Abstract:

The idea of individualized learning styles has greatly influenced Education despite criticisms (see Coffield et al. 2004 review), e.g. as there is no evidence that identifying an individual student's learning style produces better outcomes. However, well characterized and strong focused activities, where a process of self-education exists without external tutoring, seem to be a good environment to recognize individual learning patterns. Combinatorial games (CG) can produce such environments where users are highly motivated and the topic, by itself, is mathematically rich. Many CGs have associated a so-called optimal winning strategy (OWS). In detail, a solved game is a game whose outcome (win or lose) can be correctly predicted from any position, assuming that both players play perfectly.

Traditionally, the NIM game is a CG board game for two players with quite simple rules: starting with any number of counters distributed in any number of piles, two players take turns to remove any number of counters from a single pile. The winner is the player who takes the last counter (i.e. classic rule version). In particular, the NIM game is a strong solved games, so we know an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or both sides of the players.

Fortunately, for our aims, OWS are usually difficult to grasp in a short time, so players tend to use Heuristics, which work as personal rules to approach the problem solving, learning, or discovery and usually employs a practical method not guaranteed to be optimal, but sufficient for their immediate goals or which they believe is effective from previous experience.

We develop a free available Android mobile application of the NIM game, that has now more than 150K games played from all over the World, as a tool to research how an anonymous player deal with a combinatorial problem and a data gathering mechanism as source for studying Learning Patterns. Therefore, after collecting enough game log statistics, we aim to understand and somehow classify user’s learning profiles by finding the adequate mathematical techniques from data mining/deep learning, linear optimization, and/or graph theory. For such reason, we collect basic statistics about players performance and their way of playing, for which the players are alerted at installation time.

Besides the standard expected features, the app has:
(a) Four rule variations: Classic, Misére, NIM 21 and Fibonacci;
(b) Each game can have more than two players (e.g. against computer players and/or other humans) in the local multiplayer mode or in the online multiplayer mode;
(c) A ‘power-up’ is available which allows a player to break the adversary winning strategy, based on a harmonic oscillator equation;
(d) Several achievements may be unlocked by completing game related goals (i.e. gamification techniques).

This is a project of GEOMETRIX, which is a strand line of the Center for research and Development in Mathematics and Applications, with a interdisciplinary-oriented research and development focus, targeted at assorted target groups (running from primary to higher education level), committed to the study, use and creation of intelligent digital environments to promote knowledge and skills in mathematics, reflecting a transformation in the way they are grasped and applied.

Keywords:

Combinatorial games, Optimal winning strategies, Players heuristics, Mobile application, Learning patterns.