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This book explains and demonstrates the teaching strategy of asking learners to construct their own examples of mathematical objects. The authors show that the creation of examples can involve transforming and reorganizing knowledge and that, although this is usually done by authors and teachers, if the responsibility for making examples is transferred to learners, their knowledge structures can be developed and extended. A multitude of examples to illustrate this is provided, spanning primary, secondary, and college levels. Readers are invited to learn from their own past experience augmented by tasks provided in the book, and are given direct experience of constructing examples through a collection of many tasks at many levels. Classroom stories show the practicalities of introducing such shifts in mathematics education. The authors examine how their approach relates to improving the learning of mathematics and raise future research questions. *Based on the authors' and others' theoretical and practical experience, the book includes a combination of exercises for the reader, practical applications for teaching, and solid scholarly grounding. *The ideas presented are generic in nature and thus applicable across every phase of mathematics teaching and learning. *Although the teaching methods offered are ones that engage learners imaginatively, these are also applied to traditional approaches to mathematics education; all tasks offered in the book are within conventional mathematics curriculum content. Mathematics as a Constructive Activity: Learners Generating Examples is intended for mathematics teacher educators, mathematics teachers, curriculum developers, task and test designers, and classroom researchers, and for use as a text in graduate-level mathematics education courses.
A brilliant tour of mathematical thought and a guide to becoming a better thinker, How Not to Be Wrong shows that math is not just a long list of rules to be learned and carried out by rote. Math touches everything we do; It's what makes the world make sense. Using the mathematician's methods and hard-won insights-minus the jargon-professor and popular columnist Jordan Ellenberg guides general readers through his ideas with rigor and lively irreverence, infusing everything from election results to baseball to the existence of God and the psychology of slime molds with a heightened sense of clarity and wonder. Armed with the tools of mathematics, we can see the hidden structures beneath the messy and chaotic surface of our daily lives. How Not to Be Wrong shows us how--Publisher's description.
This book exposes the various manifestations of mistreatment of teachers by principals, offering practical solutions for its prevention and correction. Information comes from a study involving interviews with elementary and secondary teachers from rural, suburban, and urban areas across the United States and Canada. The book provides tools necessary to identify destructive behavior and raises awareness of this common phenomenon in order to break the cycle of abuse. Key features include real-life examples and testimonials; specific forms and indicators of mistreatment, categorized into three levels; descriptions of the effects on schools and teachers, professionally and personally; and solutions for overcoming this problem. Seven chapters focus on: (1) "The Problem of Principal Mistreatment of Teachers"; (2) "The Many Faces of Moderate Mistreatment: From Discounting Teacher to Offensive Personal Conduct"; (3) "Escalating Mistreatment of Teachers: From Spying to Criticism"; (4) "Severe Mistreatment of Teachers: From Lying to Destruction"; (5) "The Effects of Principal Mistreatment of Teachers: Lasting Wounds and Damaged Schools"; (6) "Worlds of Pain: The Undoing of Teachers"; and (7) "Overcoming the Problem of Principal Mistreatment of Teachers: What Can We Do?" (Contains approximately 225 references.) (SM).
According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.
Improve Schools and Transform Education In order for educational systems to change, we must reevaluate deep-seated beliefs about learning, teaching, schooling, and race that perpetuate inequitable opportunities and outcomes. Hatch, Corson, and Gerth van den Berg challenge the narrative when it comes to the "grammar of schooling"--or the conventional structures, practices, and beliefs that define educational experiences for so many children—to cast a new vision of what school could be. The book addresses current systemic problems and solutions as it: Highlights global examples of successful school change Describes strategies that improve educational opportunities and performance Explores promising approaches in developing new learning opportunities Outlines conditions for supporting wide-scale educational improvement This provocative book approaches education reform by highlighting what works, while also demonstrating what can be accomplished if we redefine conventional schools. We can make the schools we have more efficient, more effective, and more equitable, all while creating powerful opportunities to support all aspects of students’ development. "You won’t find a better book on system change in education than this one. We learn why schools don’t change; how they can improve; what it takes to change a system; and, in the final analysis, the possibilities of system change. Above all, The Education We Need renders complexity into clarity as the writing is so clear and compelling. A powerful read on a topic of utmost importance." ~Michael Fullan, Professor Emeritus, OISE/Universtiy of Toronto "I cannot recommend this book highly enough – Tom tackles long-standing and emerging educational issues in new ways with an impressive understanding of the challenging complexities, but also feasible possibilities, for ensuring excellence and equity for all students." ~Carol Campbell, Associate Professor, Ontario Institute for Studies in Education, University of Toronto
A game-changing book on the origins of life, called the most important scientific discovery 'since the Copernican revolution' in The Observer.
Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th-16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. 1968 edition. Bibliography.
Honesty in voting, it turns out, is not always the best policy. Indeed, in the early 1970s, Allan Gibbard and Mark Satterthwaite, building on the seminal work of Nobel laureate Kenneth Arrow, proved that with three or more alternatives there is no reasonable voting system that is non-manipulable; voters will always have an opportunity to benefit by submitting a disingenuous ballot. The ensuing decades produced a number of theorems of striking mathematical naturality that dealt with the manipulability of voting systems. This 2005 book presents many of these results from the last quarter of the twentieth century, especially the contributions of economists and philosophers, from a mathematical point of view, with many new proofs. The presentation is almost completely self-contained, and requires no prerequisites except a willingness to follow rigorous mathematical arguments. Mathematics students, as well as mathematicians, political scientists, economists and philosophers will learn why it is impossible to devise a completely unmanipulable voting system.
For textual studies relating to the ancient mathematical corpus the efforts by the Danish philologist, 1. L. Heiberg (1854-1928), are especially significant. Beginning with his doctoral dissertation, Quaestiones Archimedeae (Copen hagen, 1879), Heiberg produced an astonishing series of editions and critical studies that remain the foundation of scholarship on Greek mathematical 4 science. For comprehensiveness and accuracy, his editions are exemplary. In his textual studies, as also in the prolegomena to his editions, he carefully described the extant evidence, organized the manuscripts into stemmata, and drew out the implications for the state of the text. 5 With regard to his Archimedean work, Heiberg sometimes betrayed signs of the philologist's occupational disease - the tendency to rewrite a text deemed on subjective grounds to be unworthy. 6 But he did so less often than his prominent 7 contemporaries, and not as to detract appreciably from the value of his editions. In examining textual questions bearing on the Archimedean corpus, he attempted to exploit as much as possible evidence from the ancient commentators, and in some instances from the medieval translations. It is here that opportunities abound for new work, extending, and in some instances superseding, Heiberg's findings. For at his time the availability of the medieval materials was limited. In recent years Marshall Clagett has completed a mammoth critical edition of the medieval Latin tradition of Archimedes,8 while the bibliographical instruments for the Arabic tradition are in good order thanks to the work of Fuat Sezgin.
Properly crafted and individually tailored feedback on student work boosts student achievement across subjects and grades. In this updated and expanded second edition of her best-selling book, Susan M. Brookhart offers enhanced guidance and three lenses for considering the effectiveness of feedback: (1) does it conform to the research, (2) does it offer an episode of learning for the student and teacher, and (3) does the student use the feedback to extend learning? In this comprehensive guide for teachers at all levels, you will find information on every aspect of feedback, including • Strategies to uplift and encourage students to persevere in their work. • How to formulate and deliver feedback that both assesses learning and extends instruction. • When and how to use oral, written, and visual as well as individual, group, or whole-class feedback. • A concise and updated overview of the research findings on feedback and how they apply to today's classrooms. In addition, the book is replete with examples of good and bad feedback as well as rubrics that you can use to construct feedback tailored to different learners, including successful students, struggling students, and English language learners. The vast majority of students will respond positively to feedback that shows you care about them and their learning. Whether you teach young students or teens, this book is an invaluable resource for guaranteeing that the feedback you give students is engaging, informative, and, above all, effective.