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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. 

In this chapter, we address the problem of why blockages occur during mathematization by introducing a method for studying mathematizing based in quantitative reasoning. We report on interview data with six tertiary STEM majors as they developed models of the population dynamics of cats and birds in a backyard habitat. Our analysis focused on real-world relationships participants tried to express when using a given arithmetic operation in a predator-prey modelling task. Our results reveal the conceptions of × participants used to justify their models when constructing an expression for the decrease in the bird population. We conclude by discussing the method’s utility for studying mathematization and with conjectures on how instructors might leverage participants’ justifications to scaffold their emergent models towards a conventionally correct model. 

One reason mathematical modelling remains highly challenging for students is because it requires knowledge about both mathematics and the real-world. Recent work suggests promoting the learning of mathematical modelling as conceiving quantities and establishing relationships among quantities could help students overcome the challenges they experience. While promising, this approach may be oversimplistic in its claims. Through analyzing data collected via a teaching experiment methodology, we present one student’s (Szeth’s) work on two tasks to illustrate how Szeth’s reasoning with quantities was limited during his model construction process in the following ways: Szeth (i) used already constructed mathematical expressions to reason about how quantities vary, and (ii) did not construct a mathematically correct expression despite having reasoned with quantities. 

While a standard calculus course may include some neatly-packaged applications of rate of change or Riemann sums to problems of kinematics, majors from biology and medicine are in urgent need of mathematics taught from a modeling perspective. Yet, the art of modeling is scarce in tertiary mathematics classrooms in part because, much like in schools, many mathematicians may lack (a) the relevant real-world concepts (beyond simple physics and engineering) (b) knowledge of the mathematics from a modeling perspective or (c) confidence to change their classroom practices. To remedy this, we trialed a professional development workshop for faculty to learn to mathematically model biological contexts with dynamical systems. The workshop enacted the field’s recommendations for professional development with teachers. We observed gains in faculty’s self-reported comfort with mathematics and biology concepts and teaching mathematics with a modeling perspective. 

This paper reports a study of 10 post-secondary STEM (Science, Technology, Engineering, Mathematics) instructors’ beliefs about mathematical modelling and the role of mathematics in STEM coursework. The participants were selected from STEM disciplines that are atypical to the literature base (e.g., anthropology and geography), in order to extend what is known about STEM instructors’ beliefs to other disciplines. We conducted episodic narrative interviews to hypothesize the genesis of participants’ most salient beliefs. We then conducted a cross-case synthesis to reflect on the similarities between our participants’ beliefs and findings previously reported in STEM education literature. Our participants held many beliefs in common with typical STEM instructors with regards to how they define modelling, the role of modelling in STE (Science, Technology, Engineering) courses, and their views of students as learners of mathematics and modelling. Our analysis suggests participants’ commitments within these categories are interdependent and arise from lived experiences. Additionally, participants within the same field held competing beliefs about modelling, suggesting that constituting ‘major’ as an independent variable in future research may not be straightforward