Research exploring DGBL has become popular in the academic literature in recent years. The main reason for this is that the knowledge is presented to the students in a fun atmosphere designed. Pointing to this fact, (Prensky, 2001) emphasized that the key feature of DGBL environments is the integration of serious learning and an interactive entertainment.

Digital games were found to not only make learning fun but also enable focus on the task (Chen et al., 2012), motivate learning and increase success (Hung et al., 2014), and affect performance and attitude (Ke & Grabowski, 2007). These results imply that digital games should not be designed at random for educational goals and should include certain features. However, researchers disagree on the features that digital games should contain that will serve educational purposes.

(Mayer & Johnson, 2010) stated that an educational computer game should have four features: rule-based, responsive, challenging, and cumulative. (Prensky, 2001) explained these characteristics as follows: 1) rules, 2) goals and objectives, 3) outcomes and feedback, 4) challenge, 5) interaction, and 6) representation or story.

How can these game features be integrated into mathematics education? In answer to this question, (Yong et al., 2020) proposed four criteria for game integration into mathematics education: 1) problem-based learning, 2) process-oriented learning, 3) turning mistakes into opportunities, and 4) stories. Another study indicated that such a digital game should include initial strategies, reflection, and abstraction (Jong et al., 2008). The emphasis in these studies is to make students experience mathematical processes. In this context, mathematical processes are explained as trial and error, hypothesis assertion, hypothesis testing, proofing, problem-solving, reasoning and proving, conveying (sharing), relating, and representing (Arsac et al., 1991); (Mathematics, 2000).

Learning theories about how mathematical processes can be used to ascertain how digital games should be designed for learning environments. (Yong et al., 2020) stated that learning theories (e.g., experimental learning and constructivist theories) provide a foundation for understanding how students learn in a game designed within the scope of DGBL. Emphasizing this fact, (Gee, 2007) argued that it is more appropriate for today’s students to experience learning theories through reflections in a good digital game rather than the situations that they are taught at school. Therefore, in this study, the students experienced a suitable learning environment designed by drawing on the relationship between the Theory of Didactical Situations (TDS) and DGBL.

The TDS (Brousseau, 2002), a game-based theory, offers robust arguments for students’ acquisition of knowledge through a carefully designed game tool. The theory adopts knowledge acquisition in a student-centered environment (Laborde, 2007). Similarly, DGBL creates a learning in which students can act independently in a game designed with a balanced integration of educational content (e.g., mathematical knowledge) and game components. Based on this, the arguments of the theory offer a suitable atmosphere for DGBL.

In theory, the concept of the situation holds central importance. The situation implies the relevant conditions that require one or more students to use and learn mathematical knowledge or a project organized correctly in this context (Warfield & M., 2014); (Brousseau & Warfield, 2014). Students have to struggle with the milieu in a didactic situation. The milieu refers to the relevant characteristics of the student’s surroundings, including area, teacher, materials, and the presence or absence of other students (Warfield & M., 2014). In this regard, the milieu functions as modeling the gamers’ or students’ activities within certain cognitive, social, and physical elements (e.g., prior knowledge, group interaction, data set, and special technology) (Nickels & Cullen, 2017). As the aim is to achieve an instructional goal, the milieu should be carefully designed. Thus, digital games can be used effectively in creating these conditions.

In a digital game, mathematical knowledge can be hidden because it is masked by the game’s features. In this way, the problem situation comprising many tasks caused by some constraints and conditions can be presented to students implicitly in a digital game (Gök et al., 2020). Such learnings are within the scope of a didactic situation(Brousseau & Warfield, 2014), as the intervention of the authority (e.g., teacher) is limited, the knowledge is hidden from the students for a while, and they are organized in a manner that allows them to act independently during the game process. Each didactic process has three stages: devolution, mathematical (action, formulation, and validation), and institutionalization (Brousseau & Warfield, 2014).

In the devolution stage, students take responsibility for actions without assistance from the authority, after the criteria, such as the conditions, rules, purpose, success in the game, which are necessary to create an interesting and instructive mathematical situation (e.g., using a digital game), are explained by the authority.

In the mathematical stage, students should experience the environment and produce new things (e.g., knowledge, expressions) from the feedback they receive from the game, discuss their validity first in the group and subsequently, with the participation of the entire class, make decisions in this direction and form hypotheses and try to prove them. They should also generate models, arguments, and proofs from the assured conclusions and organize them to achieve a more general goal. Moreover, they should evaluate and correct the results of their choices on their own in this stage.

Finally, in the institutionalization stage, the authority notes students’ responses to the problem presented in the context of the game in the mathematical situation and their progress in the problem, placing them in the perspective of the curriculum. In other words, this stage provides a standard way of formulating or generalizing the pieces of knowledge that students discover in the process of the game.

These stages allow students to experience different mathematical processes. Due to limited teacher intervention, particularly in the mathematical stage, it is assumed that students can use varied mathematical processes in interaction with objects as the source of knowledge (e.g., inferences in the context of the game) and with one another. Students use mathematical processes effectively in the situations mentioned (Brousseau, 2002). For instance, (Brousseau, 2002) expressed that students can use pragmatic, semantic, and intellectual reasons at the mathematical stage. If an argument is uncovered through trial and error, it is for pragmatic reasons. If it is presented concerning a previously played game, it is for semantic reasons. If it is based on logical inference, it is for intellectual reasons.

A concept called a didactic contract is used in theory for the successful functioning of these processes. Based on knowledge, the didactic contract is the sum of the expectations of teachers and students from each other ((Brousseau, 2002); (Warfield & M., 2014)). This contract, usually based on implicit rules, occurs in negative situations, such as a student not wanting to solve a problem. More clearly, this concept is necessary for students to take ownership of solving the problem, which is implicitly presented in the game, and to move forward in the game by acting independently based on knowledge.

Although the TDS has considerable potential in digital game design, few studies have explored its suitability in this context ((Gök et al., 2020); (Nickels & Cullen, 2017). (Nickels & Cullen, 2017) noted that robotic games designed within the framework of a didactic conditions for a 14-year-old child with a critical illness (Acute Lymphoblastic Leukemia) provide strong evidence for performing robotic tasks designed in this context. Another study was conducted with pre-service teachers (Gök et al., 2020); no study exists in the literature in which mathematical knowledge is taught to middle school students in an adidactical situation by designing a digital game.

This study describes the process of experiencing a digital game designed by adidactical situations of 6th-grade students at different stages of the TDS. The unique value of the study is in how a digital game, which reflects the meaning of mathematical knowledge and the adidactical situation, enables students to use their mathematical process skills.

The research participants comprised 16 (7 girls and 9 boys) students studying in the 6th grade of a public school in Turkey’s Eastern Anatolia region. After verbal permission was obtained from the school administration, the study was conducted with the voluntary participation of 16 students from a class of 34.

The students’ mathematics teacher stated that they had no learning experience in the context of DGBL. However, the teacher revealed that the students’ socio-economic status is modest, they are skilled in using technology and are accustomed to playing different games.

Race with Numbers game is a variation of the Race to 20 game, in which (Brousseau, 2002) introduced the main arguments of the TDS. A digital game was developed by expanding the game Race to 20 to a certain extent and adapting it in a different context (balloon popping). As this game’s objective is not only to complete the game but also to discover the mathematical knowledge hidden beyond the game (Prensky, 2001), it can be evaluated in the context of DGBL. In the study, permission was obtained to use the game developed within the scope of a project (Gök et al., 2020).

Race with Numbers is a game in which players play against the computer and experience victory or defeat at the end of the game. The winning numbers in the game form a changing number sequence for each task. The game comprises easy, medium, and hard levels, and they are organized according to when the computer activates the winning number sequence in the game. For example, the computer activates a sequence of winning numbers in moves near the end of the game at the easy level. In contrast, the algorithm is activated in the first moments of the game at the hard level. Each level comprises 12 tasks.

Each task has a certain number of balloons on the game screen (e.g., 16 in the first mission, 18 in the second, and 20 in the third). Those that players can choose are in red, and these balloons are activated in two’s, three’s, and four’s each in tasks. The balloons chosen by the player are colored green, those selected by the computer are colored orange, and those that are not selected are marked light gray. In addition, 1, 2, …, n numbers are written in the center of the balloons, allowing easier monitoring of the process, giving different strategies and better discussion of strategies. The game aims to reach the largest numbered balloon on the screen. Figure 1 shows the screenshots of the digital game.

Race with Numbers comprises 36 activities, with each of the 3 levels containing 12 tasks. As the constraints and conditions change in each task, the winning numbers also change accordingly. The winning numbers are not random, and they form certain sequences. Therefore, it can be explained with certain models. Choosing each number in the winning number sequence in the game equals 1 point. For example, the winning number sequence in the game in Figure 1 is {1, 4, 7, 10, 13, 16}. As the player chose only 2 of these numbers, he earned 2 points at the end of the game. More detailed examinations regarding these explanations are given in Table 1.

During the application process, first, the classroom was arranged in a manner that encourages group work, which differed from the traditional seating arrangement. First, the students were allowed to form groups of two. Next, some of these groups (girls) formed the largest group, while the others (boys) formed the second largest group. The game application was conducted as a competition between two groups (4 or 5 rounds) from their tablets and two large groups on the smartboard. The researcher conducted the application process in line with the stages of the TDS (because the mathematics teacher of the classroom is far from the basic philosophy of TDS). Details of this process in terms of both the TDS and DGBL are given in Table 2.

The interaction shown in Table 2 indicates that the teacher mediates the students’ understanding of the game at the devolution stage. Thus, the teacher transfers the tasks in the game, its rules, and the objective to be achieved in these tasks to the students based on the story used in the game. Students interact with the digital game in the mathematical stage and challenge its constraints and conditions. In this process, through the results and feedback they receive from the game, they explore the game, produce winning strategies, and experience testing and revising them in the game context. Through a discussion of these strategies and by generalizing the proven strategies, models that enable students to

win the game are obtained. These are expressed mathematically by the teacher by taking them out of the game context during the institutionalization stage. Thus, the game aimed to reach the mathematical knowledge hidden in the digital game presented as an adidactic situation.

The research data were collected by recording the application and, then, the interviews through a video and audio recording device. These recordings were first transferred to the computer using the re-listening method. Next, the data were analyzed descriptively in line with the different stages of the TDS. Moreover, the suggested phenomenon was supported by making direct quotations regarding the critical situations for each stage. In the results section, participants were coded based on the classification of Group 1 members (9 people) as S1, S3, …, S17, Group 2 members (7 people) as S2, S4, …, S14, and the researcher as T.

The mathematical stage began with the students playing the game on their tablets or mobile phones for a certain time. At least five games were played during this period.

In these games, students started to challenge the game’s constraints and conditions as they faced either of winning or losing. Although they had some implicit inferences about the game (indicative of winning status), they did not share this among themselves. Thus, students generally acted individually in this process. A screenshot from this process is presented in Figure 2. The students’ behaviors indicated that they won with gestures, jumping up from where they were, and exclamations such as “I won, sir!” and “Yes, hurrah!” were noteworthy in this case. However, no information was shared to clarify how the students won the game and why they thought the winning strategy was correct in this process. The students lost in some games. Such situations are considered important for students to make informed decisions. The dialogue below reflects the students’ perplexity at not understanding why they lost the game.

The emergence of such a situation is of critical importance in terms of continuing the game process. If the students were successful in using the inferences they obtained at the easy level of the game in the later stages of the game, they would not want to play it anymore. Therefore, the sense of uncertainty and curiosity that kept the game replayable would have disappeared. In other words, although students adapted to the game and focused on the idea of winning, the implicit strategies they found through trials and errors in the first tasks did not work in more advanced tasks.

After the gameplay in doubles, the class was divided into two large groups; then, the group game was started. Making such a change can be explained as the dissemination and verification (approval or rejection) of the knowledge obtained in groups of two implicitly, first within the group and, then, among the groups. In addition, this approach can reveal mathematical processes such as transfer, reasoning, problem-solving and association, argument, and proofing.

The group game was played on the smartboard. Although the groups were not intended to be separated, particularly as boys and girls, no interventions were made at the students’ request (normally groups are mixed and equal in number). When the students won the game, they received the points specified in Table 1 at the end of the game each time they chose the winning numbers. Additionally, it was stated that the winning group would receive 1 point in games played on the smartboard in group games. Part of the game process performed between the groups (12 games at the easy level) is shown in Table 3.

The students completed 12 easy-level tasks by playing a total of 32 games. While G1 completed 6 of these tasks, G2 completed 5 of them. A fifth of these tasks (games ranked 11‒16 in Table 3) was skipped before being completed. In these games, the students largely choose consciously or randomly. In the case of winning the game, gaining more than 2 points indicates that a certain strategy may have been followed, while getting more than 2 points despite losing shows that the game’s limitations and conditions cannot be associated in the context of problem-solving. In this context, getting 3 points in the games won supports this and shows that the students noticed the winning number sequence. However, games numbered 4 and 16 were lost despite gaining many points. This situation resulted in students losing, although they were very close to winning. This suggested that they either played the game randomly or lost it on purpose. When these two games were examined in more detail, it was determined that the students made their first three moves correctly. However, as they did not have a specific strategy, they made their choices randomly and could not follow the winning number sequence with the activation of the algorithm. In other words, the defeat was not deliberate and on purpose. Therefore, it can be stated that students make their choices on purpose. Making conscious choices in the game process is of great importance in accessing the knowledge hidden in the game. In Figure 1, the process of the student considering the move she would make while making her choices shows that the game choices are made on purpose. This process is clearly stated in the dialogue below.

This dialogue shows that S2 makes her choices by reasoning. In this context, the student stated why she did not choose 13 and 14 after choosing 15 and leaving the numbers 16, 17, and 18 to the computer in the next move. Therefore, S2 stated that no matter which computer she chose, she could win the game by choosing 19 when it was her turn. In this process, the student makes her decisions about choosing balloons by logical inference. Therefore, there is intellectual reasoning in this step. However, the student coded S1, who wanted to be involved in the game process, was acting against the didactic contract because an intervention to the student’s game on the smartboard would undermine her thinking process, and the opportunity to make reasoning would be blocked by not being exposed to a situation she could overcome. As the teacher did not allow S1 to intervene in the game, such an obstacle did not arise. After these games, the teacher made the following explanations to reshape the student interaction in the environment to reveal winning strategies in the game.

In the next period, the environment was organized again by the teacher so that the students could attempt to glean the knowledge hidden behind the game. They were asked what the winning strategy/strategies could be in the game in this context. They were asked to discuss their approval of these strategies. This approach can serve in solving the problem comprising many tasks hidden behind the game in the transmission and dissemination of knowledge among students. Table 4 shows the students’ hypotheses in different tasks in these processes and the verification process in the game context.

In Table 4, the strategies (or hypotheses) developed during the game process can be divided into three. Those in the first category (e.g., the first number in Table 4) were won in the game context and were immediately approved. The second category (e.g., the second number in Table 4) are strategies that are generally rejected in group discussions. Because of the intellectual reasoning in the third category (e.g., fifth and sixth numbers in Table 4), the strategies through which the computer won in the game and whose accuracy was logically discovered became the knowledge of students. Although these result in a losing position because of struggle at the hard level in the game (or because the algorithm is activated in early moves), students are aware that they can reach the winning position by developing and revising these strategies. Based on this fact, the last strategy given in Table 4 is a consistent synthesis for the emergence of the game’s winning strategy by analyzing the previous ones.

Because of this strategy, a local solution emerged regarding Tasks 1 to 3 (see Table 1). While other tasks could be specified similarly, it could have been provided in time for them to emerge and students to discover local solutions for these tasks. However, as the implementation phase was conducted in a limited time, the teacher decided to move to the institutionalization stage after the local solution. The mathematical situation stage lasted approximately 65 minutes.

Institutionalization Stage

At this stage, the teacher generalized the students’ solution and presented a general strategy to be used in all tasks in the game. He also pragmatically demonstrated how this strategy worked for different tasks in the context of the game. Dialogues regarding these explanations are given in the following paragraph.

As seen from the dialogue, the teacher showed how it could be used by adapting it to different tasks’ constraints and conditions after referring to the students’ strategy. Then, he generalized it to a mathematical dimension. In this context, he explained that the mathematical structure hidden in the game is the division with remainders (Euclidean division). He also tested the division with remainders in one final task (Task 11) and demonstrated the overall solution’s accuracy. This stage lasted 5 minutes.

In the literature focus, the comments regarding the study results are given below under two subheadings.

Introduction of Mathematical Knowledge Through Digital Games Designed According to the TDS

This study showed that digital games could be used as an effective tool in preparing a student-centered environment and reaching mathematical knowledge in this environment. Based on this context, the concept of the division with remainders was introduced to 6th-grade students through a digital game. (Devlin, 2021) emphasizes that few digital games encourage conceptual understanding. The study, in which the conceptual meaning of the concept of the division with remainders is presented with a digital game, is important in terms of reflecting an application of this. This result largely coincides with the results obtained from studies on creating student-centered environments of digital games (Navarrete, 2013) and the use of digital games as a tool for educational purposes (Sousa & Costa, 2018).

It was also determined that the digital game allowed students to act independently in different stages of the TDS. In this direction, they understood the rules and purpose of the game and how winning or losing happened during the devolution stage. The mathematical situation includes discovering the digital game and developing strategies regarding the knowledge hidden in the game, testing them, and constantly revising their strategies accordingly. In this process, limiting the teacher’s interventions and their efforts to reach the knowledge behind the game from the inferences they derive from the digital game are indicators of supporting student-centered ideas. The didactic contract between teacher and students (Brousseau, 2002) can be stated to play a key role in keeping the student-centered approach alive in this process.

However, (Brousseau, 2002) stated that students might experience intermittent difficulties at the mathematical situation stage. Such blockages can be resolved in well-designed technology-based learning environments, such as digital games(Gök et al., 2020). The reason for this can be explained by the availability of feedback from multiple sources (e.g., visual, auditory, and numeric), not just a single source, in learning situations where technological tools are used.

In this study, it was determined that 6th-grade students could produce limited solutions to the tasks presented in the context of digital games. The teacher explained the general solution during the institutionalization stage. In addition, the teacher stated that the mathematical concept behind the game in the institutionalization stage was the division with remainders. In the games developed within the framework of the TDS, although the participants mostly discover the mathematical situation behind the game in the context of the game, this does not occur in a formal sense ((Brousseau, 2002); (Gök et al., 2020)). Considering the formation process of mathematical concepts, this is difficult for middle school students. In this sense, the results obtained in this study are similar to those suggested in the literature.

Contribution of the Digital Game and Student Interaction to Mathematical Processes

In this study, a problem situation presented as hidden in the digital game is divided into many tasks. The solution of these tasks includes limited solutions to the problem situation, and the solution is obtained by synthesizing these tasks. Therefore, well-designed digital games could support middle school students’ problem-solving skills.(Kiili, 2007) states that digital games enable students to use processes such as developing strategies, experimenting, interpreting, and adapting them to solve problems, and this experiential approach provides evidence for students to become problem solvers. In another study, (Lowrie, 2005) observes that digital games encourage productivity in problem-solving and motivate players to solve problems as a part of the game. This study determined that students developed many strategies in games for tasks in the digital game, proved them during in-group discussions and large-class discussions, spread the implicit models that they obtained, and transformed them into classroom knowledge. These results indicate that students use a mathematical processes strategy in their problem-solving attempts in the context of digital games.

(Tonéis, 2017) pointed out that environments designed around digital games could be used to access mathematical knowledge through reasoning. Similar to this result, this study shows that feedback from digital games plays an important role in accessing the mathematical knowledge hidden in digital games by passing through the reasoning filter. In this regard, mathematical reasoning encourages the cycle of proof of technology in the form of discovery, assumption, test/control, and proof that supports each other to reveal knowledge in the digital game (Cullen et al., 2020). That is, digital games improve students’ social skills (e.g., communication, cooperation, negotiation, and taking responsibility) and higher-order thinking skills (e.g., problem-solving and critical reasoning) (Shute & Ke, 2012).

These results indicate that digital games have the potential to develop students’ skills. To this end, (Shute & Ke, 2012) claimed that new ideas were needed for students to develop 21st-century skills and digital games could be used accordingly. The same study also emphasized that they motivated students and facilitated the presentation of complex academic content and the learning of 21st-century skills.