Professor Stephen Lerman has been a leader in the field of mathematics education for thirty years. His work is extensive, making many significant contributions to a number of key areas of research. Stephen retired from South Bank University in 2012, where he had worked for over 20 years, though he continues to work at Loughborough University. In this book several of his long standing colleagues and collaborators reflect on his contribution to mathematics education, and in so doing illustrate how some of Steve's ideas and interventions have resulted in significant shifts in the domain.

This anthology fosters an interdisciplinary dialogue between the mathematical and artistic approaches in the field where mathematical and artistic thinking and practice merge. The articles included highlight the most significant current ideas and phenomena, providing a multifaceted and extensive snapshot of the field and indicating how interdisciplinary approaches are applied in the research of various cultural and artistic phenomena. The discussions are related, for example, to the fields of aesthetics, anthropology, art history, art theory, artistic practice, cultural studies, ethno-mathematics, geometry, mathematics, new physics, philosophy, physics, study of visual illusions, and symmetry studies. Further, the book introduces a new concept: the interdisciplinary aesthetics of mathematical art, which the editors use to explain the manifold nature of the aesthetic principles intertwined in these discussions.

This book is one of very few which, address in detail and in-depth, how preservice teachers build important mathematical ideas. A detailed case study of important mathematical learning by five preservice elementary school teachers, who construct powerful mathematical ideas by working through a major influential construction personally.

This book explores the idea that mathematics educators and teachers are also problem solvers and learners, and as such they constantly experience mathematical and pedagogical disturbances. Accordingly, many original tasks and learning activities are results of personal mathematical and pedagogical disturbances of their designers, who then transpose these disturbances into learning opportunities for their students. This learning-transposition process is a cornerstone of mathematics teacher education as a lived, developing enterprise. Mathematical Encounters and Pedagogical Detours unfold the process and illustrate it by various examples. The book engages readers in original tasks, shares the results of task implementation and describes how these results inform the development of new tasks, which often intertwine mathematics and pedagogy. Most importantly, the book includes a dialogue between the authors based on the stories of their own learning, which triggers continuous exploration of learning opportunities for their students.

This book explores the concept of a map as a fundamental data type. It defines maps at three levels. The first is an abstract level, in which mathematic concepts are leveraged to precisely explain maps and operational semantics. The second is at a discrete level, in which graph theory is used to create a data model with the goal of implementation in computer systems. Finally, maps are examined at an implementation level, in which the authors discuss the implementation of a fundamental map data type in database systems. The map data type presented in this book creates new mechanisms for the storage, analysis, and computation of map data objects in any field that represents data in a map form. The authors develop a model that includes a map data type capable of representing thematic and geometric attributes in a single data object. The book provides a complete example of mathematically defining a data type, ensuring closure properties of those operations, and then translating that type into a state that is suited for implementation in a particular context. The book is designed for researchers and professionals working in geography or computer science in a range of fields including navigation, reasoning, robotics, geospatial analysis, data management, and information retrieval.

This book provides a common language for and makes connections between transfer research in mathematics education and transfer research in related fields. It generates renewed excitement for and increased visibility of transfer research, by showcasing and aggregating leading-edge research from the transfer research community. This book also helps to establish transfer as a sub-field of research within mathematics education and extends and refines alternate perspectives on the transfer of learning. The book provides an overview of current knowledge in the field as well as informs future transfer research.

THE THEORY OF LINEAR OPERATORS FROM THE STANDPOINT OF DIFFEREN TIAL EQUATIONS OF INFINITE ORDER By HAROLD T. DAVIS INDIANA UNIVERSITY AND THE COWLES COMMISSION FOR RESEARCH IN ECONOMICS THE PRINCIPIA PRESS Bloommgton, Indiana 1936 MONOGRAPH OF THE WATERMAN INSTITUTE OF INDIANA UNIVERSITY CONTRIBUTION NO. 72 THE THEORY OF LINEAR OPERATORS To Agnes, who endured so patiently the writing of it, this boo is affectionately dedicated. TABLE OF CONTENTS CHAPTER I LINEAR OPERATORS 1. The Nature of Operators ------------1 2. Definition of an Operator -----.--3 3. A Classification of Operational Methods --------7 4. The Formal Theory of Operators ----------g 5. Generalized Integration and Differentiation - - 16 6. Differential and Integral Equations of Infinite Order -----23 7. The Generatrix Calculus - - 28 8. The Heaviside Operational Calculus ---------34 9. The Theory of Functionals ------------33 10. The Calculus of Forms in Infinitely Many Variables -----4 CHAPTER II PARTICULAR OPERATORS 1. Introduction ----------------51 2. Polynomial Operators --------53 3. The Fourier Definition of an Operator ---------53 4. The Operational Symbol of von Neumann and Stone -----57 5. The Operator as a Laplace Transform ---------59 6. Polar Operators ...-60 7. Branch Point Operators ------------64 8. Note on the Complementary Function ---------70 9. Riemanns Theory - .--.--72 10. Functions Permutable with Unity ----------76 11. Logarithmic Operators ------------78 12. Special Operators --------------85 13. The General Analytic Operator ----------99 14. The Differential Operator of Infinite Order -------100 15. Differential Operators as a Cauchy Integral -------103 16. The Generatrix of Differential Operators--------104 17. Five Operators of Analysis ------------105 CHAPTER III THE THEORY OF LINEAR SYSTEMS OF EQUATIONS 1. Preliminary Remarks -------------108 2. Types of Matrices --------------109 3. The Convergence of an Infinite Determinant -------114 4. The Upper Bound of a Determinant. Hadamards Theorem - - 116 5. Determinants which do not Vanish - - - - - - - - - 123 6. The Method of the Liouville-Neumann Series -------126 7. The Method of Segments ------------130 8. Applications of the Method of Segments. --------132 9. The Hilbert Theory of Linear Equations in an Infinite Number of Variables - - - - 137 10. Extension of the Foregoing Theory to Holder Space 149 vii Vlll THE THEORY OF LINEAR OPERATORS CHAPTER IV OPERATIONAL MULTIPLICATION AND INVERSION 1. Algebra and Operators -------.. --153 2. The Generalized Formula of Leibnitz ---------154 3. Bourlets Operational Product --. 155 4. The Algebra of Functions of Composition --------159 5. Selected Problems in the Algebra of Permutable Functions - - - - 164 G. The Calculation of a Function Permutable with a Given Function - 166 7. The Transformation of Peres -----------171 8. The Permutability of Functions Permutable with a Given Function - 173 9. Permutable Functions of Second Kind - --176 10. The Inversion of Operators Bourlets Theory ------177 It. The Method of Successive Substitutions --------181 12. Some Further Properties of the Resolvent Generatrix - 185 13. The Inversion of Operators by Infinite Differentiation - 188 14. The Permutability of Linear PilYeiential Operators -----190 15. A Class of Non-permutable Operators ---------194 16. Special Examples Illustrating the Application of Operational Processes 200 CHAPTER V GRADESDEFINED BY SPECIAL OPERATORS 1. Definition ----------------211 2. The Grade of an Unlimitedly Differentiable Function - 212 3. Functions of Finite Grade ------------215 4. Asymptotic Expansions --- 222 5. The Summability of Differential Operators with Constant Coefficients 230 6. The Summability of Operators of Laplace Type ------235 CHAPTER VI DIFFERENTIAL EQUATIONS OF INFINITE ORDER WITH CONSTANT COEFFICIENTS 1. Introduction ---------------238 2. Expansion of the Resolvent Generatrix --------239 3. The Method of Cauchy-Bromwich ----------250 4...

This comprehensive volume provides teachers, researchers and education professionals with cutting edge knowledge developed in the last decades by the educational, behavioural and neurosciences, integrating cognitive, developmental and socioeconomic approaches to deal with the problems children face in learning mathematics. The neurocognitive mechanisms and the cognitive processes underlying acquisition of arithmetic abilities and their significance for education have been the subject of intense research in the last few decades, but the most part of this research has been conducted in non-applied settings and there's still a deep discrepancy between the level of scientific knowledge and its implementation into actual educational settings. Now it's time to bring the results from the laboratory to the classroom. Apart from bringing the theoretical discussions to educational settings, the volume presents a wide range of methods for early detection of children with risks in mathematics learning and strategies to develop effective interventions based on innovative cognitive test instruments. It also provides insights to translate research knowledge into public policies in order to address socioeconomic issues. And it does so from an international perspective, dedicating a whole section to the cultural diversity of mathematics learning difficulties in different parts of the world. All of this makes the International Handbook of Mathematical Learning Difficulties an essential tool for those involved in the daily struggle to prepare the future generations to succeed in the global knowledge society.

With breathtaking detail, Maria Georgiadou sheds light on the work and life of Constantin Carathéodory, who until now has been ignored by historians. In her thought-provoking book, Georgiadou maps out the mathematician’s oeuvre, life and turbulent historical surroundings. Descending from the Greek élite of Constantinople, Carathéodory graduated from the military school of Brussels, became engineer at the Assiout dam in Egypt and finally dedicated a lifetime to mathematics and education. He significantly contributed to: calculus of variations, the theory of point set measure, the theory of functions of a real variable, pdes, and complex function theory. An exciting and well-written biography, once started, difficult to put down.

This volume consists of seven papers related in various matters to the research work of Kostia Beidar, a distinguished ring theorist and professor of National Ching Kung University (NCKU). Written by leading experts in these areas, the papers also emphasize important applications to other fields of mathematics. Most papers are based on talks that were presented at the memorial conference which was held in March 2005 at NCKU.

The first book of its kind, "New Foundations in Mathematics: The Geometric Concept of Number" uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner.

"New Foundations in Mathematics" will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics.

The third edition of this popular and effective textbook provides in one volume a unified treatment of topics essential for first year university students studying for degrees in mathematics. Students of computer science, physics and statistics will also find this book a helpful guide to all the basic mathematics they require. It clearly and comprehensively covers much of the material that other textbooks tend to assume, assisting students in the transition to university-level mathematics.
Expertly revised and updated, the chapters cover topics such as number systems, set and functions, differential calculus, matrices and integral calculus. Worked examples are provided and chapters conclude with exercises to which answers are given. For students seeking further challenges, problems intersperse the text, for which complete solutions are provided. Modifications in this third edition include a more informal approach to sequence limits and an increase in the number of worked examples, exercises and problems.
The third edition of Fundamentals of university mathematics is an essential reference for first year university students in mathematics and related disciplines. It will also be of interest to professionals seeking a useful guide to mathematics at this level and capable pre-university students.
One volume, unified treatment of essential topicsClearly and comprehensively covers material beyond standard textbooksWorked examples, challenges and exercises throughout

This book explicates some of the fundamental philosophical tenets underpinning key theoretical frameworks, and demonstrates how these tenets inform particular kinds of research practice in mathematics education research. We believe that a deep understanding of significant theories from the humanities and social sciences is crucial for doing high-quality research in education. For that reason, this book focuses on six key theoretical sources, unpacking their relevance and application to specific research examples. We situate these key theorists within a larger framework pertaining to the history of thought more generally, and discuss how competing theories of teaching and learning differ in terms of their philosophical assumptions. In so doing, we offer context and motivation for particular research methods, with the agenda of helping researchers reflect on why particular approaches and not others might work for them.

This book is the first of two volumes on random motions in Markov and semi-Markov random environments. This first volume focuses on homogenous random motions. This volume consists of two parts, the first describing the basic concepts and methods that have been developed for random evolutions. These methods are the foundational tools used in both volumes, and this description includes many results in potential operators. Some techniques to find closed-form expressions in relevant applications are also presented. The second part deals with asymptotic results and presents a variety of applications, including random motion with different types of boundaries, the reliability of storage systems and solutions of partial differential equations with constant coefficients, using commutative algebra techniques. It also presents an alternative formulation to the Black-Scholes formula in finance, fading evolutions and telegraph processes, including jump telegraph processes and the estimation of the number of level crossings for telegraph processes.

This book considers the views of participants in the process of becoming a mathematician, that is, the students and the graduates. This book investigates the people who carry out mathematics rather than the topics of mathematics. Learning is about change in a person, the development of an identity and ways of interacting with the world. It investigates more generally the development of mathematical scientists for a variety of workplaces, and includes the experiences of those who were not successful in the transition to the workplace as mathematicians. The research presented is based on interviews, observations and surveys of students and graduates as they are finding their identity as a mathematician. The book contains material from the research carried out in South Africa, Northern Ireland, Canada and Brunei as well as Australia.

The book gives a systematical presentation of stochastic approximation methods for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The volume presents results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.

A groundbreaking, flexible approach to computer science anddata science The Deitels' Introduction to Python for ComputerScience and Data Science: Learning to Program with AI, Big Data and the Cloudoffers a unique approach to teaching introductory Python programming,appropriate for both computer-science and data-science audiences. Providing themost current coverage of topics and applications, the book is paired withextensive traditional supplements as well as Jupyter Notebooks supplements.Real-world datasets and artificial-intelligence technologies allow students towork on projects making a difference in business, industry, government andacademia. Hundreds of examples, exercises, projects (EEPs) and implementationcase studies give students an engaging, challenging and entertainingintroduction to Python programming and hands-on data science. The book's modular architecture enables instructors toconveniently adapt the text to a wide range of computer-science anddata-science courses offered to audiences drawn from many majors.Computer-science instructors can integrate as much or as little data-scienceand artificial-intelligence topics as they'd like, and data-science instructorscan integrate as much or as little Python as they'd like. The book aligns withthe latest ACM/IEEE CS-and-related computing curriculum initiatives and withthe Data Science Undergraduate Curriculum Proposal sponsored by the NationalScience Foundation.

This book provides a collection of chapters from prominent mathematics educators in which they each discuss vital issues in mathematics education and what they see as viable directions research in mathematics education could take to address these issues. All of these issues are related to learning and teaching mathematics. The book consists of nine chapters, seven from each of seven scholars who participated in an invited lecture series (Scholars in Mathematics Education) at Brigham Young University, and two chapters from two other scholars who are writing reaction papers that look across the first seven chapters. The recommendations take the form of broad, overarching principles and ideas that cut across the field. In this sense, this book differs from classical "research agenda projects," which seek to outline specific research questions that the field should address around a central topic.