The PRIUM qualitative framework for assessment of proof comprehension: a result of collaboration among mathematicians and mathematics educators

Abstract

While proof has been studied from different perspectives in the mathematics education literature for decades, students continue to struggle to build proof comprehension. Complicating this, the manner in which proof comprehension is assessed largely remains to be the definition-theorem-proof format in which students are asked to reproduce proofs or similar proofs that are presented in class. This approach can encourage students to memorize proofs rather than develop tools for syllogistic reasoning. This paper reports on a seven-year collaboration among research mathematicians and mathematics educators. Following a model for proof comprehension, they implemented a cycle of planning assessments, implementing them and evaluating student responses in several courses each semester for three years at two universities. The model-based assessments were designed with probing questions about proofs or subproofs to point student attention to the relationships among definitions, statements and their relationships, as well as the logic used to help students to both develop proof knowledge and demonstrate their mathematical thinking. Discrepancies between the intention of assessments and student responses led to refinements as the team reviewed the results to inform its practice. Collaboration, experience and empirical data informed the development of the new Promoting Reasoning in Undergraduate Mathematics (PRIUM) Qualitative Framework for Proof Comprehension. The paper discusses three main results: the Framework and how to use it, the Framework’s utility at the individual, course and program levels of departmental evaluation, and a collaborative research process that may be utilized between research mathematics educators and mathematicians.