In today's digital information society, mathematical and computational skills are becoming increasingly important. With the demand for mathematical and computational literacy rising, the question of how these skills can be effectively taught in schools is among the top priorities in education. Game-based learning promises to diversify education, increase students' interest and motivation, and offer positive and effective learning experiences. Especially digital game-based learning (DGBL) is considered an effective educational tool for improving education in classrooms of the future. Yet, learning is a complex psychological phenomenon and the effectiveness of digital games for learning cannot be taken for granted. This is partly due to a diversity of methodological approaches in the literature and partly due to theoretical and practical considerations. We present core elements of psychological theories of learning and derive arguments for and against DGBL and non-DGBL. We discuss previous literature on DGBL in mathematics education from a methodological point of view and infer the need for randomized controlled trials for effectiveness evaluations. To increase comparability of empirical results, we propose methodological standards for future educational research. The value of multidisciplinary research projects to advance the field of DGBL is discussed and a synergy of Affective Computing and Optimal Experimental Design (OED) techniques is proposed for the implementation of adaptive technologies in digital learning games. Finally, we make suggestions for game content, which would be suitable for preparing students for university-level mathematics and computer science education, and discuss the potential limitations of DGBL in the classroom.

What drives people's exploration in complex scenarios where they have to actively acquire information? How do people adapt their selection of queries to the environment? We explore these questions using Entropy Mastermind, a novel variant of the Mastermind code-breaking game, in which participants have to guess a secret code by making useful queries. Participants solved games more efficiently if the entropy of the game environment was low; moreover, people adapted their initial queries to the scenario they were in. We also investigated whether it would be possible to predict participants' queries within the generalized Sharma-Mittal information-theoretic framework. Although predicting individual queries was difficult, the modeling framework offered important insights on human behavior. Entropy Mastermind opens up rich possibilities for modeling and behavioral research.

Conceptual descriptions and measures of information and entropy were established in the twentieth century with the emergence of a science of communication and information. Today these concepts have come to pervade modern science and society, and are increasingly being recommended as topics for science and mathematics education. We introduce a set of playful activities aimed at fostering intuitions about entropy and describe a primary school intervention that was conducted according to this plan. Fourth grade schoolchildren (8–10 years) played a version of Entropy Mastermind with jars and colored marbles, in which a hidden code to be deciphered was generated at random from an urn with a known, visually presented probability distribution of marble colors. Children prepared urns according to specified recipes, drew marbles from the urns, generated codes and guessed codes. Despite not being formally instructed in probability or entropy, children were able to estimate and compare the difficulty of different probability distributions used for generating possible codes.

A variety of conceptualizations of psychological uncertaintyexist. From an information-theoretic perspective, probabilistic uncertainty can be formalized as mathematical entropy. Cognitive emotion theories posit that uncertainty appraisals and motivation to reduce uncertainty are modulated by emotional state. Yet little is known about how people evaluate probabilistic uncertainty, and about how emotional state modulates people’s evaluations of probabilistic uncertainty and behavior to reduce probabilistic uncertainty. We tested intuitive entropyevaluations and entropy reduction strategies across four emotion conditions in the Entropy Mastermind game. We used the unified Sharma-Mittal space of entropy measures to quantify participants’ entropy evaluations. Results suggest that many people use a heuristic strategy, focusing on the number of possible outcomes, irrespective of the probabilities in the probability distribution. This result is surprising, given that previous work suggested that people are very sensitive to the maximum probability when choosing queries on probabilistic classification tasks. Emotion induction generally increased participants’heuristic assessment. The uncertainty associated with emotional states also affected game play: participants needed fewer queries and spent less time on games in high-uncertainty than in low-uncertainty emotional states. Yet entropy perceptions were not related to subjectively reported uncertainty, numeracy or entropy knowledge, suggesting that entropy perceptions may form an independent psychological construct.