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This study explored the effective application of Maple software to reduce student teachers’ errors in Integral Calculus at Evangelical Presbyterian College of Education, Volta Region – Ghana. The study employed the quasi-experimental non-equivalent group design. Convenience and simple random sampling techniques were employed to obtain a sample of 80 students, which consisted of 40 students in the control group and 40 in the experimental group. Teacher-made Pre, -Post-Calculus Achievement Tests (CAT), and questionnaires were used to collect quantitative and qualitative data respectively. Descriptive, Independent samples t-test and paired samples t-test were used in analyzing the data. Descriptive error analysis revealed that students committed many conceptual, procedural and technical errors when solving Integral Calculus tasks. The results also indicated that there was a statistically significant difference between students of the experimental group exposed to the use of Maple software in learning integral calculus to the control group exposed to traditional methods. The researchers recommend Maple assisted instruction in the teaching and learning of Integral Calculus and also the need to employ a blended teaching approach, in which Maple software is used simultaneously with traditional teaching strategy.

In this article, some properties of neutrosophic numbers as well as neutrosophic functions are presented to develop neutrosophic integral calculus. This is the first time, the concept of neutrosophic Riemann integration has been introduced. The $$(\alpha ,\beta ,\gamma )$$ -level sets of the neutrosophic Riemann integral are presented. Some numerical examples have been given to verify the concept of neutrosophic Riemann integration, and the integration has been examined in terms of tables and figures. In one of the examples, it has been presented that how one can use trapezoidal rule to calculate neutrosophic Riemann integral and the numerical approximate result shows that here also, the numerical integral gives a neutrosophic number.

This research analyzes the quality and results of a criterial and large-scale comprehensive calculus test in the school cycles between 2014 and 2019 to a total of 5367 second-semester students of the engineering careers of a mexican public university. With the results obtained it is observed that the criterial examination of integral calculus is a valid, reliable test with satisfactory power of discrimination and with a majority load of procedural reagents. The results of the research show that the greatest difficulty for students is focused on integration techniques, especially when trigonometric functions are involved. It was also found that the success of students in the ECCI is due to the ability to resolve integrals with hyperbolic, exponential and logarithmic functions, as well as the proper application of the fundamental theorem of calculus and the technique of variable change.

This study intended to determine the effect of Group Integration Method on academic performance in Integral Calculus at the College of Engineering University of Eastern Philippines. It employed a quasi-experimental design that involved two sections of second year BSEE students enrolled in Integral Calculus in the College of Engineering. The students in these sections were comparable since their pre-test on a 1-1 scoring system were not significantly different using the t-test of independent sample at 0.05 level of significance in Integral Calculus ability of the students between the control and experimental group. The average pre-test scores per class were determined to set the experimental and control group. The former bears the lower average scores, which needs more attention through cooperative learning among group members while the latter bears the higher ones since the students can handle individual learning. The individual pre-test scores were used in the categorization of the students in both control and experimental groups. The scores were ranked and categorized into three (3): the High Performer (HP), the Average Performer (AP) and the low Performer (LP). The result showed that most of the respondents for both groups are males, first timers and average students with respect to differential calculus academic performance. As to the age bracket, the data shows majority of the students were (18-20) years old. The outcomes of the study further showed that the respondents exposed to the Integration Method and Traditional Method had the overall interpretation of failed rating at the start of the study. As the study progressed 8 respondents from the Group Integration and 12 respondents were added in Traditional method above failing stage considering the post-test administered to both groups. The T-test at 0.05 level of significance stressed out that post-test and combined tests for pre-test/post-test significantly differs for both Group Integration and Traditional methods of instruction. Finally, as to the academic performance, Group Integration is better than Traditional Methods of Instruction. This implies that the intervention aided the students to boost their skills in Integral calculus under the experimental group.

Previous studies have suggested that problem-posing activities could be used to improve the teaching, learning, and assessment of mathematics. The purpose of this study is to explore undergraduate ...

Integral calculus is a branch of mathematics concerned with the determination, properties, and application of integrals. It is predominantly used in technical applications. Technical engineers, statics, physicists and others use it in their calculations on practice. There was a requirement from practice for technical universities to include integral calculus in their curricula. The subject Mathematics for Technicians is taught at the Department of Mathematics, the Slovak University of Agriculture in Nitra. The content of this subject is to teach its students to calculate indefinite and definite integral. Our research analysed students' knowledge in counting indefinite and definite integral. We used the methodology of evaluation and comparison of test results taken in the 8th week of the term and at the end of the term. The main hypothesis saying that the results of students’ tests taken at the end of the term are better that those taken in the mid- term has confirmed to be correct.

We present elementary proofs of the inequalities of Huygens, Cusa, and Wilker, using visual proofs of several simpler inequalities and integral calculus. [Mismatch]

If f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that ∫ a c f ( x ) d x + ( c - a ) g ( c ) = ∫ c b g ( x ) d x + ( b - c ) f ( c ) . \int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right). In this paper, we study the approaching of the point c towards a, when b approaches a.

No presente estudo, tem-se como objetivo analisar as contribuições da utilização da metodologia ativa Peer Instruction no ensino e na aprendizagem de tópicos trabalhados na disciplina de Cálculo A, na qual se aborda, especificamente, o conteúdo de integração de taxas de variação e integrais definidas. Na pesquisa, desenvolvida com 41 estudantes do curso de Engenharia de Produção da Universidade Federal de Santa Maria, utilizou-se a estratégia de ensino inspirada na metodologia Peer Instruction, em que os alunos, após leitura prévia do assunto fora da sala de aula, responderam, em sala de aula, a 10 questões conceituais disponibilizadas no Moodle, discutindo suas respostas com os colegas. A partir dos dados obtidos por meio das discussões realizadas nos grupos, das respostas dos alunos às questões e à pesquisa de opinião, constatou-se o envolvimento e a satisfação dos alunos na realização das atividades e a eficiência do uso da metodologia na aprendizagem dos conceitos tratados.

This study aimed to determine the effectiveness of applying video-assisted flipped classroom learning models in integral calculus courses. The effectiveness of learning in this study was measured based on the achievement of learning outcomes, student activities during learning, and student responses. This study was a pre-experimental study with a purposive sampling technique. The data collection method used two tests, observation sheets, and two questionnaires. Data analysis techniques were conducted by using statistical and inferential descriptive analysis. The results of the analysis showed: (1) The score of student learning outcomes has increased reaching a minimum completeness score of 70 with an average score of student learning outcomes in posttest 1 and posttest 2 of 75.28 and 78.09, (2) The average student activity is in the very active category, and (3) The average percentage of student response questionnaires on two response questionnaires, namely student response to the implementation of mathematics learning is amounted to 88%, and student response to instructional video media is amounted to84%. In general, it can be concluded that the video-assisted flipped classroom learning model is effective to be applied in mathematics learning.

The difficulty of students in understanding the material in the integral calculus print book is one factor that makes student learning outcomes low. The difference in students' abilities is one of the considerations in developing process. This study aims to develop integral calculus teaching material using the aptitude treatment interaction (ATI) approach assisted by Macromedia flash. This research belongs to the development research. The instruments used in this study were questionnaires and tests. Indicators of the results of this study are teaching materials declared worthy of use and students who get optimal learning outcomes> 75%. The stages in the study consisted of ten stages that began with discovering potentials and problems to mass production. The results of the study showed that the integral calculus teaching material made was feasible to use. As many as 79% of students achieve optimal learning outcomes.

In a concise description of mathematical methods, Henri Lebesgue underlined the importance of definitions and axioms (see [47]).

This final chapter is in effect an Appendix to the book, dealing with mathematical topics that would have been too much of a detour in the text, and would have interrupted a physical development. The main feature is the Abel–Plana formula, which provides an exact correspondence between a sum and certain integrals. The Abel–Plana formula is especially useful in close approach of the sphere or cylinder pairs, when the sums we need converge slowly. There is also a section devoted to proving the mathematical equivalence of apparently different capacitance expressions. An Appendix discusses expansions in powers of a parameter that is small at close approach, and in particular two remarkable series arising in the polarizabilities of a pair of cylinders. These series each have ten successive zero coefficients.

In the usual initial teacher training, the teaching of Integral Calculus focuses on emulating school mathematics as a result of solving routine exercises, leaving in the second place the mathematical knowledge that emerges in the student. Therefore, we problematize the teaching of the integral defined to confront the permanent procedures in the daily life of people with the terminal objects that are not in their daily lives. With the Socioepistemological Theory of Educational Mathematics, a design of school activity and semi-structured interviews were staged with a community of third-year mathematics teaching students from Colombia. It was identified that the emulation of mathematical procedures favors adherence to school mathematical discourse; which prevents planning and carrying out teaching from the autonomous arguments of the learner. In contrast, the category of accumulation constituted in people's daily lives is proposed as a component of a source of meaning that generates a disciplinary identity for the teaching student.

Kinematics is the study of the motion of mechanical points, bodies and systems without consideration of their associated physical properties and the forces acting on them. The study is often referred to as the geometry of motion, and it models these motions mathematically using algebra. The systems in kinematics are modeled to calculate such things as speeds and ratios. Kinematics is very useful in the conceptual design of mechanical systems. Initial geometries and velocities of bodies are a part of the model. While kinematics can help determine whether a design is theoretically possible, there are more complexities when designing something for the real world. Without consideration of materials, and the forces acting upon them, many theoretically possible designs would be prone to failure. Kinetics, in contrast to kinematics, does consider physical properties such as the mass of the bodies or the forces driving them. Kinetics is logically deduced from kinematics by way of algebraic calculation of physical properties and forces. Kinetics takes into account physical forces and properties including material properties, like mass rigidity, and tensile or compressive strength.

In this paper, we establish general differential summation formulas for integration by parts (IBP), more importantly a powerful tool that promotes exploration and creativity.

Common student errors and misconceptions can be addressed through the method of classroom voting, in which the instructor presents a multiple-choice question to the class, and after a few minutes for consideration and small-group discussion, each student votes on the correct answer, using a clicker or a phone. If a large number of students have voted for one particular incorrect answer, the instructor can recognize and address the issue. In order to identify multiple-choice questions that are especially effective at provoking common errors and misconceptions, we recorded the percentages of students voting for each option on each question used in 25 sections of integral calculus, taught by 7 instructors, at 4 institutions, over the course of 12 years, on a collection of 172 questions. We restricted our analysis to the 115 questions which were voted on by at least 5 different classes. We present the six questions that caused the largest percentages of students to vote for a particular incorrect answer, discuss how we used these questions in the classroom, and examine the common features of these questions. Further, we look for correlations between question characteristics and the mean percentage of students voting for common errors on these questions, and we find that questions based on general cases have higher percentages of students voting for common errors.

In this paper we consider two continuous functions f, g : [a, b] → ℝ and we study for these ones, under which circumstances the intermediate point function is four order di erentiable at the point x = a and we calculate its derivative.

A simple in-class demonstration of integral Calculus for first-time students is described for straightforward whole number area magnitudes, for ease of understanding. Following the Second Fundament...

Orientador: Maria Lúcia Bontorim de Queiroz Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica Resumo: O cálculo de áreas e volumes foi um tema que desafiou os matemáticos desde os primeiros registros encontrados. Desde as formas rudimentares e práticas sem embasamento teóricos dos babilônicos e egípcios até o desenvolvimento do Cálculo Diferencial e Integral, inúmeros matemáticos contribuíram para o desenvolvimento e formalização dos conceitos e maneiras de cálculo de áreas e volumes. Neste trabalho apresentaremos conceitos formais e também aspectos históricos no desenvolvimento do cálculo de áreas e volumes, através das contribuições dos matemáticos ao longo da história da Matemática. Em especial, analisaremos os trabalhos de Cavalieri, que utilizou a ideia dos indivisíveis, ampliando conceitos utilizados pelo método da exaustão e Pappus, que contribuiu formalizando o cálculo de áreas e volumes dos sólidos de revolução através do centro de gravidade. Também destacamos a contribuição das ideias de Newton e Leibniz no desenvolvimento do Cálculo Diferencial e Integral, que permitiu significativo avanço no cálculo de áreas e volumes. Finalizamos este trabalho com algumas aplicações didáticas, visando um melhor entendimento dos alunos sobre este tema Abstract: The volume and area calculation were a theme that challenged mathematicians since the first registers that they found out. From rudimental shapes and practices without theoretical support of Babylonians and Egyptians until Differential and Integral Calculus development, a vast number of mathematicians contributed for development and systematization of concepts and forms of areas and volume calculation. In this paper has as an objective to show formal concepts and historical aspects about area and volume calculation through mathematicians¿ contributions over the Mathematic history. Particularly, it will be analyzed Cavalieri¿ projects who used the indivisible idea. He extended concepts that used exhaustion method and Pappus that contributed about formalization of area and volume calculation of solids of revolution through gravity center. Other aspect pointed in this paper are the contributions of Newton and Leibniz about Differential and Integral Calculus development that to make possible a meaningful advance about area and volume calculation. To finish this paper, some didactic applications it will be presented with the aim to make students have a clear understanding about the theme Mestrado Matemática em Rede Nacional Mestra em Matemática em Rede Nacional CAPES

The components of teachers’ knowledge have been widely conceptualized since the advent of Shulman’s teacher knowledge categorization in mid-1980s. It was widely acknowledged that the effectiveness of teachers and the success of their students rely on the strength of their subject matter content knowledge. Based on this perspective, many teacher knowledge models were developed. This paper appraised the Specialized Content Knowledge (SCK) with particular reference to Calculus. SCK is a sub-domain of Mathematical Knowledge for Teaching (MKT). MKT is a required skill and expertise of mathematics that is particularly essential to a successful instruction. SCK and its relevance were examined and reported with specific focus to the teaching of Calculus at Jubail University College., {"references":["Ball, D. l., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389 – 407.","Ball, D. L., & Bass, H. (2002). Toward a practice-based theory of mathematical Knowledge for teaching. In Proceedings of the 2002 annual meeting of the Canadian Mathematics Education Study Group, 3 – 14.","Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical Knowledge foe teaching on students' achievement. American Educational Research Journal. 42(2), 371 – 406.","Larson, R., (2018). Algebra and Trigonometry with CalcChat and CalcView, 10th Eidtion,Boston, MA, Cengage Learning.","Ma, L., (1999). Knowing and Teaching Elementary School Mathematics. Mahwah, NJ, Lawrence Erlbaum Associates.","NMAP (2008). Foundation for success. The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education.","Phelps, G., \\& Howell, H. (2016). Assessing Mathematical Knowledge for Teaching: The Role of Teaching Context. The Mathematics Enthusiast. 13(1 & 2), 52 – 70.","Shulman, L. S. (1986). Those who understand, knowledge growth in teaching, Educational Researcher, 15(2), 4 – 14.","Thomas, G. B., (2016). Thomas's Calculus Early Transcendentals, 13th Edition. Essex, England. Pearson Educational Limited.","Three P(3P) – Learning (2021). How to Factor the Real World into the Mathematics Teaching Equation. 3P Learning – The teaching moments that inspire learning, https://www.mathletics.com/blog/educators/how-to-factor-the-real-world-into-the-mathematics-teaching-equation/ ((retrieved on 10-01-2021)"]}

This study aim ed to 1) t o find out the effectiveness of the integration technique module in integral calculus learning, and 2) t o find out the increase in the mathematical creative thinking ability after implementing the integration technique module in integral calculus learning. The development of module was conducted by using ADDIE model with five stages : Analysis, Design, Development, Implementation, and Evaluation. The subjects in this study were 16 students in the 3rd semester of Mathematics Education Study Program, the Faculty of Teacher Training and Education, Universitas Riau. The research instruments were an expert validation questionnaire, a student response questionnaire, and a mathematical creative thinking ability test. The data obtained were analyzed by using quantitative and qualitative descriptive analysis methods. Based on the results of the validity test, the integration technique module was declared valid and suitable for students to use in the integral calculus course. Judging from the results of the module practicality test, the average percentage of practicality values was 77.8125 and was in the practical category. Referring to the results of the module effectiveness test, the n-gain average of students' mathematical creative thinking abilities wa s 0.64 and in the medium category. Because the integrated engineering module met the criteria of validity, practicality, and effectiveness, it could be stated that the developed integration technique module improve d students' mathematical creative thinking skills in the integral calculus course.

Contrary to common understanding, we prove that the LCOE (Levelized Cost Of Energy) definition used by many government agencies around the world is mistaken. We derive the correct definition. The paper shows how a fundamental theorem of integral calculus can help to solve a practical problem that has been misinterpreted for many years.

Due to the confinement by Covid-19, teachers were forced to teach at a distance, which necessarily leads to a change in their teaching practices. Using Trouche’s theoretical instrumental orchestration framework, this case study shows that this framework continues to be valid in this type of teaching, it presents the similarities and differences found in these instrumental orchestrations in the case of distance learning, using a teacher’s teaching practice 12th year math class in the chapter on antiderivative and integral calculus. A new type of orchestration has been found.

This study provides an example of a mathematical modelling activity that utilizes scientific reasoning to support the learning of mathematics concepts. The method of merging mathematics concepts with scientific reasoning was developed using research on and recommendations about designing effective exploratory STEM modelling activities. A calculus topic, the Fundamental Theorem of Calculus (FTC), was converted into a modelling activity. The FTC was selected due to its significance in integral calculus and calls for designing activities that would develop students’ covariational reasoning crucial to understand the theorem. A group of 21 high school students participated in this study. The students’ responses showed positive effects of this activity in understanding the FTC. Suggestions for further studies conclude this paper.

Resumo Este trabalho e um recorte de uma tese de doutorado, em desenvolvimento, cujo objetivo e de desenvolver uma alternativa pedagogica e tecnologogica, que contemple aspectos do Pensamento Matematico Avancado, bem como, aplicacoes, aprimoramento do conhecimento e do significado da Integral Definida, em contextos intramatematicos e extramatematicos, em cursos de Matematica. Neste recorte, apresentamos o percurso de nossa investigacao, abordando, inicialmente, os aspectos relacionados as dificuldades na aprendizagem do Calculo Integral, em particular, da Integral Definida. Ainda, destacamos que a exploracao de aplicacoes desse conteudo, em distintas areas, e considerada como uma possibilidade para o seu ensino e para sua aprendizagem. Aspectos teoricos, metodologicos e tecnologicos sao apresentados, como orientadores do planejamento da estrategia pedagogica. E destacado um exemplo de aplicacao teorico-pratica, na perspectiva de melhorias na qualidade do ensino e da aprendizagem desse topic o . Palavras-chave : Integral Definida; Conexoes Intramatematicas; Conexoes Extramatematicas; Educacao Matematica no Ensino Superior . Abstract This work is an doctoral thesis excerpt, under development, whose objective is to construct a pedagogical and technological alternative that contemplates Advanced Mathematical Thinking aspects and applications, improvement of knowledge and the meaning of the Definite Integral, in intramathematical and extramathematical contexts, inside the mathematical programs. In this excerpt, it is presented the research way, initially approaching the aspects related to the learning disabilities in Integral Calculus, particularly Definite Integral. Still, it is emphasized that the exploitation of applications of this content, in different areas, is considered as a possibility for its teaching and learning. Theoretical, methodological and technological aspects are presented as guide of pedagogical strategy planning. An example of theoretical-practical application is highlighted, with a view to improving the quality of teaching and learning on this topic . Keywords: Definite Integral; Intramathematics Connections; Extramathematcal Connections; Mathematics Learning in Higher Education .

The landscapes of many elementary, middle, and high school math classrooms have undergone major transformations over the last half-century, moving from drill-and-skill work to more conceptual reasoning and hands-on manipulative work. However, if you look at a college level calculus class you are likely to find the main difference is the professor now has a whiteboard marker in hand rather than a piece of chalk. It is possible that some student work may be done on the computer, but much of it contains the same type of repetitive skill building problems. This should seem strange given the advancements in technology that allow more freedom than ever to build connections between different representations of a concept. Several class activities have been developed using a combination of approaches, depending on the topic. Topics covered in the activities include Riemann Sums, Accumulation, Center of Mass, Volumes of Revolution (Discs, Washers, and Shells), and Volumes of Similar Cross-section. All activities use student note outlines that are either done in a whole group interactive-lecture approach, or in a group work inquiry-based approach. Some of the activities use interactive graphs designed on desmos.com and others use physical models that have been designed in OpenSCAD and 3D-printed for students to use in class. Tactile objects were developed because they should provide an advantage to students by enabling them to physically interact with the concepts being taught, deepening their involvement with the material, and providing more stimuli for the brain to encode the learning experience. Web-based activities were developed because the topics involved needed substantial changes in graphical representations (i.e. limits with Riemann Sums). Assessment techniques for each topic include online homework, exams, and online concept questions with an explanation response area. These concept questions are intended to measure students’ ability to use multiple representations in order to answer the question, and are not generally computational in nature. Students are also given surveys to rate the overall activities as well as finer grained survey questions to try and elicit student thoughts on certain aspects of the models, websites, and activity sheets. We will report on student responses to the activity surveys, looking for common themes in students’ thoughts toward specific attributes of the activities. We will also compare relevant exam question responses and online concept question results, including common themes present or absent in student reasoning.

Although integral is one of the important concepts in mathematics, most students have problems learning it. A well-designed lesson plan consisting generalization and its related activities can play an important role to overcome the students’ difficulties in the learning process. In this study, generalization means going from particular to something general by looking at common things and expanding their domains of validity. The main goal of this study is to investigate the rate of using generalization by mathematics instructors in teaching of integral concepts based on lesson plans. Thus, the instructors were interviewed on what generalization means to them and to find the reasons of emphasis on generalization or lack of it in their lesson plans. Findings revealed that most instructors did not emphasise the teaching of generalization in their lesson plans. Although most of the instructors expressed that generalization can be used to extend the theorems, the interview clarified that majority of the instructors did not know about the important applications of generalization in the teaching of integral. The results demonstrated that instructors have oppressed generalization in the integral teaching process.

La distribución normal es una de las más importantes en el proceso de enseñanza aprendizaje de la Bioestadística para la especialidad de Medicina. Sin embargo, la ausencia de un enfoque deductivo en la introducción al tópico, dificulta la comprensión de su origen. A partir de sistemas de distribuciones continuas de probabilidad, se propone una vía alternativa para presentarla, tanto en la forma general como en la estándar. Teniendo en cuenta los métodos analítico sintético y la revisión documental, se identifican los vacío epistémicos y las dificultades en el tratamiento del tema. Los objetivos esenciales consisten en deducir la expresión matemática de tal distribución y valorar la posibilidad de introducir los elementos cualitativos básicos del procedimiento usado, en el programa de Informática Médica II, enfatizando en las nociones del Cálculo Diferencial, en etapa previa a los elementos del Cálculo Integral. La conclusión fundamental se asocia a la contribución del enfoque sugerido, al rigor y a la armonía al conocimiento estadístico matemático en la disciplina, si bien se precisa de conceptos del Análisis Diferencial, los cuales favorecen la comprensión de las aplicaciones prácticas tanto en el ámbito de la carrera de Medicina como en otras áreas de investigación de las ciencias biomédicas.

Este artículo presenta los resultados de un proyecto de investigación desarrollado en la Facultad de Ingeniería y Ciencias Básicas de la Fundación Universitaria Los Libertadores (Bogotá, D.C.) cuyo objetivo consistió en describir y analizar los efectos de implementar una estrategia didáctica para la enseñanza de la noción de área bajo la curva, durante un curso de Cálculo Integral, a partir de la integración de teléfonos celulares como dispositivos que pueden apoyar el proceso de aprendizaje y no como artefactos nocivos o distractores y que responden a nuevas formas de reconocerse e integrarse a comunidades alternativas de aprendizaje. Se realizó un diseño experimental de cuatro grupos de Solomon con el propósito de determinar la influencia de una intervención mediada por la aplicación móvil “Calculadora Gráfica” de GeoGebra considerando también el efecto de una prueba de entrada. Los resultados encontrados señalan que los dos grupos que tuvieron la intervención mediada por la aplicación móvil obtuvieron mejor rendimiento en la prueba de salida que aquellos grupos que tuvieron una intervención con calculadora científica tradicional.

This paper provides a method that can be used to review the teaching, learning, and/or assessment of mathematics at either (or both of) the senior secondary and undergraduate levels. In this paper, how this method could be enacted is exemplified by considering the case of integral calculus. The method uses Revised Bloom’s Taxonomy (RBT) (Anderson et al., 2001) in conjunction with Efklides’s metacognition framework (Efklides, 2006, 2008) to design questions to address the different RBT cognitive processes and knowledge types. Using these two frameworks can help develop questions that target broader student thinking and a range of cognitive processes, including constructive ones, than traditional questions reach. In doing so, this method can be a starting point for Faculties seeking to reform their delivery and assessment of mathematics.

The Euler calculus -- an integral calculus based on Euler characteristic as a valuation on constructible functions -- is shown to be an incisive tool for answering questions about injectivity and invertibility of recent transforms based on persistent homology for shape characterization.

We implement Leibniz’s idea about the differential as the length of an infinitesimally small elementary interval (a monad) in a form satisfying modern standards of rigor. The concept of sequential differential introduced in this paper is shown to be in good alignment with the standard convention of the integral calculus. As an application of this concept we simplify and generalize the construction of the Perron–Stieltjes integral.

Many students do not have a deep understanding of the integral concept. This article defines what a deep understanding of the integral is in respect to integration involving one independent variable; briefly discusses factors which may inhibit such an understanding; and then describes the design of a mathematical resource for introducing students to the integral concept. The resource addresses a number of challenges when introducing the integral: (1) choosing an appropriate, intuitive context which gives meaning to the symbols in the integral expression; (2) aiding the transfer of the integral expression to different contexts via using the Riemann sum in an informal way so that students can see and interpret the rectangles which are inherent in this sum; and (3) the gradual formalizing of the Riemann sum and its linkage with the Fundamental Theorem of Calculus. The resource has been used over a number of years at this university amongst first-year undergraduate science and engineering students. Anecdotal evidence would suggest that the resource is beneficial.

The literature in mathematics education identifies a traditional formal mechanistic-type paradigm in Integral Calculus teaching which is focused on the content to be taught but not on how to teach it. Resorting to the history of the genesis of knowledge makes it possible to identify variables in the mathematical content of the curriculum that have a positive influence on the appropriation of the notions and procedures of calculus, enabling a particularised way of teaching. Objective: The objective of this research was to characterise the anthology of the integral seen from the epistemic complexity that composes it based on historiography. Design: The modelling of epistemic complexity for the definite integral was considered, based on the theoretical construct “epistemic configuration”. Analysis and results: Formalising this complexity revealed logical keys and epistemological elements in the process of the theoretical constitution that reflected epistemological ruptures which, in the organisation of the information, gave rise to three periods for the integral. The characterisation of this complexity and the connection of its components were used to design a process of teaching the integral that was applied to three groups of university students. The implementation showed that a paradigm shift in the teaching process is possible, allowing students to develop mathematical competencies.

In this paper, the knowledge dimension for Revised Bloom's taxonomy (RBT) is unpacked for integral calculus. As part of this work, the 11 subtypes of the knowledge dimension are introduced, and through document analysis of chapter 4 of the RBT handbook, these subtypes are defined. Then, by consulting materials frequently used for teaching integral calculus, each subtype is exemplified. The developed dimension may enable or enhance opportunities for dialogue between lecturers, teachers, and researchers about how to develop and align educational objectives, teaching activities, and assessments in integral calculus, or how metacognition and metacognitive knowledge could be used to support teaching and learning.

The idea to write this research paper is to introduce a new multidimensional graphical approach to visualize a large number of integrals behavior in different periods of time and spaces under the application of the Megasfinity Coordinate Space. The Megasfinity Coordinate Space is willing to mapping a large number of integrals in different dimensions and periods of time extensively in the same graphical space as a whole.

В статье рассматриваются методические аспекты обучения математике в высшей школе с использованием технологий для дистанционного обучения. Обращается внимание на неумение обучаемых переносить полученные знания, умения и навыки на решение практических задач. В процессе анализа традиционных методик обучения математике выявляется превалирование абстрактной подачи учебного материала по сравнению с образной подачей. В силу того, что у большинства обучаемых превалирует не абстрактно-теоретическое, а образное мышление, делается вывод о необходимости учета психо-физических особенностей обучаемых при выстраивании методики обучения математике. Среди новых подходов к обучению выделяется когнитивно-визуальный подход, который базируется на оптимальном использовании резервов визуального мышления обучаемых, предполагает интеграцию новых знаний с имеющимися. Показана реализация данного подхода в обучении высшей математике с использованием технологий дистанционного обучения. Предлагаются варианты методического наполнения приложений для пакетов «Первичная обработка статистических данных» и «Решение задач на рыночное равновесие». Акцентируется внимание на использовании анимации при объяснении базовых понятий математической статистики, интегрального исчисления. Показаны варианты организации двух режимов обучения: режим самообучения и режим оценки знаний. Прописано содержание смонтированного авторами развлекательно-познавательного видеофайла «О статистике в шутку и в серьез». В видеофильм включены изречения великих людей о статистике, диалоги героев образовательного комикса «Занимательная статистика», раскрывающие суть базовых понятий математической статистики. Как дополнительное средство мотивации к обучению, предлагаются ситуации выхода на решение практических задач. Видеофайл «О статистике в шутку и в серьез» содержит реальную историю, в которой показывается «работа» кривой Гаусса. В видеофайле «Сбор статистических данных» показывается процедура замера студентами веса, роста, давления, пульса для дальнейшей обработки студентами полученных статистических данных. Результаты могут быть применены при разработке курсов для дистанционного обучения математике.

Differintegral methods, namely those techniques using differential and integral operators on the same footing, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated formalism will be shown to be an effective means, allowing a link between umbral and operational methods. We merge these two points of view to get a new and efficient method to obtain integrals of special functions and the summation of the associated generating functions as well.

Resumo Este trabalho e um recorte de uma tese de doutorado em desenvolvimento e tem como tematica os processos de ensino e de aprendizagem da Integral Definida em cursos de Matematica e/ou Engenharias. Tem o objetivo de desenvolver uma alternativa pedagogica que contempla as aplicacoes, o aprimoramento do conhecimento e do significado da Integral Definida, em contextos intramatematicos e extramatematicos. Neste texto, de concepcao teorica, sao abordados aspectos relacionados as dificuldades na aprendizagem do Calculo Integral. Aspectos teoricos, metodologicos e tecnologicos sao apresentados como orientadores da estrategia pedagogica. Como resultado parcial, e apresentada uma proposta de articulacao entre a teoria e a pratica na perspectiva de melhorias na qualidade do ensino e aprendizagem da Integral Definida . Palavras-Chave: Integral Definida; Tecnologias Digitais no Ensino do Calculo; Educacao Matematica no Ensino Superior. Abstract This work is a part of a doctoral thesis in development and its theme is the teaching and learning processes of Integral Definite in Mathematics and / or Engineering courses. It has the objective of developing a pedagogical alternative that contemplates the applications, the improvement of the knowledge and the meaning of the Definite Integral, in intra mathematicals and extra mathematicals contexts. In this text, with theoretical approach, it is shown aspects related to the difficulties in learning the Integral Calculus. Theoretical, methodological and technological aspects are approached as a guide of the pedagogical strategy. Partly as a result, it is presented a purpose of articulation between theory and practice in the improvement prospects of teaching and learning quality of Definite Integral . Keywords: Definite Integral; Digital Technologies in Calculus Teaching; Mathematics Education in Higher Education .

Integral material is an important material to be mastered by mathematics education students. The purpose of this study is to produce integral material learning videos that are valid, practical, and effective for use in flipped classroom learning. This research is a development research consisting of 5 phases, namely: initial investigation; design; realization/ construction; testing, evaluation and revision; implementation. The instruments in this study were questionnaires filled with material and instructional media experts to measure validity, questionnaires filled by students using video to measure practicality, and test questions to measure the effectiveness of learning videos. The results of this study showed that the learning videos developed are valid with very good criteria, practical and effective with good criteria. The findings of this study also showed that the use of video learning on the flipped classroom can optimize the time of classroom learning to discuss material more broadly and deeply.

In the management of the regulation volumes of the water resource projected for a variety of benefits (hydroelectric uses, irrigation, drinking water, among others), it is essential to calculate the morphometric variables of the reservoirs to anticipate changes in their morphology and predict how these changes could affect projected achievement. In this document, taking into consideration the fundamental concepts of Integral Calculus, the development of an innovative methodology is presented to obtain the Cota-Volume and Cota-Area curves in reservoirs; the methodology was formulated in R programming language with the help of geographic information tools. A computational optimization was achieved for the processing of the variables of level, area and volume of a regulation body respect to the use of traditional methodologies. To validate the developed tool, the capacity curves of regulation volume of the Minas - San Francisco reservoir located in the south of the Republic of Ecuador were obtained. This reservoir was designed to dislodge its sediments with washing processes. This fact has motivated the continuous monitoring of the morphological conditions of the reservoir to plan maintenance processes due to the loss of volume and to the deposit of particles from its tributaries. In addition, an analysis based on wavelets curves was applied to the digital elevation models obtained from LiDAR techniques and bathymetric echo sounder to demonstrate the sedimentation processes that occur in this body of regulation.

This study aimed to describe students’ learning difficulties in integral calculus based on critical thinking skills. This study used descriptive exploratory methods with instruments: test, interview, documentation, and observation. Data was taken from 20 undergraduate students from the study program of mathematics education at Universitas Negeri Semarang. The results showed that: (1) there were 24 types of student difficulties identified in calculus lectures; these difficulties consist of 20 types of difficulties related to calculus material (types of difficulties number 1 to 20) and four types of difficulties that were not related to material (types of difficulties number 21 to 24), and (2) these types of difficulties were graded according to the level of student critical thinking skills. A higher level of critical thinking skills had fewer types of difficulties; on the other hand, a lower level of critical thinking skills had more types of difficulties and more fundamental. The fundamental types of difficulties based on the critical thinking skills aspect include understanding integral concepts and drawing graphics.

Some students have difficulty solving math problems in the integral calculus course, one of the materials for the volume of rotating objects. The purpose of this research is to describe the types of errors made by students of class F Mathematics Education FKIP Ahmad Dahlan University and their causes in solving the volume of rotating objects in integral calculus courses in the Polya stage. This research is a quantitative quantitative research. The research subjects are one class with a total of 20 students. For interviews, 3 students were taken with the criteria of high, medium and low abilities. Collecting research data using test material, the volume of rotating objects and student interviews about the factors that cause mistakes made. The steps for solving the problem consist of four stages, namely: understanding the problem, planning for completion, implementing the plan, and examining the solutions obtained. From the results of the research on student errors based on Polya's stages: 1) understanding the problem 55.89%, planning for completion of 50.00%, errors in implementing planning 32.36%, and errors in re-checking the process and results 44.12%; 2) the causes of students making mistakes in solving integral calculus questions on the volume of rotating objects, namely students lacking mastery in drawing function graphs, students lacking in basic concepts and volume formulas for rotating objects, and students unable to analyze the questions given.

The purpose of this article is to offer teaching ideas in the treatment of the definite integral concept and the Riemann sums in a technology-supported environment. Specifically, the article offers teaching ideas and activities for classroom for the numerical methods of approximating a definite integral via left- and right-hand Riemann sums, along with midpoint and trapezoidal Riemann sums. The activities demonstrate innovative and creative ways of integrating technology, in particular, GeoGebra dynamic software, into the pedagogy of college-level integral calculus. It also provides, among other things, interesting and original teaching ideas incorporating technology, and an evaluation of these activities by the students themselves who experienced these activities with the GeoGebra dynamic software.