The Many Uses of Algebraic Variables

1992 ◽  
Vol 85 (7) ◽  
pp. 557-561 ◽  
Author(s):  
Randolph A. Philipp
Keyword(s):  
Elementary School ◽  
Turning Point ◽  
History Of Mathematics ◽  
History Of ◽  
The Many ◽  
Experience Difficulty

The concept of variable is one of the most fundamental ideas in mathematics from elementary school through college (Davis 1964; Hirsch and Lappan 1989). This concept is so important that its invention constituted a turning point in the history of mathematics (Rajaratnam 1957). However, research indicates that students experience difficulty with the concept of variable, a difficulty that might partially be explained by the fact that within mathematics, variables can be used in many different ways (Rosnick 1981; Schoenfeld and Arcavi 1988; Wagner 1983).

Conclusion

2021 ◽  
pp. 221-224
Author(s):  
Isabella Lazzarini
Keyword(s):  
Turning Point ◽  
Parallel Processes ◽  
History Of ◽  
History Of Europe ◽  
The Many ◽  
Power And Authority

The Conclusion talks about fractures and continuities, and highlights at least two common processes. If many elements can support the idea of the 1500s as a turning point in the history of Europe, this volume is more open to continuities than to fractures. In the 1300s and 1400s, therefore, the polyphonic, vibrant, and sometimes contradictory fabric of politics, culture, and society takes centre stage. From such complexity, the legacy of this period to the following centuries is represented by two parallel processes. The institutional and constitutional framework of power and authority showed a thickening and defining of its many forms, but politics remained a field open to many contrasting solutions. And the emergence of a more defined written and spoken agency of individuals and groups that had previously been less visible created cultures and languages of power that rewrote tradition and enabled the many authors of such new languages to make themselves heard.

Painless Drilling Not Your Dentist, but the History of Mathematics

1980 ◽  
Vol 27 (8) ◽  
pp. 40-42
Author(s):  
Stephen Krulik
Keyword(s):  
History Of Mathematics ◽  
Theoretical Base ◽  
Body Of Knowledge ◽  
Practical Necessity ◽  
History Of ◽  
Points Of View ◽  
The Many ◽  
Drill And Practice

Many of our students are far from receptive to the many problems of drill materials with which some teachers provide them. And yet, most students need some drill and practice before they can successfully master a new concept or skill. The history of mathematics can play an important role in making these apparently contradictory points of view compatible. Many of the concepts and ideas in the history of mathematics were developed from practical necessity rather than from a theoretical base; it is these same ideas that offer a great deal of practice material for our students. This ancient body of knowledge can provide drill that is interesting, satisfying, challenging. And, at the same time, it offers the necessary drill to achieve competence in fundamental skills and concepts.

scholarly journals Finest frugt - om ‘den Vartou Kjællingepræst’: En guide til Grundtvig-Studier 1948-2008, I

2009 ◽  
Vol 60 (1) ◽  
pp. 20-85
Author(s):  
Aage Jørgensen
Keyword(s):  
World War Ii ◽  
Literary History ◽  
Turning Point ◽  
History Of Ideas ◽  
Anglo Saxon ◽  
World War ◽  
History Of ◽  
The Many ◽  
The Impact ◽  
Opening Section

Finest frugt - om ‘den Vartou Kjællingepræst En guide til Grundtvig- Studier 1948-2008,I[Fruit of the finest - concerning ‘the Vartov Old Biddies ’pas torA guide to Grundtvig-Studier 1948-2008, I]By Aage JørgensenAs early as 1948, the Grundtvig Society of 8 September 1947 launched an annual journal, Grundtvig-Studier (Grundtvig Studies). Since then, the journal has published a significant part of the subsequent research on Grundtvig, including important debate on the many Grundtvig dissertations that have appeared since World War II. This anniversary article reviews the content of the sixty volumes that have hitherto been issued. Despite its cross-disciplinary character, the material is here presented in a traditionally systematised format.The opening section deals with studies in bibliography, diplomatics and biography, and is followed by a series of fairly lengthy sections characterising: (1) material pertaining to the history of ideas and concerned with Grundtvig’s views on life and history together with his relation to Romantic philosophy, Northern mythology and the contemporary way of thinking as a whole; (2) material relating to literary history and aesthetics, with emphasis upon a series of exemplary contributions by particular authors (Gustav Albeck, Helge Toldberg, Jørgen Elbek, Flemming Lundgreen-Nielsen, Sune Auken and, as regards the influence of Anglo-Saxon upon Grundtvig, S. A. J. Bradley); and (3) the theological material, with focus especially upon the impact of Kaj Thaning’s designation of 1832 as a decisive turning-point in Grundtvig’s life, and upon Grundtvig’s relationship to Luther, Kierkegaard and mystic tradition. The exploration of Grundtvig's hymns and sermons is treated in separate sections; and finally there is a section concerned with illustration of his political endeavours. For reasons of space, contributions to the journal on Grundtvig’s educational deliberations and their significance for Danish schools, especially the folk high schools, together with the overall subsequent reception of his thinking (within Denmark and out in the wider world) will be reserved for discussion in Grundtvig-Studier 2010.

scholarly journals The Visit of the Shah of Iran to Washington in October 21-23, 1969: A study in the U.S Department of State’s reports

2018 ◽  
Vol 225 (2) ◽  
pp. 1-20
Author(s):  
Dr. Ahmed Shaker Abdel-Alak ◽  
Dr. Abdullah Lafteh Al-Budairi
Keyword(s):  
Turning Point ◽  
Arabian Gulf ◽  
The United States ◽  
Gulf Region ◽  
Oil Companies ◽  
The Us ◽  
History Of ◽  
The Many ◽  
The Military ◽  
The U.S

Shah Mohammad Reza Pahlavi tried in his visit to Washington to give the impression to the US administration that he was able to protect the interests of the United States in Iran and in the whole Arabian Gulf and the Middle East. The visit of the Shah represented a new turning point in the history of the Iranian - US relations. It included the discussion of issues concerning both countries, especially the issue of arms and the production and sale of Iranian oil after the announcement of the British government's desire to withdraw from the Gulf region within three years. The American leaders focused on meeting the demands of the Shah, specifically the military ones, to discuss oil production topics and methods of exporting and cooperation with US oil companies, have expressed American sympathy in dealing with the many issues.

Euler, the clothoid and

2016 ◽  
Vol 100 (548) ◽  
pp. 266-273 ◽  
Author(s):  
Nick Lord
Keyword(s):  
History Of Mathematics ◽  
Complex Function ◽  
Real Variable ◽  
Complex Function Theory ◽  
Subsequent Work ◽  
Definite Integrals ◽  
History Of ◽  
The Many ◽  
Image Position

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.

Logarithms for ten-year-olds

1968 ◽  
Vol 15 (3) ◽  
pp. 273-275
Author(s):  
Emma C. Carroll
Keyword(s):  
Elementary School ◽  
Fifth Graders ◽  
History Of Mathematics ◽  
The Other ◽  
Geometric Progression ◽  
Real Place ◽  
Moving Points ◽  
History Of ◽  
Addition And Subtraction

Great inventions from the history of mathematics are finding a real place in mathematics for the elementary school. One such idea—Napier's conception of logarithms as a comparison between two moving points, one generating an arithmetical and the other a geometric progression—developed into a challenging activity for my fourth- and fifth-graders. When they witnessed the simplicity and beauty of reducing difficult multiplication and division into easy addition and subtraction through a simple “log” table, eager experimenters took over, tried the “logs,” checked results with the more cumbersome multiplication and division, and raced home with “log” table copies to share the magic with parents.

Did Gauss Discover That, Too?

1986 ◽  
Vol 79 (4) ◽  
pp. 288-293
Author(s):  
Richard L. Francis
Keyword(s):  
History Of Mathematics ◽  
Remarkable Feature ◽  
History Of ◽  
The Many ◽  
Modern Era ◽  
Long Chain

A remarkable feature of the history of mathematics is the enormous interval that separates the posing of certain problems and their ultimate resolution. Also remarkable is the long chain of attempted solutions that spanned the many centuries. Examples of this phenomenon are the three famous problems of antiquity, which were resolved only in the modern era.

Historically Speaking, - -

1953 ◽  
Vol 46 (8) ◽  
pp. 575-577
Keyword(s):  
Binary System ◽  
High Speed ◽  
History Of Mathematics ◽  
Number System ◽  
Historical Background ◽  
Interesting Phenomenon ◽  
Information And Communication ◽  
History Of ◽  
The Many ◽  
And Algebra

The binary system as a special case of the generalized problem of scales of notation has had a sudden resurgence of popularity. This is largely due to its use in modern high-speed electronic calculators and in new developments in the theory of “information” and “communication.”1 However, this new utility of the binary system arrived at the same time that an even greater emphasis was being placed on “meaning” and “understanding” in the teaching of mathematics. In arithmetic (and algebra) many teachers have felt that understanding of our number system was enhanced, and in some cases first achieved, through a study of numbers written to some base other than ten. These two motives, utilitarian and pedagogical, have led to several articles on the history of the binary system and related topics,2 but it seems that none of them have stressed several additional pedagogical values to be derived from a proper survey of the historical background of scales of notation. This topic is not only intrinsically inter esting, but it also illustrates well the role of generalization and abstraction in mathematics, the roles of necessity and intellectual curiosity in mathematical invention, a few of the many connections between mathematics and philosophy and religion, and the interesting phenomenon of simultaneity in discovery which recurs so often in the history of mathematics.

scholarly journals Joaquim de Oliveira Santos e a modernização da aritmética da escola primária ludovicense

2021 ◽  
Vol 7 (2) ◽  
pp. 13
Author(s):  
Maria do Carmo Alves da Cruz ◽  
Neuza Bertoni Pinto
Keyword(s):  
Elementary School ◽  
Cultural History ◽  
History Of Mathematics ◽  
History Of Education ◽  
Primary Teachers ◽  
Quality Of Education ◽  
Central Question ◽  
School History ◽  
History Of

Este estudo tem como objetivo compreender as contribuições do professor Joaquim de Oliveira Santos na modernização da Aritmética da Escola Primária ludovicense. O texto busca responder ao questionamento central: quais as contribuições de Joaquim de Oliveira Santos na modernização da aritmética da escola primária ludovicense? Para tanto, vale-se do aporte teórico-metodológico ancorado na História Cultural, com base nos trabalhos de Chartier (2017), e na História da educação matemática, com base em Valente (2017) e Pinto (2017). Quanto ao corpus investigativo, as fontes incluem documentos oficiais e, principalmente, livros didáticos escritos pelo professor. A produção do professor Joaquim de Oliveira Santos, identificada nos documentos aos quais tivemos acesso, evidencia incessante busca pela modernização da Aritmética da escola primária ludovicense, bem como pela melhor qualidade da formação de professores primários maranhenses em tempos da vaga intuitiva. As análises permitem inferir que o professor Joaquim de Oliveira Santos desenvolveu trabalhos com vistas à modernização da Aritmética do ensino primário em São Luís, em harmonia com aquilo que era evidenciado no cenário educacional em níveis nacional e internacional.Joaquim de Oliveira Santos and the modernization of arithmetic in the Ludovician primary schoolThis study aims to understand the contributions of Professor Joaquim de Oliveira Santos in the modernization of Arithmetic at Escola Primária Ludovicense. The text seeks to answer the central question: what are the contributions of Joaquim de Oliveira Santos in the modernization of Arithmetic at Elementary School Ludovicense? For that, it makes use of the theoretical-methodological contribution anchored in Cultural History, based on the works of Chartier (2017), the History of Mathematics Education, Valente (2017), Pinto (2017). As for the investigative corpus, the sources include official documents, and mainly textbooks written by the professor. The production of teacher Joaquim de Oliveira Santos, identified in the documents to which we had access, shows the teacher's incessant search for the modernization of Arithmetic in the ludovician Elementary School, as well as for the better quality of education for primary teachers in Maranhão, in times of the intuitive wave. The analyzes allow us to infer that Professor Joaquim de Oliveira Santos developed works with a view to modernizing Arithmetic in primary education in São Luís, in harmony with what was evident in the educational scenario at national and international levels.Keywords: Joaquim de Oliveira Santos; Teaching math; Primary School; History of Education in Maranhão.

A 200-Year History of the Study of Childhood Language Disorders of Unknown Origin: Changes in Terminology

2020 ◽  
Vol 5 (1) ◽  
pp. 6-11 ◽  
Author(s):  
Laurence B. Leonard
Keyword(s):  
Language Impairment ◽  
Specific Language Impairment ◽  
Scientific Literature ◽  
Language Disorder ◽  
Language Disorders ◽  
Unknown Origin ◽  
Developmental Language Disorder ◽  
The Past ◽  
History Of ◽  
The Many

Purpose The current “specific language impairment” and “developmental language disorder” discussion might lead to important changes in how we refer to children with language disorders of unknown origin. The field has seen other changes in terminology. This article reviews many of these changes. Method A literature review of previous clinical labels was conducted, and possible reasons for the changes in labels were identified. Results References to children with significant yet unexplained deficits in language ability have been part of the scientific literature since, at least, the early 1800s. Terms have changed from those with a neurological emphasis to those that do not imply a cause for the language disorder. Diagnostic criteria have become more explicit but have become, at certain points, too narrow to represent the wider range of children with language disorders of unknown origin. Conclusions The field was not well served by the many changes in terminology that have transpired in the past. A new label at this point must be accompanied by strong efforts to recruit its adoption by clinical speech-language pathologists and the general public.

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