, weight, consistency, etc.

As for the practice and the development of strategies, in the electronic version, it proved to be a more passive game, in which the players waited for the tablet’s command to indicate whose turn it was to play or who won the game. In many instances, they didn’t seem to comprehend when messages to play again or capture seeds appeared. Whereas, with the physical version, the children needed to count the seeds to know who won, coordinate the direction in which to move the seeds, identify if they could play again or not and be mindful of the opponent’s plays.

Another important aspect worth noticing, concerns the problem situations triggered by the game. With the concrete version, the special rule of seed capture seed rule was added one in the third session and, even so, it was possible to see that this created a conflict, because the smaller children were not able to coordinate the basic and special rules simultaneously, being better coordinated by players over the age of 8. Two players of the age of 7 were not able to play with the capture rule because, when added, they started to disregard the basic rules. In the electronic version, the players in this same age range had to deal with all the rules since the first rounds, highlighting higher instances of automation and randomness, especially with the younger players. On top of this, to solve the problem situations involving the concepts that corresponded to the same amount, more, less, with the concrete version, the children had the possibility of coming back to the action in order to reconstruct it and manipulate the seeds to get to the result. With the electronic version, this was not possible.

In the last session, one of the proposed records was taking note of the points obtained in each round, until the end of the game, in order to compare quantities later, establishing connections between who managed to capture more seeds, who got less, how many were still needed in order to win, etc. The players of the electronic version struggled to take note of the points, because they weren’t able to establish the number of seeds that were placed in the Kalah, seeing as the total was updated automatically by the game, making it harder to reconstruct the amount that was there before. Whereas with the concrete version, the players had an easier experience because they were able to, in each round, quantify their seeds, develop strategies to know how many they had in the previous round and how many they had obtained. It is known that these innumerable actions over the objects make classifications and serializations possible, which are needed to comprehend the notion of number. The players of both modalities didn’t show signs of being disinterested in the game, therefore, there was a positive social interaction between players of both versions.

Thus, both games were attractive to the participants, involving attitudes of cooperation and motivation. There are at least two important reflections to be made in this regard. The first is that in times of significant technological appeal for children, the sustained interest in the concrete game compels us to highlight its effectiveness, as shown in other studies (Ribeiro, 2019; Wang; Zheng, 2021). The second pertains to the fact that games, whether concrete or electronic, as well as activities different from those usually used in school, are powerful instruments for working with mathematics. The study by Martins, Bianchini, and Yaegashi (2017) showed how an engaging activity, addressing mathematics, can provide cooperation, joy, and interest in 10 and 11-year-old students. The authors discuss these motivational aspects from the Piagetian perspective of affective development. Thus, what keeps the subject engaged, challenged, and persisting in a task is an important issue for the major equilibration process to be maintained, overcoming disequilibrium towards greater adaptation.

During our interventions, in both modalities of games, we also observed aspects such as cooperation, interest, interactions, as well as children’s involvement in activities like addition, subtraction, counting, comparisons, and numerical recording, performed repeatedly in a pleasurable and playful manner. However, we still emphasize that, although the difference between the experimental groups was not very significant, it did occur. This fact, in our view, indicates the need for new and different comparative studies, since there are more studies pointing to the importance of one game modality or another in the construction of specific aspects of development than those that compare them.

In any case, it is the interventions constructed by the teachers that can make a difference (Martins; Bianchini; Yaegashi, 2017), and our study indicated the greatest difference in relation to the group that did not undergo any form of intervention with the game. Regarding the possibilities of greater differences between concrete and electronic games, good teacher interventions should generate the need to adapt more reductionist or passive actions of certain electronic games.

9 Final Considerations

One of this study’s goals was evaluating pedagogical interventions’ reach and limits, through the use of electronic and concrete games. About this, what can be said it that some constructivists principles couldn’t be ensured in the intervention with digital games, as for example, Piaget and Inhelder (2013) attribute a fundamental role to the assignments and experiences done on the objects to the development of the logical mathematical structures. In this regard, the authors differentiated the physical experience from the logical mathematical experience, in the following manner:

The physical experience consists of acting on objects to grasp their properties (for example, comparing independently from their volumes); the logical mathematical experience, that acts on the objects, but to know the result of the coordination of the actions [...] (Piaget; Inhelder, 2013, p. 137).

In this case, limitations were noticed in the intervention with the use of the electronic game, once that the action on the object to discover its physical properties was centered on the perceptible aspects, being possible to identify only the objects’ most apparent traits.

As for the logical mathematical experience, on several occasions the own software automatically processed the actions that the subject could perform, such as the distribution of the seeds, considering the directionality in the Kalah game, indicating the quantity, showing messages informing whose turn it was to play.

Another limitation noticed during the electronic games’ pedagogical intervention, and that hurt the constructivists principles, concerns the interdependent relations between development and learning, in a way that “learning depends on development and not the other way around” (Mantovani de Assis, 2002, p. 20). This means that the same game might not be suitable for all ages, if offered in the same manner, without taking into consideration each subject’s interests, specificities, and cognitive basis. Through this perspective, the software used brought the game with all the basic and special rules simultaneously, making it impossible to alter them, increasing or decreasing the game’s complexity. In this regard, it is highlighted the indispensable role of the educator in the work with games, because he is the one that’s supposed to offer facilitators or new challenges gradually. About this, Albuquerque and Kern (2019) claim that the digital games bring a more rigid format to the learning opportunities provided by the act of playing, due to the amount of pre-established details, when compared to the physical games:

In the digital games, the child’s possibilities are more limited to what was pre-programed. It’s possible to illustrate this point, using [...] chess. When analog, the child can negotiate with the opponent as for the pieces beginning the game in alternative locations, or that the queen can move jumping over other pieces, as does the horse. In the digital version of the same game, this isn’t possible. The child is bound to the traditional rules, as a frame that foils or constricts their possible actions (Albuquerque; Kern, 2019, p. 667).

It’s worth noticing another essential principle extracted from the Piagetian theory that concerns the importance of the social interactions between the pair for the children’s morals, social and intellectual development. This way, the work with games is propitious to the decentering of the thought process and coordinate different points of view (Kamii; Devries, 2009).

When it comes to digital games, in its majority there is the option to play alone (against the machine) or online. When you play against a machine, hardly will the individual be able to beat it, which is a negative aspect when you take into account the motivation and interest in solving the problem posed by the game. Concerning this, Macedo and Bressan (2016) add that competing against a computer instead of a human being is very different, because the person is playing against a program that was developed by someone, but they are not present during the game. Surely, this programmer predicted game levels and a program that always wins. This way, online games can be interesting as long as they don’t let go of the interaction between the children, as was the case with this study, because the intervention was structured in such a way that it would secure that there were exchanges, dialogues, decisions between partners even during the matches that involved the electronic games, a positive aspect to be taken into account when it comes to the later experiences with this version.

Thus, on what concerns the limits of intervention with the use of electronic games, based on constructivists principles, it was confirmed that most of the issues found concerned the way in which software were programmed. Therefore, in order for there to be change, it would be necessary to create specific game apps that took those observations into account.

In regard to the effects of the pedagogic intervention in the construction of basic logic structures, taking into consideration both versions of the game, in general lines, the results obtained in the post-test indicated that the qualitative and quantitative changes in the structures of thought that were assessed were more expressive among the subjects of the experimental groups, if compared to the Control Group. This data corroborates the idea that a solicitor environment, through games with planned and intentional pedagogic intervention, is a favorable instrument for cognitive development.

Be that as it may, the present study has shown it possible to also develop a paper applying electronic games, in such a way that the use of the two versions of the game favored the development of a cognitive basis. However, when you compare the effects of the two intervention methods in the experimental groups, a difference was found, even though it wasn’t as expressive, because the subjects that participated in a context of intervention because the concrete games fared better in two of the three applied tests (conservation and serialization).

Thus, we highlight that the work with games requires intervention by a professional that plays the role of presenting problems and instigate situations, that allows the subject to become aware of their actions during the plays and challenges. This for the electronic and concrete versions, meaning, during no point is the educator’s role less important. Another relevant point of this study, that deserves to be highlighted, concerns the fact that electronic games do not replace the concrete ones in several aspects, especially those concerning physical knowledge. Finally, we consider that electronic games need interventions, and, just as in a work with concrete games, the educator’s role is essential.

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Publication Dates

• Publication in this collection
  10 Jan 2025
• Date of issue
  2024

History

• Received
  27 Feb 2024
• Accepted
  03 July 2024