Uffon’s Needle Problem: An Exciting Application of Many Mathematical Concepts

1974 ◽  
Vol 67 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Lee L. Schroeder
Keyword(s):  
Probability Theory ◽  
Definite Integral ◽  
Integral Calculus ◽  
Mathematical Concepts ◽  
Calculus I ◽  
Exciting Application

While teaching integral calculus, I have often looked for interesting applications of the calculus that are relatively easy for the students to master but yet not trite. One such problem is known as ‘Buffon’s Needle Problem.” A consideration of this problem involves the solution of a simple definite integral and requires a basic introduction to probability theory. Let’s consider the “Needle Problem.”

A Definite Integral in Probability Theory

SIAM Review ◽  
1996 ◽  
Vol 38 (3) ◽  
pp. 514-515
Author(s):  
M. Aslam Chaudhry ◽  
Asghar Qadir ◽  
M. Rafique ◽  
S. M. Zubair
Keyword(s):  
Probability Theory ◽  
Definite Integral

scholarly journals A Teaching Experience through the use of Tasks: Limits and possibilities for learning Mathematics in a university context

2020 ◽  
Vol 22 (2) ◽  
Author(s):  
Alessandro Ribeiro ◽  
Juliana Paulin
Keyword(s):  
Teaching And Learning ◽  
Teaching Practice ◽  
Teaching Experience ◽  
Analysis Data ◽  
University Teaching ◽  
Mathematical Tasks ◽  
Integral Calculus ◽  
Mathematical Concepts ◽  
Fundamental Theorem Of Calculus ◽  
The University

Context: Rethinking mathematics teaching practices in a university context is an emerging research theme. Objectives: In this article, we aim to discuss the limits and possibilities of using mathematical tasks in the teaching and learning processes of the concepts of Derivative, Integral and the Fundamental Theorem of Calculus. Design: The study is based on a qualitative-interpretative perspective of research, with methodological procedures inspired by a Design-Based Research. Environment and participants: The research was developed with students attending a Functions of a Variable class in a public university in the state of São Paulo. Data collection and analysis: Data were collected through mathematical tasks on Differential and Integral Calculus solved by students. The protocols produced were analysed, pointing out the main aspects identified, which led us to organize categories of analysis and dimensions (i) knowledges mobilized and developed by students in relation to mathematical concepts; (ii) main errors and difficulties presented by students in the development of tasks; (iii) limits and possibilities of the practice of exploratory teaching in the university context. Results: The results reveal aspects that characterize a process of resignifying the mathematical concepts discussed with the students and a deepening of their knowledge about the concepts of the DIC. Conclusions: As future notes, we suggest rethinking university teaching practice, since the study indicated possibilities and potentialities of the use of exploratory tasks in the teaching of Differential and Integral Calculus.

scholarly journals Uma proposta de articulação entre teoria e prática no estudo da integral definida

2020 ◽  
Vol 9 (2) ◽  
Author(s):  
JANINE FREITAS MOTA ◽  
CELINA APARECIDA ALMEIDA PEREIRA ABAR
Keyword(s):  
Teaching And Learning ◽  
Mathematical Thinking ◽  
Mathematics Learning ◽  
Definite Integral ◽  
Advanced Mathematical Thinking ◽  
Practical Application ◽  
Integral Calculus ◽  
Mathematical Programs ◽  
Strategy Planning

ResumoEste trabalho é um recorte de uma tese de doutorado, em desenvolvimento, cujo objetivo é de desenvolver uma alternativa pedagógica e tecnologógica, que contemple aspectos do Pensamento Matemático Avançado, bem como, aplicações, aprimoramento do conhecimento e do significado da Integral Definida, em contextos intramatemáticos e extramatemáticos, em cursos de Matemática. Neste recorte, apresentamos o percurso de nossa investigação, abordando, inicialmente, os aspectos relacionados às dificuldades na aprendizagem do Cálculo Integral, em particular, da Integral Definida. Ainda, destacamos que a exploração de aplicações desse conteúdo, em distintas áreas, é considerada como uma possibilidade para o seu ensino e para sua aprendizagem. Aspectos teóricos, metodológicos e tecnológicos são apresentados, como orientadores do planejamento da estratégia pedagógica. É destacado um exemplo de aplicação teórico-prática, na perspectiva de melhorias na qualidade do ensino e da aprendizagem desse tópico. Palavras-chave: Integral Definida; Conexões Intramatemáticas; Conexões Extramatemáticas; Educação Matemática no Ensino Superior.AbstractThis 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.   

scholarly journals Analysis and evaluation of students' results in the subject of Mathematics for Technicians

2020 ◽  
Vol 6 (2) ◽  
pp. 80-87
Author(s):  
Vladimír Matušek ◽  
Eva Matušeková
Keyword(s):  
Definite Integral ◽  
Test Results ◽  
Integral Calculus ◽  
Main Hypothesis ◽  
Analysis And Evaluation ◽  
The Subject

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.

scholarly journals Empleo de simulaciones dinámicas en matlab como parte del proceso de enseñanza-aprendizaje de las matemáticas con aplicación al cálculo diferencial e integral.

2018 ◽  
Vol 5 (1) ◽  
pp. 36-41
Author(s):  
Miguel Lema Carrera
Keyword(s):  
Learning Process ◽  
Definite Integral ◽  
Integral Calculus ◽  
Dynamic Simulations ◽  
Student Community ◽  
Basic Ideas ◽  
Definition Of ◽  
Teaching Learning ◽  
Disk Method ◽  
Geometric Definition

     La matemática en todos los tiempos ha tenido como principal fuente de inspiración la visualización, jugando un papel importante en el desarrollo de conceptos, nociones e ideas básicas del cálculo diferencial e integral. El presente trabajo proporciona herramientas y métodos básicos de uso relativamente sencillo, desarrollados en el paquete computacional MATLAB, trabajando temas como la definición geométrica de derivada, la integral definida y cálculo de volúmenes de revolución utilizando el método de discos, que permite obtener resultados muy poderosos en simulaciones dinámicas “animadas” que sirvan de soporte y recurso didáctico facilitador en el proceso de enseñanza-aprendizaje del cálculo. Modificando y renovando en una primera instancia la forma tradicional de enseñanza de esta asignatura en los primeros años del ciclo básico universitario en esta institución y porque no del país, además, se espera que este trabajo, permita desterrar el paradigma entorno a la comunidad estudiantil, que ha relacionado al cálculo matemático con una idea pura y completamente algebraizada, estática y memorística. ABSTRACT The mathematics of all time has had as the main source of inspiration the visualization, playing an important role in the development of concepts, notions and basic ideas of the differential and integral calculus. The present work provides tools and basic methods of use relatively simple, developed in the computational package Matlab, working topics such as the geometric definition of derivative, the definite integral and calculation of volumes of revolution using the disk method, which allows to obtain very powerful results in "animated" dynamic simulations that serve as support and facilitating didactic resource in the teaching-learning process of calculus. Modifying and renewing in the first instance the traditional way of teaching this subject in the first years of the basic university cycle in this institution and why not in the country, in addition, it is expected that this work, to banish the paradigm around the student community, that has related to the calculus with a pure and completely algebraic, static and rote idea.

Mathematics and Fiction: Thomas Pynchon, Gravity’S Rainbow

2018 ◽  
pp. 126-156
Author(s):  
Nina Engelhardt
Keyword(s):  
Probability Theory ◽  
Thomas Pynchon ◽  
Decisive Factor ◽  
Mathematical Concepts ◽  
Gravity’S Rainbow ◽  
Gravity's Rainbow ◽  
New Perspective ◽  
Ontological Uncertainty

Chapter 4 sets the engagement with modernist mathematics into broader context when examining the rise, fall and transformation of Enlightenment thinking and science in Pynchon’s Gravity’s Rainbow. This chapter also zeroes in on a topic that runs through all chapters: the interrelations of mathematics and fiction. The analysis focuses on illustrations of fictionality regarding the mathematical concepts of infinitesimals, the calculus, and probability theory and their philosophical and ethical consequences. The examination of interdependent ‘real’ and ‘fictional’ elements in mathematics provides a new perspective on Brian McHale’s identification of ontological uncertainty as the novel’s definitive postmodernist trait: the chapter shows that the novel’s renegotiation of mathematics is a decisive factor in its introduction of postmodernist features. As the title ‘Gravity’s Rainbow’ with its combination of a scientific and a poetical image implies, Pynchon’s novel suggests that the shared use of fictional concepts both in mathematics and in literature connects the seemingly opposed realms.

scholarly journals Epistemic Complexity of the Mathematical Object “Integral”

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2453
Author(s):  
Enrique Mateus-Nieves ◽  
Vicenç Font Moll
Keyword(s):  
Mathematics Education ◽  
Paradigm Shift ◽  
Mathematical Object ◽  
Positive Influence ◽  
Definite Integral ◽  
Theoretical Construct ◽  
Integral Calculus ◽  
Mathematical Content ◽  
Mathematical Competencies ◽  
History Of

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.

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