By Thomas Rajotte, Professor and Researcher at Université du Québec à Rimouski
This article stems from a professional experiment that I carried out in 2017 as part of the École d’été en orthopédagogie des mathématiques (Summer School in Remedial Instruction in Mathematics) at Université du Québec à Montréal (UQAM). In addition, the theoretical foundations presented in connection with the use of play for teaching mathematics are derived from my experience in writing a book (Marinova and Biron, 2016), which I did in collaboration with Krasimira Marinova, a Professor specializing in preschool education at Université du Québec en AbitibiTémiscamingue (UQAT).
Using play to engage students in their learning in mathematics
An undeniable advantage of using play to teach mathematics stems from the fact that it helps to actively engage students in complex tasks without the impression that they are carrying out an academic task. When discussing this with my university students, I usually suggest that they think about the game of Battleship. This game is highly pertinent to mathematics since it can help students to understand the Cartesian plane and its resulting coordinates. However, I am not sure that any students have ever shown an interest in the game of Battleship intending to become better at reading Cartesian planes. Students who become involved in this game tend to be interested instead in the idea of playing the role of the admiral of a large naval fleet and of being able to launch bombs to sink their opponent’s ships. It is the essence of the game that draws students to become involved in the activity.
To more deeply understand the features of games that promote learner engagement, Rajotte, Héroux and Boivin (2021) refer to the concepts of didactic tasks and playbased tasks. These two notions are briefly defined in the table below.
Table 1. Definition of the characteristics of play relating to playbased tasks and didactic tasks  
Playbased tasks  Didactic tasks 
Playbased tasks are the cornerstone of play that help to engage students in the learning goal. These tasks:

The instructional objectives are concealed behind the game. These tasks are:

By looking at the chart above, we can see that the playbased task (the driver of the activity) of the game of Battleship is the possibility of sinking the opponent’s ships. Furthermore, the didactic task (instructional objectives) is concealed behind the playbased task. This latter task is essentially accomplished when the students involved in the game master the Cartesian plane.
Students' engagement in math games necessarily stems from their interest in completing the playbased task. However, the teacher’s focus should be on the didactic task involved in the game and the specific instances of learning that could result from it.
Play: a way to foster selfesteem or selfconfidence in elementary school students
As mentioned by Marinova and Biron (2016), play is undeniably beneficial for exceptional students. Described by Bruner as the “paradox of learning through play” (1983), play encourages students to take risks, undertake actions and make errors without their selfesteem being affected.
By becoming involved in the game, the students can take ownership of their achievements and successes (intrinsic attribution factors), while attributing their failures to the character that they are playing or to another extrinsic factor (for example, their opponent’s incredibly good luck). By attributing any failures to an extrinsic factor that does not pertain to the students themselves or their cognitive abilities, exceptional students can develop their selfesteem and selfconfidence regarding mathematics.
Differentiating games in mathematics: a way to meet the specific needs of students
Learning games need to be adapted continuously to prevent students from losing interest in the activities proposed to them (Rajotte, 2021). Still, more importantly for exceptional students, changes can also be made to meet the learning needs of children.
In practical terms, differentiating a game to the needs of the student consists of modifying the underlying didactic variables for the proposed game. The goal when modifying the didactic variables is to change the strategies and reasoning that need to be implemented to complete the game, thus fostering different types of learning.
Example of a game adaptation
Game: Students take turns throwing a sixsided die. Students record each number they rolled and add it to their previous rolls. The first student to roll a total of 25 wins.
Some Grade 3 students quickly become bored due to the simplicity of the proposed task. In order to keep their interest, the teacher could adapt the game and to ask them to take turns throwing a die with 20 sides and to be the first to reach a number greater than or equal to “100.”
The primary didactic variables characterizing a game
Every game is unique and is characterized by its own didactic variables. Biron (2012) identifies three major categories of didactic variables that relate to most games accessible to students. These variables are highlighted in the table below:
Table 2. Primary didactic variables involved in a mathematical task  
Mathematical structure  Numerical data  Linguistic aspects 



The didactic variables relating to mathematical structure refer essentially to the nature of the operations that are required by the game (addition and/or multiplication structures) as well as to the number of operations needed to reach the set objective. Numerical data pertain primarily to the order of magnitude of the numbers as well as the nature of the set of numbers (whole numbers, integer numbers, rational numbers, decimal numbers, etc.). Finally, the variables associated with linguistic aspects pertain to the vocabulary and the sentence structure that allow for playing the game. While the variable classification proposed by Biron (2012) was initially applied to analyzing problemsolving activities, it can easily be considered for addressing the modalities of adapting a math game (Rajotte, 2021). Use the following table, inspired by Rajotte (2021), as an example of adaptation for each of the major categories of didactic variables.
Table 3. Example of differentiation for each of the major types of didactic variables  
Mathematical structure  Numerical data  Linguistic aspects 
To accelerate the game Snakes and Ladders, you can modify the nature of the operations involved in the task. Instead of using a single die with six sides, players could throw two dice and multiply the resulting numbers to move faster on the game board.  In the card game of War, you can modify the numbers involved to make the game more complex. For example, instead of using cards representing natural whole numbers (2, 4, 6, 8), the teaching could adapt the game to use cards with representations of fractions (½; ¾, etc.) or even cards on which decimal numbers are illustrated (1.234; 12.34 or 123.4).  In the math version of the game Headbanz, each player has to guess the geometrical figure placed on their forehead by asking the other players questions that can be answered only by “yes” or “no”. You could adapt some of the vocabulary. Instead of using pictures of commonly found twodimensional shapes (squares, rectangles, triangles), you could use threedimensional shapes (cubes, tetrahedrons, octahedrons). 
Conclusion
I heartily recommend educators use math games to intervene with exceptional students. There are numerous undeniable benefits. As mentioned previously, they can result in increased engagement and involvement by learners in math activities. In addition, on an emotional level, it is essential to take into account the sense of competence that is consolidated and developed in math students as a result of playing games. Finally, I would like to reiterate that such contributions of play on a developmental level merit consideration by all educators. Indeed, by carrying out simple adaptations of the rules, teachers can find a way to use a wide range of games to intervene with students and to meet their specific learning needs. Have a good game and have fun!
References
Biron, D. (2012). Développement de la pensée mathématique chez l’enfant : du préscolaire et au premier cycle du primaire. Les Éditions CEC, Anjou.
Bruner, J. (1983). Le développement de l’enfant : Savoirfaire, savoirdire. Presses universitaires de France, Paris.
Dorier, J.L. et Maréchal, C. (2008). Analyse didactique d’une activité sous forme de jeu en lien avec l’addition, Grand N, 82, 6989.
Marinova, K. et Biron, D. (2016). Mathématiques ludiques pour les enfants de 4 à 8 ans. Presses de l’Université du Québec, Québec.
Rajotte, T. (2021). L’adaptation des jeux en mathématiques : pour que le plaisir dure longtemps! Dossier spécial sur le jeu en mathématiques. Vivre le primaire, 34(1), 4143.
Rajotte, T.; Héroux, S. et Boivin, É (2021). Le jeu en classe de mathématiques : Engager activement les élèves et favoriser leurs apprentissages. Chenelière Éducation, Montréal.
Thomas Rajotte is a Professor at the Lévis Campus of Université du Québec à Rimouski. As a researcher in the didactics of mathematics, he is affiliated with the Centre de recherche interuniversitaire sur la formation et la profession enseignante (CRIFPE) [Interuniversity research centre on training and the teaching profession] and the Réseau de recherche et de valorisation de la recherche sur le bienêtre et la réussite (RÉVERBÈRE) [Network for research and promotion of research on wellbeing and success].