Advanced Students’ Actions for Operationalizing Multiply-quantified Relationships

Based on analysis with four graduate students’ proof productions, this paper characterizes eight ways of operationalizing quantification for reasoning about and constructing proofs. We call these quantification operationalizations quantops. These constructs describe the actions and meanings these students attributed to quantified relationships that rendered them useful for proving. We position these graduate students as experienced provers such that these quantops represent aspirational goals for the learning of more novice students that might be targeted by instruction. Some of these constructs build upon prior studies of quantification, such as fixing and varying an object (The Journal of Mathematical Behavior, 61, 100802 Sellers et al., 2021) or reasoning about dependence or independence between related objects (Educational Studies in Mathematics, 60(2), 149–172 Durand-Guerrier & Arsac, 2005). Other quantops such as cardinality, notation, or multiply-quantified packages represent novel contributions to the literature. A key feature of these quantops was how they allowed students to foreground quantification to construct meaning or to background quantification to manage working memory. We accordingly identify a foregrounding and a backgrounding form of each quantop.