Characterizing and monitoring student discomfort in upper-division quantum mechanics
Giaco Corsiglia, Tyler Garcia, Benjamin P. Schermerhorn, Gina Passante, Homeyra Sadaghiani, Steven Pollock
CCharacterizing and monitoring student discomfort in upper-division quantum mechanics
Giaco Corsiglia Department of Physics, University of Colorado, Boulder, Boulder, Colorado, 80309, USA
Tyler Garcia , Benjamin P. Schermerhorn , Gina Passante , Homeyra Sadaghiani , and Steven Pollock Department of Physics & Astronomy, California Polytechnic University Pomona, Pomona, California, 91768, USA and Department of Physics, California State University, Fullerton, Fullerton, California, 92831, USA
We investigate student comfort with the material in an upper-division spins-first quantum mechanics course. Pre-lecture surveys probing students’ comfort were administered weekly, in which students assigned themselves a“discomfort level” on a scale of 0–10 and provided a written explanation for their choice. The weekly class-wide average discomfort level was effectively constant over the semester, suggesting that the class found nosingle unit especially jarring nor especially easy. Student written responses were coded according to theirreported source of discomfort— math, math-physics connection, physics, and notation . The relative prevalenceof these categories varied significantly over the semester, indicating that students find that different units presentdifferent challenges, and also that some of these challenges fade in importance as the semester progresses. Semi-structured interviews with students in a similar quantum mechanics course at a different institution providedadditional context and insight into these results. a r X i v : . [ phy s i c s . e d - ph ] J u l . INTRODUCTION Upper-division quantum mechanics (QM) is known to be bothconceptually and mathematically challenging for students .Much research has focused on student difficulties, but re-cent work has also investigated student perceptions of thecourse—e.g., examining students’ ideas about the nature ofquantum mechanics—as a way to guide instruction and fu-ture studies . Johansson discusses students’ perceptions oftheir QM courses in light of cultural expectations, arguingthat an expectancy mismatch can affect how students identifyas physicists . Schermerhorn et al. investigated whether stu-dents considered the physics or math more challenging whenstudying spin systems or spatial wave functions . Outside ofthe QM context, Gupta et al. explored connections betweenstudents’ emotions and conceptual reasoning .We investigate students’ self-reported sense of comfortwith the material in QM. Prompting students to reflect ontheir own understanding of recently covered material and en-couraging them to identify challenging aspects has precedentin the Just-in-Time-Teaching paradigm . In this paper, wepresent the results of a survey administered on a weekly basisin an upper-division spins-first QM course, as well as com-plementary preliminary results from an interview study withstudents from a similar course at a different university. On thesurvey, students were asked to identify their level of discom-fort with the material along with reasons for that discomfort.In the interviews, students were prompted to reflect on thecontent they were learning and their sense of comfort with it.A spins-first approach is one of two common paradigmsfor upper-division QM instruction . Where a positions-first approach focuses on solving the Schrödinger equationfor continuous wave functions that describe the position- ormomentum-space distribution of a particle, a spins-first ap-proach instead begins with spin-½ systems described by a dis-crete, two-state basis before a transition is made to studyingcontinuum systems near the end of the semester. The spins-first structure emphasizes fundamental quantum mechanicalconcepts in part by postponing development of the more com-putationally intensive mathematical formalism required forcontinuous wave functions . However, one concern is thatstudents may find the discrete-to-continuous transition espe-cially difficult or jarring .A marked increase in student discomfort following thecourse’s switch to studying spatial wave functions wouldhighlight the aforementioned concern about the difficulty ofthis transition point. More generally, the degree of and rea-sons for student discomfort with the material illuminate thethings that students consider important in their own learning,which can inform instruction as well as future research. II. METHODOLOGY
This paper focuses on student responses to a weekly pre-lecture survey administered in one semester of upper-division QM at University A, a large, public, primarily undergradu-ate and Hispanic-serving institution. Preliminary analysis ofthese data also prompted an interview study with students inthe first semester upper-division QM course at University B,an R1, PhD-granting institution. The interview study was de-veloped during and ran concurrently with continued analysisof the survey data. Both courses adhered to a “spins-first” QMcurriculum using McIntyre’s text . The course at UniversityA was taught by physics education research faculty. A. Pre-lecture surveys
A pre-lecture assignment was administered online beforeeach class to students in an (in-person) upper-division QMcourse at University A, where a total of 26 students respondedat least once. Students received participation credit for com-pleting the survey, but it was not graded for correctness. Be-ginning in the semester’s third week, the first survey of eachweek opened with the same two questions asking students toreflect on their comfort with the material they were learning.Students were asked to rank their discomfort level on ascale of 0–10 as follows:
On a scale of 0–10, rate your discomfort (orcomfort) with the ideas presented in class thisweek.(We are not asking about how well you think youcan answer homework or exam questions, buthow the concepts and ideas are “sitting” withyou and your intuition about the world.)0 — (no discomfort: “all these ideas make com-plete sense and seem reasonable”)5 — (moderate discomfort: “some of the ideasseem illogical and bother me, but I can seewhat’s going on”)10 — (complete discomfort: “none of these ideasmake any sense and I’m deeply concerned”)Think of this as a “quantum pain scale.”
Students were then asked to explain their discomfort level:
If you selected a number greater than 0, givea concrete example of an idea or concept thatmakes you uncomfortable.
Not all students provided explanations on every survey, butusually they wrote one or two sentences.We examined weekly and individual-student averages ofthe numerical responses. The written responses were catego-rized using an emergent coding strategy to identify the type ofmaterial the student reported discomfort with (see Sec. III B).Once the coding scheme was agreed upon, codes were as-signed to all responses by two researchers working indepen-dently, with 75% initial agreement. Mismatches were re-solved in discussion between the two researchers until 100%greement was reached. Individual responses were assignedmultiple codes if the student reported multiple sources of dis-comfort, meaning that some disagreements were resolved byadopting the codes suggested by both researchers.
B. Semi-structured interviews
In the semester after the pre-lecture survey data were col-lected, an interview study was performed to probe the reasonsfor student discomfort in more detail. Interviewees were alldrawn from the first semester upper-division QM course atUniversity B. The same six volunteers were interviewed in-dividually several times, including during the time evolution unit in weeks 6–7, and immediately after the infinite squarewell unit—the first unit in the wave functions context—inweek 10. The volunteers were compensated for participation.The interviews were conducted in spring 2020, and the sec-ond round took place one week after University B transitionedto remote learning due to COVID-19. Instruction on the in-finite square well unit preceded the switch. (The pre-lecturesurveys were administered in an unaffected semester.)The interview protocol prompted students to share howthey were feeling overall in the course, to assess their com-fort level with material in general, and to reflect specificallyon their comfort level with certain topics, including notation,math, and physics (i.e., physical concepts). The protocol alsoincluded additional questions outside the scope of this paper.Analysis of the interview data is ongoing, but preliminary re-sults are presented in Sec. IV.
III. RESULTS
In this section, we present student responses to both questionson the pre-lecture surveys. First, we present the average self-reported discomfort level on individual-student and weeklybases. Then we explore qualitative analysis of students’ writ-ten explanations for their reported discomfort level.
A. Tracking discomfort level over the semester
Figure 1 presents the distribution of students’ discomfort lev-els averaged over the entire semester. Centered around theoverall average of 4, the distribution is reasonably normal.Individual student responses changed on a week-by-week ba-sis by ± . on average. The figure shows some correlationbetween higher discomfort score and poorer course perfor-mance, but it is unclear if this reflects students’ ability to di-agnose their own difficulties, their emotional response to thegrades they received on assignments, or another effect.This quantitative discomfort scale is, at best, calibrated onan individual-student basis. That is, we hope that a 4 / Average discomfort level N u m b e r o f s t ud e n t s Self-reported discomfort level individual-student average
Course GradeUpper 50%Lower 50%
FIG. 1. The histogram shows the distribution of students’ averageself-reported discomfort levels. Averages are calculated over all ofthe pre-lecture surveys that the given student responded to. Theindividual-student standard deviation averaged over all students is1.5. The outlying student on the right only reported their discomfortlevel on one survey, and the one on the left exclusively reported ze-roes. The darker (lighter) colored bars indicate students whose finalcourse grade fell in the upper (lower) 50% of the class. same thing to every student. This makes it difficult to com-pare the magnitude of discomfort level across students. Still,averaging over students may capture class-wide shifts in dis-comfort. Figure 2 shows the class-wide average discomfortlevel reported on each of the 12 weekly pre-lecture surveysadministered over the semester.The class-wide average discomfort level is effectively con-stant over the entire semester: it hovers close to 4 and variesby less than 1. Additionally, there is a sizable spread everyweek: the standard deviation is usually 2 or higher. Eachweek, some students report a high level of discomfort whileothers report a low level, but there is never a net trend towardshigher or lower class-wide discomfort, regardless of changesin the topical areas being covered.
B. Identifying categories of discomfort and tracking theirprevalence over the semester
Qualitative analysis focused on students’ explanations fortheir discomfort level on the pre-lecture surveys. Throughemergent coding, we identified four categories of discomfortcorresponding with different aspects of the material: math,math-physics connection, physics, and notation . We also em-ployed a fallback category, other . We define each of thesecategories as follows. The percentages indicate the frequencyof the given category amongst the 241 total assigned codes.
Math (30%): The response references either conceptualor procedural understanding of the mathematics, such as dis-comfort computing integrals. For example, “The conceptsweren’t too difficult I think I just need to practice the calcula-tions . . . .” od e r n P hy s i c s ( ) S p i n s + M a gn e ti c F i e l d s ( ) S t a t e V ec t o r s ( ) O p e r a t o r s + C h a ng i ng B a s i s ( ) O p e r a t o r s + M ea s u r e m e n t ( ) O p e r a t o r s + M ea s u r e m e n t ( ) U n ce r t a i n t y P r i n c i p l e ( ) T i m e D e p e nd e n ce ( ) R e v i e w ( ) Q u a n t u m C o m pu ti ng ( ) E n t a ng l e m e n t ( )I n f i n it e S qu a r e W e ll ( )I n f i n it e S qu a r e W e ll ( ) H a r m on i c O s c ill a t o r ( ) Week (N = Number of survey respones) D i s c o m f o r t l e v e l ( l o w t o h i gh ) Self-reported discomfort level weekly average
FIG. 2. Each dot shows the average discomfort level reported by allstudents on the pre-lecture survey taken when the most recently cov-ered topic was the one given by the corresponding label on the hor-izontal axis. The topics are arranged in the order they were taught,and the labels include the number of students that responded to thecorresponding pre-lecture. The blue (smaller) error bars show thestandard error of the mean, and the orange (larger) error bars givethe standard deviation. The dashed line shows the overall average.
Math-Physics (16%): The response references the connec-tion between the math and the physics, such as discomfortwith the physical meaning of an equation. For example, “I’mmostly uncomfortable with knowing how and when to use theequations.”
Physics (24%): The response references physical con-cepts, such as questions about measurement. For example, “I am confused but intrigued by entanglement . . . .”
This cat-egory also includes responses that reference specific contentbut are too vague for another category, such as, “I was justhaving some problems with the clicker questions for measure-ment and measuring uncertainty . . . .”
Notation (10%): The response references notation used inthe course, usually Dirac notation. For example, “The Diracnotation and braket rules are still fuzzy . . . .”
Other (20%): The response does not reference specificcontent covered in the course. Examples include, “Examsare coming and i am having anxiety . . . ,” and “Still gettingthe hang of things.”
We focus on the four content-centeredcategories in this work.Overall, more students cite discomfort with math than anyother category, with physics being the second-most common,followed by math-physics and notation . However, this rank-ing was not consistent on a week-by-week basis. Figure 3presents the distribution of discomfort categories for eachweek that the pre-lecture survey was administered.Early on, math and notation are the dominant sourcesof discomfort. While students report discomfort with math M od e r n P hy s i c s ( ) S p i n s + M a gn e ti c F i e l d s ( ) S t a t e V ec t o r s ( ) O p e r a t o r s + C h a ng i ng B a s i s ( ) O p e r a t o r s + M ea s u r e m e n t ( ) O p e r a t o r s + M ea s u r e m e n t ( ) U n ce r t a i n t y P r i n c i p l e ( ) T i m e D e p e nd e n ce ( ) R e v i e w ( ) Q u a n t u m C o m pu ti ng ( ) E n t a ng l e m e n t ( )I n f i n it e S qu a r e W e ll ( )I n f i n it e S qu a r e W e ll ( ) H a r m on i c O s c ill a t o r ( ) Week (N = Number of codes assigned) P e r ce n t OtherNotationPhysicsMath-PhysicsMath
Discomfort category weekly breakdown
FIG. 3. The vertical cross-sections show the distribution of codesassigned to students’ written explanations for their self-reported dis-comfort level on each of the pre-lecture surveys. Each cross-sectionis normalized to the total number of codes assigned to student re-sponses for the given week—this number is included in the labelson the horizontal axis. The topics in the labels are the same as inFig. 2. Some student responses were assigned multiple codes, andnot all students provided a written explanation for their reported dis-comfort level. throughout the semester, the notation category all but disap-pears by the fourth pre-lecture. Meanwhile, students reportincreasing discomfort with physical concepts as the semesterprogresses, peaking during the units on quantum computingand entanglement, where the physics category dominates.The physics category drops back to below 20% duringthe “Infinite Square Well” unit, which is when the courseswitches to discussing continuous wave functions as opposedto discrete spin systems. We also see that math-physics is thedominant source of discomfort during this unit, accountingfor about 40% of all reported discomfort, more than twice itsmaximum in any other week. We also see a small resurgenceof the notation category at this time, with two students refer-encing notation as a source of discomfort in either this unitor the following one. Finally, we note that the other categorysurpasses 20% only in the one or two weeks preceding eachexam.Most individual students’ responses included a diversityof categories over the semester. One student did not pro-vide a single written response. Of the remaining students,all 25 reported discomfort with math at least once, 24 / physics , and 19 /
25 with math-physics , although nota-tion only arose for 15 /
25 students. Only 10 /
25 students re-ported discomfort with any single category more than 50% ofthe time (3 of whom favored other ).To investigate the relationship between students’ discom-fort level and categories, we generated a version of Fig. 3herein each occurrence of each category was weighted bythe discomfort level reported by the given student in the givenweek. This did not produce a meaningful change, indicatingthat there was minimal correlation between discomfort cate-gory and discomfort level.
IV. DISCUSSION & INTERVIEW OUTCOMES
The transition from discussing discrete spin systems to con-tinuous position-space wave functions did not produce anyshift in class-wide discomfort at University A, contrary toour expectations . Although conducted with a different set ofstudents at a different school, several potential explanationsarose in the second round of interviews held at University B,which immediately followed this transition point in a simi-larly structured QM course. Discussing the material in thewave functions unit, one student remarked, It’s starting to look a little more familiar to whatI’ve done before because we’ve started goingover, like finite and infinite square wells. We’vealso started looking at waves which we’re alsodoing in E&M.
Students have often seen examples of position-space systemslike the infinite square well in previous courses, and the math-ematics of waves may also feel familiar from other courses.Another student said of the wave functions unit that, com-pared with earlier material, she did “better with the concep-tual stuff just because for me it’s—it feels more applied.”
Other interviewees echoed this sentiment: the familiar phys-ical meaning of “position” may help students make sense ofthe new system despite new mathematical formalism.Based on qualitative data, however, the transition point stillwarrants instructor attention. One interviewee remarked, “Ifeel like the material got really hard last week, very quickly,” and others reported that it took them time to find connectionsbetween the two halves of the course. The courses at bothuniversities (A and B) included similar activities at the startof the wave functions unit designed specifically to ease thetransition point, and several interviewees at University B ref-erenced this activity as helpful .Several patterns arose in the coding of students’ written ex-planations for their reported discomfort level. First, althoughDirac notation is a clear source of discomfort for studentswhen it is introduced at the beginning of the semester, dis-comfort with notation fades quickly. Interviewed studentsshared a similar sentiment, and some expressed that theyhad come to appreciate the new notation by the end of thesemester. These results suggest that the existing curriculum’sapproach to this topic worked well for students despite theirinitial discomfort with Dirac notation. We also found that more students reported discomfort withthe mathematics in the first half of the semester, despite thefact that the mathematics required in the second half tendsto be more computationally intensive. Interviewed studentsechoed this sentiment. For example, one student said he con-sidered the class to be “70% math, 30% physical physics un-derstanding” in his first interview. He, along with other in-terviewees, pointed out that they were less familiar with lin-ear algebra and matrix manipulation than with integration anddifferential equations at the start of the course.Not all interviewees felt this way, however (see also re-lated research in ). One remarked in his first interview that, “I’m way more confident in my math abilities in this coursethan I was in like diff-EQ or calc 3,” although he still consid-ered problem-solving in the course to be “all math.” His per-spective shifted in the second interview, saying, “this class ismuch more about understanding your approach to the prob-lem than the math itself.”
This appeared to track the surveyresults, where units on the uncertainty principle and time de-pendence saw increases in student discomfort with physicalconcepts and a decrease in discomfort with the mathemat-ics compared with earlier in the semester. This trend saw itspeak during the units on quantum computing and entangle-ment, suggesting that those topics were especially effective atdrawing out physical concepts.
V. CONCLUSION
Weekly surveys probing students’ comfort with the materialwere administered in an upper-division QM course. Stu-dents’ average level of discomfort remained constant overthe semester, but the reasons for that discomfort varied as thecourse progressed and as different topics were covered. Webelieve our results can guide instructors and researchers to-wards topical areas that students may feel deserve attention.We do not, however, consider student discomfort with thecourse’s material inherently bad. Our results can be seen asidentifying the aspects of the material that are top-of-mind forstudents at different points in their QM class. For example,the increase in the importance of the physics discomfort cat-egory during the quantum computing and entanglement unitsmay be an argument for teaching these topics.Similarly, discomfort with quantum mechanics is commonamong physicists and could suggest an expert-like under-standing of the material. This paper previews an ongoinganalysis of an interview study, which will also examine stu-dents’ comfort with quantum weirdness.
ACKNOWLEDGMENTS
We are grateful to the authors’ respective PER groups, andto the students in both courses included in our study. Thiswork has been supported in part by the NSF under GrantsNo. DUE-1626594, 1626280, and 1626482.
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