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1. Integrating math and science content through covariational reasoning: the case of gravity

   https://doi.org/10.1080/10986065.2020.1814977  

   Panorkou, Nicole; Germia, Erell Feb (January 2020, Mathematical Thinking and Learning)

   null (Ed.)

   Integrating mathematics content into science usually plays a supporting role, where students use their existing mathematical knowledge for solving science tasks without exhibiting any new mathematical meanings during the process. To help students explore the reciprocal relationship between math and science, we designed an instructional module that prompted them to reason covariationally about the quantities involved in the phenomenon of the gravitational force. The results of a whole-class design experiment with sixth-grade students showed that covariational reasoning supported students’ understanding of the phenomenon of gravity. Also, the examination of the phenomenon of gravity provided a constructive space for students to construct meanings about co-varying quantities. Specifically, students reasoned about the change in the magnitudes and values of mass, distance, and gravity as those changed simultaneously as well as the multiplicative change of these quantities as they changed in relation to each other. They also reasoned multivariationally illustrating that they coordinated mass and distance working together to define the gravitational force. Their interactions with the design, which included the tool, tasks, representations, and questioning, showed to be a structuring factor in the formation and reorganization of meanings that students exhibited. Thus, this study illustrates the type of design activity that provided a constructive space for students’ forms of covariational reasoning in the context of gravity. This design can be used to develop other STEM modules that integrate scientific phenomena with covariational reasoning through technology. 

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2. HOW TRANSITIONS BETWEEN RELATED ARTIFACTS SUPPORT STUDENTS’ COVARIATIONAL REASONING

   Germia, E.; York, T.; Panorkou, N. (January 2022, Proceedings of the forty-fourth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.)

   Lischka, A. E.; Dyer, E. B.; Jones, R. S.; Lovett, J. N..; Strayer, J.; & Drown, S. (Ed.)

   Many studies use instructional designs that include two or more artifacts (digital manipulatives, tables, graphs) to support students’ development of reasoning about covarying quantities. While students’ forms of covariational reasoning and the designs are often the focus of these studies, the way students’ interactions and transitions between artifacts shape their actions and thinking is often neglected. By examining the transitions that students make between artifacts as they construct and reorganize their reasoning, our study aimed to justify claims made by various studies about the nature of the synergy of artifacts. In this paper, we present data from a design experiment with a pair of sixth-grade students to discuss how their transitions between artifacts provided a constructive space for them to reason about covarying quantities in graphs. 

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3. When Am I (N)ever Going to Use This? How Engineers Use Algebra

   Istas, Brooke; Walkington, Candace; Leyva, Elizabeth (July 2021, 2021 ASEE Virtual Annual Conference)

   Mathematics is an important tool in engineering practice, as mathematical rules govern many designed systems (e.g., Nathan et al., 2013; Nathan et al., 2017). Investigations of structural engineers suggest that mathematical modelling is ubiquitous in their work, but the nature of the tasks they confront is not well-represented in the K-12 classroom (e.g., Gainsburg, 2006). This follows a larger literature base suggesting that school mathematics is often inauthentic and does represent how mathematics is used in practice. At the same time, algebra is a persistent gatekeeper to careers in engineering (e.g., Harackiewicz et al., 2012; Olson & Riordan, 2012). In the present study, we interviewed 12 engineers, asking them a series of questions about how they use specific kinds of algebraic function (e.g., linear, exponential, quadratic) in their work. The purpose of these interviews was to use the responses to create mathematical scenarios for College Algebra activities that would be personalized to community college students’ career interests. This curriculum would represent how algebra is used in practice by STEM professionals. However, our results were not what we expected. In this paper, we discuss three major themes that arose from qualitative analyses of the interviews. First, we found that engineers resoundingly endorsed the importance of College Algebra concepts for their day-to-day work, and uniformly stated that math was vital to engineering. However, the second theme was that the engineers struggled to describe how they used functions more complex than linear (i.e., y=mx+b) in their work. Students typically learn about linear functions prior to College Algebra, and in College Algebra explore more complex functions like polynomial, logarithmic, and exponential. Third, we found that engineers rarely use the explicit algebraic form of an algebraic function (e.g., y=3x+5), and instead rely on tables, graphs, informal arithmetic, and computerized computation systems where the equation is invisible. This was surprising, given that the bulk of the College Algebra course involves learning how to use and manipulate these formal expressions, learning skills like factoring, simplifying, solving, and interpreting parameters. We also found that these trends for engineers followed trends we saw in our larger sample where we interviewed professionals from across STEM fields. This study calls into question the gatekeeping role of formal algebraic courses like College Algebra for STEM careers. If engineers don’t actually use 75% of the content in these courses, why are they required? One reason might be that the courses are simply outdated, or arguments might be made that learning mathematics builds more general modelling and problem-solving skills. However, research from educational psychology on the difficulty of transfer would strongly refute this point – people tend to learn things that are very specific. Another reason to consider is that formal mathematics courses like advanced algebra have emerged as a very convenient mechanism to filter people by race, gender, and socioeconomic background, and to promote the maintenance of the “status quo” inequality in STEM fields. This is a critical issue to investigate for the future of the field of engineering as a whole. 

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4. Symbolic forms analysis of expressions for probability in Dirac and wave function notations for spins-first students

   https://doi.org/10.1103/PhysRevPhysEducRes.21.010105  

   Riihiluoma, William D; Topdemir, Zeynep; Thompson, John R (January 2025, Physical Review Physics Education Research)

   [This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] The ability to relate physical concepts and phenomena to multiple mathematical representations—and to move fluidly between these representations—is a critical outcome expected of physics instruction. In upper-division quantum mechanics, students must work with multiple symbolic notations, including some that they have not previously encountered. Thus, developing the ability to generate and translate expressions in these notations is of great importance, and the extent to which students can relate these expressions to physical quantities and phenomena is crucial to understand. To investigate student understanding of the expressions used in these notations and the ways they relate, clinical think-aloud interviews were conducted with students enrolled in an upper-division quantum mechanics course. Analysis of these interviews used the symbolic forms framework to determine the ways that participants interpret and reason about these expressions. Multiple symbolic forms—internalized connections between symbolic templates and their conceptual interpretations—were identified in both Dirac and wave function notations, suggesting that students develop an understanding of expressions for probability both in terms of their constituent pieces and as larger composite expressions. 

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5. The Role of Reasoning with Quantities in Undergraduates’ Modeling Activities

   https://doi.org/10.1007/s40753-025-00273-7  

   Kularajan, Sindura; Czocher, Jennifer_A (July 2025, International Journal of Research in Undergraduate Mathematics Education)

   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. 

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