Student Application of the Fundamental Theorem of Calculus with Graphical Representations in Mathematics and Physics

One mathematical concept frequently applied in physics is the Fundamental Theorem of Calculus (FTC). Mathematics education research on student understanding of the FTC indicates student difficulties with the FTC. Similarly, a few studies in physics education have implicitly indicated student difficulties with various facets of the FTC, such as with the definite integral and the area under the curve representation, in physics contexts. There has been no research on how students apply the FTC in graphically-based physics questions. This study investigated student understanding of the FTC and its application to graphically-based problems. Our interest spans several aspects of the FTC: student difficulties, problem-solving strategies, and visual attention. Written and interview findings revealed student difficulties common to mathematics and physics, e.g., confusion between the antiderivative difference and the function difference. Three problem-solving strategies were identified: algebraic, graphical, and integral. For a deeper analysis of problem-solving strategies, we applied the perspectives of epistemological framing (student expectations/perceptions) and epistemic games (problem-solving games). While most observed frames and epistemic games were somewhat modified versions of those previously reported, we identified one new game: the equation-based analytical game. In addition, a novel eye-tracking study was conducted to explore students’ visual attention to different parts of graphically-based FTC questions. Results indicated that students’ visual behavior was affected by the representations in the questions, such as the presence or absence of certain equation(s) and/or graphical feature(s), as well as context (math vs. physics). Because student responses seemed to be both conceptually and salient-feature driven, the results were explained using the cognitive perspectives of top-down (conceptually driven) and bottom-up (feature-driven) processes. Eye-tracking results provided support for interview findings about problem solving strategies. For many students, the absence of specific visual cues led to a particular framing of the problem that was associated with inappropriate e-games for that problem. Minor interviewer prompting often enabled students to reframe a problem and invoke relevant knowledge and strategies, suggesting that students possess knowledge of individual facets of the FTC, but this knowledge may not be elicited by a particular problem representation(s). Additionally, specific difficulties can be seen as due to inappropriate problem framing

Direct experimental evidence for quadruplex–quadruplex interaction within the human ILPR

Here we report the analysis of dual G-quadruplexes formed in the four repeats of the consensus sequence from the insulin-linked polymorphic region (ACAGGGGTGTGGGG; ILPRn=4). Mobilities of ILPRn=4 in nondenaturing gel and circular dichroism (CD) studies confirmed the formation of two intramolecular G-quadruplexes in the sequence. Both CD and single molecule studies using optical tweezers showed that the two quadruplexes in the ILPRn=4 most likely adopt a hybrid G-quadruplex structure that was entirely different from the mixture of parallel and antiparallel conformers previously observed in the single G-quadruplex forming sequence (ILPRn=2). These results indicate that the structural knowledge of a single G-quadruplex cannot be automatically extrapolated to predict the conformation of multiple quadruplexes in tandem. Furthermore, mechanical pulling of the ILPRn=4 at the single molecule level suggests that the two quadruplexes are unfolded cooperatively, perhaps due to a quadruplex–quadruplex interaction (QQI) between them. Additional evidence for the QQI was provided by DMS footprinting on the ILPRn=4 that identified specific guanines only protected in the presence of a neighboring G-quadruplex. There have been very few experimental reports on multiple G-quadruplex-forming sequences and this report provides direct experimental evidence for the existence of a QQI between two contiguous G-quadruplexes in the ILPR

Experts' understanding of partial derivatives using the partial derivative machine

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Partial derivatives are used in a variety of different ways within physics. Thermodynamics, in particular, uses partial derivatives in ways that students often find especially confusing. We are at the beginning of a study of the teaching of partial derivatives, with a goal of better aligning the teaching of multivariable calculus with the needs of students in STEM disciplines. In this paper, we report on an initial study of expert understanding of partial derivatives across three disciplines: physics, engineering, and mathematics. We report on the central research question of how disciplinary experts understand partial derivatives, and how their concept images of partial derivatives differ, with a focus on experimentally measured quantities. Using the partial derivative machine (PDM), we probed expert understanding of partial derivatives in an experimental context without a known functional form. In particular, we investigated which representations were cued by the experts' interactions with the PDM. Whereas the physicists and engineers were quick to use measurements to find a numeric approximation for a derivative, the mathematicians repeatedly returned to speculation as to the functional form; although they were comfortable drawing qualitative conclusions about the system from measurements, they were reluctant to approximate the derivative through measurement. On a theoretical front, we found ways in which existing frameworks for the concept of derivative could be expanded to include numerical approximation

Student Understanding of Definite Integrals with Relevance to Physics Using Graphical Representations

Learning physics concepts often requires fluency with the underlying mathematics concepts. Mathematics not only serves as a representational tool in physics (e.g. equations, graphs and diagrams), but it also provides logical paths to solve complex physics problems. The broad scope of this research is to understand the extent to which students’ mathematical knowledge and understanding influence their responses to physics questions. Only a few studies in physics education research (PER) have investigated connections between student difficulties with physics concepts and those with either the mathematics concepts, application of those concepts, or the representations used. One mathematical concept that is widely used across a broad spectrum of disciplines such as physics, chemistry, biology, economics, etc., is the definite integral. Particularly in physics, where most phenomena are expressed in the language of mathematics, an in-depth knowledge of definite integrals is extremely important to understanding the phenomena. Studies in mathematics education have shown student difficulties with conceptual understanding of definite integrals. We studied students’ conceptual understanding of definite integrals that are relevant in physics contexts. We also identified specific difficulties that students have with definite integrals, particularly with graphical representations. One strong focus of this work was how students reasoned about integrals that yield a negative result. In this study, two written surveys were administered in introductory calculusbased physics and multivariable calculus classes, and seven individual interviews were conducted with students from the physics survey population. In the first survey, students were asked to determine the signs and compare the magnitudes of two integrals. In interviews, students\u27 deep understanding of integrals was probed by varying representational features of the graphs from the first written survey. The second survey was administered in the following semester to test the reproducibility of some of the interview results in a larger population. Many of our findings corroborate previous results reported in the literature, including students’ using the area under the curve to reason about definite integrals, and ensuing difficulties generalizing area as always being a positive quantity. Additionally, novel results in this work include: multiple student difficulties in applying the Fundamental Theorem of Calculus in graphical situations; difficulties determining the signs of integrals that are carried out in the “negative direction” (i.e., from a larger to a smaller value of the independent variable); and student success invoking physical contexts to interpret certain aspects of definite integrals. Furthermore, we find that although students dominantly use area under the curve reasoning, including unprompted invocation of the Riemann sum, when contemplating definite integrals, their reasoning is often not sufficiently deep to help think about negative definite integrals. Overall, our results serve as one example that the connections between mathematics and physics are not trivial for students to make, and need to be explicitly pointed out. Implications for additional research as well as for instruction are discussed

Analytical derivation: An epistemic game for solving mathematically based physics problems

Problem solving, which often involves multiple steps, is an integral part of physics learning and teaching. Using the perspective of the epistemic game, we documented a specific game that is commonly pursued by students while solving mathematically based physics problems: the analytical derivation game. This game involves deriving an equation through symbolic manipulations and routine mathematical operations, usually without any physical interpretation of the processes. This game often creates cognitive obstacles in students, preventing them from using alternative resources or better approaches during problem solving. We conducted hour-long, semi-structured, individual interviews with fourteen introductory physics students. Students were asked to solve four “pseudophysics” problems containing algebraic and graphical representations. The problems required the application of the fundamental theorem of calculus (FTC), which is one of the most frequently used mathematical concepts in physics problem solving. We show that the analytical derivation game is necessary, but not sufficient, to solve mathematically based physics problems, specifically those involving graphical representations

Students’ strategies for solving a multirepresentational partial derivative problem in thermodynamics

We present an empirical analysis of students’ use of partial derivatives in the context of problem solving in upper-division thermodynamics. Task-based individual interviews were conducted with eight middle-division physics students. The interviews involved finding a partial derivative from information presented in a table and a contour plot. Using thematic analysis, we classified student problem-solving strategies into two principal categories: the analytical derivation strategy and the graphical analysis strategy. We developed a new flowchartlike analysis method: representational transformation. Our analysis of students’ strategies using this method revealed three types of transformation phenomena: translation, consolidation, and dissociation. Students in this study did not seem to have much difficulty with the concepts underlying the partial derivative; instead, they seemed to have difficulty with the transformation phenomena, particularly the consolidation of multiple representations into a single representation

Experts’ understanding of partial derivatives using the partial derivative machine

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Partial derivatives are used in a variety of different ways within physics. Thermodynamics, in particular, uses partial derivatives in ways that students often find especially confusing. We are at the beginning of a study of the teaching of partial derivatives, with a goal of better aligning the teaching of multivariable calculus with the needs of students in STEM disciplines. In this paper, we report on an initial study of expert understanding of partial derivatives across three disciplines: physics, engineering, and mathematics. We report on the central research question of how disciplinary experts understand partial derivatives, and how their concept images of partial derivatives differ, with a focus on experimentally measured quantities. Using the partial derivative machine (PDM), we probed expert understanding of partial derivatives in an experimental context without a known functional form. In particular, we investigated which representations were cued by the experts’ interactions with the PDM. Whereas the physicists and engineers were quick to use measurements to find a numeric approximation for a derivative, the mathematicians repeatedly returned to speculation as to the functional form; although they were comfortable drawing qualitative conclusions about the system from measurements, they were reluctant to approximate the derivative through measurement. On a theoretical front, we found ways in which existing frameworks for the concept of derivative could be expanded to include numerical approximation

Assessment of Palatal Plane and Occlusal Plane for Determining Anteroposterior Jaw Relation.

INTRODUCTION: Sagittal jaw relationship is an important parameter for orthodontic treatment planning. Angular and linear measurements both have been proposed and used in orthodontic cephalometrics to assess the sagittal jaw relationships. However, angular measurement has been questioned over the years for its reliability as a result of changes in facial height, jaw inclination and the variable positions of Nasion. So, the objective of our study was to assess the linear anteroposterior jaw relation in a sample of Nepali population using occlusal (Wits appraisal) and palatal planes as reference lines. METHODS: A descriptive cross-sectional study was conducted using the lateral cephalogram of 101 individuals visiting the Department of Orthodontics, Kantipur Dental College, Kathmandu, Nepal. Individuals with Class I skeletal relation were selected using convenience sampling method. Radiographs were standardised and traced. Occlusal and palatal planes were drawn that were bisected by the perpendicular lines from Point A and Point B. The linear distances between the intersections were measured to determine sagittal jaw relations. RESULTS: In Nepali individuals with normal ANB angle (3.05°±2.511°), the sagittal jaw relation with reference to occlusal (Wits appraisal) and palatal planes were found to be 0.203±3.343mm and 3.574±4.074mm respectively. CONCLUSIONS: Various methods has been proposed and used to assess the sagittal jaw relation and each method has its own strength and limitations. So, it is well advised to use additional cephalometric analysis whenever possible before arriving at any diagnosis and treatment plans