Historically Speaking—: Identification of Napier's Inequalities

1970 ◽  
Vol 63 (1) ◽  
pp. 67-71
Author(s):  
Sidney G. Hacker

The discovery of logarithms by John Napier (1550-1617) is a well known facet in the history of mathematics. His singular accomplishment in defining the logarith mic function of a real variable by providing a numerical description of it, over a wide range of its argument, at small intervals and to several (decimal) places, antedated by many yeara the development of funda mental concepts which the modern stu dent regards as necessary to achieve even the same limited goals. Napier success fully bridged, solely in regard to this function, these lacunae in the mathematical knowledge of bis day. It has long been of interest to identify the concepts which he intuitively invoked. This is not done, it should be clearly said, with any idea of assigning to him some kind of priority for them, but merely in the interests of a elearer appreciation of the ingenuity he displayed and the power of his methods. Two inequalities that he obtained are the key to his numerical resolution of the problem and his consequent table of logarithms. The analytical identification of these inequalities appears to have been overlooked. Before exhibiting this identi-fication we shall speak of the fundamental role that these inequalities played. In the interests of intelligibility we first recollect a few familiar facts regarding Napier's formulation of the problem.

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 349
Author(s):  
María José Madrid ◽  
Alexander Maz-Machado ◽  
Fernando Almaraz-Menéndez ◽  
Carmen León-Mantero

Different studies consider the possibility of including history of mathematics in the classroom. However, its inclusion in the teaching and learning of mathematics depends on the conceptions of it that teachers have, among other factors. This study displays a comparative analysis between the opinions of primary education teachers-to-be and the opinions of mathematics teachers-to-be at secondary school and A-levels after the realization of an activity related to two historical or unusual multiplication methods. These trainee teachers were asked to identify the differences between these methods and the multiplication algorithm usually used in Spain. We collected these data and conducted an exploratory, descriptive and qualitative study. In order to analyse the information obtained, we used the technique content analysis. The answers given by these trainee teachers show their lack of knowledge about other multiplication methods and the various differences which they observed. These differences are mainly related to the structure of each method, the procedure and application of these methods and the mathematical processes carried out for each method. The comparison between the opinions of the teachers-to-be at different levels shows similarities but also some differences, probably due to the different mathematical knowledge they have.


PARADIGMA ◽  
2020 ◽  
pp. 212-239
Author(s):  
Ligia Arantes Sad ◽  
Claudia Alessandra C. de Araujo Lorenzoni

El texto discute el potencial y las contribuciones de la Historia de las Matemáticas en las prácticas de enseñanza de la Educación Matemática, ilustrada por dos episodios específicos de la práctica pedagógica de los autores. Toma como notas teóricas estudios como los de Ferreira, D'Ambrosio, Barbin, Jankivist y Vianna sobre los argumentos, implicaciones y sugerencias dirigidas al uso didáctico de la historia de las matemáticas. Los fundamentos de los autores se basan en un diálogo con la etnomatemática, entendiendo así las matemáticas escolares o las matemáticas, vistas en una forma occidental dominante, como una entre otras posibilidades de hacer y pensar matemáticamente. A lo largo del texto, la historia se destaca como un subsidio para la creación, tanto individual como colectiva, de explicaciones, relaciones de significados, objetos y significados que no se constituyeron hasta entonces. La creatividad, desde la perspectiva de Karwowski, Jankovska y Szwajkowski, y la investigación, en la línea de Ponte, se presentan como elementos relevantes en el proceso de enseñanza para estimular en cada estudiante una relación de construcción y apropiación del conocimiento matemático escolar de manera participativa. , interrogador y productor de nuevos conocimientos. Como resultado, señalamos: ser capaces de unir teorías e ideas científicas al analizar el potencial y las contribuciones de la Historia de las Matemáticas en las prácticas de enseñanza de la enseñanza de las matemáticas en la escuela, involucrando investigación y creatividad en metodologías y contextos híbridos de diferentes culturas.Palabras clave: Historia de las matemáticas. Educación matemática escolar. Juegos indígenas tradicionales. Investigación y creatividad.  HISTORY OF MATHEMATICS IN MATHEMATICAL EDUCATION, A ROUTE OF RESEARCH, CREATIVITY AND CULTURAL DIVERSITY AbstractThe text discusses the potential and contributions of the History of Mathematics in teaching practices in Mathematical Education, illustrated by two specific episodes of the authors' pedagogical practice. It takes as theoretical notes studies like those of Ferreira, D'Ambrosio, Barbin, Jankivist and Vianna about the arguments, implications and suggestions directed to the didactic use of the history of mathematics. The authors' foundations are based on a dialogue with Ethnomathematics, thus understanding school mathematics or mathematics - seen in a dominant Western way - as one among other possibilities of doing and thinking mathematically. Throughout the text, history stands out as a subsidy for the creation, both individual and collective, of explanations, relations of meanings, objects and meanings that were not constituted until then. Creativity, from the perspective of Karwowski, Jankovska and Szwajkowski, and research, in the Ponte line, are presented as relevant elements in the teaching process in order to stimulate in each student a relationship of construction and appropriation of school mathematical knowledge in a participatory way, questioner and producer of new knowledge. As a result we point out - being able to unite theories and scientific ideas when analyzing potentialities and contributions of the History of Mathematics in teaching practices of school Mathematics teaching, involving research and creativity in hybrid methodologies and contexts of different cultures.Keywords: History of Mathematics. School mathematical education. Traditional indigenous games. Research and creativity. HISTÓRIA DA MATEMÁTICA NA EDUCAÇÃO MATEMÁTICA, UMA VIA DE INVESTIGAÇÃO, CRIATIVIDADE E DIVERSIDADE CULTURAL ResumoO texto discute potencialidades e contribuições da História da Matemática em práticas docentes da Educação Matemática, ilustradas por dois episódios específicos da prática pedagógica das autoras. Toma como apontamentos teóricos estudos como os de Ferreira, D’Ambrosio, Barbin, Jankivist e Vianna acerca dos argumentos, implicações e sugestões direcionadas ao uso didático da história da matemática. Os fundamentos das autoras são alicerçados em diálogo com a Etnomatemática, entendendo, assim, a matemática ou a matemática escolar - vista de modo ocidental dominante – como uma entre outras possibilidades do fazer e pensar matematicamente. Ao longo do texto, destaca-se a história como subsídio para criação, tanto individual quanto coletiva, de explicações, relações de significados, objetos e sentidos que não estavam até então constituídos. A criatividade, na perspectiva de Karwowski, Jankovska e Szwajkowski, e a investigação, na linha de Ponte, são apresentadas como elementos relevantes no processo de ensino a fim de estimular em cada estudante uma relação de construção e apropriação do conhecimento matemático escolar de forma participativa, questionadora e produtora de novos conhecimentos. Como resultado apontamos - poder unir teorias e ideias científicas ao analisar  potencialidades e contribuições da História da Matemática em práticas docentes do ensino da Matemática escolar, envolvendo a investigação e a criatividade em metodologias híbridas e contextos de diferentes culturas.Palavras-chave: História da Matemática. Educação matemática escolar. Jogos tradicionais indígenas. Investigação e criatividade.


2021 ◽  
Vol 5 (8) ◽  
pp. 148-154
Author(s):  
Linfeng Han ◽  
Qian Tao

History and Pedagogy of Mathematics (HPM) is one of the important research fields in mathematics education, which has received widespread attention from the mathematics education community because of its educational value. Modern mathematics education technology plays an important auxiliary role in mathematics teaching. Hawgent is a dynamic mathematics software that can present abstract mathematical knowledge visually and static mathematical knowledge dynamically. In view of this, this research takes “the recognition of circle” as an example to conduct a research on HPM teaching supported by Hawgent Dynamic Mathematics Software in three aspects: analyze the contents and uncover the history of mathematics, make the products and show the history of mathematics, design the teaching and integrate the history of Mathematics.


Conceptus ◽  
2009 ◽  
Vol 38 (94) ◽  
Author(s):  
Adrian Frey

SummaryKitcher’s philosophy of mathematics rests on the idea that a philosopher who tries to understand mathematical knowledge ought to take its historical development into consideration. In this paper, I take a closer look at Kitcher’s reasons for proposing such a historical turn. I argue that, whereas a historical account is indeed an essential part of the standpoint advanced in The Nature of Mathematical Knowledge, this is no longer the case for the position defended in the manuscript Mathematical Truth? The Wittgensteinian account of mathematics advocated in that manuscript does not force us to take a historical turn.


2018 ◽  
Vol 17 (1) ◽  
pp. 245
Author(s):  
Maria Célia Leme Da Silva

O estudo busca responder às questões: De que maneira o conhecimento da história da educação matemática pode contribuir para as reflexões e desafios postos nos documentos atuais? Para tanto, analisa-se como a medida de superfícies em dois momentos históricos: final do século XIX, período caracterizado pela pedagogia moderna e início do século XXI no âmbito do Plano Nacional de Alfabetização. As fontes examinadas são: Caderno do PNAIC (2014), Parecer de Rui Barbosa (1883) e a Proposta de Gabriel Prestes (1895, 1896). Propõe-se pensar e conhecer os saberes matemáticos elementares do passado em seu contexto histórico, perceber que a institucionalização da expertise participa poderosamente da produção de novos saberes no campo pedagógico, porém seu processo de legitimação, de reconhecimento por seus pares é longo, complexo e conflituoso.Palavras-chave: PNAIC, Rui Barbosa, Gabriel Prestes. Expertise. AbstractThe study seeks to answer the questions: How can the knowledge of the history of mathematics education contribute to the reflections and challenges posed in the current documents? To this end, it is analyzed as the measurement of surfaces in two historical moments: the end of the nineteenth century, a period characterized by modern pedagogy and the beginning of the 21st century within the scope of the National Literacy Plan. The sources examined are: Notebook of the PNAIC (2014), Opinion of Rui Barbosa (1883) and the Proposal of Gabriel Prestes (1895, 1896). It is proposed to think and know the elementary mathematical knowledge of the past in its historical context, to realize that the institutionalization of expertise participates powerfully in the production of new knowledge in the pedagogical field, but its process of legitimation, recognition by its couple is long, complex and conflicting.Keywords: PNAIC, Rui Barbosa, Gabriel Prestes. Expertise.ResumenEl estudio busca responder a las preguntas: ¿De qué manera el conocimiento de la historia de la educación matemática puede contribuir a las reflexiones y desafíos planteados en los documentos actuales? Para ello, se analiza como la medida de superficies en dos momentos históricos: final del siglo XIX, período caracterizado por la pedagogía moderna e inicio del siglo XXI en el marco del Plan Nacional de Alfabetización. Las fuentes examinadas son: Cuaderno del PNAIC (2014), Dictamen de Rui Barbosa (1883) y la Propuesta de Gabriel Prestes (1895, 1896). Se propone pensar y conocer los saberes matemáticos elementales del pasado en su contexto histórico, percibir que la institucionalización de la expertise participa poderosamente de la producción de nuevos saberes en el campo pedagógico, pero su proceso de legitimación, de reconocimiento por sus pares es largo, complejo y complejo, conflicto.Palabras clave: PNAIC, Rui Barbosa, Gabriel Prestes. Expertise.Recebido  


2016 ◽  
Vol 100 (548) ◽  
pp. 266-273 ◽  
Author(s):  
Nick Lord

One of the many definite integrals that Euler was the first to evaluate was(1)He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title,On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recentGazettearticle [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.


1998 ◽  
Vol 29 (3) ◽  
pp. 357-363
Author(s):  
Richard S. Kitchen ◽  
Joanne Rossi Becker

Arthur B. Powell and Marilyn Frankenstein's new book, Ethnomathematics: Challenging Eurocentrism in Mathematics Education, illuminates for our consideration a body of very practical mathematical knowledge largely discounted in the traditional mathematical community when compared with the abstract, theoretical mathematical knowledge typically valued highly by mathematicians. Ethnomathematics has caused us to call into question which mathematical knowledge really counts and thus has come to signify more than just “the study of mathematical ideas of nonliterate peoples” (a definition first offered by Marcia and Robert Ascher in the early 1980s in their paper, “Ethnomathematics,” reprinted as chapter 2 of this volume, p. 26). Editors Powell and Frankenstein use, instead, the broader definition of ethnomathematics provided in the book's opening chapter, “Ethnomathematics and Its Place in the History and Pedagogy of Mathematics,” by Ubiratan D'Ambrosio, a Brazilian mathematics educator whom many consider the intellectual progenitor of ethnomathematics. D'Ambrosio defines ethnomathematics as the mathematics that all cultural groups engage in, including “national tribal societies, labor groups, children of a certain age bracket, professional classes, and so on” (p. 16). Each group, including mathematicians, has its own mathematics. From D'Ambrosio's perspective, ethnomathematics exists at the confluence of the history of mathematics and cultural anthropology, overcoming the Egyptian/Greek differentiation between practical and academic mathematics.


2016 ◽  
Vol 29 (3) ◽  
pp. 273-304 ◽  
Author(s):  
François Lê

ArgumentThis paper challenges the use of the notion of “culture” to describe a particular organization of mathematical knowledge, shared by a few mathematicians over a short period of time in the second half of the nineteenth century. This knowledge relates to “geometrical equations,” objects that proved crucial for the mechanisms of encounters between equation theory, substitution theory, and geometry at that time, although they were not well-defined mathematical objects. The description of the mathematical collective activities linked to “geometrical equations,” and especially the technical aspects of these activities, is made on the basis of a sociological definition of “culture.” More precisely, after an examination of the social organization of the group of mathematicians, I argue that these activities form an intricate system of patterns, symbols, and values, for which I suggest a characterization as a “cultural system.”


Sign in / Sign up

Export Citation Format

Share Document