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Mathematics in Language
cognitive semantics 6 (2020) 243-278 brill.com/cose Mathematics in Language Richard Hudson University College London, London, England [email protected] Abstract Elementary mathematics is deeply rooted in ordinary language, which in some respects anticipates and supports the learning of mathematics, but which in other re- spects hinders this learning. This paper explores a number of areas of arithmetic and other elementary areas of mathematics, considering for each area whether it helps or hinders the young learner: counting and larger numbers, sets and brackets, algebra and variables, zero and negation, approximation, scales and relationships, and probability. The conclusion is that ordinary language anticipates the mathematics of counting, arithmetic, algebra, variables and brackets, zero and probability; but that negation, approximation and probability are particularly problematic because mathematics de- mands a different way of thinking, and different mental capacity, compared with ordi- nary language. School teachers should be aware of the mathematics already built into language so as to build on it; and they should also be able to offer special help in the conflict zones. Keywords numbers – grammar – semantics – negation – probability 1 Introduction1 This paper is about the area of language that we might call ‘mathematical language’, and has a dual function, as a contribution to cognitive semantics and as an exercise in pedagogical linguistics. Unlike earlier discussions of 1 I should like to acknowledge detailed comments on an earlier version of this paper from Neil Sheldon, as well as those of an anonymous reviewer. © Richard Hudson, 2020 | doi:10.1163/23526416-bja10005 This is an open access article distributed under the terms of the CC BY 4.0Downloaded License. -
Algorithmic Factorization of Polynomials Over Number Fields
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3). -
Chapter 4 Mathematics As a Language
Chapter 4 Mathematics as a Language Peculiarities of Mathematical Language in the Texts of Pure Mathematics What is mathematics? Is mathematics, represented in all its modern variety, a single science? The answer to this clear-cut question was at- tempted by the French mathematicians who sign their papers with the name Nicolas Bourbaki. In their article "Architecture of Mathematics" published in Russian as an appendix to The History of Mathematics' (Bourbaki, 1948), they stated that mathematics (and certainly it is only pure mathematics that is referred to) is a uniform science. Its uniformity is given by the system of its logical constructions. A chracteristic feature of mathematics is the explicit axiomatico-deductive method of construct- ing judgments. Any mathematical paper is first of all characterized by containing a long chain of logical conclusions. But the Bourbaki say that such chaining of syllogisms is no more than a transforming mechanism. It may be applied to any system of premises. It is just an outer sign of a system, its dressing; it does not yet display the essential system of logical constructions given by the postulates. The system of postulates in mathematics is not in the least a colorful mosaic of separate initial statements. The peculiarity of mathematics lies in the ability of the system of postulates to form special concepts, mathematical structures, rich with logical consequences which may be derived from them deductively. Mathematics is principally an axioma- ' This is one of the volumes of the unique tractatus TheElernenrs of Marhernorrcs, which wos ro give the reader the fullest impression of modern mathematics, organized from the standpoint of one of the largest modern schools. -
The Utility of Mathematics
THE UTILITY OF MATHEMATICS It is not without great surprise that we read of the efforts of modern "educators" to expunge the study of mathematics from the scholastic curriculum. It is logical, and consistent with an already incomplete, un satisfactory program, for the State to exclude religious instruc tion from the free public schools. The proscription of the study of the German language in the schools is consonant with the existing antipathy for all things German. But who has weighed mathematics in the balance of honest investigation and found it wanting? Two American soldiers training for the great conflict "over there" visited a book shop. One purchased a German grammar. His companion severely reprimanded him for his apparent weak ness, and eloquently proclaimed the uselessness of studying Ger man. "Well," said the first soldier, "you will be in an awful fix when you reach Berlin." Without a working knowledge of the German language the victorious Allies will, indeed, be at a great disadvantage when the present drive shall have led them into Germany on the road to Berlin. But what a rocky road to Ber lin! The vandalism of the retreating Hun is obstructing that road by the wanton destruction of cities, towns and villages, with their churches, bridges and every other architectural triumph of constructive engineering. The damage must be repaired when the war is· over. It will require the careful work of skilled engineers. Was there ever an engineer who attained success without the aid of mathemat ics? The war has wrought havoc and destruction wherever the tents of battle have been pitched. -
Some Properties of the Discriminant Matrices of a Linear Associative Algebra*
570 R. F. RINEHART [August, SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A LINEAR ASSOCIATIVE ALGEBRA* BY R. F. RINEHART 1. Introduction. Let A be a linear associative algebra over an algebraic field. Let d, e2, • • • , en be a basis for A and let £»•/*., (hjik = l,2, • • • , n), be the constants of multiplication corre sponding to this basis. The first and second discriminant mat rices of A, relative to this basis, are defined by Ti(A) = \\h(eres[ CrsiCij i, j=l T2(A) = \\h{eres / ,J CrsiC j II i,j=l where ti(eres) and fa{erea) are the first and second traces, respec tively, of eres. The first forms in terms of the constants of multi plication arise from the isomorphism between the first and sec ond matrices of the elements of A and the elements themselves. The second forms result from direct calculation of the traces of R(er)R(es) and S(er)S(es), R{ei) and S(ei) denoting, respectively, the first and second matrices of ei. The last forms of the dis criminant matrices show that each is symmetric. E. Noetherf and C. C. MacDuffeeJ discovered some of the interesting properties of these matrices, and shed new light on the particular case of the discriminant matrix of an algebraic equation. It is the purpose of this paper to develop additional properties of these matrices, and to interpret them in some fa miliar instances. Let A be subjected to a transformation of basis, of matrix M, 7 J rH%j€j 1,2, *0). -
The Language of Mathematics: Towards an Equitable Mathematics Pedagogy
The Language of Mathematics: Towards an Equitable Mathematics Pedagogy Nathaniel Rounds, PhD; Katie Horneland, PhD; Nari Carter, PhD “Education, then, beyond all other devices of human origin, is the great equalizer of the conditions of men, the balance wheel of the social machinery.” (Horace Mann, 1848) 2 ©2020 Imagine Learning, Inc. Introduction From the earliest days of public education in the United States, educators have believed in education’s power as an equalizing force, an institution that would ensure equity of opportunity for the diverse population of students in American schools. Yet, for as long as we have been inspired by Mann’s vision, our history of segregation and exclusion reminds us that we have fallen short in practice. In 2019, 41% of 4th graders and 34% of 8th graders demonstrated proficiency in mathematics on the National Assessment of Educational Progress (NAEP). Yet only 20% of Black 4th graders and 28% of Hispanic 4th graders were proficient in mathematics, compared with only 14% of Black 8th grades and 20% of Hispanic 8th graders. Given these persistent inequities, how can education become “the great equalizer” that Mann envisioned? What, in short, is an equitable mathematics pedagogy? The Language of Mathematics: Towards an Equitable Mathematics Pedagogy 1 Mathematical Competencies To help think through this question, we will examine the language of mathematics and the role of language in math classrooms. This role, of course, has changed over time. Ravitch gives us this picture of math instruction in the 1890s: Some teachers used music to teach the alphabet and the multiplication tables..., with students marching up and down the isles of the classroom singing… “Five times five is twenty-five and five times six is thirty…” (Ravitch, 2001) Such strategies for the rote learning of facts and algorithms once seemed like all we needed to do to teach mathematics. -
The Story of 0 RICK BLAKE, CHARLES VERHILLE
The Story of 0 RICK BLAKE, CHARLES VERHILLE Eighth But with zero there is always an This paper is about the language of zero The initial two Grader exception it seems like sections deal with both the spoken and written symbols Why is that? used to convey the concepts of zero Yet these alone leave much of the story untold. Because it is not really a number, it The sections on computational algorithms and the is just nothing. exceptional behaviour of zero illustrate much language of Fourth Anything divided by zero is zero and about zero Grader The final section of the story reveals that much of the How do you know that? language of zero has a considerable historical evolution; Well, one reason is because I learned it" the language about zero is a reflection of man's historical difficulty with this concept How did you learn it? Did a friend tell you? The linguistics of zero No, I was taught at school by a Cordelia: N a thing my Lord teacher Lear: Nothing! Do you remember what they said? Cordelia: Nothing Lear: Nothing will come of They said anything divided by zero is nothing; speak again zero Do you believe that? Yes Kent: This is nothing, fool Why? Fool: Can you make no use of Because I believe what the school says. nothing, uncle? [Reys & Grouws, 1975, p 602] Lear: Why, no, boy; nothing can be made out of nothing University Zero is a digit (0) which has face Student value but no place or total value Ideas that we communicate linguistically are identified by [Wheeler, Feghali, 1983, p 151] audible sounds Skemp refers to these as "speaking sym bols " 'These symbols, although necessary inventions for If 0 "means the total absence of communication, are distinct from the ideas they represent quantity," we cannot expect it to Skemp refers to these symbols or labels as the surface obey all the ordinary laws. -
The Academic Language of Mathematics
M01_ECHE7585_CH01_pp001-014.qxd 4/30/13 5:00 PM Page 1 chapter 1 The Academic Language of Mathematics Nearly all states continue to struggle to meet the academic targets in math for English learners set by the No Child Left Behind Act. One contributing factor to the difficulty ELs experience is that mathematics is more than just numbers; math education involves terminology and its associated concepts, oral or written instructions on how to complete problems, and the basic language used in a teacher’s explanation of a process or concept. For example, students face multiple representations of the same concept or operation (e.g., 20/5 and 20÷5) as well as multiple terms for the same concept or operation 6920_ECH_CH01_pp001-014.qxd 8/27/09 12:59 PM Page 2 2 (e.g., 13 different terms are used to signify subtraction). Students must also learn similar terms with different meanings (e.g., percent vs. percentage) and they must comprehend mul- tiple ways of expressing terms orally (e.g., (2x ϩ y)/x2 can be “two x plus y over x squared” and “the sum of two x and y divided by the square of x”) (Hayden & Cuevas, 1990). So, language plays a large and important role in learning math. In this chapter, we will define academic language (also referred to as “academic Eng- lish”), discuss why academic language is challenging for ELs, and offer suggestions for how to effectively teach general academic language as well as the academic language specific to math. Finally, we also include specific academic word lists for the study of mathematics. -
A Historical Survey of Methods of Solving Cubic Equations Minna Burgess Connor
University of Richmond UR Scholarship Repository Master's Theses Student Research 7-1-1956 A historical survey of methods of solving cubic equations Minna Burgess Connor Follow this and additional works at: http://scholarship.richmond.edu/masters-theses Recommended Citation Connor, Minna Burgess, "A historical survey of methods of solving cubic equations" (1956). Master's Theses. Paper 114. This Thesis is brought to you for free and open access by the Student Research at UR Scholarship Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. A HISTORICAL SURVEY OF METHODS OF SOLVING CUBIC E<~UATIONS A Thesis Presented' to the Faculty or the Department of Mathematics University of Richmond In Partial Fulfillment ot the Requirements tor the Degree Master of Science by Minna Burgess Connor August 1956 LIBRARY UNIVERStTY OF RICHMOND VIRGlNIA 23173 - . TABLE Olf CONTENTS CHAPTER PAGE OUTLINE OF HISTORY INTRODUCTION' I. THE BABYLONIANS l) II. THE GREEKS 16 III. THE HINDUS 32 IV. THE CHINESE, lAPANESE AND 31 ARABS v. THE RENAISSANCE 47 VI. THE SEVEW.l'EEl'iTH AND S6 EIGHTEENTH CENTURIES VII. THE NINETEENTH AND 70 TWENTIETH C:BNTURIES VIII• CONCLUSION, BIBLIOGRAPHY 76 AND NOTES OUTLINE OF HISTORY OF SOLUTIONS I. The Babylonians (1800 B. c.) Solutions by use ot. :tables II. The Greeks·. cs·oo ·B.c,. - )00 A~D.) Hippocrates of Chios (~440) Hippias ot Elis (•420) (the quadratrix) Archytas (~400) _ .M~naeobmus J ""375) ,{,conic section~) Archimedes (-240) {conioisections) Nicomedea (-180) (the conchoid) Diophantus ot Alexander (75) (right-angled tr~angle) Pappus (300) · III. -
501 Algebra Questions 2Nd Edition
501 Algebra Questions 501 Algebra Questions 2nd Edition ® NEW YORK Copyright © 2006 LearningExpress, LLC. All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by LearningExpress, LLC, New York. Library of Congress Cataloging-in-Publication Data: 501 algebra questions.—2nd ed. p. cm. Rev. ed. of: 501 algebra questions / [William Recco]. 1st ed. © 2002. ISBN 1-57685-552-X 1. Algebra—Problems, exercises, etc. I. Recco, William. 501 algebra questions. II. LearningExpress (Organization). III. Title: Five hundred one algebra questions. IV. Title: Five hundred and one algebra questions. QA157.A15 2006 512—dc22 2006040834 Printed in the United States of America 98765432 1 Second Edition ISBN 1-57685-552-X For more information or to place an order, contact LearningExpress at: 55 Broadway 8th Floor New York, NY 10006 Or visit us at: www.learnatest.com The LearningExpress Skill Builder in Focus Writing Team is comprised of experts in test preparation, as well as educators and teachers who specialize in language arts and math. LearningExpress Skill Builder in Focus Writing Team Brigit Dermott Freelance Writer English Tutor, New York Cares New York, New York Sandy Gade Project Editor LearningExpress New York, New York Kerry McLean Project Editor Math Tutor Shirley, New York William Recco Middle School Math Teacher, Grade 8 New York Shoreham/Wading River School District Math Tutor St. James, New York Colleen Schultz Middle School Math Teacher, Grade 8 Vestal Central School District Math Tutor -
Mathematics, Language and Translation Sundar Sarukkai
Document généré le 23 mai 2020 16:12 Meta Journal des traducteurs Mathematics, Language and Translation Sundar Sarukkai Volume 46, numéro 4, décembre 2001 Résumé de l'article La langue mathématique utilise la langue naturelle. Les symboles se URI : https://id.erudit.org/iderudit/004032ar rapportent eux aussi aux termes de la langue naturelle. Les textes DOI : https://doi.org/10.7202/004032ar mathématiques relèvent de combinaisons de symboles, de langue naturelle, de représentations graphiques, etc. Pour arriver à une lecture cohérente de ces Aller au sommaire du numéro textes, il faut passer par la traduction. Les mathématiques appliquées, comme la physique, passent continuellement d'une langue (et culture) à une autre, et par conséquent, elles sont mieux comprises quand elles relèvent du domaine Éditeur(s) de la traduction. Les Presses de l'Université de Montréal ISSN 0026-0452 (imprimé) 1492-1421 (numérique) Découvrir la revue Citer cet article Sarukkai, S. (2001). Mathematics, Language and Translation. Meta, 46 (4), 664–674. https://doi.org/10.7202/004032ar Tous droits réservés © Les Presses de l'Université de Montréal, 2001 Ce document est protégé par la loi sur le droit d’auteur. L’utilisation des services d’Érudit (y compris la reproduction) est assujettie à sa politique d’utilisation que vous pouvez consulter en ligne. https://apropos.erudit.org/fr/usagers/politique-dutilisation/ Cet article est diffusé et préservé par Érudit. Érudit est un consortium interuniversitaire sans but lucratif composé de l’Université de Montréal, l’Université Laval et l’Université du Québec à Montréal. Il a pour mission la promotion et la valorisation de la recherche. -
The Evolution of Equation-Solving: Linear, Quadratic, and Cubic
California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work.