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Symbolic Data Analysis: Conceptual Statistics and Data Mining (Wiley Series in Computational Statistics #654)
by Lynne Billard Edwin DidayWith the advent of computers, very large datasets have become routine. Standard statistical methods don’t have the power or flexibility to analyse these efficiently, and extract the required knowledge. An alternative approach is to summarize a large dataset in such a way that the resulting summary dataset is of a manageable size and yet retains as much of the knowledge in the original dataset as possible. One consequence of this is that the data may no longer be formatted as single values, but be represented by lists, intervals, distributions, etc. The summarized data have their own internal structure, which must be taken into account in any analysis. This text presents a unified account of symbolic data, how they arise, and how they are structured. The reader is introduced to symbolic analytic methods described in the consistent statistical framework required to carry out such a summary and subsequent analysis. Presents a detailed overview of the methods and applications of symbolic data analysis. Includes numerous real examples, taken from a variety of application areas, ranging from health and social sciences, to economics and computing. Features exercises at the end of each chapter, enabling the reader to develop their understanding of the theory. Provides a supplementary website featuring links to download the SODAS software developed exclusively for symbolic data analysis, data sets, and further material. Primarily aimed at statisticians and data analysts, Symbolic Data Analysis is also ideal for scientists working on problems involving large volumes of data from a range of disciplines, including computer science, health and the social sciences. There is also much of use to graduate students of statistical data analysis courses.
Symbolic Data Analysis and the SODAS Software
by Edwin Diday Monique Noirhomme-FraitureSymbolic data analysis is a relatively new field that provides a range of methods for analyzing complex datasets. Standard statistical methods do not have the power or flexibility to make sense of very large datasets, and symbolic data analysis techniques have been developed in order to extract knowledge from such data. Symbolic data methods differ from that of data mining, for example, because rather than identifying points of interest in the data, symbolic data methods allow the user to build models of the data and make predictions about future events. This book is the result of the work f a pan-European project team led by Edwin Diday following 3 years work sponsored by EUROSTAT. It includes a full explanation of the new SODAS software developed as a result of this project. The software and methods described highlight the crossover between statistics and computer science, with a particular emphasis on data mining.
Symmetry Analysis of Differential Equations: An Introduction
by Daniel J. ArrigoA self-contained introduction to the methods and techniques of symmetry analysis used to solve ODEs and PDEsSymmetry Analysis of Differential Equations: An Introduction presents an accessible approach to the uses of symmetry methods in solving both ordinary differential equations (ODEs) and partial differential equations (PDEs). Providing comprehensive coverage, the book fills a gap in the literature by discussing elementary symmetry concepts and invariance, including methods for reducing the complexity of ODEs and PDEs in an effort to solve the associated problems. Thoroughly class-tested, the author presents classical methods in a systematic, logical, and well-balanced manner. As the book progresses, the chapters graduate from elementary symmetries and the invariance of algebraic equations, to ODEs and PDEs, followed by coverage of the nonclassical method and compatibility. Symmetry Analysis of Differential Equations: An Introduction also features: Detailed, step-by-step examples to guide readers through the methods of symmetry analysis End-of-chapter exercises, varying from elementary to advanced, with select solutions to aid in the calculation of the presented algorithmic methods Symmetry Analysis of Differential Equations: An Introduction is an ideal textbook for upper-undergraduate and graduate-level courses in symmetry methods and applied mathematics. The book is also a useful reference for professionals in science, physics, and engineering, as well as anyone wishing to learn about the use of symmetry methods in solving differential equations.
Symmetry Analysis of Differential Equations: An Introduction
by Daniel J. ArrigoA self-contained introduction to the methods and techniques of symmetry analysis used to solve ODEs and PDEsSymmetry Analysis of Differential Equations: An Introduction presents an accessible approach to the uses of symmetry methods in solving both ordinary differential equations (ODEs) and partial differential equations (PDEs). Providing comprehensive coverage, the book fills a gap in the literature by discussing elementary symmetry concepts and invariance, including methods for reducing the complexity of ODEs and PDEs in an effort to solve the associated problems. Thoroughly class-tested, the author presents classical methods in a systematic, logical, and well-balanced manner. As the book progresses, the chapters graduate from elementary symmetries and the invariance of algebraic equations, to ODEs and PDEs, followed by coverage of the nonclassical method and compatibility. Symmetry Analysis of Differential Equations: An Introduction also features: Detailed, step-by-step examples to guide readers through the methods of symmetry analysis End-of-chapter exercises, varying from elementary to advanced, with select solutions to aid in the calculation of the presented algorithmic methods Symmetry Analysis of Differential Equations: An Introduction is an ideal textbook for upper-undergraduate and graduate-level courses in symmetry methods and applied mathematics. The book is also a useful reference for professionals in science, physics, and engineering, as well as anyone wishing to learn about the use of symmetry methods in solving differential equations.
Synergetic Agents: From Multi-Robot Systems to Molecular Robotics
by Hermann Haken Paul LeviThis book addresses both multi robot systems and miniaturization to the nanoscale from a unifying point of view, but without leaving aside typical particularities of either. The unifying aspect is based on the concept of information minimization whose precise formulation is the Haken-Levi-principle. The authors introduce basic concepts of multi-component self-organizing systems such as order parameters (well known from equilibrium and non-equilibrium phase transitions) and the slaving principle (which establishes a link to dynamical systems). Among explicit examples is the docking manoeuvre of two robots in two and three dimensions. The second part of the book deals with the rather recently arising field of molecular robotics. It is particularly here where nature has become a highly influential teacher for the construction of robots. In living biological cells astounding phenomena occur: there are molecules (proteins) that literally walk on polymer strands and transport loads that are heavier than their carriers, or molecules that, by joint action, contract muscles. The book provides the reader with an insight into these phenomena, especially by a detailed theoretical treatment of the molecular mechanism of muscle contraction. At the molecular level, for an appropriate approach the use of quantum theory is indispensable. The authors introduce and use it in a form that avoids all the clumsy calculations of wave-functions. They present a model which is based on an elementary version of quantum field theory and allows taking into account the impact of the surrounding on the quantum mechanical activity of a single molecule. By presenting explicit and pedagogical examples, the reader gets acquainted with the appropriate modelling of the walking behaviour of single molecular robots and their collective behaviour. The further development of multi-robot systems and particularly of molecular robots will require the cooperation of a variety of disciplines. Therefore the book appeals to a wide audience including researchers, instructors, and advanced graduate students.
Synergetic Agents: From Multi-Robot Systems to Molecular Robotics
by Hermann Haken Paul LeviThis book addresses both multi robot systems and miniaturization to the nanoscale from a unifying point of view, but without leaving aside typical particularities of either. The unifying aspect is based on the concept of information minimization whose precise formulation is the Haken-Levi-principle. The authors introduce basic concepts of multi-component self-organizing systems such as order parameters (well known from equilibrium and non-equilibrium phase transitions) and the slaving principle (which establishes a link to dynamical systems). Among explicit examples is the docking manoeuvre of two robots in two and three dimensions. The second part of the book deals with the rather recently arising field of molecular robotics. It is particularly here where nature has become a highly influential teacher for the construction of robots. In living biological cells astounding phenomena occur: there are molecules (proteins) that literally walk on polymer strands and transport loads that are heavier than their carriers, or molecules that, by joint action, contract muscles. The book provides the reader with an insight into these phenomena, especially by a detailed theoretical treatment of the molecular mechanism of muscle contraction. At the molecular level, for an appropriate approach the use of quantum theory is indispensable. The authors introduce and use it in a form that avoids all the clumsy calculations of wave-functions. They present a model which is based on an elementary version of quantum field theory and allows taking into account the impact of the surrounding on the quantum mechanical activity of a single molecule. By presenting explicit and pedagogical examples, the reader gets acquainted with the appropriate modelling of the walking behaviour of single molecular robots and their collective behaviour. The further development of multi-robot systems and particularly of molecular robots will require the cooperation of a variety of disciplines. Therefore the book appeals to a wide audience including researchers, instructors, and advanced graduate students.
System Theory -- A Modern Approach, Volume 1: Linear Ordinary and Functional Differential Equations
by Henri BourlèsThe theory of dynamic systems is addressed in this book in accordance with the “modern” approach, heir to algebraic analysis, which has been implemented since the last decade of the 20th century. After a reminder of the evolution of the representation of systems based on transfer functions or matrices, the duality of controllability and observability is revisited, and new results are produced concerning time-varying discrete-time systems. To complete and improve the existing analyses, the poles and zeros of linear systems and their interconnections are presented in a new way, as well as the problem of systems governed by functional differential equations (of retarded or neutral type) and their stabilization. This book also proposes known and original mathematical complements.
System Theory -- A Modern Approach, Volume 1: Linear Ordinary and Functional Differential Equations
by Henri BourlèsThe theory of dynamic systems is addressed in this book in accordance with the "modern" approach, heir to algebraic analysis, which has been implemented since the last decade of the 20th century. After a reminder of the evolution of the representation of systems based on transfer functions or matrices, the duality of controllability and observability is revisited, and new results are produced concerning time-varying discrete-time systems. To complete and improve the existing analyses, the poles and zeros of linear systems and their interconnections are presented in a new way, as well as the problem of systems governed by functional differential equations (of retarded or neutral type) and their stabilization. This book also proposes known and original mathematical complements.
Systematic Reviews in the Social Sciences: A Practical Guide
by Mark Petticrew Helen RobertsSuch diverse thinkers as Lao-Tze, Confucius, and U.S. Defense Secretary Donald Rumsfeld have all pointed out that we need to be able to tell the difference between real and assumed knowledge. The systematic review is a scientific tool that can help with this difficult task. It can help, for example, with appraising, summarising, and communicating the results and implications of otherwise unmanageable quantities of data. This book, written by two highly-respected social scientists, provides an overview of systematic literature review methods: Outlining the rationale and methods of systematic reviews; Giving worked examples from social science and other fields; Applying the practice to all social science disciplines; It requires no previous knowledge, but takes the reader through the process stage by stage; Drawing on examples from such diverse fields as psychology, criminology, education, transport, social welfare, public health, and housing and urban policy, among others. Including detailed sections on assessing the quality of both quantitative, and qualitative research; searching for evidence in the social sciences; meta-analytic and other methods of evidence synthesis; publication bias; heterogeneity; and approaches to dissemination.
Systems Biology: A Textbook
by Edda Klipp Wolfram Liebermeister Christoph Wierling Axel KowaldThis advanced textbook is tailored for an introductory course in Systems Biology and is well-suited for biologists as well as engineers and computer scientists. It comes with student-friendly reading lists and a companion website featuring a short exam prep version of the book and educational modeling programs. The text is written in an easily accessible style and includes numerous worked examples and study questions in each chapter. For this edition, a section on medical systems biology has been included.
Systems Biology: A Powerful Tool In Systems Biology (Advanced Biotechnology #6)
by Sang Yup Lee J. Nielsen Gregory StephanopoulosComprehensive coverage of the many different aspects of systems biology, resulting in an excellent overview of the experimental and computational approaches currently in use to study biological systems. Each chapter represents a valuable introduction to one specific branch of systems biology, while also including the current state of the art and pointers to future directions. Following different methods for the integrative analysis of omics data, the book goes on to describe techniques that allow for the direct quantification of carbon fluxes in large metabolic networks, including the use of 13C labelled substrates and genome-scale metabolic models. The latter is explained on the basis of the model organism Escherichia coli as well as the human metabolism. Subsequently, the authors deal with the application of such techniques to human health and cell factory engineering, with a focus on recent progress in building genome-scale models and regulatory networks. They highlight the importance of such information for specific biological processes, including the ageing of cells, the immune system and organogenesis. The book concludes with a summary of recent advances in genome editing, which have allowed for precise genetic modifications, even with the dynamic control of gene expression. This is part of the Advances Biotechnology series, covering all pertinent aspects of the field with each volume prepared by eminent scientists who are experts on the topic in question.
Systems Biology: A Textbook
by Edda Klipp Wolfram Liebermeister Christoph Wierling Axel KowaldThis advanced textbook is tailored for an introductory course in Systems Biology and is well-suited for biologists as well as engineers and computer scientists. It comes with student-friendly reading lists and a companion website featuring a short exam prep version of the book and educational modeling programs. The text is written in an easily accessible style and includes numerous worked examples and study questions in each chapter. For this edition, a section on medical systems biology has been included.
Systems Biology (Advanced Biotechnology)
by Sang Yup Lee J. Nielsen Gregory StephanopoulosComprehensive coverage of the many different aspects of systems biology, resulting in an excellent overview of the experimental and computational approaches currently in use to study biological systems. Each chapter represents a valuable introduction to one specific branch of systems biology, while also including the current state of the art and pointers to future directions. Following different methods for the integrative analysis of omics data, the book goes on to describe techniques that allow for the direct quantification of carbon fluxes in large metabolic networks, including the use of 13C labelled substrates and genome-scale metabolic models. The latter is explained on the basis of the model organism Escherichia coli as well as the human metabolism. Subsequently, the authors deal with the application of such techniques to human health and cell factory engineering, with a focus on recent progress in building genome-scale models and regulatory networks. They highlight the importance of such information for specific biological processes, including the ageing of cells, the immune system and organogenesis. The book concludes with a summary of recent advances in genome editing, which have allowed for precise genetic modifications, even with the dynamic control of gene expression. This is part of the Advances Biotechnology series, covering all pertinent aspects of the field with each volume prepared by eminent scientists who are experts on the topic in question.
Systems Biology Modelling and Analysis: Formal Bioinformatics Methods and Tools
by Elisabetta De MariaSystems Biology Modelling and Analysis Describes important modelling and computational methods for systems biology research to enable practitioners to select and use the most suitable technique Systems Biology Modelling and Analysis provides an overview of state-of-the-art techniques and introduces related tools and practices to formalize models and automate reasoning for systems biology. The authors present and compare the main formal methods used in systems biology for modelling biological networks, including discussion of their advantages, drawbacks, and main applications. Each chapter includes an intuitive presentation of the specific formalism, a brief history of the formalism and of its applications in systems biology, a formal description of the formalism and its variants, at least one realistic case study, some applications of formal techniques to validate and make deep analysis of models encoded with the formalism, and a discussion on the kind of biological systems for which the formalism is suited, along with concrete ideas on its possible evolution. Edited by a highly qualified expert with significant experience in the field, some of the methods and techniques covered in Systems Biology Modelling and Analysis include: Petri nets, an important tool for studying different aspects of biological systems, ranging from simple signaling pathways to metabolic networks and beyond Pathway Logic, a formal, rule-based system and interactive viewer for developing executable models of cellular processes Boolean networks, a mathematical model which has been widely used for decades in the context of biological regulation networks Answer Set Programming (ASP), which has proven to be a strong logic programming paradigm to deal with the inherent complexity of biological models For systems biologists, biochemists, bioinformaticians, molecular biologists, pharmacologists, and computer scientists, Systems Biology Modelling and Analysis is a comprehensive all-in-one resource to understand and harness the field’s current models and techniques while also preparing for their potential developments in coming years with the help of the author’s expert insight.
Systems Biology Modelling and Analysis: Formal Bioinformatics Methods and Tools
by Elisabetta De MariaSystems Biology Modelling and Analysis Describes important modelling and computational methods for systems biology research to enable practitioners to select and use the most suitable technique Systems Biology Modelling and Analysis provides an overview of state-of-the-art techniques and introduces related tools and practices to formalize models and automate reasoning for systems biology. The authors present and compare the main formal methods used in systems biology for modelling biological networks, including discussion of their advantages, drawbacks, and main applications. Each chapter includes an intuitive presentation of the specific formalism, a brief history of the formalism and of its applications in systems biology, a formal description of the formalism and its variants, at least one realistic case study, some applications of formal techniques to validate and make deep analysis of models encoded with the formalism, and a discussion on the kind of biological systems for which the formalism is suited, along with concrete ideas on its possible evolution. Edited by a highly qualified expert with significant experience in the field, some of the methods and techniques covered in Systems Biology Modelling and Analysis include: Petri nets, an important tool for studying different aspects of biological systems, ranging from simple signaling pathways to metabolic networks and beyond Pathway Logic, a formal, rule-based system and interactive viewer for developing executable models of cellular processes Boolean networks, a mathematical model which has been widely used for decades in the context of biological regulation networks Answer Set Programming (ASP), which has proven to be a strong logic programming paradigm to deal with the inherent complexity of biological models For systems biologists, biochemists, bioinformaticians, molecular biologists, pharmacologists, and computer scientists, Systems Biology Modelling and Analysis is a comprehensive all-in-one resource to understand and harness the field’s current models and techniques while also preparing for their potential developments in coming years with the help of the author’s expert insight.
Systems Dependability Assessment: Benefits of Petri Net Models
by Jean-Francois Aubry Nicolae Brinzei Mohammed-Habib MazouniPetri Nets were defined for the study of discrete events systems and later extended for many purposes including dependability assessment. In our knowledge, no book deals specifically with the use of different type of PN to dependability. We propose in addition to bring a focus on the adequacy of Petri net types to the study of various problems related to dependability such as risk analysis and probabilistic assessment. In the first part, the basic models of PN and some useful extensions are briefly recalled. In the second part, the PN are used as a formal model to describe the evolution process of critical system in the frame of an ontological approach. The third part focuses on the stochastic Petri Nets (SPN) and their use in dependability assessment. Different formal models of SPN are formally presented (semantics, evolution rules…) and their equivalence with the corresponding class of Markov processes to get an analytical assessment of dependability. Simplification methods are proposed in order to reduce the size of analytical model and to make it more calculable. The introduction of some concepts specific to high level PN allows too the consideration of complex systems. Few applications in the field of the instrumentation and control (l&C) systems, safety integrated systems (SIS) emphasize the benefits of SPN for dependability assessment.
Systems Dependability Assessment: Benefits of Petri Net Models
by Jean-Francois Aubry Nicolae Brinzei Mohammed-Habib MazouniPetri Nets were defined for the study of discrete events systems and later extended for many purposes including dependability assessment. In our knowledge, no book deals specifically with the use of different type of PN to dependability. We propose in addition to bring a focus on the adequacy of Petri net types to the study of various problems related to dependability such as risk analysis and probabilistic assessment. In the first part, the basic models of PN and some useful extensions are briefly recalled. In the second part, the PN are used as a formal model to describe the evolution process of critical system in the frame of an ontological approach. The third part focuses on the stochastic Petri Nets (SPN) and their use in dependability assessment. Different formal models of SPN are formally presented (semantics, evolution rules…) and their equivalence with the corresponding class of Markov processes to get an analytical assessment of dependability. Simplification methods are proposed in order to reduce the size of analytical model and to make it more calculable. The introduction of some concepts specific to high level PN allows too the consideration of complex systems. Few applications in the field of the instrumentation and control (l&C) systems, safety integrated systems (SIS) emphasize the benefits of SPN for dependability assessment.
Systems with Delays: Analysis, Control, and Computations
by A. V. Kim A. V. IvanovThe main aim of the book is to present new constructive methods of delay differential equation (DDE) theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays. Referred to as “systems with delays” in this volume, this class of differential equations is also called delay differential equations (DDE), time-delay systems, hereditary systems, and functional differential equations. Delay differential equations are widely used for describing and modeling various processes and systems in different applied problems At present there are effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). There are many applications for these types of equations, because of this progress, but there are not as many methodologies in systems with delays that are easily applicable for the engineer or applied mathematician. there are no methods of finding solutions in explicit forms, and there is an absence of generally available general-purpose software packages for simulating such systems. Systems with Delays fills this void and provides easily applicable methods for engineers, mathematicians, and scientists to work with delay differential equations in their operations and research.
Systems with Delays: Analysis, Control, and Computations
by A. V. Kim A. V. IvanovThe main aim of the book is to present new constructive methods of delay differential equation (DDE) theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays. Referred to as “systems with delays” in this volume, this class of differential equations is also called delay differential equations (DDE), time-delay systems, hereditary systems, and functional differential equations. Delay differential equations are widely used for describing and modeling various processes and systems in different applied problems At present there are effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). There are many applications for these types of equations, because of this progress, but there are not as many methodologies in systems with delays that are easily applicable for the engineer or applied mathematician. there are no methods of finding solutions in explicit forms, and there is an absence of generally available general-purpose software packages for simulating such systems. Systems with Delays fills this void and provides easily applicable methods for engineers, mathematicians, and scientists to work with delay differential equations in their operations and research.
Tableau For Dummies
by Jack A. HymanDiscover how visualization turns data into action Tableau gives you the power to understand your data and put it in a format that is appealing and meaningful for everyone who needs to see it. Tableau For Dummies walks you through the steps to turn your data into a story that inspires action. This easy-to-understand guide offers insights from an enterprise data pro on how to transform data into a clear and memorable visual presentation. Navigate the Tableau user interface and connect to data sources Use drag-and-drop features to create stunning visualizations Work with templates, add graphs, and create clear charts Export your visualizations to multiple formats for easy sharingThis is the perfect Dummies software guide for business professionals who need to better derive value from that all-important data.
Tableau For Dummies
by Jack A. HymanDiscover how visualization turns data into action Tableau gives you the power to understand your data and put it in a format that is appealing and meaningful for everyone who needs to see it. Tableau For Dummies walks you through the steps to turn your data into a story that inspires action. This easy-to-understand guide offers insights from an enterprise data pro on how to transform data into a clear and memorable visual presentation. Navigate the Tableau user interface and connect to data sources Use drag-and-drop features to create stunning visualizations Work with templates, add graphs, and create clear charts Export your visualizations to multiple formats for easy sharingThis is the perfect Dummies software guide for business professionals who need to better derive value from that all-important data.
Tableau Your Data!: Fast and Easy Visual Analysis with Tableau Software
by Daniel G. MurrayTransform your organization's data into actionable insights with Tableau Tableau is designed specifically to provide fast and easy visual analytics. The intuitive drag-and-drop interface helps you create interactive reports, dashboards, and visualizations, all without any special or advanced training. This all new edition of Tableau Your Data! is your Tableau companion, helping you get the most out of this invaluable business toolset.Tableau Your Data! shows you how to build dynamic, best of breed visualizations using the Tableau Software toolset. This comprehensive guide covers the core feature set for data analytics, and provides clear step-by-step guidance toward best practices and advanced techniques that go way beyond the user manual. You'll learn how Tableau is different from traditional business information analysis tools, and how to navigate your way around the Tableau 9.0 desktop before delving into functions and calculations, as well as sharing with the Tableau Server. Analyze data more effectively with Tableau Desktop Customize Tableau's settings for your organization's needs with detailed real-world examples on data security, scaling, syntax, and more Deploy visualizations to consumers throughout the enterprise - from sales to marketing, operations to finance, and beyond Understand Tableau functions and calculations and leverage Tableau across every link in the value chain Learn from actual working models of the book's visualizations and other web-based resources via a companion website Tableau helps you unlock the stories within the numbers, and Tableau Your Data! puts the software's full functionality right at your fingertips.
Tableau Your Data!: Fast and Easy Visual Analysis with Tableau Software
by Daniel G. MurrayTransform your organization's data into actionable insights with Tableau Tableau is designed specifically to provide fast and easy visual analytics. The intuitive drag-and-drop interface helps you create interactive reports, dashboards, and visualizations, all without any special or advanced training. This all new edition of Tableau Your Data! is your Tableau companion, helping you get the most out of this invaluable business toolset.Tableau Your Data! shows you how to build dynamic, best of breed visualizations using the Tableau Software toolset. This comprehensive guide covers the core feature set for data analytics, and provides clear step-by-step guidance toward best practices and advanced techniques that go way beyond the user manual. You'll learn how Tableau is different from traditional business information analysis tools, and how to navigate your way around the Tableau 9.0 desktop before delving into functions and calculations, as well as sharing with the Tableau Server. Analyze data more effectively with Tableau Desktop Customize Tableau's settings for your organization's needs with detailed real-world examples on data security, scaling, syntax, and more Deploy visualizations to consumers throughout the enterprise - from sales to marketing, operations to finance, and beyond Understand Tableau functions and calculations and leverage Tableau across every link in the value chain Learn from actual working models of the book's visualizations and other web-based resources via a companion website Tableau helps you unlock the stories within the numbers, and Tableau Your Data! puts the software's full functionality right at your fingertips.
Teach Your Children Tables: How To Blitz Tests And Succeed In Mathematics For Life
by Bill HandleyBill Handley is well known for making maths fun! The first edition of Teach Your Children Tables challenged over 20000 readers -- and Bill has been inundated with letters of thanks since from parents whose children have quickly become proficient in maths and problem solving. This not only pleases their teachers but does wonders for a child's self-esteem. In this new, fully revised edition, rewritten for clearer understanding, Bill expands the sections on explaining multiplication to young children, on problem solving, and the correlation between the multiplication method and subtraction.
Teach Your Children Tables
by Bill HandleyBill Handley is well known for making maths fun! The first edition of Teach Your Children Tables challenged over 20000 readers -- and Bill has been inundated with letters of thanks since from parents whose children have quickly become proficient in maths and problem solving. This not only pleases their teachers but does wonders for a child's self-esteem. In this new, fully revised edition, rewritten for clearer understanding, Bill expands the sections on explaining multiplication to young children, on problem solving, and the correlation between the multiplication method and subtraction.