Investigating Student Understanding of Vector Calculus in Upper-Division Electricity and Magnetism: Construction and Determination of Differential Element in Non-Cartesian Coordinate Systems

Differential length, area, and volume elements appear ubiquitously over the course of upper-division electricity and magnetism (E&M), used to sum the effects of or determine expressions for electric or magnetic fields. Given the plethora of tasks with spherical and cylindrical symmetry, non-Cartesian coordinates are commonly used, which include scaling factors as coefficients for the differential terms to account for the curvature of space. Furthermore, the application to vector fields means differential lengths and areas are vector quantities. So far, little of the education research in E&M has explored student understanding and construction of the non-Cartesian differential elements used in applications of vector calculus. This study contributes to the research base on the learning and teaching of these quantities. Following course observations of junior-level E&M, targeted investigations were conducted to categorize student understanding of the properties of these differentials as they are constructed in a coordinate system without a physics context and as they are determined within common physics tasks. In general, students did not have a strong understanding of the geometry of non-Cartesian coordinate systems. However, students who were able to construct differential area and volume elements as a product of differential lengths within a given coordinate system were more successful when applying vector calculus. The results of this study were used to develop preliminary instructional resources to aid in the teaching of this material. Lastly, this dissertation presents a theoretical model developed within the context of this study to describe students’ construction and interpretation of equations. The model joins existing theoretical frameworks: symbolic forms, used to describe students’ representational understanding of the structure of the constructed equation; and conceptual blending, which has been used to describe the ways in which students combine mathematics and physics knowledge when problem solving. In addition to providing a coherent picture for how the students in this study connect contextual information to symbolic representations, this model is broadly applicable as an analytical lens and allows for a detailed reinterpretation of similar analyses using these frameworks

Characterizing and monitoring student discomfort in upper-division quantum mechanics

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 no single unit especially jarring nor especially easy. Student written responses were coded according to their reported source of discomfort---math, math-physics connection, physics, and notation. The relative prevalence of these categories varied significantly over the semester, indicating that students find that different units present different 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 provided additional context and insight into these results.Comment: 4 pages, 3 figures, to be published in 2020 Physics Education Research Conference Proceeding

Results of the ANCHOR prospective, multicenter registry of EndoAnchors for type Ia endoleaks and endograft migration in patients with challenging anatomy

ObjectiveProximal attachment site complications continue to occur after endovascular repair of abdominal aortic aneurysms (EVAR), specifically type Ia endoleak and endograft migration. EndoAnchors (Aptus Endosystems, Sunnyvale, Calif) were designed to enhance endograft proximal fixation and sealing, and the current study was undertaken to evaluate the potential benefit of this treatment.MethodsDuring the 23-month period ending in December 2013, 319 subjects were enrolled at 43 sites in the United States and Europe. EndoAnchors were implanted in 242 patients (75.9%) at the time of an initial EVAR procedure (primary arm) and in 77 patients with an existing endograft and proximal aortic neck complications (revision arm). Technical success was defined as deployment of the desired number of EndoAnchors, adequate penetration of the vessel wall, and absence of EndoAnchor fracture. Procedural success was defined as technical success without a type Ia endoleak at completion angiography. Values are expressed as mean ± standard deviation and interquartile range.ResultsThe 238 male (74.6%) and 81 female (25.4%) subjects had a mean age of 74.1 ± 8.2 years. Aneurysms averaged 58 ± 13 (51-63) mm in diameter at the time of EndoAnchor implantation (core laboratory measurements). The proximal aortic neck averaged 16 ± 13 (7-23) mm in length (42.7% <10 mm and 42.7% conical) and 27 ± 4 mm (25-30 mm) in diameter; infrarenal neck angulation was 24 ± 15 (13-34) degrees. The number of EndoAnchors deployed was 5.8 ± 2.1 (4-7). Technical success was achieved in 303 patients (95.0%) and procedural success in 279 patients (87.5%), 217 of 240 (89.7%) and 62 of 77 (80.5%) in the primary and revision arms, respectively. There were 29 residual type Ia endoleaks (9.1%) at the end of the procedure. During mean follow-up of 9.3 ± 4.7 months, 301 patients (94.4%) were free from secondary procedures. Among the 18 secondary procedures, eight were performed for residual type Ia endoleaks and the others were unrelated to EndoAnchors. There were no open surgical conversions, there were no aneurysm-related deaths, and no aneurysm ruptured during follow-up.ConclusionsUse of EndoAnchors to treat existing and acute type Ia endoleaks and endograft migration was successful in most cases. Prophylactic use of EndoAnchors in patients with hostile aortic neck anatomy appears promising, but definitive conclusions must await longer term follow-up data

Physics students’ construction and checking of differential volume elements in an unconventional spherical coordinate system

In upper-division physics courses, students’ use of differential line, area, and volume elements and their facility with the various multivariable coordinate systems consistently go hand in hand. As part of an effort to investigate student understanding of the structure of non-Cartesian coordinate systems and the associated differential elements, we interviewed students (mostly in pairs) in junior-level electricity and magnetism courses at two universities. In a sequence of tasks, students were asked to construct a differential length vector and a differential volume element in an unconventional spherical coordinate system. None of the students were able to arrive at a correct differential length element initially. This work addresses the construction and checking of the volume element. Volume element construction occurred by either combining associated lengths, an attempt to determine sides of a differential cube, or mapping from the existing spherical coordinate system. Students who constructed volume elements from differential length components corrected their length element terms as a result of checking the volume element expression by integration. Other students who relied heavily on spherical coordinates displayed further difficulty connecting dimensionality and projection ideas to differential construction

Physics students’ construction of differential length vectors in an unconventional spherical coordinate system

Vector calculus and multivariable coordinate systems play a large role in the understanding and calculation of much of the physics in upper-division electricity and magnetism. Differential vector elements represent one key mathematical piece of students’ use of vector calculus. In an effort to examine students’ understanding of non-Cartesian differential length elements, students in junior-level electricity and magnetism were interviewed in pairs and asked to construct a differential length vector for an unconventional spherical coordinate system. One aspect of this study identified symbolic forms invoked by students when building these vector expressions, some previously identified and some novel, given the vector calculus context. Analysis also highlighted several common ideas in students’ concept images of a non-Cartesian differential length vector as they determined their expressions. As no interview initially resulted in the construction of an appropriate differential, analysis addresses the role of the evoked concept images and symbolic forms on students’ performance

Students’ use of symbolic forms when constructing equations of boundary conditions

In an effort to examine students’ understanding about the structure of boundary conditions in upper-division courses, think-aloud interviews were conducted in the context of both quantum mechanics and electricity and magnetism. In the quantum mechanics task, students were presented with a standard potential step scenario. In the electricity and magnetism task, students were given a classical question about electromagnetic waves at the boundary of two media. The analysis of the interviews was guided by the use of a symbolic forms perspective. In this paper, we report the symbolic forms identified in students’ work and provide examples of students’ reasoning. Generally speaking, students were able to generate the correct symbolic templates while having difficulties with the underlying conceptual ideas. We also discuss instructional implications from this study

Vascular Quality Initiative Assessment of Compliance with Society for Vascular Surgery Clinical Practice Guidelines on the Management of Extracranial Cerebrovascular Disease

OBJECTIVES: Compliance with SVS clinical practice guidelines (CPGs) is associated with improved outcomes for treatment of abdominal aortic aneurysm, but this has not been assessed for carotid artery disease. The VQI registry was used to examine compliance with the SVS CPGs for management of extracranial cerebrovascular disease and its impact on outcomes. METHODS: The 2021 SVS extracranial cerebrovascular disease CPGs were reviewed for evaluation by VQI data. Compliance rates by center and provider were calculated, and the impact of compliance on outcomes was assessed using logistic regression with inverse probability-weighted risk adjustment for each CPG recommendation, allowing for clustering by center. Our primary outcome was a composite endpoint of in-hospital stroke/death. As a secondary analysis, compliance with the 2021 SVS carotid Implementation Document recommendations and associated outcomes were also assessed. RESULTS: Of the 11 carotid CPG recommendations, 4 (36%) could be evaluated using VQI registry data. Median center-specific CPG compliance ranged from 38-95%, and mean provider-specific compliance ranged from 36-100%. After adjustment, compliance with 2 of the recommendations was associated with lower rates of in-hospital stroke/death: first, use of best medical therapy (antiplatelet and statin therapy) in low/standard surgical risk patients undergoing CEA for \u3e70% asymptomatic stenosis (event rate in compliant vs non-compliant cases 0.59% vs 1.3%; adjusted odds ratio[aOR] 0.44, 95% confidence interval 0.29-0.66); and second, CEA over tfCAS in low/standard surgical risk patients with \u3e50% symptomatic stenosis (1.9% vs 3.4%; aOR 0.55, 0.43-0.71). Of the 132 Implementation Document recommendations, only 10 (7.6%) could be assessed using VQI data, with median center- and provider-specific compliance rates ranging from 67-100%. The impact of compliance on outcomes could only be assessed for 6 (4.5%) of these recommendations, and compliance with 6 of the recommendations was associated with lower stroke/death. CONCLUSION: Few SVS recommendations could be assessed in the VQI due to incongruity between the recommendations and the VQI data variables collected. While guideline compliance was extremely variable among VQI centers and providers, compliance with most of these recommendations was associated with improved outcomes after carotid revascularization. This finding confirms the value of guideline compliance, which should be encouraged for centers and providers. Optimization of VQI data to promote evaluation of guideline compliance and distribution of these findings to VQI centers and providers will help facilitate quality improvement efforts in the care of vascular patients