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Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aims to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. “Applications of Combinatorial Optimization” is presenting a certain number among the most common and well-known applications of Combinatorial Optimization.
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Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aims to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. “Paradigms of Combinatorial Optimization” is divided in two parts: • Paradigmatic Problems, that handles several famous combinatorial optimization problems as max cut, min coloring, optimal satisfiability tsp, etc., the study of which has largely contributed to both the development, the legitimization and the establishment of the Combinatorial Optimization as one of the most active actual scientific domains; • Classical and New Approaches, that presents the several methodological approaches that fertilize and are fertilized by Combinatorial optimization such as: Polynomial Approximation, Online Computation, Robustness, etc., and, more recently, Algorithmic Game Theory.
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Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aim to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. Concepts of Combinatorial Optimization, is divided into three parts: – On the complexity of combinatorial optimization problems, presenting basics about worst-case and randomized complexity; – Classical solution methods, presenting the two most-known methods for solving hard combinatorial optimization problems, that are Branch-and-Bound and Dynamic Programming; – Elements from mathematical programming, presenting fundamentals from mathematical programming based methods that are in the heart of Operations Research since the origins of this field.
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Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aim to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. Concepts of Combinatorial Optimization, is divided into three parts: – On the complexity of combinatorial optimization problems, presenting basics about worst-case and randomized complexity; – Classical solution methods, presenting the two most-known methods for solving hard combinatorial optimization problems, that are Branch-and-Bound and Dynamic Programming; – Elements from mathematical programming, presenting fundamentals from mathematical programming based methods that are in the heart of Operations Research since the origins of this field.
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13721.77 руб. или Купить в рассрочку!

Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aims to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. Concepts of Combinatorial Optimization, is divided into three parts: On the complexity of combinatorial optimization problems, that presents basics about worst-case and randomized complexity; Classical solution methods, that presents the two most-known methods for solving hard combinatorial optimization problems, that are Branch-and-Bound and Dynamic Programming; Elements from mathematical programming, that presents fundamentals from mathematical programming based methods that are in the heart of Operations Research since the origins of this field.
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Combinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aim to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. Concepts of Combinatorial Optimization, is divided into three parts: – On the complexity of combinatorial optimization problems, presenting basics about worst-case and randomized complexity; – Classical solution methods, presenting the two most-known methods for solving hard combinatorial optimization problems, that are Branch-and-Bound and Dynamic Programming; – Elements from mathematical programming, presenting fundamentals from mathematical programming based methods that are in the heart of Operations Research since the origins of this field.
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Praise for the First Edition “This excellent text should prove a useful accoutrement for any developing mathematics program . . . it’s short, it’s sweet, it’s beautifully written.” —The Mathematical Intelligencer “Erickson has prepared an exemplary work . . . strongly recommended for inclusion in undergraduate-level library collections.” —Choice Featuring a modern approach, Introduction to Combinatorics, Second Edition illustrates the applicability of combinatorial methods and discusses topics that are not typically addressed in literature, such as Alcuin’s sequence, Rook paths, and Leech’s lattice. The book also presents fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise questions and observations. Many important combinatorial methods are revisited and repeated several times throughout the book in exercises, examples, theorems, and proofs alike, allowing readers to build confidence and reinforce their understanding of complex material. In addition, the author successfully guides readers step-by-step through three major achievements of combinatorics: Van der Waerden’s theorem on arithmetic progressions, Pólya’s graph enumeration formula, and Leech’s 24-dimensional lattice. Along with updated tables and references that reflect recent advances in various areas, such as error-correcting codes and combinatorial designs, the Second Edition also features: Many new exercises to help readers understand and apply combinatorial techniques and ideas A deeper, investigative study of combinatorics through exercises requiring the use of computer programs Over fifty new examples, ranging in level from routine to advanced, that illustrate important combinatorial concepts Basic principles and theories in combinatorics as well as new and innovative results in the field Introduction to Combinatorics, Second Edition is an ideal textbook for a one- or two-semester sequence in combinatorics, graph theory, and discrete mathematics at the upper-undergraduate level. The book is also an excellent reference for anyone interested in the various applications of elementary combinatorics.
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This book is the first of a series which focuses on the interpolation and extrapolation of optimal designs, an area with significant applications in engineering, physics, chemistry and most experimental fields. In this volume, the authors emphasize the importance of problems associated with the construction of design. After a brief introduction on how the theory of optimal designs meets the theory of the uniform approximation of functions, the authors introduce the basic elements to design planning and link the statistical theory of optimal design and the theory of the uniform approximation of functions. The appendices provide the reader with material to accompany the proofs discussed throughout the book.
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Combinatorial auctions are auctions in which each bid can be placed on a set of items, as opposed to standard auctions, in which each bid is placed on a single item. The winner determination problem for combinatorial auctions is known to be NP-complete. One of the approaches to cope with the hardness of the problem is to identify tractable classes of combinatorial auctions by means of hypertree decompositions. The winner determination problem is tractable on the class of instances with corresponding dual hypergraphs having hypertree width bounded by a fixed natural number. This book describes an optimal algorithm, called ComputeSetPackingK, for solving the winner determination problem based on these ideas. The algorithm was implemented, and experimental results are also presented.
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The International Workshop on Combinatorial Algorithms was established in 1989 as the Australasian Workshop on Combinatorial Algorithms. As a consequence of the workshop's success in attracting mathematicians and computer scientists from around the world, it was decided at the 2006 meeting to go global, to change the workshop's name, and to hold it in appropriate venues around the world. The workshop supports basic research on the interface between mathematics and computing, specifically * Algorithms & Data Structures * Complexity Theory * Algorithms on Graphs & Strings * Combinatorial Optimization * Cryptography & Information Security * Computational Biology * Communications Networks and many other related areas. This is Volume 2 in the series of IWOCA proceedings. See
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The 14 chapters in this monograph provide support for mathematics teacher educators in both their Practical Knowledge and their Professional Knowledge. Individually, these articles provide insights into advancing our thinking about professional development, teacher preparation, and program development. Collectively, they have the potential to help the field of mathematics teacher education move forward in framing effective practices in mathematics teacher education and developing a focused, cohesive research agenda. ATME's Monograph 5, therefore, is a superb resource for mathematics teacher education.
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Introductory textbook on number-theoretic combinatorics. Combinatorial problems of distribution and occupancy are studied using a number-theoretic viewpoint. This book is best suited for advanced undergraduates or beginning graduate students in mathematics.
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Engage students in mathematics using growth mindset techniques The most challenging parts of teaching mathematics are engaging students and helping them understand the connections between mathematics concepts. In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the fourth-grade level through visualization, play, and investigation. During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same message—that they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So the authors designed Mindset Mathematics around the principle of active student engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to improve student test scores, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks in Mindset Mathematics reflect the lessons from brain science that: There is no such thing as a math person – anyone can learn mathematics to high levels. Mistakes, struggle and challenge are the most important times for brain growth. Speed is unimportant in mathematics. Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics. With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics, Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.
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Engage students in mathematics using growth mindset techniques The most challenging parts of teaching mathematics are engaging students and helping them understand the connections between mathematics concepts. In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the sixth-grade level through visualization, play, and investigation. During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same message—that they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So the authors designed Mindset Mathematics around the principle of active student engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to improve student test scores, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks in Mindset Mathematics reflect the lessons from brain science that: There is no such thing as a math person – anyone can learn mathematics to high levels. Mistakes, struggle and challenge are the most important times for brain growth. Speed is unimportant in mathematics. Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics. With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics, Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.
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Curriculum materials are among the most pervasive and powerful influences on schoolmathematics. In many mathematics classes, student assignments, the questions theteacher asks, the ways students are grouped, the forms of assessment, and much moreoriginate in curriculum materials. At the same time, teachers have considerable latitude inhow they use their curriculum materials. Two classes making use of the same materialsmay differ markedly in what mathematics content is emphasized and how students are engaged in learning that content.This volume considers a variety of research tools for investigating the enactment of mathematics curriculum materials,describing the conceptualization, development, and uses of seven sets of tools. Mathematics education researchers,curriculum developers, teacher educators, district supervisors, teacher leaders, and math coaches will find insights that canimprove their work, and guidance for selecting, adapting, and using tools for understanding the complex relationshipbetween curriculum materials and their enactment in classroom instruction.
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The mathematics curriculum - what mathematics is taught, to whom it is taught, and when it is taught - is the bedrock to understanding what mathematics students can, could, and should learn. Today's digital technology influences the mathematics curriculum in two quite different ways. One influence is on the delivery of mathematics through hardware such as desktops, laptops, and tablets. Another influence is on the doing of mathematics using software available on this hardware, but also available on the internet, calculators, or smart phones.These developments, rapidly increasing in their availability and decreasing in their cost, raise fundamental questions regarding a mathematics curriculum that has traditionally been focused on paper-and-pencil work and taught in many places as a set of rules to be practiced and learned.This volume presents the talks given at a conference held in 2014 at the University of Chicago, sponsored by the Center for the Study of Mathematics Curriculum. The speakers - experts from around the world and inside the USA - were asked to discuss one or more of the following topics:• changes in the nature and creation of curricular materials available to students• transformations in how students learn and how they demonstrate their learning• rethinking the role of the teacher and how students and teachers interact within a classroom and across distances from each otherThe result is a set of articles that are interesting and captivating, and challenge u...
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Engage students in mathematics using growth mindset techniques The most challenging parts of teaching mathematics are engaging students and helping them understand the connections between mathematics concepts. In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the third-grade level through visualization, play, and investigation. During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same message—that they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So the authors designed Mindset Mathematics around the principle of active student engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to improve student test scores, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks in Mindset Mathematics reflect the lessons from brain science that: There is no such thing as a math person – anyone can learn mathematics to high levels. Mistakes, struggle and challenge are the most important times for brain growth. Speed is unimportant in mathematics. Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics. With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics, Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.
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A perennial discussion about teacher development is the optimal content background for teachers. Inrecent years, that discussion has taken center stage in the work of mathematics education researchers,mathematicians, mathematics professional developers, and mathematics education policymakers. Muchof the existing and prior work in this area has been directed toward mathematical knowledge forteaching at the elementary level. The work described in this volume takes a sometimes-neglectedapproach, focusing on the dynamic nature of mathematical understanding rather than on a stabledescription of mathematical knowledge, and on mathematics for secondary teaching rather thanmathematics for teaching at the elementary level.The work reported in Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations is a practice-basedresponse to the question of what mathematical understandings secondary teachers could productively use in their teaching. For each of more than 50events, our team of almost 50 mathematics educators who were experienced mathematics teachers developed descriptions of the mathematics thatteachers could use-each of those descriptions (consisting of the event and the mathematics related to the event) is what we call a Situation. Wedeveloped our Framework for Mathematical Understanding for Secondary Teaching (MUST) based on an analysis of our entire set of Situations. Wecall the work practice-based because the MUST framework is based on actu...
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Provides timely applications, modifications, and extensions of experimental designs for a variety of disciplines Design and Analysis of Experiments, Volume 3: Special Designs and Applications continues building upon the philosophical foundations of experimental design by providing important, modern applications of experimental design to the many fields that utilize them. The book also presents optimal and efficient designs for practice and covers key topics in current statistical research. Featuring contributions from leading researchers and academics, the book demonstrates how the presented concepts are used across various fields from genetics and medicinal and pharmaceutical research to manufacturing, engineering, and national security. Each chapter includes an introduction followed by the historical background as well as in-depth procedures that aid in the construction and analysis of the discussed designs. Topical coverage includes: Genetic cross experiments, microarray experiments, and variety trials Clinical trials, group-sequential designs, and adaptive designs Fractional factorial and search, choice, and optimal designs for generalized linear models Computer experiments with applications to homeland security Robust parameter designs and split-plot type response surface designs Analysis of directional data experiments Throughout the book, illustrative and numerical examples utilize SAS®, JMP®, and R software programs to demonstrate the discussed techniques. Related data sets and software applications are available on the book's related FTP site. Design and Analysis of Experiments, Volume 3 is an ideal textbook for graduate courses in experimental design and also serves as a practical, hands-on reference for statisticians and researchers across a wide array of subject areas, including biological sciences, engineering, medicine, and business.
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Written by two well-known scholars in the field, Combinatorial Reasoning: An Introduction to the Art of Counting presents a clear and comprehensive introduction to the concepts and methodology of beginning combinatorics. Focusing on modern techniques and applications, the book develops a variety of effective approaches to solving counting problems. Balancing abstract ideas with specific topical coverage, the book utilizes real world examples with problems ranging from basic calculations that are designed to develop fundamental concepts to more challenging exercises that allow for a deeper exploration of complex combinatorial situations. Simple cases are treated first before moving on to general and more advanced cases. Additional features of the book include: • Approximately 700 carefully structured problems designed for readers at multiple levels, many with hints and/or short answers • Numerous examples that illustrate problem solving using both combinatorial reasoning and sophisticated algorithmic methods • A novel approach to the study of recurrence sequences, which simplifies many proofs and calculations • Concrete examples and diagrams interspersed throughout to further aid comprehension of abstract concepts • A chapter-by-chapter review to clarify the most crucial concepts covered Combinatorial Reasoning: An Introduction to the Art of Counting is an excellent textbook for upper-undergraduate and beginning graduate-level courses on introductory combinatorics and discrete mathematics.
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Professionals in all areas – business; government; the physical, life, and social sciences; engineering; medicine, etc. – benefit from using statistical experimental design to better understand their worlds and then use that understanding to improve the products, processes, and programs they are responsible for. This book aims to provide the practitioners of tomorrow with a memorable, easy to read, engaging guide to statistics and experimental design. This book uses examples, drawn from a variety of established texts, and embeds them in a business or scientific context, seasoned with a dash of humor, to emphasize the issues and ideas that led to the experiment and the what-do-we-do-next? steps after the experiment. Graphical data displays are emphasized as means of discovery and communication and formulas are minimized, with a focus on interpreting the results that software produce. The role of subject-matter knowledge, and passion, is also illustrated. The examples do not require specialized knowledge, and the lessons they contain are transferrable to other contexts. Fundamentals of Statistical Experimental Design and Analysis introduces the basic elements of an experimental design, and the basic concepts underlying statistical analyses. Subsequent chapters address the following families of experimental designs: Completely Randomized designs, with single or multiple treatment factors, quantitative or qualitative Randomized Block designs Latin Square designs Split-Unit designs Repeated Measures designs Robust designs Optimal designs Written in an accessible, student-friendly style, this book is suitable for a general audience and particularly for those professionals seeking to improve and apply their understanding of experimental design.
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Through the chapters in this volume we learn about the research foci and/ or questions that these classroom teachers are interested in examining,the mathematics content through which they engaged their students in these explorations, the data sources they used to make sense oftheir focus and questions, and their roles in the research.An overarching theme through all the chapters in this volume is the learning and professional development that occurs through teacherresearch. What these authors learned about student learning and their own teaching practice far exceeded the focus of their particularresearch questions. For some, the research validated their beliefs and instructional practices; for others, it deepened or extended their understandingof mathematics, or raised their expectations of students' capabilities. For all, it is fair to say, their research increased their awarenessof how students come to know and understand mathematics, and enabled them to gain insight into the complexity of teaching. Lampert(2001) noted, "One reason teaching is a complex practice is that many of the problems a teacher must address to get students to learn occursimultaneously, not one after another" (p. 2). We, as readers, gain a window into these teachers' research within and about the complexity ofclassroom teaching.
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A UNIQUE ENGINEERING AND STATISTICAL APPROACH TO OPTIMAL RESOURCE ALLOCATION Optimal Resource Allocation: With Practical Statistical Applications and Theory features the application of probabilistic and statistical methods used in reliability engineering during the different phases of life cycles of technical systems. Bridging the gap between reliability engineering and applied mathematics, the book outlines different approaches to optimal resource allocation and various applications of models and algorithms for solving real-world problems. In addition, the fundamental background on optimization theory and various illustrative numerical examples are provided. The book also features: An overview of various approaches to optimal resource allocation, from classical Lagrange methods to modern algorithms based on ideas of evolution in biology Numerous exercises and case studies from a variety of areas, including communications, transportation, energy transmission, and counterterrorism protection The applied methods of optimization with various methods of optimal redundancy problem solutions as well as the numerical examples and statistical methods needed to solve the problems Practical thoughts, opinions, and judgments on real-world applications of reliability theory and solves practical problems using mathematical models and algorithms Optimal Resource Allocation is a must-have guide for electrical, mechanical, and reliability engineers dealing with engineering design and optimal reliability problems. In addition, the book is excellent for graduate and PhD-level courses in reliability theory and optimization.
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This book provides examples of the ways in which 9-12 grade mathematics teachers from across North America are engaging inresearch. It offers a glimpse of the questions that capture the attention of teachers, the methodologies that they use to gather data, and theways in which they make sense of what they find. The focus of these teachers' investigations into mathematics classrooms ranges from students'understanding of content to pedagogical changes to social issues. Underlying the chapters is the common goal of enabling students todevelop a deep understanding of the mathematics they learn in their classrooms.By opening their analysis of their classroom practice to our inspection, these courageous teachers have invited us to think alongwith them and to learn more about our own teaching as a result. By sharing their work, they have given the mathematics education communityan important opportunity. Everyone who reads this book-teachers, researchers, teacher-researchers, policy makers, administrators, andothers interested in mathematics education-can learn from the findings and the light that they shed on issues important to mathematics education.This book, and the series of which it is a part, also provides the opportunity to step back and reflect on what can be learned aboutresearch from teachers who have engaged in the process. Areas of insight include: (a) the importance of collaboration and participation incommunities that value research, (b) the potential of teache...
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The driving forces behind mathematics learning trajectories is the need to understand how children actuallylearn and make sense of mathematics-how they progress from prior knowledge, through intermediateunderstandings, to the mathematics target understandings-and how to use these insights to improveinstruction and student learning. In this book, readers will come to understand what learning trajectories are,the research and methodology that are necessary for developing them, and gain insight into potentialapplications of learning trajectories.A synthesis and research outcome in their own right, learning trajectories provide detailed description ofinstructionally-grounded development of mathematical concepts and reasoning from the perspective of student learning, and, overall, building ondecades of accumulated experience in mathematics education research. However, their greater importance may lie in their potential as frameworks thatcontribute an unprecedented coherence across classroom instruction, professional development, standards, and assessment, by focusing squarely onconceptual understanding and reasoning instead of assessment-driven procedural knowledge. This potential was sufficiently compelling as anorganizing framework to have been cited as a basis for the Common Core mathematics standards, the new mathematics learning expectations that arenow consistent across most of the United States. (Among the conference attendees were the writers of the Common Core State Stand...
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This book shows how mathematics is used in developing economic theory and in applied economic analysis. The text gradually develops the mathematical skills needed by students and allows them to progress at their own pace. A wide variety of examples shows how, and why, the application of mathematics has become essential to economists.
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How many ways do exist to mix different ingredients, how many chances to win a gambling game, how many possible paths going from one place to another in a network ? To this kind of questions Mathematics applied to computer gives a stimulating and exhaustive answer. This text, presented in three parts (Combinatorics, Probability, Graphs) addresses all those who wish to acquire basic or advanced knowledge in combinatorial theories. It is actually also used as a textbook. Basic and advanced theoretical elements are presented through simple applications like the Sudoku game, search engine algorithm and other easy to grasp applications. Through the progression from simple to complex, the teacher acquires knowledge of the state of the art of combinatorial theory. The non conventional simultaneous presentation of algorithms, programs and theory permits a powerful mixture of theory and practice. All in all, the originality of this approach gives a refreshing view on combinatorial theory.
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A new and unique way of understanding the translation of concepts and natural language into mathematical expressions Transforming a body of text into corresponding mathematical expressions and models is traditionally viewed and taught as a mathematical problem; it is also a task that most find difficult. The Language of Mathematics: Utilizing Math in Practice reveals a new way to view this process—not as a mathematical problem, but as a translation, or language, problem. By presenting the language of mathematics explicitly and systematically, this book helps readers to learn mathematics¿and improve their ability to apply mathematics more efficiently and effectively to practical problems in their own work. Using parts of speech to identify variables and functions in a mathematical model is a new approach, as is the insight that examining aspects of grammar is highly useful when formulating a corresponding mathematical model. This book identifies the basic elements of the language of mathematics, such as values, variables, and functions, while presenting the grammatical rules for combining them into expressions and other structures. The author describes and defines different notational forms for expressions, and also identifies the relationships between parts of speech and other grammatical elements in English and components of expressions in the language of mathematics. Extensive examples are used throughout that cover a wide range of real-world problems and feature diagrams and tables to facilitate understanding. The Language of Mathematics is a thought-provoking book of interest for readers who would like to learn more about the linguistic nature and aspects of mathematical notation. The book also serves as a valuable supplement for engineers, technicians, managers, and consultants who would like to improve their ability to apply mathematics effectively, systematically, and efficiently to practical problems.
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As we enter the new millennium, combinatorial chemistry is providing significant impetus to new innovations in synthetic chemistry. Combinatorial chemistry has rapidly become the rising star among research methods, allowing scientists to efficiently test the feasibility of a multitude of new compounds. The pursuit of new drugs is but one challenging field in which these combinatorial methods are particularly advantageous, helping researchers meet the modern-day demands of a highly competitive environment. This book emphasises that modern combinatorial synthesis is possible not only in the solid phase, but also in solutions. Moreover, it discusses computer-assisted methods as well as the apparatus and instrumentation required for the combinatorial method. Successful and experienced researchers in the leading pharmaceutical companies and most renowned research institutes offer a solid insight and perspective into this diverse field. A 'must' for every scientist in the area of pharmaceutical research
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A volume in Research in Mathematics EducationSeries Editor Barbara J. Dougherty, University of MississippiThis volume contains the proceedings of the First International Curriculum Conference sponsored by theCenter for the Study of Mathematics Curriculum (CSMC). The CSMC is one of the National Science FoundationCenters for Learning and Teaching (Award No. ESI-0333879).The countries-China, Japan, Korea, and Singapore (in alphabetical order, which also happens to be theorder of their populations)-have each been in the news because of their performance on international testsand/or their economic performance and potential. They also have centralized education ministries that createa single mathematics curriculum framework followed in the entire country.
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A new and unique way of understanding the translation of concepts and natural language into mathematical expressions Transforming a body of text into corresponding mathematical expressions and models is traditionally viewed and taught as a mathematical problem; it is also a task that most find difficult. The Language of Mathematics: Utilizing Math in Practice reveals a new way to view this process—not as a mathematical problem, but as a translation, or language, problem. By presenting the language of mathematics explicitly and systematically, this book helps readers to learn mathematics¿and improve their ability to apply mathematics more efficiently and effectively to practical problems in their own work. Using parts of speech to identify variables and functions in a mathematical model is a new approach, as is the insight that examining aspects of grammar is highly useful when formulating a corresponding mathematical model. This book identifies the basic elements of the language of mathematics, such as values, variables, and functions, while presenting the grammatical rules for combining them into expressions and other structures. The author describes and defines different notational forms for expressions, and also identifies the relationships between parts of speech and other grammatical elements in English and components of expressions in the language of mathematics. Extensive examples are used throughout that cover a wide range of real-world problems and feature diagrams and tables to facilitate understanding. The Language of Mathematics is a thought-provoking book of interest for readers who would like to learn more about the linguistic nature and aspects of mathematical notation. The book also serves as a valuable supplement for engineers, technicians, managers, and consultants who would like to improve their ability to apply mathematics effectively, systematically, and efficiently to practical problems.
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In this paper we show that for the D-optimal design, departures from the design are much less important than a departure from the model. As a consequence, we propose, based on D-optimality, a rule for choosing the regression degree. We also study different types of departures from the model to define a new class of D-optimal designs, which is robust and more efficient than the uniform one.
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Graphing calculators are a mainstay in the U.S. high school mathematics curriculum and because of that considerable research has been done on their effect in the mathematics classroom. While much of this work has focused on the impact of graphing calculator use on achievement and attitude, little has been done to address how students use the graphing calculator when they are working in independent situations or their perceptions of how the graphing calculator impacts their mathematical experience. As such, the work in this book provides a new lens through which to view graphing calculator use, the lens of the student. Six case studies of high school calculus students, their words and their actions, provide insight into the reasons why and the ways with which these students incorporate graphing calculator use in their independent mathematics activity. The results of the study will be of interest to both mathematics teachers and mathematics education researchers who are interested in the role that technology plays in students' mathematical activity.
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Macmillan Mathematics is a six-level Primary Mathematics course designed to meet the needs of international learners. The Teacher's Guide includes facsimiles of the Pupil's Book pages, answers to all the exercises, and supportive lesson notes to ensure that teachers get the most out of the course and their pupils.
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Mathematics teacher education has a critical role to play in preparing teachers to put at center stage goals tosupport equity in mathematics education and to diversify student interest and participation in mathematics.These goals must also resonate with broader public interest goals to improve educational and socialconditions both in the U.S. and abroad. The Mathematics Teacher Education in the Public Interest book aimsto support mathematics teacher educators to prepare teachers with new knowledge and skills to support allstudents to learn mathematics and to become informed, engaged, and critical citizens within theircommunity, nation, and world. While internationally there is considerable interest among mathematicseducators in issues of equity and social justice, the literature on mathematics teacher education for equity andsocial justice thus far has been very limited.The book provides theoretical discussions on the need for equity and social justice emphases inmathematics teacher education, as well as practical examples from mathematics teacher educators, documenting their own professional efforts tocenter practices on equity and social justice. Section emphases include critical perspectives on mathematics teacher education, the use of equity andsocial justice-themed activities in mathematics teacher preparation courses, and issues of identity and community and cultural contexts in mathematicsteacher education. In addition syntheses of major ideas of the book are offered b...
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Macmillan Mathematics is a six-level Primary Mathematics course designed to meet the needs of international learners. The Teacher's Guide includes facsimiles of the Pupil's Book pages, answers to all the exercises, and supportive lesson notes to ensure that teachers get the most out of the course and their pupils.
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Macmillan Mathematics is a six-level Primary Mathematics course designed to meet the needs of international learners. The Teacher's Guide includes facsimiles of the Pupil's Book pages, answers to all the exercises, and supportive lesson notes to ensure that teachers get the most out of the course and their pupils.
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With a growing number of scientists and engineers using JMP software for design of experiments, there is a need for an example-driven book that supports the most widely used textbook on the subject, Design and Analysis of Experiments by Douglas C. Montgomery. Design and Analysis of Experiments by Douglas Montgomery: A Supplement for Using JMP meets this need and demonstrates all of the examples from the Montgomery text using JMP. In addition to scientists and engineers, undergraduate and graduate students will benefit greatly from this book. While users need to learn the theory, they also need to learn how to implement this theory efficiently on their academic projects and industry problems. In this first book of its kind using JMP software, Rushing, Karl and Wisnowski demonstrate how to design and analyze experiments for improving the quality, efficiency, and performance of working systems using JMP. Topics include JMP software, two-sample t-test, ANOVA, regression, design of experiments, blocking, factorial designs, fractional-factorial designs, central composite designs, Box-Behnken designs, split-plot designs, optimal designs, mixture designs, and 2 k factorial designs. JMP platforms used include Custom Design, Screening Design, Response Surface Design, Mixture Design, Distribution, Fit Y by X, Matched Pairs, Fit Model, and Profiler. With JMP software, Montgomery's textbook, and Design and Analysis of Experiments by Douglas Montgomery: A Supplement for Using JMP, user...
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Japan, South Korea, Singapore and India are the countries that have strong and dynamic economy in the 21st century. One of the most important factors which contributes to these countries to come to the existing position is due to the curriculum that they have designed in mathematics and science subjects for their young generations. The authors have tried to go through the curriculum 0f these countries starting from kindergarten to high school. The book shows how they have laid foundation in mathematics and science in the ladder of their education system. We encourage scholars to go through the book and check what their country education looks like by taking the indicators that we used to compare. After reading the book, they will have an opportunity to learn from these countries and forward some valuable intellectual inputs to their countries. These days it has become common to call South Korea by the name of "Dynamic Korea". This is the result of the science and mathematics curriculum they have designed for their competency. Finally we strongly encourage all developing and developed nations who are attempting to progress and bring sustainable development have to take time and lear
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This book focuses on shame: a reaction to other people's criticisms and an emotional response to knowing and doing mathematics. It is suggested that shame is a crucial feature of primary teachers' professional identity, since it appears to disrupt their relationship with mathematics and a sense of disconnectedness to it. Three factors are being explored as regards to their action on inducing shame, namely the nature of school mathematics, teachers' personal experiences with mathematics and the gendering of mathematics. Furthermore, a discussion for turning shame on its head is attempted. In specific, it is stressed out that there is a need for restructuring primary teachers' professional identity by creating conditions under which mathematics is demystified, in order for teachers to (re)connect with the subject. Participating in communities of practice and developing different kinds of relationships are suggested to be productive ways of influencing primary teachers' beliefs, attitudes and emotions about mathematics.
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A volume in Research on Technology and the Teaching and Learningof Mathematics: Syntheses, Cases, and PerspectivesAccording to NCTM's Principles and Standards for School Mathematics, "Technology isessential in teaching and learning of mathematics; it influences the mathematics that is taughtand it enhances students' learning." How does research inform this clarion call for technology inmathematics teaching and learning? In response to the need to craft appropriate roles fortechnology in school mathematics new technological approaches have been applied to theteaching and learning of mathematics, and these approaches have been examined by researchersworld-wide.The second volume has a dual focus: cases and perspectives. It features descriptive cases thatprovide accounts of the development of technology-intensive curriculum and tools. In thesecases the writers describe and analyze various roles that research played in their developmentwork and ways in which research, curriculum development, and tool development can informeach other. These thoughtful descriptions and analyses provide documentation of how this processcan and does occur. The remaining chapters in the second volume address research-related issues and perspectives on the use oftechnology in the teaching and learning of mathematics. The lessons learned from the research presented in these volumes are lessonsabout teaching and learning that can be applied more broadly than solely in technological setting...
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Praise for the Second Edition «An amazing assemblage of worldwide contributions in mathematics and, in addition to use as a course book, a valuable resource . . . essential.» —CHOICE This Third Edition of The History of Mathematics examines the elementary arithmetic, geometry, and algebra of numerous cultures, tracing their usage from Mesopotamia, Egypt, Greece, India, China, and Japan all the way to Europe during the Medieval and Renaissance periods where calculus was developed. Aimed primarily at undergraduate students studying the history of mathematics for science, engineering, and secondary education, the book focuses on three main ideas: the facts of who, what, when, and where major advances in mathematics took place; the type of mathematics involved at the time; and the integration of this information into a coherent picture of the development of mathematics. In addition, the book features carefully designed problems that guide readers to a fuller understanding of the relevant mathematics and its social and historical context. Chapter-end exercises, numerous photographs, and a listing of related websites are also included for readers who wish to pursue a specialized topic in more depth. Additional features of The History of Mathematics, Third Edition include: Material arranged in a chronological and cultural context Specific parts of the history of mathematics presented as individual lessons New and revised exercises ranging between technical, factual, and integrative Individual PowerPoint presentations for each chapter and a bank of homework and test questions (in addition to the exercises in the book) An emphasis on geography, culture, and mathematics In addition to being an ideal coursebook for undergraduate students, the book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the history of mathematics.
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In this book, Thom Poole explores the psychology, perceptions, mythology and business of colours through the ages and around the world. Using this understanding, brand marketers and graphic designers can better adapt their designs to influence and engage their audiences.
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The author of this book presents a general, robust, and easy-to-use method that can handle many design parameters efficiently. Following an introduction, Chapter 1 presents the general concepts of truss layout optimization, starting from topology optimization where structural component sizes and system connectivity are simultaneously optimized. To fully realize the potential of truss layout optimization for the design of lightweight structures, the consideration of geometrical variables is then introduced. Chapter 2 addresses truss geometry and topology optimization by combining mathematical programming and structural mechanics: the structural properties of the optimal solution are used for devising the novel formulation. To avoid singularities arising in optimal configurations, this approach disaggregates the equilibrium equations and fully integrates their basic elements within the optimization formulation. The resulting tool incorporates elastic and plastic design, stress and displacement constraints, as well as self-weight and multiple loading. The inherent slenderness of lightweight structures requires the study of stability issues. As a remedy, Chapter 3 proposes a conceptually simple but efficient method to include local and nodal stability constraints in the formulation. Several numerical examples illustrate the impact of stability considerations on the optimal design. Finally, the investigation on realistic design problems in Chapter 4 confirms the practical applicability of the proposed method. It is shown how we can generate a range of optimal designs by varying design settings.
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A volume in Research on Technology and the Teaching and Learningof Mathematics: Syntheses, Cases, and PerspectivesAccording to NCTM's Principles and Standards for School Mathematics, "Technology isessential in teaching and learning of mathematics; it influences the mathematics that is taughtand it enhances students' learning." How does research inform this clarion call for technology inmathematics teaching and learning? In response to the need to craft appropriate roles fortechnology in school mathematics new technological approaches have been applied to theteaching and learning of mathematics, and these approaches have been examined by researchersworld-wide.The first volume provides insight into what research suggests about the nature of mathematicslearning in technological environments. Included in this volume are syntheses of research ontechnology in the learning of rational number, algebra, elementary and secondary geometry,mathematical modeling, and calculus. Additional chapters synthesize research on technology in the practice of teaching and on equityissues in the use of technology in mathematics instruction. Instead of simply reporting achievement scores of students who use technologyin their learning, authors provide thoughtful analyses of bodies of research with the goal of understanding the ways in which technologyaffects what and how students learn. Each of the chapters in this volume is written by a team of experts whose own research hasprovided i...
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A Volume in International Perspectives on Mathematics Education - Cognition, Equity & SocietySeries Editor Bharath Sriraman, The University of Montana and Lyn English, Queensland University of TechnologyOver the past two decades, the theoretical interests of mathematics educators have changed substantially-as any brief look at the titles and abstracts ofarticles shows. Largely through the work of Paul Cobb and his various collaborators, mathematics educators came to be attuned to the intricaterelationship between individual and the social configuration of which she or he is part. That is, this body of work, running alongside more traditionalconstructivist and psychological approaches, showed that what happens at the collective level in a classroom both constrains and affords opportunitiesfor what individuals do (their practices). Increasingly, researchers focused on the mediational role of sociomathematical norms and how these emergedfrom the enacted lessons.A second major shift in mathematical theorizing occurred during the past decade: there is an increasing focus on the embodied and bodilymanifestation of mathematical knowing (e.g., Lakoff & Núñez, 2000). Mathematics educators now working from this perspective have come to theirposition from quite different bodies of literatures: for some, linguistic concerns and mathematics as material praxis lay at the origin for their concerns;others came to their position through the literature on the situated nature of cogniti...
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Have you ever wondered why students too often have only a rudimentary understanding of mathematics, why even rich and exciting hands-on learning does not always result in "real" learning of new concepts? The answer lies in whether students have actually learned mathematical concepts, rather than merely memorizing facts and formulas.Concept-Rich Mathematics Instruction is based on the constructivist view that concepts are not simply facts to be memorized and later recalled, but rather knowledge that learners develop through an active process of adapting to new experiences.The teacher's role is critical in this process. When teachers prompt students to reflect on their experiences and report and answer questions verbally, students must re-examine and even revise their concepts of reality.Meir Ben-Hur offers expert guidance on all aspects of Concept-Rich Mathematics Instruction, including• Identifying the core concepts of the mathematics curriculum.• Planning instructional sequences that build upon concepts that students already understand.• Designing learning experiences that provoke thoughtful discussions about new concepts and prepare students to apply these concepts on their own.• Identifying student errors, particularly those caused by preconceptions, as important sources of information and as key instructional tools.• Conducting classroom dialogues that are rich in alternative representations.• Using a variety of formative assessment methods to reveal the state...
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Mathematics research papers provide a forum for all mathematics enthusiasts to exercise their mathematical experience, expertise and excitement. The research paper process epitomizes the differentiation of instruction, as each student chooses their own topic and extends it as far as their desire takes them. The features and benefits of the research paper process offer a natural alignment with all eight Common Core State Standards for Mathematical Practice.Writing Math Research Papers serves both as a text for students and as a resource for instructors and administrators. It systematically describes the steps involved in creating a mathematics research paper and an oral presentation. The chapters offer tips on technical writing, formatting, and preparing visual aids. For instructors and administrators, the book covers the logistics necessary in setting up a mathematics research program in a high school setting. This program received the1997 Chevron Best Practices in Education Award as the premier high school mathematics course in the United States.
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This book presents examples of the scholarship of Brazilian mathematics educators. The authors who contributed to this collection situate their work within the construct of creative insubordination. With this collection we hope to increase the participation of Brazilian scholars in the international conversations of research on mathematics teaching and learning.
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Description: The work contains an analytical review of modern wind energy, based on different physical principles. We consider wind installations with horizontal-axial and vertical-axial configurations, innovative designs for wind turbines. Also, modern methods and possibilities for scaling and optimal regulation of wind parks are considered. Mathematical models for the interaction of airflow with blades of collinear and orthogonal plants have been developed, and methods for optimal regulation have been identified that increase the energy efficiency of plants. A number of patent developments represent promising directions for the development of wind energy: horizontal and vertical collinear installations, vertical-axial orthogonal devices with unregulated and self-regulating blades and with optimal control systems based on the author methods of optimizing.
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This study was designed to make a comparative analysis of mathematics achievement between girls and boys at lower secondary education level. It was also aimed at investigating the extent of association between girls' mathematics achievement and school type. Besides, students' attitude towards mathematics, parents' attitude and teachers' attitude on students' mathematics performance based on their gender were analysed. The study has employed both quantitative and qualitative data gathering methods. Instruments were applied to 1280 students drawn from 10 Addis Ababa lower cycle of secondary schools (grade nine and ten). This book will help other professionals as stepping stone for further study in the area by considering more variables which don't included under this book.
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Presents a uniquely balanced approach that bridges introductory and advanced topics in modern mathematics An accessible treatment of the fundamentals of modern mathematics, Principles of Mathematics: A Primer provides a unique approach to introductory andadvanced mathematical topics. The book features six main subjects, whichcan be studied independently or in conjunction with each other including: settheory; mathematical logic; proof theory; group theory; theory of functions; andlinear algebra. The author begins with comprehensive coverage of the necessary building blocks in mathematics and emphasizes the need to think abstractly and develop an appreciation for mathematical thinking. Maintaining a useful balance of introductory coverage and mathematical rigor, Principles of Mathematics: A Primer features: Detailed explanations of important theorems and their applications Hundreds of completely solved problems throughout each chapter Numerous exercises at the end of each chapter to encourage further exploration Discussions of interesting and provocative issues that spark readers’ curiosity and facilitate a better understanding and appreciation of the field of mathematics Principles of Mathematics: A Primer is an ideal textbook for upper-undergraduate courses in the foundations of mathematics and mathematical logic as well as for graduate-level courses related to physics, engineering, and computer science. The book is also a useful reference for readers interested in pursuing careers in mathematics and the sciences. Vladimir Lepetic, PhD, is Professor in the Department of Mathematical Sciences at DePaul University. His research interests include mathematical physics, set theory, foundations of mathematics, and the philosophy of mathematics.
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From rainbows, river meanders, and shadows to spider webs, honeycombs, and the markings on animal coats, the visible world is full of patterns that can be described mathematically. Examining such readily observable phenomena, this book introduces readers to the beauty of nature as revealed by mathematics and the beauty of mathematics as revealed in nature. Generously illustrated, written in an informal style, and replete with examples from everyday life, Mathematics in Nature is an excellent and undaunting introduction to the ideas and methods of mathematical modeling. It illustrates how mathematics can be used to formulate and solve puzzles observed in nature and to interpret the solutions. In the process, it teaches such topics as the art of estimation and the effects of scale, particularly what happens as things get bigger. Readers will develop an understanding of the symbiosis that exists between basic scientific principles and their mathematical expressions as well as a deeper appreciation for such natural phenomena as cloud formations, halos and glories, tree heights and leaf patterns, butterfly and moth wings, and even puddles and mud cracks. Developed out of a university course, this book makes an ideal supplemental text for courses in applied mathematics and mathematical modeling. It will also appeal to mathematics educators and enthusiasts at all levels, and is designed so that it can be dipped into at leisure.
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Mathematics and Science education have both grown in fertile directions indifferent geographic regions. Yet, the mainstream discourse in international handbooks does not lend voice todevelopments in cognition, curriculum, teacher development, assessment, policy and implementation ofmathematics and science in many countries. Paradoxically, in spite of advances in information technology andthe "flat earth" syndrome, old distinctions and biases between different groups of researcher's persist. In additionlimited accessibility to conferences and journals also contribute to this problem.The International Sourcebooks in Mathematics and Science Education focus on under-represented regions of theworld and provides a platform for researchers to showcase their research anddevelopment in areas within mathematics and science education.The First Sourcebook on Asian Research in Mathematics Education: China,Korea, Singapore, Japan, Malaysia and India provides the first synthesizedtreatment of mathematics education that has both developed and is nowprominently emerging in the Asian and South Asian world. The book isorganized in sections coordinated by leaders in mathematics education inthese countries and editorial teams for each country affiliated with them. Thepurpose of unique sourcebook is to both consolidate and survey theestablished body of research in these countries with findings that haveinfluenced ongoing research agendas and informed practices in Europe,North America (...
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Полный вариант заголовка: «Twelve beautiful designs for farm-houses : with their proper offices, and estimates of the whole... / by William Halfpenny».
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With a refreshing perspective, The Art of Pop-Up shows the designs of pop-up creators all around the world with some of their most innovative designs and practical examples...
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Presents a clear bridge between mathematics and the liberal arts Mathematics for the Liberal Arts provides a comprehensible and precise introduction to modern mathematics intertwined with the history of mathematical discoveries. The book discusses mathematical ideas in the context of the unfolding story of human thought and highlights the application of mathematics in everyday life. Divided into two parts, Mathematics for the Liberal Arts first traces the history of mathematics from the ancient world to the Middle Ages, then moves on to the Renaissance and finishes with the development of modern mathematics. In the second part, the book explores major topics of calculus and number theory, including problem-solving techniques and real-world applications. This book emphasizes learning through doing, presents a practical approach, and features: A detailed explanation of why mathematical principles are true and how the mathematical processes work Numerous figures and diagrams as well as hundreds of worked examples and exercises, aiding readers to further visualize the presented concepts Various real-world practical applications of mathematics, including error-correcting codes and the space shuttle program Vignette biographies of renowned mathematicians Appendices with solutions to selected exercises and suggestions for further reading Mathematics for the Liberal Arts is an excellent introduction to the history and concepts of mathematics for undergraduate liberal arts students and readers in non-scientific fields wishing to gain a better understanding of mathematics and mathematical problem-solving skills.
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Document from the year 2012 in the subject Education - Educational Tests & Measurements, , language: English, abstract: Mathematics is an important discipline of learning at the secondary stage. It predominantly contributes to the development of precision, rational and analytical thinking, reasoning and scientific temper. One of the basic aims of teaching Mathematics at secondary stage is to inculcate the skill of quantification of experience gathered by the learner. But still some of the students find Mathematics as very difficult subject. Among different branches of Mathematics, Algebra is used to enable the learner to apply its knowledge to other branches of Mathematics. The pre-requisite test can help the students to know their weaknesses in the basic concepts and it also helps the teachers to know the level of their students. The authors had constructed a pre-requisite test covering all units of Algebra of standard X and implemented on the randomly selected standard X students of the twelve non-granted Gujarati medium schools of Vadodara, Gujarat, India. Based on the findings of the study, the authors discussed and suggested some ways for the teachers teaching Mathematics at secondary school level.
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A volume in the series: Research in Mathematics Education. Series Editor(s): Barbara J. Dougherty, Iowa State UniversityIn response to No Child Let Behind, states have developed mathematics curriculum frameworks that outline their intended curriculum for grades K-8. While some have indicated that districts or individual schools may use their framework as a model for specific curricular programs, others have taken a more prescriptive or even mandatory stance. Collectively, these frameworks present a sense of the national mathematics program and what we expect students learn.This volume follows The Intended Curriculum as Represented in State Mathematics Curriculum Standards: Consensus or Confusion? (Reys). While the Reys volume focused on number and operations, algebra and reasoning strands, the Smith volume analyzes geometry, measurement, probability, and statistics strands. It also presents an analysis what verbs used tell us about the cognitive demand of grade level expectations. This volume, even more than the Reys volume, emphasizes the theme of variability in the content, expression, and clarity of grade level expectations across the states.As the nation moves toward implementation of the Common Core Standards, this volume highlights some of the challenges teachers and other school personnel face in interpreting mathematics grade-level standards as goals for classroom teaching. The shift from 50 state standards to one document does not resolve this basic challenge.
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Scandinavians are exceptionally gifted in design. They are world-famous for their inimitable, democratic designs which bridge the gap between crafts and industrial... Praise for the First Edition “This excellent text should prove a useful accoutrement for any developing mathematics program . . . it’s short, it’s sweet, it’s beautifully written.” —The Mathematical Intelligencer “Erickson has prepared an exemplary work . . . strongly recommended for inclusion in undergraduate-level library collections.” —Choice Featuring a modern approach, Introduction to Combinatorics, Second Edition illustrates the applicability of combinatorial methods and discusses topics that are not typically addressed in literature, such as Alcuin’s sequence, Rook paths, and Leech’s lattice. The book also presents fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise questions and observations. Many important combinatorial methods are revisited and repeated several times throughout the book in exercises, examples, theorems, and proofs alike, allowing readers to build confidence and reinforce their understanding of complex material. In addition, the author successfully guides readers step-by-step through three major achievements of combinatorics: Van der Waerden’s theorem on arithmetic progressions, Pólya’s graph enumeration formula, and Leech’s 24-dimensional lattice. Along with updated tables and references that reflect recent advances in various areas, such as error-correcting codes and combinatorial designs, the Second Edition also features: Many new exercises to help readers understand and apply combinatorial techniques and ideas A deeper, investigative study of combinatorics through exercises requiring the use of computer programs Over fifty new examples, ranging in level from routine to advanced, that illustrate important combinatorial concepts Basic principles and theories in combinatorics as well as new and innovative results in the field Introduction to Combinatorics, Second Edition is an ideal textbook for a one- or two-semester sequence in combinatorics, graph theory, and discrete mathematics at the upper-undergraduate level. The book is also an excellent reference for anyone interested in the various applications of elementary combinatorics.