Curvature in the Calculus Curriculum

This article presents a study in which applications were integrated in the Multivariable Calculus course at the Technion in the framework of supplementary tutorials. The purpose of the study was to test the opportunity of extending the conventional curriculum by optional applied problem-solving activities and get initial evidence on the possible impact of the tutorials on students’ beliefs about the value of learning mathematics with applications. The study lasted three semesters and consisted of three experiments in which supplementary tutorials were offered in different forms: a weekly evening meeting for interested students, a weekly extra hour added to the conventional calculus class, and a workshop which introduces mathematics concepts from the application perspective. The study reveals: (1) significant positive effect of the tutorials on the students’ beliefs (in all three experiments); (2) statistically significant advantage of the group involved in the tutorials in relation to the group that learned in the conventional way (second experiment); (3) students’ positive evaluation of the workshops for better understanding the course lectures (third experiment). Grounding on the study experience, we propose for further discussion a stage model of the applied problem solving cycle.

This study aims to determine students' perceptions of the use of multimedia learning software GeoGebra in Multivariable Calculus course. Students' perceptions of the use of media or multimedia learning are closely related to their achievement. Media or multimedia learning in accordance with the desire of students will make them motivated. This research is qualitative data analysis was done descriptively. The data obtained in this study comes from researchers as the main instrument, the data questionnaire and interview data is unstructured. The results are students' perceptions of the use of multimedia learning software GeoGebra in Multivariable Calculus courses are in the positive category.

This note describes an interdisciplinary project designed to enhance students’ knowledge of the basic techniques taught in a multivariable calculus course. The note discusses the four main requirements of the project and then the solutions for each requirement. Concepts covered include differentials, gradients, Lagrange multipliers, constrained and unconstrained optimization, level curves (isoquants and isocosts), marginal revenue and cost, the law of diminishing returns, marginal productivity, and inventory control.

<em>The aim of this research is to analyze students' mathematical representation abilities in terms of cognitive style in the multivariable calculus course taught in semester 4 of the IKIP PGRI Pontianak Mathematics Education Study Program. This research is a qualitative descriptive research in the form of case study research. Research subjects were selected based on their cognitive style which was measured using the geft instrument. Selected subjects were then asked to work on mathematical representation ability questions and then interviewed according to these questions. Based on the research results, it was found that students who had a field dependent cognitive style in symbolic representation were able to link symbols with mathematical concepts, in graphical representation they were able to create images in the form of requested graphs, while in verbal representation they were less able to convey ideas effectively orally. or written. Meanwhile, students with an independent cognitive style in symbolic representation have been able to link symbols with mathematical concepts, in graphic representation they have been able to create images in the form of requested graphics, and in verbal representation they have been able to convey ideas effectively.</em>

Penelitian ini adalah penelitian deskriptif kualitatif untuk menganalisis kesalahan pemahaman konsep peserta didik dalam menyelesaikan soal-soal integral lipat dua pada koordinat polar. Tiga puluh peserta didik yang mengambil mata kuliah kalkulus peubah banyak menjadi subjek dalam penelitian ini. Data dikumpulkan dengan menggunakan tes dan dokumentasi. Hasil jawaban tes peserta didik serta dokumentasi hasil pengerjaan peserta didik, dikoreksi, dianalisis berdasarkan indikator pemahaman konsep dan didisplay data. Analisis kesalahan jawaban peserta didik berdasarkan indikator pemahaman konsep yakni (1) menyatakan ulang konsep integral. (2) merepresentasikan konsep ke bentuk lain, (3) memilih dan menggunakan prosedur serta (4) menggunakan konsep dan algoritma pemecahan masalah. Dalam menyatakan ulang konsep integral peserta didik langsung menggunakan batas atas dan batas bawah integrasi tanpa menuliskan terlebih dahulu hasil pengintegralan. Kesalahan merepresentasikan fungsi ke grafik adalah, fungsi langsung disketsakan tanpa membuat table, memplot titik dan membuat kurva. Kesalahan dalam memilih dan menggunakan prosedur dan penggunaan algoritma pemecahan masalah dikarenakan kurangnya pengetahuan awal peserta didik mengenai materi prasyarat yakni, integral trigonometri dan fungsi trigonometri.

In this article we analyze publications written about different teaching modalities and evaluate how each applies to a calculus class during the on-going COVID-19 pandemic. We focus on the positives and negatives of teaching and learning in a virtual, classroom, or HyFlex environment. Although arguments could be made for each environment, especially given different institutional objectives, this work aims to explain why we eventually preferred teaching our Fall 2020 multivariable calculus course in a face-to-face classroom setting at the United States Military Academy at West Point. We also offer measures of performance to compare the current COVID-19 semester with previous semesters. The results support two major conclusions drawn from our decision to teach in-person under in a time constrained environment: learning modality matters in mathematics and this pandemic will influence student-teacher interaction for semesters to come.

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.

Calculus is typically one of the first college courses encountered by science, technology, engineering, and mathematics (STEM) majors. Calculus often presents major challenges affecting STEM student persistence, particularly for students from groups historically underrepresented in STEM. For life sciences majors, calculus courses may not offer content that is relevant to biological systems or connect with students’ interests in biology. We developed a transformative approach to teaching college-level math, using a dynamical systems perspective that focuses first on demonstrating why students need math to understand living systems, followed by providing quantitative and computational skills, including concepts from calculus, that students need to build and analyze mathematical models representing these systems. We found that students who complete these new math courses perform better in subsequent science courses than their counterparts who take traditional calculus courses. We also provide evidence that the new math curriculum positively impacts students’ academic performance, with data that show narrowing of the achievement gap, based on students’ math grades, between student subgroups in the new math courses. Moreover, our results indicate that students’ interest in the concepts and skills critical to the quantitative preparation of 21st-century life sciences majors increases after completing the new contextualized math curriculum.

Calculus in 3D is an accessible, well-written textbook for an honors course in multivariable calculus for mathematically strong first- or second-year university students. The treatment given here carefully balances theoretical rigor, the development of student facility in the procedures and algorithms, and inculcating intuition into underlying geometric principles. The focus throughout is on two or three dimensions. All of the standard multivariable material is thoroughly covered, including vector calculus treated through both vector fields and differential forms. There are rich collections of problems ranging from the routine through the theoretical to deep, challenging problems suitable for in-depth projects. Linear algebra is developed as needed. Unusual features include a rigorous formulation of cross products and determinants as oriented area, an in-depth treatment of conics harking back to the classical Greek ideas, and a more extensive than usual exploration and use of parametrized curves and surfaces. Zbigniew Nitecki is Professor of Mathematics at Tufts University and a leading authority on smooth dynamical systems. He is the author of Differentiable Dynamics, MIT Press; Differential Equations, A First Course (with M. Guterman), Saunders; Differential Equations with Linear Algebra (with M. Guterman), Saunders; and Calculus Deconstructed, MAA Press.

In this paper, we present how spatial visualization skills can be developed in engineering students, using augmented reality and remote virtual environments in calculus courses. Two tools have been specifically developed for this purpose: AVRAM (Remote Virtual Environments for the Learning of Mathematics), which allows the visualization and manipulation of surfaces in a virtual three-dimensional space and ARC (Augmented Reality in Calculus), which uses activity cards for each multivariable calculus topic. This paper describes the type of skills that can be developed with the use of these two apps in the dynamic visualization framework and presents some activities that have the purpose of developing logical thinking and problem solving and spatial abstraction skills. More than 1200 students have used these apps. The results of their use in calculus classes indicate an increase in students' engagement, visualization skills and a significant improvement in final grades.

In this study, we redesigned paper-and-pencil activities and adapted them to an introductory multivariate calculus course supported by 3D dynamic geometry software. We used semi-structured interviews and students’ written productions during the semester to analyse the use of technology in supporting students’ learning. Considering the work of two students, we find that the software has transformative potential but that it did not promote students’ learning as expected. We discuss possible reasons for this.

Condensation Calorimeters for the Measurement of Latent and Specific Heats. It is shown how an ordinary vacuum jacket flask can be converted into a suitable condensation calorimeter. Errors arising from loss of heat and wetness of the vapour are almost eliminated by making two experiments. Another condensation calorimeter is constructed on a different plan with a view to the elimination of the same errors. The Thermal Conductivity of a Narrow Metal Bar. Gray's apparatus for the measurement of the conductivity of a narrow bar has been modified for the purpose of rendering the loss of heat relatively small and reducing the time taken in the measurement, thus introducing the experiment to the elementary student. The Measurement of Poisson's Ratio for a Rectangular Lath. Two mirrors are attached to opposite sides of the bent lath for the measurement of the anti-clastic radius of curvature, and Poisson's ratio is then deduced from the observations in the usual manner. A Method of Measuring the Coincidence Period of a Kater and a Clock Pendulum. The Kater and the clock pendulums are electrically connected so that when at the centres of their swings a momentary current passes through a telephone receiver. A Differential Telephone Receiver. An ordinary receiver is connected to the secondary of a simple differentially wound transformer, and it is thus converted into a differential receiver for the purpose of electrical measurement. Experiment on Interference Fringes. The fringes produced by a Lloyd's mirror in white light are nearly achromatised by a simple grating on smoked glass. Other suggestive interference effects with the grating are also described.

Based on the forthcoming book, Time for Science Education (Matthews, in press), this article notes that the US National Science Education Standards advocate liberal, contextual, or cultural goals for science education, including the expectation that students will understand a certain amount of the historical and cultural significance of science. After mentioning something of the rich role played by the pendulum in the foundation of modern science, in solving the longitude problem, in establishing a universal length standard, and in allowing the creation of an accurate timekeeper, as well as the pendulum clock's function in philosophy and theology, the article draws attention to the scant treatment given the pendulum in the Standards. Opportunities are thus lost for realizing the Standards' laudable goals for US science education. Finally, it is claimed that realizing these cultural goals for science education requires more routine incorporation of the history and philosophy of science into preservice and inservice courses for science teachers.

The article describes the areas of application of roller machines, trends in the development of their designs, research work on the modernization and creation of new roller machines. Special attention given to the roller machines with a variable center distance of the working shafts. The descriptions of the gear-lever differential transmission mechanisms created by authors are given. In particular, the correct applications of the created mechanisms in roller machines of six types are shown; they are 1. One of the working shafts has the ability to rotate around its own axis, and the second working shaft, in addition to rotation around its own axis, moves relative to the first working shaft along the line, passing through the center of the axes of rotation of both working shafts; 2. Both working shafts have the ability to rotate around their own axes and move symmetrically along a line passing through the same axes of rotation relative to each other; 3. One of the working shafts has the ability to rotate around its own axis, and the second working shaft, in addition to rotation around its own axis, moves relative to the first working shaft along an arc with a certain radius of curvature; 4. Both working shafts can rotate around their own axes and move symmetrically relative to each other along an arc with some symmetrical centers of rotation; 5. One of the working shafts has the ability to rotate around its own axis, and the second working shaft, in addition to rotation around its own axis, moves relative to the first working shaft along a complex trajectory; 6. Both working shafts can rotate around their own axes and move symmetrically relative to each other along a complex trajectory.

The wafer-level production of Fused Silica microlens arrays is limited by systematic process non-uniformities. The common molten resist-reflow process with subsequent dry-etching allows for manufacturing of microlens arrays on 200 mm wafers. A thorough process review yielded one free parameter. By switching from the geometrical lens description via radius of curvature and conic constant to a functional assessment via the optical design figure of merit we can describe microlens via their optical quality for the intended application with one parameter for a wide variety of cases. Leveraging these points we show improvements on the uniformity of microlens arrays by a factor of 2 for Fused Silica microlens arrays bigger than 100 mm by 100 mm.