Reflections from course design supported by games, simulation and history: the oldest probability problem

Abstract

The rapid development of technology has affected the field of education and approaches to mathematics learning and teaching in particular. Considering the diversity of learner characteristics, diversification of teaching has become essential for educators. In this study, a course was designed based on the problem that led to the emergence of probability theory. The course provides a learning environment that blends game, simulation and history of mathematics with prediction, observation and explanation (POE) techniques. A case study took place to examine the reflections of a group of prospective teachers (n = 38) enrolled in this course from experimenting with this learning environment. The construction stages of the simulation models for the problem are presented in the article based on the POE approach adopted by the course. Participants’ reflections about the learning environment were analyzed using the thematic analysis approach. Findings indicate that the alignment of historical context, technology and interactive learning offers a robust framework for mathematics education that can serve as a blueprint for promoting prospective teachers’ deeper conceptual understanding of probability and teaching readiness.

1 Introduction

In today’s technology-driven society, characterized by constant change, education’s primary stakeholders—students and teachers—are directly impacted. As society evolves, educational programs and teaching methods must adapt continuously to meet shifting demands (Imbernón, 2014). The rapid advancement and widespread integration of technology into education have profoundly reshaped the skills and competencies required of both teachers and learners. Among these essential competencies, mathematical reasoning emerges as a cornerstone of 21st-century education, underpinning higher order analytical and logical thinking skills that are essential for addressing genuine, multidimensional problems in an ever-changing world. Fostering mathematical reasoning enhances students’ overall problem-solving skills, but also equips them to effectively analyze complex real-life situations, make informed and timely decisions, construct logical arguments and draw systematic conclusions. Consequently, mathematics occupies a central position in contemporary curricula across all educational levels.

Probability, a core branch of mathematics, quantifies uncertainty to facilitate effective decision-making in daily life and professional contexts (Metz, 2010). Serving as the theoretical foundation of statistics, probability has diverse applications across various fields, including sports, weather forecasting and games of chance (Sharma, 2015; Candelario-Aplaon, 2017). Over the past two decades, its teaching has gained prominence, as it imparts critical skills necessary for navigating uncertainty in an increasingly data-driven and complex world (Everitt, 1999; Bennett, 2003). Despite its importance, probability is widely regarded as one of the most difficult subjects to teach and learn (Batanero et al., 2004; Leavy & Hourigan, 2014; Koparan, 2015). A major challenge lies in the misalignment between the intrinsic nature of probability and statistics and the instructional methods typically employed (Koparan, 2022a). This issue is compounded by several factors, including the inadequate use of suitable instructional materials, limited opportunities for hands-on experimentation and observation and teachers’ insufficient training and experience in probabilistic concepts (Koparan, 2015). The inherent complexities of the discipline, such as the clash between intuitive thinking and probabilistic reasoning (Meletiou-Mavrotheris, 2007; Meletiou-Mavrotheris et al., 2019), further exacerbate these challenges.

To address these difficulties and foster sound probabilistic reasoning, both teachers and students require targeted training and support (Batanero et al., 2010; Koparan, 2019a; Koparan, 2022b). Traditional classroom approaches focused on solving mathematical problems through prescribed methods to arrive at a single correct answer (Koparan, 2019a) should be replaced by contemporary learning environments. These environments should incorporate authentic real-world tasks, concrete materials, educational games and technological tools to engage students effectively. Additionally, improving instructional effectiveness necessitates high-quality professional development for teachers, emphasizing both the content of probability and statistics and the pedagogical strategies needed to teach these subjects successfully.

Despite extensive research exploring innovative approaches to teaching probability, students continue to encounter significant difficulties with fundamental probabilistic concepts (Mezhennaya & Pugachev, 2018; Koparan, 2022b). Fischbein & Gazit (1984) emphasized the importance of learners’ probabilistic intuitions, underscoring how nurturing sound intuitions about the stochastic process can lead to a more meaningful integration of statistical ideas. To facilitate this process, educators should create learning environments where students can confront and refine their misleading or false intuitions about the stochastic process (Koparan, 2022a). Equally crucial is the provision of engaging learning environments that foster student motivation, defined as the desire to undertake a task, influencing both goal setting and the effort to achieve it (Keller, 2007) since low motivation toward probability and statistics remains a pressing concern (Peiró-Signes et al., 2020).

The research literature highlights a range of methods and tools for teaching probability and statistics. Among the most frequently used resources are tangible items such as dice, coins and cards (Nilsson, 2014), which not only facilitate hands-on experimentation but also help sustain students’ engagement and motivation. Another widely emphasized instructional strategy is the use of real-world problems, considered among the most effective ways to motivate learners (Konold, 1994; Zetterqvist, 2017; Sheikh, 2019; Cai et al., 2020). Interest in simulations and other technological tools is also increasing in probability and statistics classrooms (Koparan, 2022a). Simulations, in particular, can effectively address challenges in traditional instructional settings, such as conflicts between students’ intuitions and probabilistic reasoning, practical difficulties with repeated physical experiments and inadequate visualization of experimental outcomes (Koparan & Kaleli Yılmaz, 2015). They offer unique opportunities for mathematical modelling, allowing students to adjust parameters, conduct extensive experiments that are otherwise impractical in a classroom, generate results applicable to real-life situations and present them in visual formats (Koparan & Kaleli Yılmaz, 2015; Koparan, 2016; Koparan & Taylan Koparan, 2019).

A growing body of research indicates that appropriate and strategic integration of simulations, and other digital tools can have a positive impact on both student attitudes and learning of mathematical concepts and processes (Koparan, 2022b) including key statistical and probabilistic ideas (Meletiou-Mavrotheris, 2013), while at the same time, helping students develop the critical thinking, problem-solving and innovation skills required to cope with the demands of the complex digital era (Meletiou-Mavrotheris & Prodromou, 2016; Durdu & Dag, 2017). Using digital tools can help students concretize abstract concepts (Fabian & Topping, 2019; Koparan & Rodríguez-Alveal, 2022), explore mathematical relationships (Hoyles, 2018), take an active role in the learning process (Bray & Tangney, 2016), boost confidence in mathematics and develop cooperative skills (Fabian & Topping, 2019). Yet, despite the growing use of subject-specific software and other digital tools (Oztop & Buluc, 2022), effectively incorporating digital resources into the classroom remains challenging (Koparan, 2022b), in part because many educators lack sufficient preparation to harness ICT’s transformative potential (Ertmer et al., 2012; Meletiou-Mavrotheris et al., 2019) Consequently, further research is needed on how technological resources can be most effectively integrated into probability and statistics education to enhance student engagement and learning outcomes.

In response to the well-documented and persistent challenges students face in learning probability, this study developed and pilot tested an alternative learning environment that integrates games, simulations and the history of mathematics with prediction, observation and explanation techniques. This technology-enhanced learning environment was designed to help the prospective teachers participating in the study express, confront and refine their intuitive understandings of probability through authentic, real-world tasks. The study aimed to illuminate the nature of participants’ interactions with the learning environment’s technological and pedagogical tools. In particular, it explored how these interactions shaped the prospective teachers’ conceptual understanding of probability and informed their future instructional practices. Specifically, the study sought to answer the following research question: What are the prospective teachers’ interactions with a learning environment that blends games, simulations and the history of mathematics, and how do these interactions contribute to their conceptual understanding of probability and their preparation for teaching probability?

1.1 Theoretical framework

This study explored how prospective teachers interacted with the specially designed learning environment depicted in Fig. 1.

The learning activities were grounded in the prediction-observation-explanation (POE) technique (Joyce, 2006), which aligns with constructivist learning theories. In the context of probability instruction, the POE approach facilitates the exploration of learners’ initial conceptions regarding the problem and/or game at hand, stimulates discussion around these ideas and enhances students’ engagement with the lesson (Joyce, 2006; Koparan, 2019a). Exposure to unpredictable or surprising outcomes encourages students to reassess and refine their personal theories. The POE process involves three steps: students first predict the outcome of an event or experiment, then conduct the experiment (or observe the event) and make observations, and finally provide explanations for the observed outcome. Requiring students to justify their predictions enables teachers to gain valuable insights into students’ cognitive processes, helping to identify misunderstandings or gaps in their reasoning—particularly when observations deviate from predictions. Simultaneously, as students articulate and test their own reasoning while engaging with the explanations of their peers, they develop critical evaluation skills and construct deeper, more robust conceptual understandings.

1.1.1 Game-based learning and its applications in teaching probability

Game-based learning extends far beyond mere play and entertainment. By immersing students in ‘productive struggles’ (Koparan, 2019a), games can enhance engagement, improve attitudes toward learning, and offer social, cognitive and emotional benefits (Spires, 2015). Numerous studies reveal that games enrich the learning environment, boost performance, enhance motivation and foster collaboration (Davies, 1995; Nisbet & Williams, 2009; Koparan, 2019a; Koparan, 2021). In mathematics education specifically, games have always played an important role in cultivating mathematical thinking (Kamii & Rummelsburg, 2008), promoting computational fluency (Rutherford, 2015) and facilitating discovery-based learning (Chow et al., 2011). Research does caution, however, that while game-based approaches can be more effective than traditional teaching methods for improving students’ mathematics achievement (Randel et al., 1992), games are not a panacea. Evidence suggests that their impact on long-term learning may be limited when used in isolation (Bragg, 2012), whereas integrating games with other effective pedagogical strategies produces more substantial benefits (Young-Loveridge, 2004; Koparan, 2019a, 2019b; Koparan, 2021; Rodríguez-Alveal & Koparan, 2023).

Because probability typically involves random or uncertain phenomena, it naturally lends itself to game-based learning. Indeed, the formal development of probability can be traced to the study of games of chance, which were popular as early as the Roman Empire and flourished in 17th century France. A gambling puzzle posed to Pascal by a nobleman named Mere, and Pascal’s subsequent exchange with Fermat, ultimately led to the emergence of probability theory (Borovcnik & Kapadia, 2014; Williams, 2021). Drawing on these historical roots, scholars such as Góngora (2011) advocate a game-based approach to introducing probability concepts, contending that integrating games of chance especially those rooted in historical or cultural contexts can make learning both enjoyable and meaningful. In addition to offering historical insights, this approach can help learners examine how mathematical ideas intersect with real-life contexts. Moreover, it can expose and address students’ misleading beliefs or intuitions about random events (Koparan, 2019a; Sharma et al., 2021).

Students must be able to interpret probabilities across diverse contexts (Jones et al., 2007) and presenting multiple contexts and interpretations can foster deeper conceptual understanding. Games provide an effective avenue for such exploration: Batanero et al. (2004) point out that engaging prospective teachers in varied game-based activities equips them with knowledge and skills vital for future practice, while Corbalán (2002) argues that games should precede formal theoretical instruction. Moreover, the National Council of Teachers of Mathematics (NCTM, 2000) recommends having students make predictions, perform experiments and compare their predictions with the experimental outcomes, thereby motivating them to investigate discrepancies. Simple, cost-effective materials like coins, dice and cards (Vásquez & Alsina, 2014) not only maintain student interest but also help them develop skills such as observation, reasoning and inference (Edo et al., 2007). Through repeated trials, learners deepen their understanding of sample spaces, experimental versus theoretical probability and the law of large numbers (Wan de Walle et al., 2014; Koparan, 2019a). Although teachers should not expect one or two games to dramatically transform probabilistic reasoning (Garfield & Ben-Zvi, 2009), providing multiple, carefully designed game-based opportunities can meaningfully strengthen students’ engagement and conceptual growth in probability (Koparan, 2019a; Koparan, 2021).

In sum, teaching probability through games presents a low-cost, accessible and motivating approach. Historical and cultural contexts can heighten relevance, while tangible materials (e.g., dice, coins, cards) enable experiential learning. When integrated into a broader, research-informed pedagogical framework, game-based probability activities can significantly enhance students’ interest, engagement and conceptual understanding.

1.1.2 Simulation-based learning

Another effective way to enhance students’ probabilistic thinking is through simulation-based learning. Simulations provide teachers and learners with flexible environments that support students’ reasoning about the stochastic process through multiple trials and tangible insights into probability concepts (Koparan & Kaleli Yılmaz, 2015; Koparan, 2022a). As classroom technologies improve, using computer simulations of real-world probabilistic scenarios is increasingly recommended, as such experimental observations can deepen students’ theoretical understanding (Batanero et al., 2005), spark their motivation and strengthen their probabilistic intuitions (Borovcnik & Kapadia, 2009). Simulations also enable students to investigate probability problems that are impractical or impossible to replicate physically (Batanero & Diaz, 2007) and to focus on conceptual understanding rather than ROTE calculations (Borovcnik & Kapadia, 2009). They can further enrich students’ grasp of statistical concepts (Konold et al., 2007), particularly in chance experiments (Koparan, 2022a). By modelling probability problems, learners can draw meaningful connections between real-world contexts and formal mathematical frameworks (Greer & Mukhopadhyay, 2005).

There are two main approaches to using models in probability and statistics education. The first involves constructing a model from experimental, survey or existing data to describe and explain variability (Garfield & Ben-Zvi, 2008). The second entails designing or selecting suitable models to generate data for addressing a specific research question. In this study, participants created game-based models by following a step-by-step guide and also used provided models to generate data and gain insights into the problem solution. By merging these two approaches, the study presents a novel perspective on integrating technology into probability teaching and learning.

1.1.3 History of mathematics

Pre-service and in-service mathematics teacher’s benefit from being well versed in the historical development of the subject matter they teach, as doing so can enrich both their own understanding and their students’ learning experiences. The history of mathematics can serve as a powerful pedagogical resource, illustrating that mathematics is not static but a continuously evolving field shaped by experimentation, application and discovery (Baki, 2006). It also highlights the influence of mathematics on human progress, other disciplines and technological developments, revealing how concepts, methods and proofs have emerged over time. Introducing historical elements into teaching can enhance student motivation, as it includes emotionally compelling narratives and highlights cultural connections that capture learners’ interest and spark curiosity (Sipsak et al., 2025). By presenting significant historical events and famous mathematicians’ biographies, educators can create an engaging classroom atmosphere that helps learners appreciate the human dimension of mathematical progress. Moreover, historical problems can serve as catalysts for deeper thinking, illustrating that activities such as formulating hypotheses, experimenting, clarifying and generalizing have been vital to the growth of mathematics. In the realm of probability, focusing on the historical roots of the discipline can illuminate its real-world significance and provide context for key concepts.

2 Method

The study adopted an exploratory case study design to investigate the prospective mathematics teachers’ interactions with a learning environment that blended games, simulations and the history of mathematics. The exploratory case study approach was chosen to gain in-depth insights into how these interactions influenced participants’ conceptual understanding of probability and informed their preparation for future teaching practices. It was judged that this methodology was particularly suited for studying the complex, context-specific dynamics of prospective teachers’ learning experiences, allowing for a detailed exploration of their engagement with both the technological and pedagogical elements of the learning environment (Merriam, 1998; Yin, 2014).

Rather than testing specific hypotheses, the study aimed to generate rich, qualitative descriptions of prospective teachers’ interactions with the learning environment and reflective engagement with the learning process. These findings offered a detailed perspective on how such interactions contributed to the participants’ readiness to teach probability effectively in their future classrooms. While the findings are not intended to be broadly generalizable, they provide valuable insights that can inform similar educational contexts and guide future research on innovative pedagogical strategies in probability education.

2.1 Context and participants

The study was conducted at a university in the Western Black Sea Region of Turkey during the spring semester of the 2021–2022 academic year. The case under study was a class of 38 third-year prospective mathematics teachers enrolled in a compulsory, theoretical course on probability teaching and learning. The participants had completed one method course on statistics and another on probability theory in the preceding year. The first author served as the course instructor.

The course lasted 14 weeks, with 3 hours of instruction per week. This study focuses on a two-week segment of the course. During the first week, the course and the TinkerPlots software were introduced. In the second week, the designed learning environment was implemented. The prospective teachers were introduced to the problem that historically led to the emergence of probability theory. The two games related to this problem were presented to the participants:

Game 1: The game is won when a dice is rolled four times and a six appears at least once.

Game 2: The game is won if a pair of dice is rolled 24 times and a (6,6) appears at least once.

Prospective teachers were asked to determine whether their chances of winning were higher in the first or second game, or if their chances of winning were equal in both games. They were then required to justify their responses and apply their knowledge of probability in doing so.

The learning environment was structured around the POE technique (Joyce, 2006), in alignment with constructivist learning theories. This approach was chosen with the expectation that it would help students better observe and reflect on the entire learning process. The POE approach facilitated active learner engagement through the following stages:

• Prediction stage: participants made predictions about the outcomes of the two games before playing them.

• Observation stage: participants played the games, initially using physical dice to observe first-hand outcomes over a limited number of trials. They then transitioned to the dynamic statistics software TinkerPlots, which allowed them to model and simulate the games over a large number of trials, facilitating extensive data collection and analysis (Konold & Miller, 2004).

  • TinkerPlots, a versatile software suitable for use across educational levels from primary school to university, was chosen for its capacity to seamlessly integrate exploratory data analysis with probabilistic modeling. The software offers a range of tools for data analysis and probability modeling, allowing users to generate data (e.g., by drawing samples from constructed models) and actively engage in experimentation (e.g., refining models and conducting simulations).

• Explanation stage: in the final stage, prospective teachers calculated the theoretical probabilities underlying the two games and compared these with the experimental results. This was followed by discussions on the historical importance of the problem in the development of probability theory.

Prospective teachers were given a brochure with instructions to guide them in creating the models for the games. This brochure is provided as an Appendix.

2.2 Data collection and analysis

To examine how participants’ interactions with the learning environment and its tools influenced their conceptual understanding of probability and their preparation for teaching it, data collection focused on documenting participants’ engagement with the physical games and their simulations and their responses to tasks designed to connect theoretical and experimental probability concepts. Specifically, to address the research question, What are the prospective teachers’ interactions with a learning environment that blends games, simulations, and the history of mathematics, and how do these interactions contribute to their conceptual understanding of probability and their preparation for teaching probability?, it utilized two primary data sources:

• Instructor’s reflective notes: The instructor observed participants’ interactions with the learning environment, including their engagement with both technological tools and physical activities, and maintained reflective notes after each class session.

• Students’ written responses: Prospective teachers’ written answers to problems related to the historical emergence of probability theory, as well as their reflections during the three stages of the POE process, were collected and analyzed.

The prospective teachers’ written responses were qualitatively analyzed by the first author using a thematic analysis approach (Braun & Clarke, 2006) to uncover their thought processes and reflections at each stage of the POE process. The analysis began with a thorough review of all data, including written responses and the instructor’s reflective notes, to identify initial patterns and familiarize researchers with the content. Data were then systematically coded, focusing on participants’ reasoning during the POE stages, their engagement with technological and physical tools and their reflections on the historical context of probability.

Emerging codes were grouped into themes (e.g., conceptual understanding of probability, pedagogical insights into teaching probability, engagement with games, engagement with simulations). To ensure rigor, coding consistency was verified by the second author. Representative quotations were selected to illustrate key findings, capturing both individual variations and shared experiences. This repeated process provided detailed understanding of how the blended learning environment participants’ conceptual and pedagogical development.

3 Results

The study findings are next presented within the framework of the POE approach.

3.1 Prediction

Before playing the games, prospective teachers were asked to reflect on the games and make predictions about whether one of the games offered an advantage or if the chances of winning were equal. Table 1 provides an overview of participants’ predictions during this initial phase.

As seen in Table 1, 20 of the prospective teachers predicted that the probability of winning would be higher in the first game, 14 believed it would be higher in the second game and 4 predicted equal probabilities for both games. Of the 20 prospective teachers who anticipated a higher probability of winning for the first game, only two provided explanations grounded in a sound theoretical basis. An analysis of the initial predictions revealed that some were not based on probability calculations, while others relied on flawed calculations that focused solely on the number of dice, ignoring the broader situation context.

3.2 Observation

During the observation phase, prospective teachers were provided with dice and instructed to conduct experiments on the games and record the results. They were given the option to change their initial predictions if they wished. It was observed that some prospective teachers chose to revise their predictions based on their experimental findings during this phase.

PT 11: I thought you had a better chance of winning in the first game. When I used the dice, my decision changed. In the first game, there was no 6 in 4 shots. In the second game, a 6-6 came once in 24 rolls of the dice.

PT29: I thought the second game was more advantageous. When I tried using the material, I decided that my guess was not correct.

Other prospective teachers opted to maintain their initial predictions without making any changes.

PT2: I experimented with the dice and won the second game. But I still think there is a higher chance of winning in the first game.

PT32: The chances are even. If I have to choose one, I experiment with the dice once and choose the winning game in the experiment.

PT5: In the first game, 6 came 2 times in 4 trials. In the second game, there was no 6-6 in 24 trials.

PT35: In the second game, I never scored 6-6. But 6 came in the first game. The first game is more advantageous.

PT21: Using the dice did not change my mind, it supported it.

PT14: There are more cases in double dice. Less chance of winning in the second game. My decision hasn't changed.

PT26: The second game has a higher chance of winning. There was no 6 in any of the 4 shots in the first game. In the second game, I scored 6-6 in the 14th shot.

PT37: Our first thought is always right. So I'm not changing my initial guess.

At the conclusion of the dice experiments, prospective teachers were asked if they were confident in their final decisions. Nearly, all of them (35 out of 38) expressed uncertainty regarding their choices.

PT8: In the first game, I got a 6 on the second shot. In the second game, it came 6-6. It's difficult to decide.

PT27: According to my experiment, the chances of winning in both games are almost the same. I'm undecided.

PT36: In the first game, there was no 6 in 4 shots. A 6-6 came once in the second game. I do not know.

PT1: The desired outcome may not occur in either game. I cannot tell for sure.

PT19: The experiments I have run make me think that there is a high probability of achieving the desired results in both games.

PT14: The first game won more often when I used the dice. But still I am not sure.

The dice experiments conducted by the prospective teachers appeared to increase their confusion. While some changed their original prediction that the first game had a higher chance of winning, others chose to maintain their initial decision. The primary factor influencing these decisions was the outcomes observed during their experiments. Some participants used the experimental results to validate their predictions, whereas others observed outcomes that contradicted their expectations. Despite conducting these experiments, nearly all prospective teachers remained uncertain about which game was more advantageous. This persistent uncertainty can likely be attributed to the limited number of experimental trials. One participant explicitly suggested that conducting additional trials would be necessary to gain a clearer and more confident understanding.

PT15: Sometimes the first game wins, sometimes the second game wins. I wasn't sure as the number of trials was small.

After this stage, prospective teachers were given the opportunity to use simulation models.

Using simulations, prospective teachers independently decided on the number of repetitions for their experiments, exploring outcomes from 10, 100, 1000 or even 100 000 trials. By examining the experimental results generated by the models as displayed in Fig. 2, they were able to draw meaningful inferences. Notably, some prospective teachers reconsidered and revised their initial decisions based on the insights gained from these simulations.

PT12: I saw more experimental results with the simulation, my opinion changed.

PT2: I've seen in the simulation that the chances of winning are lower in the second game. I was expecting a 20-30% chance of winning in the second game. If we didn't use the simulation, I would never have thought of a 50% probability. Simulation made me change my mind on this issue.

All prospective teachers reported that after seeing the simulation outputs, they were able to make a definite decision about which game had a higher probability of winning.

PT7: I observed that the chance of winning in the first game is between 51% and 52%. I thought that there was a probability of winning closer to 52% since it was mostly 52% in the trials. In the second game, I observed 50% of the cases where it did not come in 24 shots (6.6). Therefore, I thought that the remaining situations, namely the probability of coming at least once (6.6), could be close to 50%. This value may be slightly more or slightly less than 50%. Thanks to the resulting simulation, I was experimentally sure that the first of two games was more likely to win.

The following are selected excerpts from prospective teachers’ perspectives on how simulations contributed to their understanding:

PT24: Thanks to the simulation, we achieved a result in a short time (Saving on time)

PT28: Thanks to the simulation, we were able to see 100 000 test results. We couldn't have done this much experimentation. (Lots of possibilities to experiment)

PT19: I couldn't calculate the test results while I was experimenting. The simulation test results were presented to us as frequency and percentage. (Ease of calculation)

PT5: My vague thoughts became definite thanks to the simulation. The simulation presents the test outputs by grouping and facilitates decision making. Simulation results are more understandable (Comprehensive data representation)

PT33: Simulation outputs were colourful and clear (Visualization)

PT1: Experimental results are instantly reflected on the graphics (Dynamism)

PT37: The simulation made me realize that my own probability experiments were insufficient (Truth)

PT15: My theoretical calculations did not work, I could not obtain the result, but thanks to the simulation, we obtained experimental results (Experimental solution)

PT4: There are 1 or 2% differences between the two games. I couldn't understand this difference with few experiments. The simulation revealed this delicate distinction (Sensitivity)

PT11: The fact that there are simulation models for two games and the number of experiments can be changed makes the simulation different and important (Experimental variables can be changed)

PT38: I wasn't sure of my guess. The simulation clarified the situation (Embodiment)

PT20: The simulation made me change my mind. The chances of winning in the games were unequal (Facing mistakes)

PT27: I noticed that the change in the results was less as the number of experiments increased. Thanks to simulation, I understood the relationship between experimental probability and theoretical probability (Experimental and theoretical probability relationship).

3.3 Explanation

After concluding with the observation stage, the prospective teachers were asked to calculate the theoretical probabilities and to compare them with the empirical results. While not all participants well able to adequately respond to this task, approximately half demonstrated correct reasoning about theoretical probability. Many, however, reported experiencing difficulties in calculating the probability for the second game, citing computational challenges. To address this challenge, the researcher took precautionary measures and encouraged the use of the advanced calculator Derive 6 software to facilitate these calculations.

PT13: Let's find the probability of getting a 6 at least once in the experiment of rolling a dice four times in the first game. Instead, if we find the probability that no 6 appears in the experiment of rolling a dice 4 times and subtract it from 1, we shall conclude. In other words, the probability of never getting 6 in an experiment of rolling a dice is 4 times:

In the experiment of rolling a dice 4 times, the probability of a 6 is obtained as:

PT31: In the second game, let's find the probability that 6-6 appears at least once when the double dice is rolled 24 times. If we subtract from 1, the probability that no 6-6 will appear in the experiment of rolling the two dices 24 times can be found.

The probability of 6-6 not appearing in one roll of the double dice is |$\frac{35}{36}.$|

The probability that no 6-6 will appear in the double-dice-roll experiment is calculated as

If this is continued, the probability that no 6-6 will appear in the experiment of rolling the double dice 24 times is obtained as

Prospective teachers required additional support for their calculations, as standard and mobile-based calculators were insufficient for the task. Anticipating this challenge, the researchers had taken proactive measures to address the issue and transform it into a learning opportunity. Participants were encouraged to use Derive 6 software to facilitate their calculations and deepen their understanding of the process.

The probability of getting at least one 6-6 when rolling the double dice 24 times is

After the theoretical calculations, the researcher explained the place and importance of the problem in the history of mathematics as follows.

R: By the 1650s, gambling became very common in France, and games such as dice, cards, toss and roulette were highly developed. The increase in the need for money brought the idea that gambling chances can be calculated with some formulas. We can say that the motivation behind the emergence of probability theory is the ambition to make money. Antonie Gombaud, a respected Frenchman, although not a knight, was an amateur mathematician, called as Chevalier de Méré by his friends, who liked to play with numbers and bet. Chevalier de Méré was obsessed with increasing his wealth by gambling. It was a turning point in the development of probability theory when he encountered a problem that he had trouble understanding while gambling. The rule in the game he played ‘Whoever rolls a 6 at least once in four rolls wins’ is in the form. But Chevalier, in order to win more, changes the rules of the game to ‘It is won if two dices are rolled 24 times and they roll at least once (6,6)’. But he observed that this rule paid less than the first game, not more. He asked his friend Blaise Pascal why that was the case. Pascal, one of the mathematicians of that period, searched for the answer to the question posed to him. In his solution, he calculated Chevalier's chance of winning 51.8% in the first game and 49.1% in the second game, revealing why he lost more in the second game. Today, probability theory has entered many areas of human life. For example, probability theory is used in many fields such as economy, sports, games of chance, weather forecasts, prediction of newborn baby gender and genetics, banking and insurance, quantum mechanics, and kinetic theory of gases. Who would have thought that the occupations that started with betting problems in the 17th century would become a sub-branch of mathematics and take such a prominent place in modern life?

Some quotes that illustrate how the preservice teachers responded to the instructor’s explanation of the problem’s significance in the history of mathematics are presented below.

PT11: Wow. I didn't know that probability theory emerged from a game of chance. I was greatly inspired by this story.

PT6: I found it interesting that probability theory emerged in the 1650s. I thought it might have emerged much earlier.

PT32: I was surprised that probability theory emerged from a game of chance. I thought probability came from a more pure field.

PT15: Scientists were writing letters at that time. The correspondence between Pascal and Fermat must have been quite interesting and enjoyable.

PT5: I can engage my students by telling these stories in class. Historical anecdotes can make math more fun.

As can be seen from the reactions, the prospective teachers found the anecdote mentioned by the educator astonishing, intriguing, disappointing, entertaining and inspiring. Some prospective teachers stated that they had a new application idea that they could tell about the history of mathematics in their classes.

Prospective teachers were visibly surprised to learn that the dice activities, which began as simple games, were in fact linked to a historical problem that had been addressed by mathematicians. Their amazement grew when they discovered that the theory of probability itself emerged from efforts to solve this problem. Their excitement was evident through the gleam in their eyes and the smiles on their faces. Some students remarked that this revelation helped them understand that mathematics is not a pre-existing entity that ‘fell from the sky’, but rather a discipline that developed gradually through human activities, with contributions from various scientists and cultures shaping its evolution over time.

4 Discussion

In recent years, statistics education researchers and practitioners have increasingly emphasized the importance of incorporating real-life problems, technological tools, authentic data and dedicating additional classroom time to data-based discussions to enhance students’ intuitions about the stochastic process (Koparan, 2019a, 2019b; Koparan, 2022a). Building on these principles, the learning environment designed and pilot-tested in this study adopted a student-centred approach. It actively engaged prospective teachers with historically significant game-based problems, simulations and hands-on activities. The study examined how these interactions influenced the participants’ conceptual understanding of probability and their preparation for teaching the subject. The findings underscore the value of integrating historical context, technological tools and interactive learning strategies, such as games, to deepen conceptual understanding and enhance prospective teachers’ readiness for teaching mathematics effectively.

The inclusion of a historical problem, namely the origins of probability theory, allowed participants to explore how mathematical concepts emerged through real-world challenges faced by mathematicians. By examining how mathematicians approached uncertainty and used mathematical models to solve complex problems, participants gained insights into the problem-solving processes that drive the evolution of mathematical thought. This approach not only deepened their conceptual understanding of probability but also reshaped their perceptions of mathematics as a dynamic, human endeavour that develops over time through problem-solving and cultural contributions. The overwhelmingly positive feedback from prospective teachers highlights the effectiveness of embedding historical context into mathematics education. This approach fosters curiosity, enhances engagement and encourages more positive attitudes toward both probability and the broader field of mathematics.

The study’s findings highlight the potential of student-centred, technology-enhanced approaches to replace traditional teacher-centred methods, as suggested by Clark-Wilson et al. (2020). The integration of concrete materials and simulation-supported activities allowed prospective teachers to engage deeply with the content by making predictions about the game outcomes, testing them through observation and reflecting on the results. This interactive environment encouraged them to construct their own knowledge, aligning with the shift in educational paradigms toward more active and inquiry-based learning.

Simulations, in particular, offered prospective teachers valuable opportunities to explore diverse problem-solving strategies while applying probabilistic thinking to real-world scenarios (Koparan, 2022a). As Kadijevich (2007) emphasizes, the strategic use of technology in modelling problems can significantly enhance learning outcomes. In this study, prospective teachers were exposed to an empirical approach to one of the foundational problems in probability theory. The integration of simulations enabled them to experiment, test predictions and draw conclusions based on empirical evidence. These activities encouraged participants to move beyond ROTE calculations, enabling them to connect theoretical probability with real-world applications. Such interactions align with recommendations in the literature emphasizing the importance of technological tools in fostering inquiry-based learning and deepening understanding (Clark-Wilson et al., 2020). By independently setting the number of trials in their simulations and observing the results, prospective teachers were able to experience the iterative nature of mathematical reasoning. This autonomy not only strengthened their conceptual understanding but also prepared them to implement similar strategies in their future classrooms.

The findings also highlight the dual challenges and opportunities presented by computational tasks. While many participants struggled with probability calculations, particularly for the second game, these difficulties became opportunities for learning. By incorporating Derive 6 software, the course provided a technological scaffold that helped participants manage computational complexity while emphasizing the importance of accurate mathematical modelling. This experience underscored the role of technology in supporting learners as they engage with challenging content and develop problem-solving skills relevant to modern mathematics teaching (Kadijevich, 2007).

The integration of computer-based simulations and other technological tools played a pivotal role in fostering the foundations of these future teachers’ reasoning about the stochastic. Through activities that involved modelling and simulating games, solving problems and making decisions based on both empirical data and formal stochastic reasoning, participants engaged with core principles of modern statistics and mathematics education (GAISE, 2016; NCTM, 2000; Bargagliotti et al., 2020). These activities allowed prospective teachers to observe how engaging, real-world contexts, such as games and experiments, can be effectively used to teach probability in ways that resonate with students. By interacting with technology to solve historically significant game-based problems, they experienced first-hand how interactive learning environments enable learners to explore, test and refine their ideas, leading to a deeper understanding of mathematical concepts. This innovative approach not only enhanced their conceptual understanding of probability but also aligned with contemporary educational goals that emphasize active engagement, critical thinking and decision-making grounded in mathematical reasoning. Furthermore, it prepared them to apply these methods in their future classrooms, equipping them to create dynamic and interactive teaching practices that encourage exploration and critical inquiry.

In response to our research question—examining how prospective teachers interact with a learning environment that integrates games, simulations and the history of mathematics, and how these interactions shape their conceptual understanding of probability and their preparation for teaching—the findings of this study highlight the transformative potential of such an approach. By engaging with historically significant probability problems and experimenting with simulations, participants moved beyond procedural calculations to develop deeper probabilistic reasoning and a contextualized understanding of the subject. These interactions reinforced probability’s relevance to real-world problem-solving while fostering an appreciation for the iterative nature of mathematical discovery. Additionally, technology-supported activities promoted active, inquiry-driven learning, equipping prospective teachers with strategies to create engaging, student-centred classrooms. Ultimately, this interdisciplinary approach, which intertwines historical context, technological tools and interactive learning, provides a robust framework for enhancing both conceptual mastery and pedagogical readiness in the teaching of probability.

5 Concluding remark

This research was exploratory in nature, and as such, the findings should be considered suggestive rather than conclusive, warranting further investigation. The exploratory case-study design, qualitative methodology, small scale and geographically limited scope of the study caution against broad generalizations. However, the findings do suggest that the approach adopted in the study has significant potential for enhancing the teaching and learning of probability and statistics. The learning environment, which blended games, simulations and the history of mathematics, enabled prospective teachers to engage in data-driven discussions centred on foundational probability concepts and ideas. It also introduced them to an unconventional, simulation-based problem-solving approach, distinct from the traditional paper-and-pencil process they were accustomed to. It is thought that the use of such simulations in mathematics teaching will be useful for prospective teachers to understand the potential of technology and how to effectively integrate it into classroom activities. It is recommended that teachers and researchers collaborate in designing-learning environments that incorporate contemporary approaches, such as those used in this study, to create more engaging and effective mathematics instruction. Such initiatives have the potential to create engaging and impactful teaching practices that support deeper understanding and application of stochastical concepts.

AI statement

No AI was employed during the research conduct or initial drafting of this paper. During the manuscript refinement phase, ChatGPT was used selectively for language editing when deemed beneficial. This assistance was aimed at enhancing the clarity, coherence and style of the text, while ensuring the integrity of the original content was preserved.

Funding

No funding was received for conducting this study.

Conflict of interest statement

All authors declare that they have no conflicts of interest.

Data availability

All data generated or analyzed during this study are included in this published article [see Appendix].

Ethics approval

This research was reviewed and approved by the Human Research Ethics Committee of Zonguldak Bulent Ecevit University. Ethics Committee Name: Zonguldak Bulent Ecevit University Human Research Ethics Committee (Approval Code: 96985 and Approval Date: 04.11.2021).

Author contribution

T.K.: Conceptualization, Methodology, Formal Analysis and investigation, Writing—original draft preparation and Writing—review and editing and M.M.-M.: Supervision.

References

A Appendix: A guide with instructions to help prospective teachers create game models.

TinkerPlots steps for the first game (At least one 6 in 4 dice experiments)

• Step TP1 Click on the “Sampler” in TinkerPlots.

• Step TP2 Choose the Mixer device.

• Step TP3 Add numbers from 1 to 6 using (+)

• Step TP4 Enter the Repeat value as 10000 and Draw 4.

• Step TP5 Change Attr1, Attr2, Attr3, and Attr4 as Die1, Die2, Die3, Die4 in Figure 3 and then press Run.

• Step TP6 Type “Result” in the first cell of column 5 in Figure 4

• Step TP7 Right-click in the same cell and select edit formula.

• Step TP8 Type the following formula in the window that opens and then press OK.

  

• Step TP9 With the Result column selected, click on the Plot and see if there are 6’s or not in the 4 dice experiments for 10000 trials. In Figure 5, 0 indicates the number of cases where no 6 appears, and 1 the number of cases where 6 appears at once.

TinkerPlots steps for the second game (At least one 6-6 in 24 rolls of the double dice)

• Step TP1 Click on the “Sampler” in TinkerPlots.

• Step TP2 Choose the Mixer device twice

• Step TP3 Add numbers from 1 to 6 using (+)

• Step TP4 Enter the Repeat value as 24 and Draw 2, then press Run in Figure 6.

• Step TP5 Click on “Options” from Result of Sampler 1, click “Result Attributes” and choose “Join” in Figure 7.

• Step TP6 Type “Result” in the first cell of column 4 in Figure 6.

• Step TP7 Right-click in the same cell and select edit formula.

• Step TP8 Type the following formula in the window that opens and then press OK.

  

• Step TP9 With the Results column selected, click on the Plot and see if there are 6-6 or not in 2 dice experiments for 24 trials.In Figure 8, 0 indicates the number of cases where no 6-6 appears, and 1 indicates the number of cases where it appears at least once.

• Step TP10 Use “History of Results of Sampler 1” and click the number in the top right in Figure 8.

• Step TP11 Enter the number of trials (1000) and click “Collect”.

• Step TP12 With the count Result selected, click the Plot and visualize outputs for 1000 trials in Figure 9.In Figure 10, one can see that in 51 % of the trials, 6-6 did not appear at all, while in 49% of the trials it appeared at least once.

Dr. Timur Koparan has a Ph.D. in Mathematics Education from Karadeniz Technical University and an M.Sc. in Mathematics from Pamukkale University. He has participated in extensive research and has numerous publications in international scientific journals and congresses. Dr. Timur Koparan is a faculty member at Zonguldak Bülent Ecevit University Education Faculty of Eregli. His research focuses on probability teaching, teaching with games, using simulation and learning environments and using ICT tools in mathematics/statistics teaching and learning. (https://www.researchgate.net/profile/Timur-Koparan)

Dr. Maria Meletiou-Mavrotheris is a professor at European University Cyprus and Director of the ICT Enhanced Education Research Laboratory. She has a Ph.D. in Mathematics Education, an M.Sc. in Statistics, an M.Sc. in Engineering and a B.A. in Mathematics from the University of Texas at Austin, and an M.A. in Open and Distance Learning from UK Open University. She has engaged extensively in research and has numerous publications in scholarly international journals and books. Her research focuses on technology-enabled education, specifically on the use of digital games and other ICT tools in mathematics/statistics teaching and learning. (https://www.researchgate.net/profile/Maria-Meletiou-Mavrotheris)