The potential of the realization tree as an assessment tool for exploring students’ learning of multivariable calculus: the case of integrals

Integral calculus is a fundamental topic within multivariable calculus, playing an important role in modeling phenomena across many fields, including engineering. This study investigates students’ discourse on single and multivariable integrals through the lens of commognition theory, particularly the realization tree. Guided by a phenomenological approach, we conducted semi-structured interviews with 18 volunteer students who had completed both single-variable and multivariable calculus during their first year of study. By analyzing students’ realization trees alongside their group discussions, we identified distinct graphical, symbolic, and verbal realizations of single and multivariable integrals and revealed instances of weak and strong incommensurable discourses manifested through commognitive conflict. The study also explores how students extend their realizations from single-variable to multivariable integrals and highlights coherence or fragmentation in their discourse as they moved from single to multivariable contexts. The findings underscore the potential of realization trees as an assessment tool for investigating students’ mathematical discourse at the tertiary level and point to directions for future research as well as implications for teaching multivariable integrals.