Representational disfluency in algebra: evidence from student gestures and speech

In this investigation, we analyzed US middle school students’ (grades 6–8) gestures and speech during interviews to understand students’ reasoning while interpreting quantitative patterns represented by Cartesian graphs. We studied students’ representational fluency, defined as their abilities to work within and translate among representations. While students translated across representations to address task demands, they also translated to a different representation when reaching an impasse, where the initial representation could not be used to answer a task. During these impasse events, which we call representational disfluencies, three categories of behavior were observed. Some students perceived the graph to be bounded by its physical and numerical limits, and these students were categorized as physically grounded. A second, related, disfluency was categorized as spatially grounded. Students who were classified as spatially grounded exhibited a bounded view of the graph that limited their ability to make far predictions until they physically altered the spatial configuration of the graph by rescaling or extending the axes. Finally, students who recovered from one or more of these disfluencies by translating the quantitative information to alternative but equivalent representations (i.e., exhibiting representational fluency), while retaining the connection back to the linear pattern as graphed, were categorized as interpretatively grounded. Understanding the causes and varieties of representational fluency and disfluency contributes directly to our understanding of mathematics knowledge, learning and adaptive forms of reasoning. These findings also provide implications for mathematics instruction and assessment.