## PERC 2021 Abstract Detail Page

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Abstract Title: | Considering covariational reasoning in math and physics |
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Abstract: | Characterizing change is at the heart of physics. Mathematics and physics education researchers grapple with how change is conceptualized and quantified. The ways in which students make sense of how two quantities are related -- how changes in one quantity affect changes in another -- have been an integral part of physics education research on scaling, proportional reasoning, and modeling. In mathematics education research, this is called covariational reasoning, and is often described as a central skill for precalculus and calculus curriculum. These courses are typically considered pre- or co-requisites for introductory physics courses, suggesting that students may be assumed to have strong, reliable covariational reasoning skills as an outcome of their math courses. Recent work in both physics and mathematics education indicates that covariational reasoning involves high levels of sophisticated thinking, and by considering it prerequisite we may unintentionally enforce a barrier for students coming into physics. As mathematics education is highly variable across U.S. high schools, this prerequisite thinking may also disproportionately affect students from low socioeconomic backgrounds. In this parallel session, we bring together physics and mathematics education experts to explore the overlap of covariational reasoning in our disciplines. We will consider the ways that we collectively ask students to reason about related quantities and discuss modes of student reasoning about change and rates of change. In addition, as a mechanism for broadening access and differentiating instruction in our courses, we aim to foster a discussion about supporting instructors (and students) to shift the narrative towards framing covariational reasoning as an emerging and developing skill during college physics and mathematics courses. The session will begin with talks from each of the researchers, and conclude with time for a larger discussion and questions for speakers. |

Abstract Type: | Talk Symposium |

Session Time: | Parallel Sessions Cluster I |

## Author/Organizer Information | |

Primary Contact: |
Charlotte Zimmerman University of Washington Seattle, WA 98105 |

Co-Author(s) and Co-Presenter(s) |
Paul Emigh, Oregon State University Cameron Byerley, University of Georgia Michael Loverude, California State University Fullerton Darío González, Universidad de Chile |

## Symposium Specific Information | |

Presentation 1 Title: | Assessing Covariational Reasoning in College Calculus and Physics Courses: The Use of Graphs |

Presentation 1 Authors: | Cameron Byerley (Inspired by ideas of my co-authors James Drimalla and Brady Tyburski) |

Presentation 1 Abstract: | Both calculus and physics were invented to describe how quantities change together. For example, calculus can help us specify precisely the difference between the rate of change of a quadratic function and the rate of change of an exponential function. College instructors use many methods to help students represent relationships between quantities and to help students make sense of new ideas such as rate of change functions (derivatives). Graphs are often used in calculus to convey new concepts and covariational relationships to students. For example, teachers often use graphical representations to convey why derivatives find the rate of change of one quantity with respect to another. Additionally, graphs are often part of assessments in calculus and physics class. For example, given a graph that shows the amount of a quantity at any moment in time, can the students produce a graph that shows the rate of change of that quantity at any moment in time. In both of these teaching situations-using students' graphs to assess their understanding and using teachers graphs to convey new concepts-there is often an implicit assumption that students understand graphs as representing and emergent trace of the covariation of two quantities (Moore et. al., 2017). If students do not understand a graph as representing covariation of two quantities, and the teacher primarily explains the meaning of a derivative graphically, the students are not likely to learn the meaning for derivative the teacher intended to convey. Our research team has also found that calculus students are often able to describe covariational relationships using words and gestures that they are not able to correctly represent graphically on assessments of covariational reasoning (Tyburski et. al., 2021; Drimalla et. al., 2020). This talk will discuss the importance of helping calculus and physics students' understand graphs as covariational relationships between two quantities. |

Presentation 2 Title: | How Students Reason about Changes when Finding Partial Derivatives |

Presentation 2 Authors: | Paul J. Emigh, Ian W. Founds, & Corinne A. Manogue |

Presentation 2 Abstract: | Partial derivatives are among the many mathematical tools used throughout physics to help describe the physical world. We have been investigating how students are able to determine and interpret partial derivatives both symbolically and from contour graphs with mathematical, electrostatic, and thermodynamic contexts. While many students had strong procedural abilities for calculating derivatives, most did not strongly link this process (of finding a partial derivative) with the idea of "holding a variable constant," especially when dealing with a thermodynamics context. We found this to be true regardless of the representation in which students were asked to work. These results suggest a deeper theoretical framework is needed for understanding how students think about partial derivatives. |

Presentation 3 Title: | Connecting mathematics education and climate change education through covariational reasoning |

Presentation 3 Authors: | Darío A. González |

Presentation 3 Abstract: | The present study examined the covariational reasoning of three preservice mathematics teachers (PSTs) as they make sense of a simple model to introduce climate change. The PSTs worked on a mathematical task that required PSTs to activate their covariational reasoning abilities in order to explore two key notions related to climate change: (i) the balance of energy between the components of the climate system (Sun, surface, and atmosphere), and (ii) the link between carbon dioxide (CO2) pollution and global warming. The PSTs completed the task during an individual, task-based interview of approximately 60 minutes. The analysis revealed that, in order to develop a operative understanding the climate system model, the PSTs' covariational reasoning must support, at a minimum, the ability to: (i) understand how the balance between the energy inflow into and the energy outflow from the climate system regulates the variation of the plante's mean surface temperature over time (dynamic relationships), and (ii) conceptualize the energy exchange between the surface and the atmosphere interms of two energy flows changing in tandem and obeying a circular relationship as time elapses (feedback loop). The analysis also revealed that the differences in the PSTs' covariational reasoning abilities coincided with two ways of thinking about the link between CO2 pollution and global warming: a productive way of thinking that had the potential of making visible the role of human activities in driving global climate change, and a less productive way that had the potential of promoting misconceptions about regarding that environmental challenge. The study's results show that PSTs can engage in covariational reasoning while they learn about climate change and that covariational reasoning (and in general mathematics education) can have a role in promoting climate change education. |

Presentation 4 Title: | Prompting sense-making with bidirectional reasoning using a convention-breaking representation in kinematics |

Presentation 4 Authors: | Michael Loverude, Henry Taylor |

Presentation 4 Abstract: | The function concept is central to college math instruction. Introductory physics courses approach many of the same topics and use what seem to be the same mathematical ideas, but the level of emphasis on the function concept in most introductory physics courses is minimal. We describe efforts to examine student physics learning using ideas adopted from the Research in Undergraduate Mathematics Education (RUME) community, including student understanding of functions as well as covariation [Carlson et al 2002]. In particular we highlight work inspired by the RUME literature [Moore et al 2014, 2015] in which student sense-making is prompted by tasks that break typical representational conventions. We will describe student responses to tasks posed in interviews in which we used comparison tasks drawn from the existing PER and RUME literature as well as a novel, convention-breaking task intended to probe an aspect of covariation known as bidirectionality [Thompson 1996, Moore 2015]. Supported in part by NSF grants PHY#1406035 and PHY#1912660 as well as the Black Family Foundation. |

Presentation 5 Title: | Emerging covariational reasoning student resources in physics lab courses |

Presentation 5 Authors: | Charlotte Zimmerman, Jared Canright, Suzanne White Brahmia |

Presentation 5 Abstract: | Students and experts engage in a plethora of sophisticated ways of thinking during modeling tasks, blending between mathematical and physical reasoning. This integrated way of reasoning about quantities and how they are related is ubiquitous to "thinking like a physicist." One facet of this process is covariational reasoning -- reasoning about how changes in one quantity relate to changes in another quantity. In laboratory courses that engage students in experimental design, students commonly use covariational reasoning as they model changing quantities and the relationships between them. However, those without strong mathematical reasoning skills may unintentionally be perceived as also having less nuanced physics reasoning skills. To investigate the multifaceted ways students model in physics lab courses, we examined student discussions during remote, student-designed labs in which students interact with physical, but fictitious, matter in a virtual reality (VR) simulation. The VR environment allows for students to observe and measure novel situations for which the "answer," in this case a mathematical model about the particles' behavior, is not known. Using a conceptual blend analysis, we examine two lab groups that take different approaches to the task, despite beginning with the same observations and reporting on the same final model. One group demonstrates expert-like mathematical reasoning that is typically rewarded in physics courses, while the other engages in early signs of expert-like physics reasoning. Our analysis suggests that there are productive ideas emerging in both groups, and that of the less mathematically rigorous may be going unnoticed. We suggest that characterizing these features of physics covariational reasoning may provide an opportunity to move beyond data analysis techniques in lab courses towards incorporating more physics expert-like techniques. In doing so, instructors may recognize hidden strengths that may otherwise go undetected. |