Analysing the Mathematical Experience: Posing the ‘What Is Mathematics?’ Question
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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. -
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. -
The Theorem of Pythagoras
The Most Famous Theorem The Theorem of Pythagoras feature Understanding History – Who, When and Where Does mathematics have a history? In this article the author studies the tangled and multi-layered past of a famous result through the lens of modern thinking, look- ing at contributions from schools of learning across the world, and connecting the mathematics recorded in archaeological finds with that taught in the classroom. Shashidhar Jagadeeshan magine, in our modern era, a very important theorem being Iattributed to a cult figure, a new age guru, who has collected a band of followers sworn to secrecy. The worldview of this cult includes number mysticism, vegetarianism and the transmigration of souls! One of the main preoccupations of the group is mathematics: however, all new discoveries are ascribed to the guru, and these new results are not to be shared with anyone outside the group. Moreover, they celebrate the discovery of a new result by sacrificing a hundred oxen! I wonder what would be the current scientific community’s reaction to such events. This is the legacy associated with the most ‘famous’ theorem of all times, the Pythagoras Theorem. In this article, we will go into the history of the theorem, explain difficulties historians have with dating and authorship and also speculate as to what might have led to the general statement and proof of the theorem. Vol. 1, No. 1, June 2012 | At Right Angles 5 Making sense of the history Why Pythagoras? Greek scholars seem to be in Often in the history of ideas, especially when there agreement that the first person to clearly state PT has been a discovery which has had a significant in all its generality, and attempt to establish its influence on mankind, there is this struggle to find truth by the use of rigorous logic (what we now call mathematical proof), was perhaps Pythagoras out who discovered it first. -
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. -
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 Language of Mathematics in Science
The Language of A Guide for Mathematics in Teachers of Science 11–16 Science A note on navigating around within this PDF file Introduction: the language of mathematics in science Background About this publication Mathematics and science Thinking about purposes and using judgement Understanding the nature of data Assessment Process of development of this publication Further references on terminology and conventions Overview of chapters 1 Collecting data 1.1 Measuring and counting 1.2 Measurement, resolution and significant figures 1.3 Characteristics of different types of data 1.4 Naming different types of data 1.5 Where do data come from? 2 Doing calculations and representing values 2.1 Calculations and units THE LANGUAGE OF MATHEMATICS IN SCIENCE A Guide for Teachers of 11–16 Science The Association for Science Education (ASE) is the largest subject association in the UK. As the professional body for all those involved in The Association science education from pre-school to higher education, the ASE provides a national network supported by a dedicated staff team. Members include for Science Education teachers, technicians and advisers. The Association plays a significant role in promoting excellence in teaching and learning of science in schools and colleges. More information is available at www.ase.org.uk. The Nuffield Foundation is an endowed charitable trust that aims to improve social well-being in the widest sense. It funds research and innovation in education and social policy and also works to build capacity in education, science and social science research. The Nuffield Foundation has funded this project, but the views expressed are those of the authors and not necessarily those of the Foundation. -
Learning the Language of Mathematics 45
Learning the Language of Mathematics 45 Learning the Language of Mathematics Robert E. Jamison Clemson University Just as everybody must strive to learn language and writing before he can use them freely for expression of his thoughts, here too there is only one way to escape the weight of formulas. It is to acquire such power over the tool that, unhampered by formal technique, one can turn to the true problems. — Hermann Weyl [4] This paper is about the use of language as a tool for teaching mathematical concepts. In it, I want to show how making the syntactical and rhetorical structure of mathematical language clear and explicit to students can increase their understanding of fundamental mathematical concepts. I confess that my original motivation was partly self-defense: I wanted to reduce the number of vague, indefinite explanations on home- work and tests, thereby making them easier to grade. But I have since found that language can be a major pedagogical tool. Once students understand HOW things are said, they can better understand WHAT is being said, and only then do they have a chance to know WHY it is said. Regrettably, many people see mathematics only as a collection of arcane rules for manipulating bizarre symbols — something far removed from speech and writing. Probably this results from the fact that most elemen- tary mathematics courses — arithmetic in elementary school, algebra and trigonometry in high school, and calculus in college — are procedural courses focusing on techniques for working with numbers, symbols, and equations. Although this formal technique is important, formulae are not ends in themselves but derive their real importance only as vehicles for expression of deeper mathematical thoughts. -
Fall 2007 T and Puzzle-Connois- University of Massachusetts Lowell Seur Peter Winkler by Students
A Review of The Art of Mathematics: Coffee Time in Memphis by Béla Bollobás. xvi + 359 pp. Cambridge University Press, 2006. BY JAMES PROPP Here is one of my favorites: This proof exhibits economy Thanks For the Suppose 10 chairs are arranged (the chief idea is contained in Tangents he mathematician Tangents in a circle, half of them occupied the five-word hint), surprise Contributions! News from the Department of Mathematical Sciences Fall 2007 T and puzzle-connois- University of Massachusetts Lowell seur Peter Winkler by students. Show that there (who would think to bring ran- once joked, with a exists some whole number n domness and probability into Our thanks to all who have nod to Isaac Newton, between 1 and 9 such that if solving a problem like this?) contributed to the Department “If I have seen far- each of the 5 students moves n and fecundity (the probabilistic of Mathematical Sciences over ther than others, it is because chairs clockwise in the circle, or method has been an enormously the past few years. Your more of them will end up sitting powerful tool in the hands of I have stood on the shoulders generosity has allowed us in a previously occupied chair. Erdös and others). Bollobás’s 2007 Alumni-Awards Banquet A review of of Hungarians.” One of these to make purchases, award Hungarians is the late Paul This is not how Bollobás book is full of tasty little morsels scholarships, and engage in Next Banquet set for April 25, 2008 God Created Erdös, famous within mathemat- actually poses the problem in his like this one, puzzles whose ics for his contributions to num- book; in problem 3 (in a series solution requires attacking them activities that would otherwise the Integers of 157 problems), he asks the from some unexpected angle. -
The Mathematical Breakthroughs That Changed History
GOD CREATED THE INTEGERS: THE MATHEMATICAL BREAKTHROUGHS THAT CHANGED HISTORY Author: Stephen Hawking Number of Pages: 1184 pages Published Date: 07 Sep 2006 Publisher: Penguin Books Ltd Publication Country: London, United Kingdom Language: English ISBN: 9780141018782 DOWNLOAD: GOD CREATED THE INTEGERS: THE MATHEMATICAL BREAKTHROUGHS THAT CHANGED HISTORY God Created the Integers: The Mathematical Breakthroughs That Changed History PDF Book Tips are provided for making the ideas relevant to every child, whatever their learning style or individual needs and additional ideas for involving parents are also included. Inside you'll find key art, character concepts, enemy designs, sprite sheets, unused ideas, and an all-new Shovel Knight tribute art gallery. Such was the strength of solar tradition that the Emperor Constantine remained loyal nearly unto death, and up to medieval times Christ in Europe was still known as Sol Resurrectionus. This book should be on every mathematics teacher's bookshelf. Treasured Possessions: Indigenous Interventions into Cultural and Intellectual PropertyThe law of the sea provides for the regulation, management and governance of the ocean spaces that cover over two-thirds of the Earth's surface. Non-homeostatic drives such as electrical self-stimulation of the brain are also considered. The material basis of the present work consists of collections now in the herbarium of the State University of Iowa. " Stephen Parrott, Birkbeck, University of London Criminal Justice by Ursula Smartt is part of an exciting new series from SAGE. Reid, The Washington Post 0201433125B04062001 The HP-UX System Administrator's How to BookThis colorful two-book kit is designed for those helping friends and family select, set up, and use a new PC or those new to the task themselves.