Exploring Undergraduate Students’ Brackets Use in Integration and Differentiation Tasks: Mental and Unconventional Brackets

Abstract

Structure sense is central to learning mathematics. This study analyses undergraduate students' mathematical writing, focusing on their use of brackets to infer aspects of their structure sense. Using students' written responses to integration and differentiation problems, we examine two nonstandard bracketing practices: mental brackets (omitted but implicitly present) and unconventional brackets (additional brackets used to emphasise structure). Drawing on 353 student scripts from Greece and Turkiye, collected through two questionnaires, the analysis shows that both forms of nonstandard brackets occur frequently. Although these uses do not align with formal conventions or curriculum expectations, they occur consistently in students' solutions and provide insight into their structure sense. Mental brackets are mostly evident in integration tasks, where students appear to prioritise calculation over formal presentation. Unconventional brackets, meanwhile, function as structural supports, with additional notation indicating that students may use them to manage complexity and make structure visible when applying rules or substitutions. These findings suggest that nonstandard bracket use should not be viewed simply as error or redundancy, but as evidence of meaningful engagement with mathematical structure. We relate the results to research at earlier educational levels, highlighting both continuities and differences and considering implications for how instructors interpret and respond to students' notation.