An embodied approach to covariational reasoning: a hand-tracking study

Abstract

Covariational reasoning is critical for understanding functional relationships in mathematics. Yet, many students struggle to understand varying quantities and to conceptualize covariation. This study explores the research question of how bimanual movements within a digital-embodied learning environment can support students’ covariational reasoning, appropriating Thompson and Carlson’s (2017) covariational reasoning taxonomy. The teaching experiment consisted of three lessons involving 76 Grade 9 students. Data included hand-tracking data, posttask probes, and pretest and posttest results. We examined the relationship between students’ bimanual coordination and their covariational reasoning levels, addressing two hypotheses: (H1) Higher levels of covariational reasoning correlate with a shorter time to reach fluency in the bimanual coordination and (H2) higher levels of learning gains in covariational reasoning correlate with longer time spent on fluently performing the bimanual coordination. As a result, the pretest and posttest scores demonstrated significant improvements in students’ covariational reasoning after the experiment. Quantitative analyses of hand-tracking data indicated that students with higher initial covariational reasoning levels achieved fluent bimanual movement more quickly than others (supporting H1), while those with greater learning gains spent more time consolidating their understanding in fluency phases (supporting H2). Qualitative findings showed how the interplay of perception–action loops, attentional anchors, and real-time feedback facilitated the internalization of covariational relationships. This study highlights the potential of an embodied approach to fostering covariational reasoning and introduces a framework for analyzing embodied learning through the integration of hand tracking, probes, and assessments.