A Study Of Number

In this groundbreaking and highly practical book,Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3, author Jessica Shumway proposes that all children have innate number sense which can be developed through daily exercise. Shumway createda series of math routines designed to help young students strengthen and build their facility with numbers. These quick 5, 10, or 15 minute exercises are easy to implement as an add-on to any elementary math curriculum. Understanding Number Sense: Students with strong number sense understand numbers, how to subitize, relationships among numbers, and number systems. They make reasonable estimates, compute fluently, use reasoning strategies, and use visual models to solve problems. Number Sense Routines supports the early learner by instilling the importance of daily warm-ups and explains how they benefit developing math minds for long-term learning. Real Classroom Examples: Shumway compiled her classroom observations from around the country. She includes conversations among students who practice number sense routines to illustrate them in action, how children's number sense develops with daily use, and math strategies students learn as they develop their numerical literacy through self-paced practice. Assessment Strategies: Number Sense Routines demonstrates the importance of listening to your students and knowing what to look for. Teachers will gain a deeper understanding of the underlying math skills and strategies students learn as they develop numerical literacy. Shumway writes, As you read, you will step into various classrooms and listen in on students' conversations, which I hope will give you insight into the power of number sense routines and the impact they have on students' number sense development. My hope is that going into the classroom, into students' conversations, and into their thought processes, you will come away with new ideas and tools to use in your own classroom.

This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles’ resolution of Fermat’s Last Theorem.

This book is a revised and greatly expanded version of our book Elements of Number Theory published in 1972. As with the first book the primary audience we envisage consists of upper level undergraduate mathematics majors and graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract algebra. A large portion of Chapters 1-11 can be read even without such background with the aid of a small amount of supplementary reading. The later chapters assume some knowledge of Galois theory, and in Chapters 16 and 18 an acquaintance with the theory of complex variables is necessary. Number theory is an ancient subject and its content is vast. Any intro ductory book must, of necessity, make a very limited selection from the fascinat ing array of possible topics. Our focus is on topics which point in the direction of algebraic number theory and arithmetic algebraic geometry. By a careful selection of subject matter we have found it possible to exposit some rather advanced material without requiring very much in the way oftechnical background. Most of this material is classical in the sense that is was dis covered during the nineteenth century and earlier, but it is also modern because it is intimately related to important research going on at the present time.

A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

Critical Acclaim for Pi and the AGM: "Fortunately we have the Borwein's beautiful book . . . explores in the first five chapters the glorious world so dear to Ramanujan . . . would be a marvelous text book for a graduate course."--Bulletin of the American Mathematical Society "What am I to say about this quilt of a book? One is reminded of Debussy who, on being asked by his harmony teacher to explain what rules he was following as he improvised at the piano, replied, "Mon plaisir." The authors are cultured mathematicians. They have selected what has amused and intrigued them in the hope that it will do the same for us. Frankly, I cannot think of a more provocative and generous recipe for writing a book . . . (it) is cleanly, even beautifully written, and attractively printed and composed. The book is unique. I cannot think of any other book in print which contains more than a smidgen of the material these authors have included.--SIAM Review "If this subject begins to sound more interesting than it did in the last newspaper article on 130 million digits of Pi, I have partly succeeded. To succeed completely I will have gotten you interested enough to read the delightful and important book by the Borweins."--American Mathematical Monthly "The authors are to be commended for their careful presentation of much of the content of Ramanujan's famous paper, 'Modular Equations and Approximations to Pi'. This material has not heretofore appeared in book form. However, more importantly, Ramanujan provided no proofs for many of the claims that he made, and so the authors provided many of the missing details . . . The Borweins, indeed have helped us find the right roads."--Mathematics of Computation

`At last a book is written by teachers for teachers based on sound research that will generate enquiry based learning. It is essential for every classroom with lots of mathematical activities. These will purposefully engage children and allow for differentiation for those who require additional support to understand the number system and the more able children who require to be challenged. Mathematical standards in our schools will improve tremendously following these instructional activities' - Carole Cannon, Development Officer for Mathematics Recovery 'This book ‘Teaching Number in the Classroom with 4-8 year olds’ is an absolute "must have" for all educators involved in early number. Based on sound theoretical foundations, it offers a wealth of down-to-earth, tried and tested, effective approaches to teaching early number concepts and skills. It is a clearly a book written by teachers for teachers. Every single activity in the book is a nugget. Engaging with these activities will change your whole approach to teaching early number' - Noreen O’Loughlin, Associate Vice-President/Lecturer in Maths Education, Mary Immaculate College, University of Limerick, Ireland. 'The authors prove it is possible to write a teacher friendly/teacher useful mathematics book that connects theory and practice. This book may become the primary teacher's "Math Bible"' - Angela Giglio Andrews, Primary Intervention Specialist and Coordinator, and Assistant Professor of Mathematics Education, National Louis University 'Teaching Number in the Classroom translates years of research into a very understandable and comprehensive approach for teaching children how the number system is structured and how to think like a mathematician. For too many years there has been the perception that children who are struggling with mathematics don't know the basic facts. The reality is that these children lack number knowledge and skills. Teaching Number in the Classroom will guide the educational professional through the steps of understanding the development of "number sense", identifying the current levels of knowledge and providing instruction that helps children use the "framework of mathematics" to solve number problems. Teaching Number in the Classroom is a thinking skills approach to mathematics. Children are taught a variety of strategies for solving mathematical problems. The teacher using this book will be able to help all children develop a strong foundation of mathematical understanding' - Carol Meland, K-3rd Grade Principal for the School District of Milton Wisconsin, USA Teaching Number in the Classroom with 4-8 year olds is an absolute "must-have" for all educators involved in early number. Based on sound theoretical foundations, it offers a wealth of down-to-earth, tried and tested, effective approaches to teaching early number concepts and skills. It is a clearly a book written by teachers for teachers. Every single activity in the book is a nugget. Engaging with these activities will change your whole approach to teaching early number' - Noreen O'Loughlin, Associate Vice-President/Lecturer in Maths Education, Mary Immaculate College, University of Limerick Following the success of their previous bestselling titles, Early Numeracy and Teaching Number, the authors of this brand-new text now bring the principles and practice of their acclaimed Mathematics Recovery Programme to whole-class teaching. Central to the book is the concept of an inquiry-based approach to classroom instruction, and topics covered range from beginning number and early counting strategies to multi-digit addition and subtraction right through to multiplication and division. As world leaders in the field of Mathematics Recovery, this book's authors have drawn on their vast experience to create a user-friendly, practical guide focusing on classroom teaching. With its step-by-step approach, the text can be used as a training manual and course reference by teachers everywhere. Key features which make the book such a valuable tool include: - Real-life examples from classroom work - Teaching activities - Assessment tasks - Guidance on classroom organization and teaching specific topics - Activities for parents to do with children An invaluable resource for experienced mathematics recovery teachers, as well as all primary classroom teachers, from kindergarten level to Year three, this text will also be of use to classroom assistants and learning support personnel. Primary mathematics advisors, numeracy consultants and educational psychologists will also find it helpful.

"A multimedia professional learning resource"--Cover.