Better knowledge of learning trajectories contributes to better assertive interventions during learning. New technological opportunities contribute to a better understanding of these trajectories and represent benefits for the design of tools that consider diverse populations and diverse expressions of language during learning. The paper explores the trajectories generated by users during the solution of the Hanoi Towers game. The trajectories' data of the game, which exceed twenty thousand records, were obtained from the virtual implementation of the game. The data's representation was done using heat maps. The data shows trends in the exploration of the problem space and differences in the exploration according to the number of attempts made. The document raises two hypotheses about learning trajectories whose analysis is based on the behavior of large volumes of data and recognizes the potential of Big Data in the field of mathematics education.