Numerical Analysis Course Research Articles

Incorporation of Flipped Learning and Problem-Based Learning Methods for the Numerical Analysis Course Design

Incorporation of Flipped Learning and Problem-Based Learning Methods for the Numerical Analysis Course Design

A Semester-Long Bottle Design Project for an Undergraduate Numerical Analysis Course

This paper presents a semester-long bottle design project for an undergraduate Numerical Analysis course. Students implement numerical methods in MATLAB to design and 3D print a bottle. Employing mathematical methods to compute features of their designs enhances students' understanding of numerical techniques in design applications. Student feedback highlights the project's efficacy in demonstrating the intersection of mathematics and design, fostering creativity, and instilling a deeper appreciation for mathematical principles in real-world contexts.

The Practice and Exploration of the "Dual-Line Integration Teaching" Model in the Course of Numerical Analysis

Combining the advantages of online teaching and offline teaching, this paper constructs a "dual-line integration teaching" model, which is a complete set of "teaching-learning-evaluation" integrated closed-loop teaching mode. At the same time, the teaching mode is applied to the teaching practice of the course "Numerical Analysis", and good teaching results have been obtained, which not only improves the pass rate and excellent rate of the course, but also greatly improves the students' independent learning ability and practical innovation ability, so as to achieve the training goal of applied innovative talents.

Assessing Numerical Analysis Performance with the Practi Mobile App

Numerical analysis is a unique combination of mathematical and computing skills. It facilitates a deeper understanding of data analytics and machine learning software libraries, which are exploding in use and importance. However, it is a topic that continues to challenge students because it requires a confluence of conceptual, procedural, and computational skills and associated pedagogies. Therefore, it is valuable to identify effective pedagogies and tools to enhance and assess student numerical analysis skills. Despite the proliferation of mobile technology in postsecondary education, its role in the context of numerical analysis is largely unknown. This quasi-experimental pilot study used Practi, an educational mobile app designed to assess numerical analysis performance and promote both retrieval practice and deliberate practice, which have been shown to help improve performance and develop expertise. Participants were 32 undergraduate students enrolled in a second-year introductory Numerical Analysis course at a large North American university. They were prompted to use Practi to solve quizzes on a regular basis throughout the course, before and after each lecture, to promote deliberate practice and spaced retrieval. Results of a paired t-test analysis showed that Practi was able to detect improvement in student quiz performance after the lectures compared to before the lectures. Additionally, performance on the Practi quizzes after the lectures was positively associated with the overall course performance. This suggests that mobile apps supporting deliberate and retrieval practice can complement more traditional means of instruction and assessment of numerical analysis in postsecondary mathematics education.

Swirling flow of two immiscible fluids in a cylindrical container: Lattice Boltzmann and volume-of-fluid study

An experimental and numerical study of a multicomponent swirl flow of a liquid in a closed cylinder is carried out for various values of the relative cylinder elongation and the Reynolds number. The experimental technique for flow characteristics measurement is based on the PIV (particle image velocimetry) technique. To study the flow characteristics in detail, we simulated the problem numerically using the Palabos and Basilisk software open-source packages. The current implementation of the Palabos package uses the LBM (lattice Boltzmann method) approach, in which the collision integral is determined by the MRT (multiple-relaxation-time) approximation, and the intercomponent interaction is established according to the Shan–Chen pseudopotential approximation. The Basilisk package uses VOF (volume-of-fluid) approach to approximate the fluid interface. In this paper, for the first time, the data considering the emergence conditions for the zone of axial isolated recirculation in a multicomponent vortex flow were obtained at different viscosity ratios of two fluids. It is shown that with a decrease in the viscosities ratio, the recirculation zone existence curve shifts closer to that corresponding to the case of a one-component flow. In the course of numerical analysis, we found that both numerical implementations of this problem describe flow characteristics with high accuracy. Both methods recreate the recirculation zone on the cylinder axis observed experimentally. However, the velocity shift on the interface is observed only in the LBM approach.

Integrating Traditional Chinese Culture into Teaching Practice and Exploration – Taking the Numerical Analysis Course as an Example

China’s excellent traditional culture has a history of thousands of years, which will have a significant influence on students’ thinking and behavior if integrated into teaching. In this paper, the Numerical Analysis course is taken as the research subject, and the main problems and causes of incorporating traditional culture as well as ideological and political education into teaching are analyzed. At the same time, as a way of integrating science into education, the strategy and specific implementation plan for the integration of traditional culture as well as ideological and political elements into numerical calculation methods are proposed. This provides a new idea and experience for basic mathematical courses in colleges and universities, which plays a role in traditional cultural, ideological, and political education curricula.

Development of soft skills while learning numerical analysis

Many authors argue that effective communication in any professional training is a factor that positively influences future job success. Based on these ideas, in the Numerical Analysis course at the Facultad Regional San Nicolás, Universidad Tecnológica Nacional from Argentina decided to design didactic sequences that try to promote the development of communicative competence, both oral and written, while students learn the different numerical methods. The main objective of this work is to show some of the didactic sequences that were designed to contribute to the training of students in terms of communicative competence and to present the rubrics that were developed, based on the evaluation criteria that were established. Assessment instruments were also designed to analyze the degree of development of skills in students. The didactic sequences presented in this paper show what can be done from different subjects of engineering careers to contribute to the development of soft skills in students. Keywords: Careers; Communication; Engineering; Mathematics; Soft Skills.

Esquemas de diferencias finitas en el procesamiento de imágenes

Digital Image processing has been a research area of interest in the last decades, standing out for its applications in the analysis of diagnostic images and astronomical images. In this paper, we perform an overview of edge detection methods through finite-difference to present edge detection as a problem-based learning strategy for numerical differentiation, in order to improve the students’ skills in modeling and algorithmic thinking in numerical analysis courses. In addition, we present image restoration through finite-difference as a problem involving partial differential equations and software tools.

An Open-Access Clicker Question Bank for Numerical Analysis

ABSTRACT We present a question bank consisting of over 250 multiple-choice and true–false questions covering a broad range of material typically taught in an introductory undergraduate course in numerical analysis or computational science. The questions are ideal for polling students during lectures by means of a student response system that uses clicker remotes or smartphones running a suitable app. We describe our experiences implementing these clicker questions in a recent class and provide evidence of their effectiveness in terms of testing students' prior knowledge, gauging understanding of new material, increasing participation, and especially improving student satisfaction. Our conclusions are supported by a mid-semester student survey as well as anecdotal observations. The question bank has been released as an open-access educational resource under a Creative Commons license (CC BY-NC-SA) for free use by the mathematics community.

A semester of home-classroom

The 2020 academic year began in an unusual way. After a week of traditional classes at San Nicolas Regional School, National Technological University, Argentina, teachers and students had to go into a mandatory lockdown due to the COVID-19 pandemic. The objective of this paper is to discuss the experience and show the obtained results in a Numerical Analysis course of an engineering career. An online alternative was designed for the Numerical Analysis courses, and the second week of the course, the tasks in the virtual classroom started. The content of each module was presented in a series of lessons and weekly assignments were proposed throughout the semester. Students had to sit for two assessments, covering two topics each, as was usually done in face-to-face classes. The experience was demanding, for both professors and students, but good outcomes were achieved. This paper recommends that everyone involved in teaching and learning has had to adapt, given the circumstances. Keywords: Numerical Analysis, online course, assessment, home-classroom, semester.

The students’ view for teaching numerical analysis in the form of distance education

With COVID-19 pandemic, which was started in Wuhan, Chine, and is still affecting the whole world as well as our country, education offered distantly. This research aims to determine the students' opinions and their behaviors towards teaching Numerical Analysis in the form of distance education. The research was carried out with a group of students studying at the Faculty of Engineering in a private university in Northern Cyprus during the 2019-2020 academic year. In the research, an interview form was applied to get students’ opinions about distance education and to understand whether it is appropriate to give a numerical analysis course over distance education or not. At the end of the research, students’ positive and negative views about distance education emerged. Most of the students argued the idea that distance education platform is not able to provide education as effectively as the traditional classroom environment. It is supported that it is more accurate to give the form of traditional education because of the application of a Numerical Analysis course is challenging.
 Keywords: Distance Education, Numerical Analysis, Student, Distance Learning, Information and Communication

Euler–Maclaurin expansions without analytic derivatives

The Euler–Maclaurin (EM) formulae relate sums and integrals. Discovered nearly 300 years ago, they have lost none of their importance over the years, and are nowadays routinely taught in scientific computing and numerical analysis courses. The two common versions can be viewed as providing error expansions for the trapezoidal rule and for the midpoint rule, respectively. More importantly, they provide a means for evaluating many infinite sums to high levels of accuracy. However, in all but the simplest cases, calculating very high-order derivatives analytically becomes prohibitively complicated. When approximating such derivatives with finite differences (FD), the choice of step size typically requires a severe trade-off between errors due to truncation and to rounding. We show here that, in the special case of EM expansions, FD approximations can provide excellent accuracy without the step size having to go to zero. While FD approximations of low-order derivatives to high orders of accuracy have many applications for solving ODEs and PDEs, the present context is unusual in that it relies on FD approximations to derivatives of very high orders. The application to infinite sums ensures that one can use centred FD formulae (which are not subject to the Runge phenomenon).

Evaluation of numerical analysis course problem-solving processes of engineering students based on different dimensions

In this study, the aim is to evaluate the problem-solving processes in the understanding of problems and to determine and provide strategies, demonstration, expression and problem-writing dimensions in a numerical analysis course for engineering students. The quantitative data and qualitative data were interpreted using exploratory sequential method. The study group consisted of 20 students who took numerical analysis courses in the engineering faculty in a private university in Northern Cyprus. The data of the study were collected by means of an interview form consisting of common question problem solutions and open-ended questions that were asked to these students. The problem-solving processes of the students participating in the research were evaluated along with the solutions they derived for common problems and were analysed on the basis of the four different dimensions mentioned above. As a result of these analyses, it was concluded that the students understood the problems at an intermediate level and did not have sufficient infrastructure in the strategy determination, provision, demonstration and expression dimensions, and the majority could not write problems. In addition, in the light of the findings obtained from the interview form, it was emphasised that the numerical analysis course should be supported in the laboratory.
 Keywords: Numerical analysis, mathematics, problem-solving process, engineering students.

УПРАВЛІННЯ САМОСТІЙНОЮ РОБОТОЮ МАЙБУТНІХ ВЧИТЕЛІВ В ПРОЦЕСІ НАВЧАННЯ ІНФОРМАТИКИ З ВИКОРИСТАННЯМ СИСТЕМ КОМП’ЮТЕРНОЇ МАТЕМАТИКИ

Самостійна робота студентів є основною та однією з важливих форм організації освітнього процесу у вищій школі, а тому й потребує особливої уваги. З іншого боку, підготовка фахівця, здатного до самостійного професійного самовдосконалення впродовж життя є вимогою сучасного швидкозмінного інформатизованого суспільства. Одним з методичних напрямів вивчення самостійної роботи студентів є дослідження питань управління нею. Адже характерними ознаками самостійної роботи студентів є їх самостійність за умови опосередкованого спрямування, оперативного та систематичного контролю, навчально-методичного супроводу з боку викладача. Стаття присвячена дослідженню дидактичних можливостей використання програмних засобів комп’ютерної математики стосовно управління самостійною роботою майбутніх вчителів в процесі навчання інформатики, зокрема чисельних методів. Для аналізу було обрано наступні безкоштовні програмні засоби, що отримали високу науково-методичну популярність, як серед вітчизняних так і закордонних дослідників: програми-розв’язувачі Gran1, Gran-2D; обчислювальна система знань Wolfram|Alpha, хмаро орієнтована система комп’ютерної математики SageMathCloud. Окрім того, розглянуто використання мови програмуванням Python, що вважається простою і легкою для навчання та містить потужні засоби для виконання наукових та інженерних розрахунків. Подано основні напрямки застосування в курсі чисельних методів розглянутих програмних засобів, а також методи, що покладені в основу виконання деяких команд програмних засобів. Особливості управління самостійною роботою студентів розглянуто на прикладі навчання курсу чисельних методів математики, який відрізняється сильними міжпредметними зв’язками, а також навчання якого сприяє розвитку творчо-дослідницьких умінь майбутніх вчителів. Наскрізно розглянуто приклад розв’язування задачі інтерполювання функції, як одного з найбільш розповсюджених методів для отримання поліноміальних наближень в курсі чисельних методів. Запропоновані програмні засоби можуть використовуватись на різних етапах управління самостійною роботою майбутніх вчителів в процесі навчання чисельних методів.

面向工科研究生的数值分析课程教学设计案例探索

面向工科研究生的数值分析课程教学设计案例探索

Engaging non-majors in MATLAB programming through a flipped classroom approach

ABSTRACTBackground and Context: Flipped classrooms are becoming more widely adopted across engineering higher education contexts. In degree programs where enrollment is increasing and undergraduate curricula are packed with content, pedagogies that allow more time for actively participate in classroom activities are being highly sought after to aid in increases in retention and persistence.Objective: The following study analyzed the impact of a flipped classroom intervention in a numerical analysis course teaching MATLAB programming taught to students not majoring in a computer-related engineering discipline.Method: Performance in the flipped classroom was compared to historical performance in the traditional classroom. Participants’ intrinsic motivation was also investigated to try to understand the difference seen between semester offerings of the course.Findings: Student performance improved among students participating in the flipped classroom approach. Intrinsic motivation did not impact performance in the flipped classroom.Implications: Our work provides further evidence of the efficacy of flipped classroom approaches for teaching introductory programming. This work also highlights the importance of considering incoming knowledge and maturity when selecting flipped classrooms for educational contexts.

PENERAPAN MEDIA SIMULASI MATLAB BERBASIS INTERACTIVE CONCEPTUAL UNTUK MENINGKATAN PEMAHAMAN KONSEP MAHASISWA

This study aims to know the extent to which the use of Matlab simulation media based conceptual Interactive can overcome the difficulties of students in understanding the Linear Equation System in the course of Numerical Analysis and improve student understanding. This study is a classroom action research, which is carried out in two cycles. Research subjects as many as 25 students grade 7C Prodi Mathematics Education University PGRI Madiun academic year 2016/2017. Research instruments are researchers, learning observation sheets, concept comprehension tests, student response questionnaires and interview guides. Based on the result of the research, it can be concluded that the application of Matlab simulation media based on conceptual interactive in learning can increase the students understanding on Linear Equation System. This is shown by the improvement of students' ability in understanding the system of linear equations in each cycle. The data is also supported by a positive response given by students in learning.

A First Experience of Flipped Classroom in Numerical Analysis

Flipped classroom is a pedagogical model that modifies the way that time is spent both in and outside the class, working with active learning, centered on the students. Do students accept flipping the classroom? What do they think about learning with recorded classes? These questions arise when the flipped model is to be implemented. A first experience of flipped classroom was carried out in a numerical analysis course at Facultad Regional San Nicolás, and students’ opinions about this experience were gathered, so as to determine the acceptability of this methodology. The experience showed a high degree of acceptance by the students with respect to the methodology of the flipped class. Students were satisfied with the videos presented, and in general they preferred to work on their own in class, with the assistance of the faculty.

Education

Euler's method appears early and often in the undergraduate curriculum, not only as a useful method for approximating solutions to differential equations, but also as a scaffold for introducing students to the notion of consistency and accuracy. Interestingly, most students walk away with the impression that Euler, while simple and of first order, is bulletproof in the sense that as long as one is willing to wait, choosing a very small timestep will lead to a reasonable approximation of the solution. Authors Corless and Jankowski douse that notion using a leaky bucket of water in this issue of the Education section. Specifically, they use the relatively mild ODE describing the flow of water from a hole in a bucket as a model problem to explore the method of modified equations. In “Variations on a Theme of Euler,” the authors use their model problem to reintroduce readers to the residual and, from there, the concept of optimal backward error and the method of modified equations for the explicit (forward) Euler and implicit (backward) Euler methods. Using the method of modified equations, one can recast the iterative equation to show that the numerical scheme more closely approximates a modified equation that is different from the original governing equation on which the model is based. In other words, one can think of errors arising from numerical integration as errors that emerge in the modeling process. This article is a perfect module for an undergraduate modeling or numerical analysis course. Since a solid course in either area usually involves both subjects, this article is well aimed and suitable for an audience with broad interests.

Using the Web as Part of the Classroom in Numerical Analysis Courses

Virtual environments are highlighted as a good option to be used as part of the classes and to extend classes outside the classroom. The research group GIE at Facultad Regional San Nicolás from the Universidad Tecnológica Nacional of Argentina, has designed and published several websites of issues of different subjects. The use of these tools in the learning process allows students to become more active, autonomous and reflective. This fact generates adequate conditions for students to understand the basic ideas presented. The aim of this paper is to present the different sites designed for Numerical Analysis Courses and the impact they had on the subject during 2014.