One reason mathematical modeling 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 modeling 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. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

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 modeling 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 toward a conventionally correct model. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

Theories of quantitative reasoning have taken precedence as an analytical tool to interpret and describe students' mathematical reasonings, especially as students engage in mathematical modeling tasks. These theories are particularly useful to describe how students construct new quantities as they model. However, while using this lens to analyze Differential Equations students' construction of mathematical models of dynamic situations, we found cases of quantity construction that were not fully characterized by extant concepts. In this theory-building paper, we present five examples of such cases. Additionally, we introduce a new construct-quantitative operators-as an extended analytical tool to characterize those cases. Our findings suggest that quantitative operators may be viewed as an extension for theories of quantity construction and complementary to symbolic forms, when localizing theories of quantity construction for mathematical modeling, especially at the undergraduate differential equation level.

Students exit calculus with understandings of change that want for conceptual depth and are disconnected from real-world contexts. In this paper, we present a problem that will develop their skills in using change concepts for learning differential equations through modelling. The problem comes from a qualitative study of how STEM majors approach and think about differential equations as models for real-world scenarios. Our purpose is to inform faculty who are using open, authentic, scenario-driven instructional materials about the ways their students think about what are often taken-for-granted concepts in advanced mathematics and to support faculty in ensuring their students get the most from their innovative instructional materials.

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 hypothesise 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.

We studied aspects of undergraduate STEM majors' mathematical reasoning as they engaged in mathematically modeling a predator-prey scenario. The study used theoretical viewpoints on quantitative reasoning to inform scaffolding moves that would assist modelers in overcoming blockages to their mathematization of real-world problems. Our contribution is a set of four scaffolding moves contingent upon and responsive to participants' in-the-moment quantitative reasoning that guided them toward a meaningful model for a predator-prey scenario.

As part of a larger project focused on exploring development of mathematical modelling competencies among post-secondary STEM majors enrolled in advanced mathematics, we developed a pair of parallel multiple-choice modelling competencies assessments. In this chapter, we provide a technical report of item development, scale calibration, and validation of the assessment. We used multiple statistical approaches, including classical test theory (CTT), item response theory (IRT), and principal component analysis (PCA) to document item behaviours, scale properties, and dimensionality of a developing multiple-choice assessment of mathematical modelling competencies designed for post-secondary STEM majors. We share analyses and inferences, making recommendations for the field in pursuing such assessments. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Mathematical modelling is endorsed as both a means and an end to learning mathematics. Despite its utility and inclusion as a curricular objective, one of many questions remaining about learners' modelling regards how modelers choose relevant situational attributes and express mathematical relationships in terms of them. Research on quantitative reasoning has informed the field on how individuals quantify attributes and conceive of covariational relationships among them. However, this research has not often attended to modelers' mathematization in open modelling tasks, an endeavor that invites further attention to theoretical and methodological details. To this end, we offer a synthesis of existing theories to present a cognitive account of mathematical model construction through a quantity-oriented lens. Second, we use empirical data to illustrate why it is productive for theories of modelling to attend to and account for students' quantitative reasoning while modelling. Finally, we identify remaining challenges to coordinating different theoretical accounts of model construction.

This paper addresses two aspects of integrating mathematics education with engineering education that may address persistence of engineering majors (and STEM majors more broadly): an emphasis on modeling as a vehicle for more authentic learning activity (Niss et al. 2007), and the need for measures that can support academic units' efforts to collect local data about student attainment of program goals. In this paper, we contribute: (1) a measure for modeling self-efficacy and its corresponding design process; (2) a measure for modeling competency and its corresponding design process; (3) a preliminary analysis of the relationship between modeling competency and self-efficacy. We argue that such instruments address a genuine need of engineering departments (as well as STEM education researchers) to have a means for collecting local data on students' modeling self-efficacy and competency.