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Abstract Mathematical modeling is challenging for learners across all grade bands. Given its crucial role in STEM education, it is essential for the field to design learning environments that foster the optimal learning of modeling. One promising strategy is to support the learning of modeling as conceiving a situation consisting of quantities and quantitative relationships. However, before promoting the learning of modeling in this way, the field first needs accounts of how quantitative reasoningispresent in students’ mathematical modeling activities. Drawing data from individual teaching experiments with three undergraduate STEM majors, we analyzed the ways in which reasoning with quantities manifest as they mathematically modeled dynamic situations. Through our analysis, we illustrate eight types of manifestations, making explicit connections between the modeling activities and participants' reasonings with quantities during each manifestation. Our findings indicate that learners’ modeling activities are afforded through their reasonings with quantities and suggest that students’ quantitative reasoning can be leveraged to support the learning of mathematical modeling. 

Abstract To effectively teach modeling, instructors need to select tasks that allow students to learn which mathematical operations are appropriate for modeling different scenarios. Naturally, instructors might select tasks using a priori task classification systems—ones that group real-world problems for a given operation based on mathematical formalisms (e.g., as reported by Vergnaud (in: Hiebert (ed) Number concepts and operations in the middle grades, National Council of Teachers of Mathematics, 1988). In this paper, we critique the robustness of a priori task classifications systems for guiding the selection of modeling tasks. We investigated the meanings undergraduate STEM majors attributed to multiplication while modeling a predator–prey system using differential equations. Through analysis of task-based clinical interviews with 23 participants, six distinct justifications forwhymultiplication was an appropriate operation for modeling the scenario were identified. These six justifications confirm that learners’ assimilation of scenarios to operations may differ from how educators classify problems using a priori classification schemes. Our findings challenge the use of a priori task classification systems for guiding the pedagogical selection of real-world scenarios to model because classifying real-world scenarios using a priori systems can overlook nuances in modelers’structuringandvalidating. We highlight the importance of these nuances for generating task trajectories that would leverage learners existing meanings for mathematical operations to build new associations between mathematical operations and novel problems. We end by suggesting a shift towards reasoning-based classification systems for selecting real-world scenarios to model—ones that are based on students reasoning about and within a scenario.