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Adaptive expertise is a highly sought after, but difficult to achieve, outcome of mathematics education. Many teaching methods appear to support the development of adaptive expertise only in a small proportion of students. Game-based learning environments may be useful for supporting adaptive expertise. Therefore, we carried out a quasi-experimental classroom intervention to examine the possibility of using a game-based learning environment to promote adaptive rational number knowledge, a potential indicator of adaptive expertise in the domain of rational numbers. The Number Line Elaboration and Exploration learning environment relied on the increasing elaboration of the number line analogy as a means for students to explore connections between multiple aspects of rational number knowledge. Our results show the game-based learning environment was successful in promoting adaptive rational number knowledge. These results provide directions for the development and examination of how learning environments may be able to support adaptive expertise. publishedVersion

We give a new proof for the decidability of the D0L ultimate periodicity problem based on the decidability of p-periodicity of morphic words adapted to the approach of Harju and Linna., In Proceedings WORDS 2011, arXiv:1108.3412

Une version préalable est parue sous le titre "Restricted square property and infinite partial words", Technical Report of the Turku Centre for Computer Science, Turku, Finlande, 02/2009, 13 pages (ISSN 1239-1891); International audience; We prove that there exist infinitely many infinite overlap-free binary partial words containing at least one hole. Moreover, we show that these words cannot contain more than one hole and the only hole must occur either in the first or in the second position.We define that a partial word is k-overlap-free if it does not contain a factor of the form xyxyx where the length of x is at least k. We prove that there exist infinitely many 2-overlap-free binary partial words containing an infinite number of holes.