This book reflects the author's experience in teaching a mathematics content course for pre-service elementary teachers. The book addresses a number of recommendations of the Conference Board of the Mathematical Sciences for the preparation of teachers demonstrating how abstract mathematical concepts can be motivated by concrete activities. Such an approach, when enhanced by the use of technology, makes it easier for the teachers to grasp the meaning of generalization, formal proof, and the creation of an increasing number of concepts on higher levels of abstraction. A strong experiential component of the book made possible by the use of manipulative materials and digital technology such as spreadsheets, The Geometer's Sketchpad, Graphing Calculator 3.5 (produced by Pacific Tech), and Kid Pix Studio Deluxe makes it possible to balance informal and formal approaches to mathematics, allowing the teachers to learn how the two approaches complement each other. Classroom observations of the teachers' learning mathematics as a combination of theory and experiment confirm that this approach elevates one's mathematical understanding to a higher ground. The book not only shows the importance of mathematics content knowledge for teachers but better still, how this knowledge can be gradually developed in the context of exploring grade-appropriate activities and tasks and using computational and manipulative environments to support these explorations. Most of the chapters are motivated by a problem/activity typically found in the elementary mathematics curricula and/or standards (either National or New York State - the context in which the author prepares teachers). By exploring such problems in depth, the teachers can learn fundamental mathematical concepts and ideas hidden within a seemingly mundane problem/activity. The need to have experience in going beyond traditional expectations for learning is due to the constructivist orientation of contemporary mathematics pedagogy that encourages students to ask questions about mathematics they study. Each chapter includes an activity set that can be used for the development of the variety of assignments for the teachers. The material included in the book is original in terms of the approach used to teach mathematics to the teachers and it is based on a number of journal articles published by the author in the United States and elsewhere. Mathematics educators who are interested in integrating hands-on activities and digital technology into the teaching of mathematics will find this book useful. Mathematicians who teach mathematics to the teachers as part of their teaching load will be interested in the material included in the book as it connects childhood mathematics content and mathematics for the teachers.

This book promotes the experimental mathematics approach in the context of secondary mathematics curriculum by exploring mathematical models depending on parameters that were typically considered advanced in the pre-digital education era. This approach, by drawing on the power of computers to perform numerical computations and graphical constructions, stimulates formal learning of mathematics through making sense of a computational experiment. It allows one (in the spirit of Freudenthal) to bridge serious mathematical content and contemporary teaching practice. In other words, the notion of teaching experiment can be extended to include a true mathematical experiment. When used appropriately, the approach creates conditions for collateral learning (in the spirit of Dewey) to occur including the development of skills important for engineering applications of mathematics. In the context of a mathematics teacher education program, thebook addresses a call for the preparation of teachers capable of utilizing modern technology tools for the modeling-based teaching of mathematics with a focus on methods conducive to the improvement of the whole STEM education at the secondary level. By the same token, using the book's pedagogy and its mathematical content in a pre-college classroom can assist teachers in introducing students to the ideas that develop the foundation of engineering profession."

The book is written to share ideas stemming from technology-rich K-12 mathematics education courses taught by the author to American and Canadian teacher candidates over the past two decades. It includes examples of problems posed by the teacher candidates using computers. These examples are analyzed through the lenses of the theory proposed in the book.Also, the book includes examples of computer-enabled formulation as well as reformulation of rather advanced problems associated with the pre-digital era problem-solving curriculum. The goal of the problem reformulation is at least two-fold: to make curriculum materials compatible with the modern-day emphasis on democratizing mathematics education and to find the right balance between positive and negative affordances of technology.The book focuses on the use of spreadsheets, Wolfram Alpha, Maple, and The Graphing Calculator (also known as NuCalc) in problem posing. It can be used by pre-service and in-service teachers interested in K-12 mathematics curriculum development in the digital era as well as by those studying mathematics education from a theoretical perspective.

'What one takes away from this book is the notion that thereaEURO (TM)s a lot of potential to do more with these students, and the book stands as a resource for anyone who shares that opinion ... Books like AbramovichaEURO (TM)s are a welcome addition to our options as we try to do our best by these students, and by extension, their future students.'MAA ReviewsThe book is written to enhance the preparation of elementary teacher candidates by offering teaching ideas conducive to the development of deep understanding of concepts fundamental to the mathematics curriculum they are to teach. It intends to show how the diversity of teaching methods stems from the knowledge of mathematics content and how the appreciation of this diversity opens a window to the teaching of extended content.The book includes material that the author would have shared with teacher candidates should there have been more instructional time than a 3 credit hour master's level course, 'Elementary Mathematics: Content and Methods', provides. Thus the book can supplement a basic textbook for such a course by extending content and diversifying methods.Also, the book can support graduate level mathematics education programs which have problem-solving seminars/courses in their curriculum. The book is well-informed with (available in English) the mathematical standards and recommendations for teachers from Australia, Canada, Chile, England, Japan, Korea, Singapore, and the United States.

'What one takes away from this book is the notion that thereaEURO (TM)s a lot of potential to do more with these students, and the book stands as a resource for anyone who shares that opinion ... Books like AbramovichaEURO (TM)s are a welcome addition to our options as we try to do our best by these students, and by extension, their future students.'MAA ReviewsThe book is written to enhance the preparation of elementary teacher candidates by offering teaching ideas conducive to the development of deep understanding of concepts fundamental to the mathematics curriculum they are to teach. It intends to show how the diversity of teaching methods stems from the knowledge of mathematics content and how the appreciation of this diversity opens a window to the teaching of extended content.The book includes material that the author would have shared with teacher candidates should there have been more instructional time than a 3 credit hour master's level course, 'Elementary Mathematics: Content and Methods', provides. Thus the book can supplement a basic textbook for such a course by extending content and diversifying methods.Also, the book can support graduate level mathematics education programs which have problem-solving seminars/courses in their curriculum. The book is well-informed with (available in English) the mathematical standards and recommendations for teachers from Australia, Canada, Chile, England, Japan, Korea, Singapore, and the United States.

The goal of the book is to technologically enhance the preparation of mathematics schoolteachers using an electronic spreadsheet integrated with Maple and Wolfram Alpha - digital tools capable of sophisticated symbolic computations. The content of the book is a combination of mathematical ideas and concepts associated with pre-college problem solving curriculum and their extensions into more advanced mathematical topics.The book provides prospective and practicing teachers with a foundation for developing a deep understanding of many concepts fundamental to the teaching of school mathematics. It also provides the teachers with a technical expertise in designing spreadsheet-based computational environments.Consistent with the current worldwide guidelines for technology-enhanced teacher preparation, the book emphasizes the integration of context, mathematics, and technology as a method for teaching mathematics. Throughout the book, a number of mathematics education documents developed around the world (Australia, Canada, England, Japan, Singapore, United States) are reviewed as appropriate.

The goal of the book is to technologically enhance the preparation of mathematics schoolteachers using an electronic spreadsheet integrated with Maple and Wolfram Alpha - digital tools capable of sophisticated symbolic computations. The content of the book is a combination of mathematical ideas and concepts associated with pre-college problem solving curriculum and their extensions into more advanced mathematical topics.The book provides prospective and practicing teachers with a foundation for developing a deep understanding of many concepts fundamental to the teaching of school mathematics. It also provides the teachers with a technical expertise in designing spreadsheet-based computational environments.Consistent with the current worldwide guidelines for technology-enhanced teacher preparation, the book emphasizes the integration of context, mathematics, and technology as a method for teaching mathematics. Throughout the book, a number of mathematics education documents developed around the world (Australia, Canada, England, Japan, Singapore, United States) are reviewed as appropriate.

This book promotes the experimental mathematics approach in the context of secondary mathematics curriculum by exploring mathematical models depending on parameters that were typically considered advanced in the pre-digital education era. This approach, by drawing on the power of computers to perform numerical computations and graphical constructions, stimulates formal learning of mathematics through making sense of a computational experiment. It allows one (in the spirit of Freudenthal) to bridge serious mathematical content and contemporary teaching practice. In other words, the notion of teaching experiment can be extended to include a true mathematical experiment. When used appropriately, the approach creates conditions for collateral learning (in the spirit of Dewey) to occur including the development of skills important for engineering applications of mathematics. In the context of a mathematics teacher education program, the book addresses a call for the preparation of teachers capable of utilizing modern technology tools for the modeling-based teaching of mathematics with a focus on methods conducive to the improvement of the whole STEM education at the secondary level. By the same token, using the book’s pedagogy and its mathematical content in a pre-college classroom can assist teachers in introducing students to the ideas that develop the foundation of engineering profession.

This textbook is for prospective teachers of middle school mathematics. It reflects on the authors' experience in offering various mathematics education courses to prospective teachers in the US and Canada. In particular, the content can support one or more of 24-semester-hour courses recommended by the Conference Board of the Mathematical Sciences (2012) for the mathematical preparation of middle school teachers. The textbook integrates grade-appropriate content on all major topics in the middle school mathematics curriculum with international recommendations for teaching the content, making it relevant for a global readership. The textbook emphasizes the inherent connections between mathematics and real life, since many mathematical concepts and procedures stem from common sense, something that schoolchildren intuitively possess. This focus on teaching formal mathematics with reference to real life and common sense is essential to its pedagogical approach. In addition, the textbook stresses the importance of being able to use technology as an exploratory tool, and being familiar with its strengths and weaknesses. In keeping with this emphasis on the use of technology, both physical (manipulatives) and digital (commonly available educational software), it also explores e.g. the use of computer graphing software for digital fabrication. In closing, the textbook addresses the issue of creativity as a crucial aspect of education in the digital age in general, and in mathematics education in particular.

The book is intended to serve as a brief companion for mathematical educators of elementary teacher candidates who learn mathematics within a college of education both at the undergraduate and graduate levels. Being informed by mathematics teaching and learning standards of the United States, Australia, Canada, Chile, England, Japan, Korea, Singapore, and South Africa, the book can be used internationally.The teaching methods emphasize the power of visualization, the use of physical materials, and support of computer technology including spreadsheet, Wolfram Alpha, and the Geometer's Sketchpad.The basic ideas include the development of the concepts of number, base-ten system, problem solving and posing, the emergence of fractions in the context of simple real-life activities requiring the extension of whole number arithmetic, decimals, percent, ratio, geoboard geometry, elements of combinatorics, probability and data analysis.The book includes historical aspects of elementary school mathematics. For example, readers would be interested to know that two-sided counters stem from the binary system with its genesis in the 1st millennium BC China of which Leibnitz (17th century) was one of the first notable proponents. The genesis of the base-ten arithmetic is in the Egyptian mathematics of the 4th millennium BC, enriched with the positional notation with the advent of Hindu-Arabic numerals in the 12th century Europe.

The material of this book stems from the idea of integrating a classic concept of Fibonacci numbers with commonly available digital tools including a computer spreadsheet, Maple, Wolfram Alpha, and the graphing calculator. This integration made it possible to introduce a number of new concepts such as: Generalized golden ratios in the form of cycles represented by the strings of real numbers; Fibonacci-like polynomials the roots that define those cycles dependence on a parameter; the directions of the cycles described in combinatorial terms of permutations with rises, as the parameter changes on the number line; Fibonacci sieves of order k; (r, k)-sections of Fibonacci numbers; and polynomial generalizations of Cassinis, Catalans, and other identities for Fibonacci numbers.The development of these concepts was motivated by considering the difference equation f_(n+1)=af_n+bf_(n-1),f_0=f_1=1, and, by taking advantage of capabilities of the modern-day digital tools, exploring the behavior of the ratios f_(n+1)/f_n as n increases. The initial use of a spreadsheet can demonstrate that, depending on the values of a and b, the ratios can either be attracted by a number (known as the Golden Ratio in the case a = b = 1) or by the strings of numbers (cycles) of different lengths. In general, difference equations, both linear and non-linear ones serve as mathematical models in radio engineering, communication, and computer architecture research. In mathematics education, commonly available digital tools enable the introduction of mathematical complexity of the behavior of these models to different groups of students through the modern-day combination of argument and computation.The book promotes experimental mathematics techniques which, in the digital age, integrate intuition, insight, the development of mathematical models, conjecturing, and various ways of justification of conjectures. The notion of technology-immune/technology-enabled problem solving is introduced as an educational analogue of the notion of experimental mathematics. In the spirit of John Dewey, the book provides many collateral learning opportunities enabled by experimental mathematics techniques. Likewise, in the spirit of George Polya, the book champions carrying out computer experimentation with mathematical concepts before offering their formal demonstration.The book can be used in secondary mathematics teacher education programs, in undergraduate mathematics courses for students majoring in mathematics, computer science, electrical and mechanical engineering, as well as in other mathematical programs that study difference equations in the broad context of discrete mathematics.

This two-volume collection of diverse international experiences and perspectives on the current use of computers in education is a joint effort of fifty-seven authors from sixteen countries. Most of the chapters describe current work of the authors on the use of computers in their respective areas of education. The material of this volume should be of interest to teacher educators in the areas of science, technology, engineering and mathematics (STEM) education involved in the use of computers as teaching tools; tertiary education faculty in STEM areas; curriculum developers; policy makers at different levels; and college students in STEM-related programs.

This book addresses core recommendations by the Conference Board of the Mathematical Sciences -- an umbrella organisation consisting of sixteen professional societies in the United States-regarding the mathematical preparation of the teachers. According to the Board, the concept of a capstone course in a mathematics education program includes teachers' learning to use commonly available educational software with the goal to reach a certain depth of the mathematics curriculum through appropriately designed computational experiments. In turn, the notion of experiment in the teaching of mathematics sets up a path toward enhancing the "E" component of the teachers' literacy in the STEM disciplines. This book discusses experiences in teaching a computer-enhanced capstone course for prospective teachers of high school mathematics.

This book reflects the author's experience in teaching a mathematics content course for pre-service elementary teachers. The book addresses a number of recommendations of the Conference Board of the Mathematical Sciences for the preparation of teachers demonstrating how abstract mathematical concepts can be motivated by concrete activities. Such an approach, when enhanced by the use of technology, makes it easier for the teachers to grasp the meaning of generalization, formal proof, and the creation of an increasing number of concepts on higher levels of abstraction. A strong experiential component of the book made possible by the use of manipulative materials and digital technology such as spreadsheets, The Geometer's Sketchpad, Graphing Calculator 3.5 (produced by Pacific Tech), and Kid Pix Studio Deluxe makes it possible to balance informal and formal approaches to mathematics, allowing the teachers to learn how the two approaches complement each other. Classroom observations of the teachers' learning mathematics as a combination of theory and experiment confirm that this approach elevates one's mathematical understanding to a higher ground. The book not only shows the importance of mathematics content knowledge for teachers but better still, how this knowledge can be gradually developed in the context of exploring grade-appropriate activities and tasks and using computational and manipulative environments to support these explorations. Most of the chapters are motivated by a problem/activity typically found in the elementary mathematics curricula and/or standards (either National or New York State - the context in which the author prepares teachers). By exploring such problems in depth, the teachers can learn fundamental mathematical concepts and ideas hidden within a seemingly mundane problem/activity. The need to have experience in going beyond traditional expectations for learning is due to the constructivist orientation of contemporary mathematics pedagogy that encourages students to ask questions about mathematics they study. Each chapter includes an activity set that can be used for the development of the variety of assignments for the teachers. The material included in the book is original in terms of the approach used to teach mathematics to the teachers and it is based on a number of journal articles published by the author in the United States and elsewhere. Mathematics educators who are interested in integrating hands-on activities and digital technology into the teaching of mathematics will find this book useful. Mathematicians who teach mathematics to the teachers as part of their teaching load will be interested in the material included in the book as it connects childhood mathematics content and mathematics for the teachers.

This two-volume collection of diverse international experiences and perspectives on the current use of computers in education is a joint effort of fifty-seven authors from sixteen countries. Most of the chapters describe current work of the authors on the use of computers in their respective areas of education. The first volume includes chapters describing history and future trends of educational computing, technological infrastructure of Brazilian public education, the use of e-textbooks by college students in the United States, on-line leaning at the middle school level in Australia, theoretical aspects of knowledge sharing in technology-supported environments, the interplay between learning theories and technology, the integration of Web 2.0 tools in education, knowledge authoring in e-learning environments, application of technology to special education, and the use of computers in the teaching of physical education.