ABSTRACT
In the literature on mathematics education, it is often claimed that the ability to use external representations facilitates mathematical problem solving (Dienes, 1960; Even, 1998; Yerushalmy, 2006). The literature also mentions several skills that students need to possess in order to benefit from using external representations. These skills can be categorised in two groups. The first group of skills can be called representational fluency. It involves
the ability to interpret or construct representations (Ainsworth, Bibby, & Wood, 1998), as well as the ability to translate and switch between representations (on demand) accurately and quickly (Even, 1998). In sum, representational fluency refers to the efficiency (in terms of accuracy and speed) with which students can interpret, construct, translate, and switch between external representations. The second group of skills involves making appropriate representational
choices in a given problem-solving or learning situation. The way in which ‘appropriateness’ is conceptualised varies across studies. In some studies (e.g., Larkin & Simon, 1987; Schnotz & Bannert, 2003), a representational choice is considered appropriate if it is in line with the demands of the task at hand. In some studies from the strategy choice literature (e.g., Verschaffel et al., 2007), a choice is considered appropriate if it matches not only task demands, but also the characteristics of the subject that has to make the choice, and sometimes even the context in which the choice takes place. The main difference between these two conceptualisations is that the first one is (purely) based on a rational evaluation of the to-be-solved task, which results in a series of taskto-representation(s) matches which are expected to benefit the resolution of the task at hand, whereas the second conceptualisation also takes into account the individual’s capacity to use the different representations and the context in which the choice takes place. What counts as an appropriate or flexible choice
varies greatly across studies (for a review, see Warner, 2005), but for the purpose of this study we will focus on the two conceptualisations described above. In this chapter, students’ ability to make appropriate representational choices
will be referred to as representational adaptivity or flexibility1, and this skill will be the focus of this study. The first group of skills described above (and more specifically: representational efficiency) will be considered insofar as it relates to representational adaptivity (i.e. choosing the most appropriate representation may mean taking into account one’s efficiency in using particular representations).
