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Using a completeness (or closure) relation is a neat math trick in quantum theory. Completeness relations capture the fundamental property that a basis set spans a vector space, and conceptually can be thought of as a special form of the identity operator. In quantum mechanics, completeness relations can be used to express a quantum state in a particular basis, change which basis a state is expressed in, as well as mathematically model the effect of a quantum interferometer. We have analyzed students&#39; written responses to homework and final exam questions related to completeness relations from a junior-level, introduction to quantum mechanics course. In this poster, we will present student reasoning and difficulties using completeness relations.

PERC SM 21

Student Reasoning with Completeness Relations in Quantum Mechanics

sensemaking

quantum mechanics

representations

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Using a completeness (or closure) relation is a neat math trick in quantum theory. Completeness relations capture the fundamental property that a basis set spans a vector space, and conceptually can be thought of as a special form of the identity operator. In quantum mechanics, completeness relations can be used to express a quantum state in a particular basis, change which basis a state is expressed in, as well as mathematically model the effect of a quantum interferometer. We have analyzed students' written responses to homework and final exam questions related to completeness relations from a junior-level, introduction to quantum mechanics course. In this poster, we will present student reasoning and difficulties using completeness relations.

poster

**Making Physics More Inclusive and Eliminating Exclusionary Practices in Physics**

As a gateway discipline to STEM majors and careers, physics plays a central role in deciding who does, and who doesn't, get to discover and pursue scientific careers. Technical and scientific professionals are rewarded with economic privilege and status which affords them both social and cultural power. Therefore, social justice will be realized only when the opportunity, and the choice, to excel in physics is open to all students. The culture and practices of the physics classroom reflect the culture of physics at large, which raises the question of how our professional physics communities can embrace, rather than inadvertently repel, a diverse group of academics that can help improve physics culture.  To this end, the 2021 Physics Education Research Conference will focus on defining inclusive spaces and practices. Additionally, we will explore strategies for eliminating exclusionary practices in our classrooms, research spaces, and professional organizations. We call on the community to propose sessions that will drive conversations related to these initiatives throughout PERC 2021.

Please download the full program [here. ](https://drive.google.com/file/d/1TRMC_SYJ3PUqXyP7sskA7LRSA8ow0eRE/view?usp=sharing)

August 4, 2021 - August 5, 2021 in a Virtual Conference Format

Making Physics More Inclusive and Eliminating Exclusionary Practices in Physics

This study investigated the impact of an instructional intervention on student scientific reasoning skills and general attitudes toward science. The intervention consisted of nine PhET simulations introduced to a general education physics class. Students manipulated the simulations while completing worksheets testing their comprehension of the concepts and the scientific reasoning being displayed. Completion of each PhET simulation activity took students about 50 minutes to complete. Lawson’s Classroom Test of Scientific Reasoning was administered to assess improvement in student scientific reasoning skills, and the CLASS was used to assess changes in student attitudes towards science. Pre-to post Lawson results show gains in scientific reasoning. CLASS pre to post indicated overall shifts to expert-like responses. Qualitative data in the form of student surveys was collected and showed approval of use PhET simulations in the class.

Effect of PhET simulations in a general education physics course

Homework lets students practice the problem-solving process. The focus on process, rather than final answer, is particularly important. Unfortunately, many online homework systems, and some traditional pencil-and-paper strategies, grade based on the correct answer rather than students' written thought process. This can shift student focus away from the process. We have scaled a homework structure that emphasizes the problem-solving process and metacognitive reflection. Students are required to follow a problem-solving format developed previously, which has been shown to increase student metacognition.* In our procedure, students self-grade the completeness of their homework, and instructors perform a specific spot-check to increase student accountability. The spot-check takes minimal instructor time to administer and therefore can be scaled to large courses. Additionally, this homework structure shows high student satisfaction by the end of the term.

Scalable, pencil-and-paper homework with metacognitive student outcomes and minimal instructor time

Research-validated clicker questions as instructional tools for formative assessment are relatively easy to implement and can provide effective scaffolding when developed and implemented in a sequence. We present findings from the implementation of a research-validated Clicker Question Sequence (CQS) on student understanding of the time-development of two-state quantum systems. This study was conducted in an advanced undergraduate quantum mechanics course. The effectiveness of the CQS was determined by evaluating students’ performance after traditional lecture-based instruction and comparing it to their performance after engaging with the CQS.

Using a clicker question sequence to teach time-development in quantum mechanics

Two-year colleges (TYCs) have unique opportunities to support their students, for example through close student-teacher relationships due to small class sizes, and we studied one such class in a cohort program at a TYC for natural science majors intending to transfer. This course was a STEM research methods course that we observed by taking field notes, administering written journal reflections to the students throughout the course, and by interviewing some of the students at the end of the course. In this poster, we will discuss the ways instructor support helped the students feel higher self-efficacy for scientific research and a greater tolerance for obstacles.

Methods for Observing and Characterizing a TYC Research Methods Course

An emerging body of research suggests that, even after research-based instruction, poor student performance on certain physics tasks may stem primarily from domain-general reasoning phenomena rather than from a lack of conceptual understanding. The reasoning patterns (and inconsistencies) reported in these studies may be explained by dual-process theories of reasoning (DPToR). In order to help students strengthen their reasoning skills and support increased cognitive reflection, there is a need to design and test instructional intervention strategies that leverage DPToR and that may ultimately guide the development of research-based curricular materials that attend to the nature of human reasoning more explicitly. This investigation focused on an intervention designed to support analytical processing in which students were asked to set aside their own reasoning and engage in alternative lines of reasoning. In the intervention, students first responded to a qualitative physics task, then constructed reasoning chains in support of answers to that task given by two fictitious students, and finally revisited the original physics task. Analysis revealed that this intervention was successful at improving student performance. Furthermore, it appears to have supported students regardless of their cognitive reflection skills, and its effectiveness may potentially be correlated with the quality of reasoning chains generated in support of the correct fictitious student's response.

Supporting student construction of alternative lines of reasoning

Engaging with models has been considered central to the practice of doing science as it facilitates sensemaking of the world around us. Therefore, engaging students in the practice of using models is an important component of their science education. But to do so effectively, we also need to understand how students use models in their work. Consequently, we require a way to analyzing students' use of models. In the current work, we present an analytical framework which characterizes students' use of models by considering common themes from the existing literature on modeling in physics. These themes present themselves as five components: (i) Presence of a real-world phenomenon, (ii) Use of representation(s) depicting the phenomenon, (iii) Invoking of conceptual knowledge organized around representation(s), (iv) Presence of explanation/prediction about the phenomenon and (v) Linking the explanation/prediction to a representation through appropriate reasoning. Analysis of students' written and verbal responses to physics problems through these components indicate that students seldom link the predictions made to the representations through reasoning, and, when they do, representations are often mathematical equations even though diagrams are present in their solution.

Towards an analytic framework for characterizing student use of models

Upper division thermodynamics courses require students to develop proficiency with several combinatorial topics. While little research has been done to establish student comfort and proficiency with these topics within physics, anecdotally, instructors of these courses report that students experience unaddressed challenges when working with combinatorics. Through examining student solutions to questions on homework assignments and examinations, we identified some common student difficulties with combinatorics in the context of an upper-division thermodynamics and statistical mechanics course at a large research university. Some of these common challenges include identifying how to work with distinguishability and order, checking formulas against a specific case, and combining the multiplicities of multiple systems.

Understanding combinatorics challenges in an upper-division thermodynamics course

Selection and application of an appropriate equation is a key step in instructional problem-solving strategies. But what if one cannot exactly recall the relevant equation? We are currently investigating how coherence between physical and mathematical reasoning opens multiple solution pathways for qualitative physics problems. In one-on-one interviews, community college students, 4-year college undergraduates, and graduate students solved qualitative introductory-level E&M questions. Analysis of 18 interviews showed that while many students had initial difficulty in recalling a relevant equation, they demonstrated strategies for making progress. We present two cases from the data corpus. One interviewee was unsure whether the time constant for an RC circuit is RC or 1/(RC) and sought coherence with qualitative reasoning to determine the correct expression. Another student switched to reasoning qualitatively from a physical model after noting that they could not recall a relevant equation. These examples show how mathematical and physical reasoning is opportunistically incorporated in students’ problem solving.

Seeking coherence and switching reasoning after forgetting an equation

The entanglement between physical and mathematical reasoning can be exploited for problem solving, making physical/mathematical coherence seeking a desirable learning goal. We endeavor to enrich the empirical knowledge of students’ coherent physical/mathematical reasoning through a video analysis of problem solving. Drawing from a corpus of 18 interviews with students solving qualitative E&M problems, we discuss a case study of a student who undertakes multiple contradicting lines of reasoning and ultimately reconciles them. We discuss the types of coherent reasoning that they recruit and the central role of physical/mathematical coherence seeking in their problem solving and learning.

Seeking physical/mathematical coherence by recruiting and reconciling reasoning: A case study in E&M

Sequential synthesis problems involve multiple, chronologically occurring events. Successful problem solvers need apply relevant concepts consecutively to match these events. We explored students’ sensemaking of the situations presented in two sequential synthesis tasks by studying their eye fixation durations on each problem diagram. We also recorded students’ verbal descriptions of their problem-solving strategies. Results indicate that students fixated heavily on the parts of each diagram that require generating new information than on those that require reading given information. In addition, they seemed capable of invoking proper concepts and intermediate variables to handle the problems. According to the eye-mind framework, these outcomes indicate that students’ mental efforts went beyond comprehending given information. This study suggests that the spatiotemporal separation of multiple events in sequential synthesis problems may facilitate student sensemaking processes.

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Student Reasoning with Completeness Relations in Quantum Mechanics

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