Review of student difficulties in upper-level quantum mechanics

Written quiz responses of 653 students in three separate courses are analyzed in detail. There has been relatively little research on student learning of thermodynamics in physics courses at the university level. A recent study by Loverude et al. has made it evident that students at the introductory level (and beyond) face many significant difficulties in learning fundamental thermodynamic concepts such as the first law of thermodynamics. I have been engaged in an ongoing project with T. J. Greenbowe to investigate student learning of thermodynamics in both physics and chemistry courses. As part of that investigation, a short diagnostic quiz has been administered over the past two years in the calculus-based introductory physics course at Iowa State University (ISU). This quiz focuses on heat, work, and the first law of thermodynamics. At ISU, thermodynamics is studied at the end of the second semester of the twosemester sequence in calculus-based introductory general physics. This course is taught in a traditional manner, with large lecture classes (up to 250 students), weekly recitation sections (about 25 students), and weekly labs taught by graduate students. Homework is assigned and graded every week. Thermal physics comprises 18-20% of the course coverage, and includes a wide variety of topics such as calorimetry, heat conduction, kinetic theory, laws of thermodynamics, heat engines, entropy, etc. The diagnostic quiz used in this study is shown below; it has been administered in three separate classes. The version shown here was administered in May 2001; the other two versions (December 1999 and December 2000) had very minor variations from the one shown here. (There were one or two additional questions on these quizzes which are not discussed here.) The 1999 and 2000 classes were taught by the same instructor, using a different textbook in each course. The 2001 course was taught by a different instructor, using the same text that was employed in the 1999 course. Both instructors are very experienced and have taught introductory physics at ISU for many years. The quiz was administered in two different ways: in 1999 and 2001, it was given as a practice quiz in the final recitation session (last week of class). In almost all cases it was ungraded; one instructor used it as a graded quiz. In 2000 the quiz was administered as an ungraded practice quiz in the very last lecture class of the year. This p-V diagram represents a system consisting of a fixed amount of ideal gas that undergoes two different processes in going from state A to state B:

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Compared with introductory physics, relatively little is known about the development of expertise in advanced physics courses, especially in the case of quantum mechanics. Here, we describe a framework for understanding the patterns of student reasoning difficulties and how students develop expertise in quantum mechanics. The framework posits that the challenges many students face in developing expertise in quantum mechanics are analogous to the challenges introductory students face in developing expertise in introductory classical mechanics. This framework incorporates both the effects of diversity in upper-level students’ prior preparation, goals, and motivation in general (i.e., the facts that even in upper-level courses, students may be inadequately prepared, have unclear goals, and have insufficient motivation to excel) as well as the “paradigm shift” from classical mechanics to quantum mechanics. The framework is based on empirical investigations demonstrating that the patterns of reasoning, problem-solving, and self-monitoring difficulties in quantum mechanics bear a striking resemblance to those found in introductory classical mechanics. Examples from research in quantum mechanics and introductory classical mechanics are discussed to illustrate how the patterns of difficulties are analogous as students learn to unpack the respective principles and grasp the formalism in each knowledge domain during the development of expertise. Embracing such a framework and contemplating the parallels between the difficulties in these two knowledge domains can enable researchers to leverage the extensive literature for introductory physics education research to guide the design of teaching and learning tools for helping students develop expertise in quantum mechanics.Received 29 September 2014DOI:https://doi.org/10.1103/PhysRevSTPER.11.020119This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical Society

The lack of diversity and the under-performance of underrepresented students in STEM courses have been the focus of researchers in the last decade. In particular, many hypotheses have been put forth for the reasons for the under-representation and under-performance of women in physics. Here, we present a framework for helping all students learn in science courses that takes into account four factors: (1) the characteristics of instruction and learning tools, (2) student characteristics, (3) implementation of instruction and learning tools, and (4) the students’ environments. While there has been much research on factor 1 (characteristics of instruction and learning tools), there has been less focus on factor 2 (students’ characteristics, and in particular, motivational factors). Here, we focus on the baseline characteristics of introductory physics students obtained from survey data to inform factor 2 of the framework. A longitudinal analysis of students’ motivational characteristics in two-semester introductory physics courses was performed by administering pre- and post-surveys that evaluated students’ self-efficacy, grit, fascination with physics, value associated with physics, intelligence mindset, and physics epistemology. We found that female students reported lower levels of self-efficacy, fascination, and value associated with physics, and held a more “fixed” view of intelligence in the context of physics compared to male students. Female students’ fascination and value associated with physics decreased significantly more than males’ after an introductory physics course sequence. In addition, females’ view of physics intelligence became more “fixed” compared to males’ by the end of an introductory physics course sequence. Grit was the only factor on which females reported averages that were equal to or higher than males throughout introductory physics courses. The findings inform the framework and have implications for the development and implementation of effective pedagogies and learning tools to help all students learn.

In this study we conduct an integrated assessment of first-year performance in introductory calculus and physics courses. We use well-accepted conceptual assessment instruments, initial state data such as the SAT, and our own recently developed instruments designed to measure aptitude in mathematics to develop a predictive model for student performance. The analyzed population was composed of undergraduate students at Clarkson University, cross-enrolled in the calculus-based introductory physics course and the introductory calculus course from 2004 to 2007. Both the mathematics and physics classes are large-enrollment, lecture-based courses. By analyzing the combined data sets, we found the expected strong correlation between performance (final grade) in the mathematics course and performance in the physics course. We analyze not only the predictive nature of the collected data, but also the impact of the various corrective measures on final grade. This analysis contributes to an integrated evaluation of the current programs which could lead to an assessment-based initiative to offer strategic guidance to incoming students, better placing them for academic and career success in their selected STEM disciplines.

Students’ learning performance and its correlation with exam scores are examined using a unique factor: the day the online quiz was submitted. By collecting information on quizzes through an online learning management system for each subject in an introductory physics course for first-grade medical students, student learning performance is thoroughly analysed and compared with the final exam score. The results show a statistically significant negative correlation between the exam score and quiz submission day. Furthermore, by combining several parameters that are potentially related to the exam score, we are able to evaluate the student’s learning attitude in more detail. Based on the results, we discuss the negative effects of procrastinating behaviour and successful learning practices.

Student-to-student interactions are foundational to many active learning environments, but are most often studied using qualitative methods. Network analysis tools provide a quantitative complement to this picture, allowing researchers to describe the social interactions of whole classrooms as systems. Past results from introductory physics courses have suggested a sharp division in the formation of social structure between large lecture sections and small studio classroom environments. Extending those results, this study focuses on calculus-based introductory physics courses at a large public university with a heavily commuter and nontraditional student population. Community detection network methods are used to characterize pre- and post-course collaborative structure in several sections, and differences are considered between small and large classes. These results are compared with expectations from earlier findings, and comment on implications for instruction and further study.

Students’ genuine understanding (conceptual and operational) of physics is the goal of the majority of physics educators, who agree that the introductory physics course should possess the following features: reflect the currently adopted understanding of nature (the state in the subject matter), employ scientific method (philosophy of science) and expose it to the learner, and reflect our current understanding of the learning process taking place (research in physics education). We bring here two examples of suggested changes, the result of research in physics education, geared to improve the content of the introductory physics course in accordance with the above mentioned principles. More detailed information is available in the quoted publications.

This study investigates how the beliefs and values of physics faculty influence their choice of physics problems for their students in an introductory physics course. The study identifies the goals these instructors have for their students, the problem features they believe facilitate those goals, and how those features correspond to problems they choose to use in their classes. This analysis comes from an artifact-based interview of 30 physics faculty teaching introductory calculus-based physics at a wide variety of institutions. The study concludes that instructors' goals and the problem features they believe support those goals align with research-based curricular materials intended to develop competent problem solvers. However, many of these instructors do not use the beneficial problem features because they believe these features conflict with a more powerful set of values concerned with clarity of presentation and minimizing student stress, especially on exams.

This mixed methods study focuses on the use of instructional videos in an undergraduate physics classroom. Previous research has explored the effects of videos on student performance however, limited research exists on the effects of science instructional videos with embedded quiz questions on student achievement. This study compares students’ performance when a problem-solving physics video is presented to students with embedded quiz questions versus videos with quiz questions presented after the video. Additionally, the study explores students’ attitudes and preferences toward the positioning of questions in videos when learning independently. A total of 111 college students enrolled in the introductory physics course at the large Midwestern University were randomly assigned into two groups to participate in the intervention. Students completed pre and post surveys, watched instructional video and completed quiz that covered content discussed in the video that either was embedded or was presented post-video watching. The quantitative analysis did not result in statistical significance in students’ performance when groups were compared. However, qualitative analysis demonstrated strong student preferences toward the use of video in STEM classrooms when scientific problem-solving content is taught, stronger engagement with content when quizzes were embedded, and desire for immediate feedback. The study has implications for faculty who use videos and quizzing in their classroom or consider using videos in the future.

Approaches for improving students' understanding of quantum mechanics

Taking notes during lectures is one of the required skills, among many others, that students need (i) to master the topic covered in the lecture, (ii) to actively engage in the learning process with minimal to no distractions, (iii) to retain learned knowledge and skills for a longer time, and (iv) in securing higher letter grades. To learn the role of notetaking in learning undergraduate-level introductory physics courses, we present a comparative study of students’ achievement in a mid-terminal exam at a historically black Fayetteville State University (FSU) (i) when students were taught effective notetaking strategies, motivated them to prepare notes and let them use their self-prepared notes in a terminal exam versus (ii) no notetaking scheme was implemented, keeping all other conditions the same. The no-notetaking scheme was used in 4 different sections over a few semesters, and the notetaking scheme was used at the same level of an introductory physics course in 3 different sections in other semesters. Students’ scores in one of the mid-term exams are taken as measurement tools. Grade analysis indicates that effective notetaking enhances students’ letter grades and lowers failure rates.

When learning physics, students need more than just an understanding of mathematical and physical concepts. Integrating the two fields is crucial, as research indicates that students often struggle even when they have a strong grasp of both. In this paper, we use the heat equation as an example from higher education. Given the importance of the gradient and the Laplacian in the heat equation, as well as fluency and flexibility in using multiple external representations, this study investigates students’ reasoning about the gradient and the Laplacian of a function within different function representations in mathematics and physics contexts. To do so, we designed isomorphic open-response questions, administered as paper-and-pencil tests to 190 first-year students who were taking calculus-based introductory physics courses and multivariable calculus courses during their second semester. These students were enrolled in physics, mathematics, engineering, and bioengineering programs. We found that students face conceptual challenges with both the gradient and the Laplacian. Student difficulties with interpreting the gradient may be related to an incomplete understanding of steepness in graphical representations of functions of two independent variables. Moreover, their understanding of the Laplacian is limited. While context did not significantly affect students’ reasoning in terms of the types of difficulties they encountered, the proportion of students facing these difficulties, and their tendency to link or discuss more than one aspect of a concept, we did find context-specific prevalence of graphical, algebraic-symbolic, or linguistic descriptions. Students used linguistic descriptions more frequently in a physics context. We observed varying abilities to express the meaning of concepts in different answer representations. Linguistic descriptions of concepts proved particularly challenging for both concepts and contexts. Our findings provide insight into students’ understanding of two concepts related to the 2D heat equation.

Tests of formal operational reasoning and mathematics knowledge have been correlated with performance measures in introductory physics courses. A score based on collected homework was also used in our analyses in an attempt to control for the confounding effects of study effort. Course totals were broken down into separate scores for performance on test problems and performance on conceptual questions. A multiple-regression analysis indicates that the test of formal operational reasoning is a stronger predictor of course performance than the mathematics pre-test, particularly for the conceptual portion of the course totals. Possible reasons for failure in some studies to find significant correlations of course performance with measures of formal reasoning are discussed.

Since 1983, the Maryland University Project in Physics and Educational Technology (M.U.P.P.E.T.) has been investigating the implication of including student programming in an introductory physics course for physics majors. Many significant changes can result. One can rearrange some content to be more physically appropriate, include more realistic problems, and introduce some contemporary topics. One can begin training the student in professional research-related skills at an earlier stage than is traditional. An interesting point to note is that the inclusion of carefully considered computer content requires an increased emphasis on qualitative and analytic thinking.

Teaching and learning quantum mechanics is one of the most demanding educational and conceptual challenges, in particular in secondary schools where students do not possess an adequate mathematical background to effectively support the description of quantum behaviour. Educational research shows that traditional approaches, generally based on historical and narrative perspectives, are only partially effective. The reason is that they do not address in depth those basic quantum concepts that radically question the fundamentals of classical physics. A research-based educational program has been proposed to two final-year classes of an Italian scientific high school. In order to build the main concepts of quantum mechanics and their formal basic representation via real and simulated experiments, the program uses the light polarization as a context. A quantum game was then integrated in the educational program, to support students’ learning. Their conceptual paths, monitored by means of tutorials and questionnaires, show significant student learning especially on the concept of state and on appropriating the formalism meaning, whereas students more frequently referred to the geometrical vector representation instead of the algebraic-analytic formula. The quantum game has emerged to support intuition and operative experience in distinguishing the foundational concepts of superposition and entanglement.