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We report on a study of the effects of laboratory activities that model fictitious laws of physics in a virtual reality environment on (1) students' epistemology about the role of experimental physics in class and in the world; (2) students' self-efficacy; and (3) the quality of student engagement with the lab activities. We create opportunities for students to practice physics as a means of creating and validating new knowledge by simulating real and fictitious physics in virtual reality (VR). This approach seeks to steer students away from a confirmation mindset in labs by eliminating any form of prior or outside models to confirm. We refer to the activities using this approach as Novel Observations in Mixed Reality (NOMR) labs. We examined NOMR's effects in 100-level and 200-level undergraduate courses. Using pre-post measurements we find that after NOMR labs, students in both populations were more expertlike in their epistemology about experimental physics and held stronger self-efficacy about their abilities to do the kinds of things experimental physicists do. Through the lens of the psychological theory of flow, we found that students engage as productively with NOMR labs as with traditional hands-on labs. This engagement persisted after the novelty of VR in the classroom wore off, suggesting that these effects are due to the pedagogical design rather than the medium of the intervention. We conclude that these NOMR labs offer an approach to physics laboratory instruction that centers the development of students' understanding of and comfort with the authentic practice of science., Comment: 20 pages, 9+1 figures (Table III is an image because the first author was defeated by LaTeX's tables). This is the version sent for a second round of reviews at Physical Review: Physics Education Research for the Special Collection on Laboratory Instruction, having been edited in response to reviewers' suggestions and critiques from the first round

Covariational reasoning--considering how changes in one quantity affect another, related quantity--is a foundation of quantitative modeling in physics. Understanding quantitative models is a learning objective of introductory physics instruction at the college level. Prior work suggests that covariational reasoning in physics contexts differs in important ways from reasoning about functions and graphs in purely mathematical contexts; this reasoning is effortful in physics even for mathematically well-prepared students. In order to help students learn to reason covariationally in physics contexts, we need to characterize what we mean by physics covariational reasoning. To this end, we present a framework of covariational reasoning in physics contexts, to describe the ways that covariational reasoning is used in physics modeling. The framework can be used as a tool by which instructors can recognize physics covariational reasoning patterns and researchers can analyze student reasoning. The framework can also help inform the development of effective instructional materials and methods.

Pre-college mathematics modeling instruction often frames mathematics as being separated from reasoning about the real world -- and commonly treats reasoning mathematically and reasoning about the real-world context as separate stages of a modeling cycle. In this paper, we present evidence that helps characterize how experts use mathematics in physics contexts while developing graphical models. An important finding is that there was essentially no evidence of experts reasoning in a context-free way with these tasks, but instead they used physical reasoning -- either grounded in the context of the task or from abstract physical models -- to guide their mathematics. The difference in approach of physics instructors and students may lead to a mismatch in expectations, and frustration, for both parties. This work contributes to the body of knowledge about how mathematical reasoning appears in physics, and can help researchers and instructors recognize the connections to the physical world in expert mathematical reasoning. This can help instructors and researchers be better equipped to develop materials and methods that can help students start building those connections as well.

Developing and making sense of quantitative models is a core practice of physics. Covariational reasoning -- considering how the changes in one quantity affect changes in another, related quantity -- is an essential part of modeling quantitatively. Covariational reasoning has been studied widely in mathematics education research, but has only begun to be used in physics education research. We present evidence from three studies of 25 individual interviews with physics experts, in which the experts were asked to reason out loud while generating graphical models. We analyze the studies through the lens of covariational reasoning frameworks from mathematics education research, and determine that the frameworks are useful but do not completely describe the covariational reasoning of the physics experts we interviewed. From our data, we identified reasoning patterns that are not described in the mathematics education research that, together with the mathematics covariational reasoning frameworks, begin to characterize physics covariational reasoning.

Multiple-choice multiple-response (MCMR) items (i.e., multiple-choice questions for which more than one response may be selected) can be a valuable tool for assessment. Like traditional multiple-choice single-response questions, they are easy to score, but MCMR items can provide more information about student thinking by probing multiple reasoning facets in a single problem context. In this paper, we discuss differences in performance on MCMR items that result from differences in administration method (paper vs. online). In particular, we find a tendency for ``clickiness'' in online administration: students choose more responses to MCMR items when taking the electronic version of the assessment. Based on these findings, we recommend that online administration be compared only to other online administrations and not to paper administrations when comparing student results on MCMR items. These results also suggest that MCMR items provide a unique opportunity to probe differences in online and on-paper administration of low-stakes assessments., Comment: arXiv admin note: text overlap with arXiv:2009.10865

One desired outcome of introductory physics instruction is that students will develop facility with reasoning quantitatively about physical phenomena. Little research has been done regarding how students develop the algebraic concepts and skills involved in reasoning productively about physics quantities, which is different from either understanding of physics concepts or problem-solving abilities. We introduce the Physics Inventory of Quantitative Literacy (PIQL) as a tool for measuring quantitative literacy, a foundation of mathematical reasoning, in the context of introductory physics. We present the development of the PIQL and evidence of its validity for use in calculus-based introductory physics courses. Unlike concept inventories, the PIQL is a reasoning inventory, and can be used to assess reasoning over the span of students' instruction in introductory physics. Although mathematical reasoning associated with the PIQL is taught in prior mathematics courses, pre/post test scores reveal that this reasoning isn't readily used by most students in physics, nor does it develop as part of physics instruction--even in courses that use high-quality, research-based curricular materials. As has been the case with many inventories in physics education, we expect use of the PIQL to support the development of instructional strategies and materials--in this case, designed to meet the course objective that all students become quantitatively literate in introductory physics.

Covariational reasoning -- reasoning about how changes in one quantity relate to changes in another quantity -- has been examined extensively in mathematics education research. Little research has been done, however, on covariational reasoning in introductory physics contexts. We explore one aspect of covariational reasoning: ``goes like'' reasoning. ``Goes like'' reasoning refers to ways physicists relate two quantities through a simplified function. For example, physicists often say that ``the electric field goes like one over r squared.'' While this reasoning mode is used regularly by physicists and physics instructors, how students make sense of and use it remains unclear. We present evidence from reasoning inventory items which indicate that many students are sense making with tools from prior math instruction, that could be developed into expert ``goes like'' thinking with direct instruction. Recommendations for further work in characterizing student sense making as a foundation for future development of instruction are made., Comment: under review for Physics Education Research Conference Proceedings 2020

Electromagnetism (E&M) is often challenging for students enrolled in introductory college-level physics courses. Compared to mechanics, the mathematics of E&M is more sophisticated and the representations are more abstract. Furthermore, students may lack productive intuitions they had with force and motion. In this article, we explore the mathematization of electric charge. Specifically, we explore how difficulties with positive and negative signs can arise for learners who approach integers primarily as positions on a number line., Comment: Accepted by The Physics Teacher

We have developed the Physics Inventory of Quantitative Literacy (PIQL) as a tool to measure students' quantitative literacy in the context of introductory physics topics. We present the results from various quantitative analyses used to establish the validity of both the individual items and the PIQL as a whole. We show how examining the results from classical test theory analyses, factor analysis, and item response curves informed decisions regarding the inclusion, removal, or modification of items. We also discuss how the choice to include multiple-choice/multiple-response items has informed both our choices for analyses and the interpretations of their results. We are confident that the most recent version of the PIQL is a valid and reliable instrument for measuring students' physics quantitative literacy in calculus-based introductory physics courses at our primary research site. More data are needed to establish its validity for use at other institutions and in other courses., Comment: accepted for publication: 2020 Physics Education Research Conference

We are developing a new research based assessment (RBA) focused on quantitative reasoning -- rather than conceptual understanding -- in physics contexts. We rapidly moved administration of the RBA online in Spring 2020 due to the COVID-19 pandemic. We present our experiences with online, unproctored administration of an RBA in development to students enrolled in a large-enrollment, calculus-based, introductory physics course. We describe our attempts to adhere to best practices on a limited time frame, and present a preliminary analysis of the results, comparing results from the online administration to earlier results from in-person, proctored administration. We include discussion of online administration of multiple-choice/multiple-response (MCMR) items, which we use on the instrument as a way to probe multiple facets of student reasoning. Our initial comparison indicates little difference between online and paper administrations of the RBA, consistent with previous work by other researchers., Comment: PERC 2020

This paper is concerned with the leader-following output consensus problem in the framework of distributed nonlinear observers. In stead of certain hypotheses on the leader system, a group of geometric conditions is put forward to develop a novel distributed observer strategy with less conservatism, thereby definitely improving the applicability of the existing results. To be more specific, the improved distributed observer can precisely handle consensus problems for some nonlinear leader systems which are invalid for the traditional strategies with the certain assumption, such as Elastic Shaft Single Linkage Manipulator (ESSLM) systems and most of first-order nonlinear systems. We prove the sufficient conditions for the exponential stability of our distributed observer's error dynamic by proposing two pioneered lemmas to show the relationship between the maximum eigenvalues of two matrices appearing in Lyapunov type matrices. Then, a partial feedback linearization method with zero dynamic proposed in differential geometry is employed to design a purely decentralized control law for the affine nonlinear multi-agent system. With this advancement, the existing results can be regarded as a specific case owing to that the followers can be chosen as an arbitrary minimum phase affine smooth nonlinear system. At last, the novel distributed observer and the improved purely decentralized control law are applied in the distributed control framework to construct a closed-loop system. We also prove the stability of closed-loop system to achieve leader-following consensus, i.e., the distributed control framework is proved to satisfy certainty equivalence principle. Our method is illustrated by ESSLM system and Van der Pol system as leader.

Covariational reasoning -- how one thinks about the way changes in one quantity affect another quantity -- is essential to calculus and physics instruction alike. As physics is often centered on understanding and predicting changes in quantities, it is an excellent discipline to develop covariational reasoning. However, while significant work has been done on covariational reasoning in mathematics education research, it is only beginning to be studied in physics contexts. This work presents preliminary results from an investigation into expert physicists' covariational reasoning in a replication study of Hobson and Moore's 2017 investigation of covariational reasoning modes in mathematics graduate students. Additionally, we expand on this work to include results from a study that uses slightly more complex physics-context questions. Two behavioral modes were identified across contexts that appear distinct from those articulated in the Hobson and Moore study: the use of compiled relationships and neighborhood analysis., Comment: 6 pages, 4 figures, under review for the Research for Undergraduate Mathematics Education Conference Proceedings 2020

Relating two quantities to describe a physical system or process is at the heart of "doing physics" for novices and experts alike. In this paper, we explore the ways in which experts use covariational reasoning when solving introductory physics graphing problems. Here, graduate students are considered experts for the introductory level material, as they often take the role of instructor at large research universities. Drawing on work from Research in Undergraduate Mathematics Education (RUME), we replicated a study of mathematics experts' covariational reasoning done by Hobson and Moore with physics experts [N. L. F. Hobson and K. C. Moore, in RUME Conference Proceedings, pp. 664-672 (2017)]. We conducted think-aloud interviews with 10 physics graduate students using tasks minimally adapted from the mathematics study. Adaptations were made solely for the purpose of participant understanding of the question, and validated by preliminary interviews. Preliminary findings suggest physics experts approach covariational reasoning problems significantly differently than mathematics experts. In particular, two behaviors are identified in the reasoning of expert physicists that were not seen in the mathematics study. We introduce these two behaviors, which we call Using Compiled Relationships and Neighborhood Analysis, and articulate their differences from the behaviors articulated by Hobson and Moore. Finally, we share implications for instruction and questions for further research., Comment: 4 pages, 3 figures, under review for the Physics Education Research Conference Proceedings 2019

Mathematical reasoning flexibility across physics contexts is a desirable learning outcome of introductory physics, where the math world and physical world meet. Physics Quantitative Literacy (PQL) is a set of interconnected skills and habits of mind that support quantitative reasoning about the physical world. The Physics Inventory of Quantitative Literacy (PIQL), which we are currently refining and validating, assesses students proportional reasoning, covariational reasoning, and reasoning with signed quantities in physics contexts. In this paper, we apply a Conceptual Blending Theory analysis of two exemplar PIQL items to demonstrate how we are using this theory to help develop an instrument that represents the kind of blended reasoning that characterizes expertise in physics. A Conceptual Blending Theory analysis allows for assessment of hierarchical partially correct reasoning patterns, and thereby holds potential to map the emergence of mathematical reasoning flexibility throughout the introductory physics sequence.

The Physics Inventory of Quantitative Literacy (PIQL), a reasoning inventory under development, aims to assess students' physics quantitative literacy at the introductory level. The PIQL's design presents the challenge of isolating types of mathematical reasoning that are independent of each other in physics questions. In its current form, the PIQL spans three principle reasoning subdomains previously identified in mathematics and physics education research: ratios and proportions, covariation, and signed (negative) quantities. An important psychometric objective is to test the orthogonality of these three reasoning subdomains. We present results from exploratory factor analysis, confirmatory factor analysis, and module analysis that inform interpretations of the underlying structure of the PIQL from a student viewpoint, emphasizing ways in which these results agree and disagree with expert categorization. In addition to informing the development of existing and new PIQL assessment items, these results are also providing exciting insights into students' quantitative reasoning at the introductory level., Comment: submitted to 2019 PERC Proceedings

Mathematical reasoning skills are a desired outcome of many introductory physics courses, particularly calculus-based physics courses. Positive and negative quantities are ubiquitous in physics, and the sign carries important and varied meanings. Novices can struggle to understand the many roles signed numbers play in physics contexts, and recent evidence shows that unresolved struggle can carry over to subsequent physics courses. The mathematics education research literature documents the cognitive challenge of conceptualizing negative numbers as mathematical objects--both for experts, historically, and for novices as they learn. We contribute to the small but growing body of research in physics contexts that examines student reasoning about signed quantities and reasoning about the use and interpretation of signs in mathematical models. In this paper we present a framework for categorizing various meanings and interpretations of the negative sign in physics contexts, inspired by established work in algebra contexts from the mathematics education research community. Such a framework can support innovation that can catalyze deeper mathematical conceptualizations of signed quantities in the introductory courses and beyond.

In an effort to improve the quality of citizen engagement in workplace, politics, and other domains in which quantitative reasoning plays an important role, Quantitative Literacy (QL) has become the focus of considerable research and development efforts in mathematics education. QL is characterized by sophisticated reasoning with elementary mathematics. In this project, we extend the notions of QL to include the physics domain and call it Physics Quantitative Literacy (PQL). We report on early stage development from a collaboration that focuses on reasoning inventory design and data analysis methodology for measuring the development of PQL across the introductory physics sequence. We have piloted a prototype assessment designed to measure students' PQL in introductory physics: Physics Inventory of Quantitative Literacy (PIQL). This prototype PIQL focuses on two components of PQL: proportional reasoning, and reasoning with negative quantities. We present preliminary results from approximately 1,000 undergraduate and 20 graduate students.

Modeling the creative mathematical sensemaking that characterizes expert thinking in physics is typically a struggle for new learners. To help students learn to reason this way, we created a set of supplemental activities called Physics Invention Tasks (PITs)*. PITs engage students in quantification, the process of mathematically generating quantities central to modeling in physics. PITs depict a scenario in which students need to engage in decision-making associated with creating quantities in order to resolve a problem. Students make decisions about the arithmetic construction, the magnitude and associated unit, and in some cases spatial direction of a new quantity. The design of PITs is informed by Inventing with Contrasting Cases, an instructional approach shown to foster generativity, i.e., creativity with mathematical structures. In this paper, we describe the theoretical foundation of PITs, and their structure and implementation. We share preliminary observations of the impact of PITs in one course, averaged over two years, as measured by the FCI and CLASS. We see an improvement in the FCI normalized gain corresponding to the introduction of PITS and we see CLASS pre- to post-test gains that may be the highest reported from a large enrollment calculus-based physics course. We discuss future research into the learning mechanisms and instructor influence associated with PITs. *available at http://inventiontasks.physics.rutgers.edu, Comment: This paper is currently under review at the American Journal of Physics

University students taking introductory physics are generally successful executing mathematical procedures in context, but often struggle with the use of mathematical concepts for sense making. Physics instructors note that their students experience difficulty with basic algebraic reasoning, a foundation on which more advanced mathematical thinking rests. However, little systematic research has been done to measure and categorize such difficulties in this population. This paper describes an investigation of trends in student reasoning with proportions, quantification, and symbolizing in introductory calculus-based physics. Although the assessment items used to probe student thinking require mathematical reasoning typically taught at the middle school level in mathematics courses, we find success rates of about 50% among calculus-based physics students. For many of these students, numerical complexity and the level of abstraction of the quantities interfere with basic arithmetic reasoning. We argue that the algebraic thinking of physicists stems from an idiosyncratic cognitive blend, which is not addressed in prerequisite algebra courses. We suggest that for more students to understand and adopt the mathematical thinking characteristic of physics, instructors and education researchers must explore how to make mathematization a more explicit part of the curriculum., Comment: This paper is currently under review for the American Journal of Physics