Digging for fractions: using video games to uncover cognitive processes and explore implementations of fraction equivalence

In my PhD project I pursued the question of how is fraction equivalence processed, and how can it be learnt using games? As such, aspects of fraction processing (from a mathematical cognition perspective) and game-based learning define and direct my research work. To bridge fractions and games, I chose the concept of a number line as the vessel to research, due to its important conceptual and educational properties, as well as gameplay affordances. My thesis comprises seven research papers (five published to this day) that are grouped into three chapters. Building on each other, each chapter addresses specific aspects, contributing to answer the overarching research question—(I) starting with associations between fractions and games, (II) further focusing on equivalent fraction processing, (III) and finally presenting mathematical and game design ideas towards an equivalent fraction learning game.

Chapter I focuses on the essential connections between fractions and games, examining the practical details and affordances of mathematical games. It explores how the mathematical content is portrayed in games but also how players approach and access the content, through the use of game elements that shape play and mathematical learning. The first study made use of quantitative data from 12-year-olds to uncover associations between different aspects of fractions (e.g., magnitude, part-whole) assessed in a paper-pencil test with specific in-game metrics from a number line estimation game. Significant associations were found for aspects of fraction understanding closely matching game mechanics and in-game metrics—such as stars awarded as virtual incentives. The second study used audio and screen recordings from the same game, to investigate the development and adaptation of problem-solving approaches by young players when estimating fractions on the number line. From a problem-solving perspective, results from the qualitative analysis of participants playing the game for the first time highlighted the efficiency and affordances of such games, that structure and present mathematical content.

Studies of Chapter II address the cognitive foundations of processing fraction equivalence with the idea to inform the debate of holistic versus componential processing of fractions. These studies make use of carefully selected stimuli, unique tasks and methods to add to the literature regarding the processing of fraction equivalence. Instead of the commonly used magnitude comparison tasks, Study 3 employed number line estimation to evaluate whether pairs of equivalent fractions are estimated at the same location on the number line or not. Results indicated that equivalent fractions were not estimated at exactly the same location on the number line, and that the magnitude of fraction numerator and denominator significantly influences their estimation—providing evidence for fraction componential processing. This was further supported by results of Study 4, which used a fraction same-different task. Unit fractions were selected to form pairs with their equivalent expansions by multiples of 2, 3 and 4 (e.g., 1/2, 2/4, 3/6, 4/8). Results revealed that performance decreased when fraction components increased in multiples of 2, 3, and 4—again suggesting componential processing of (equivalent) fractions. Furthermore, a case study of a 13-old participant, Study 5 introduced a new method to evaluate the process of estimating equivalent fractions on the number line. Given the name representative snapshots, this method makes use of still frames from a video recording, to trace the estimation movements/pauses of the participant on the in-game number line. This unique analysis further substantiated componential processing of (equivalent) fractions, clearly indicating a systematic change in the use of reference points due to multiples present in equivalent fractions.

Finally, informed by the research presented in the previous two chapters, Chapter III serves as a practical conclusion to my thesis, showcasing game design implementations, intrinsically integrating the learning content of equivalent fractions into selected game elements. In both, Study 6 and Study 7, I presented game prototypes showing possible practical applications. Picking up on the evidence of componential processing of equivalent fractions, Study 6 introduces a new, animated, interactive fraction representation, the wheel of fractions, integrating all three: symbolic, location on the number line, and part-whole aspects of fractions. The prototype implementation presented in this study proposed how the wheel representation could be used as part of a game. Study 7 goes a step further, introducing the idea of collectible content, a learning game design approach for intrinsically integrating learning content with game elements. The characteristics of collectible content are defined, and applied to a number line estimation game prototype, that showed how informed design choices can conceptually enrich a game task to encompass and enrichen learning content (e.g., towards fraction equivalence).

In summary, the results of my studies indicate that game mechanics and in-game metrics are associated with specific aspects of fractions, and fraction number line estimation is influenced by participants problem-solving strategies. Moreover, I observed that equivalent fractions are processed componentially using new tasks and analyses. Finally, I suggested how learning equivalent fractions may be implemented in new learning prototypes, using the wheel of fractions as the central game mechanic and collectible content to enrich the gameplay. Taken together, this answers how equivalent fractions are processed as well as suggests how they can be learnt in a game-based way.