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.
Oh Yeah? Proof It! enhances a student’s understanding of the importance of proofreading. Presented as a school newspaper, the student has the opportunity to edit a variety of newspaper articles, including school news, play reviews, advertisements, and even comics! The variety of activities not only helps the student identify mistakes in written text, but also allows the student to become more confident in his or her own writing, reinforcing essential grammar, spelling and other language arts skills. A complete answer key is provided for the student or teacher to check their work. Oh Yeah? Proof It! provides the student an opportunity to grow their language arts skills, all while being Editor-In-Chief!
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.
Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.
A fresh look at a “robot-proof” education in the new age of generative AI. In 2017, Robot-Proof, the first edition, foresaw the advent of the AI economy and called for a new model of higher education designed to help human beings flourish alongside smart machines. That economy has arrived. Creative tasks that, seven years ago, seemed resistant to automation can now be performed with a simple prompt. As a result, we must now learn not only to be conversant with these technologies, but also to comprehend and deploy their outputs. In this revised and updated edition, Joseph Aoun rethinks the university’s mission for a world transformed by AI, advocating for the lifelong endeavor of a “robot-proof” education. Aoun puts forth a framework for a new curriculum, humanics, which integrates technological, data, and human literacies in an experiential setting, and he renews the call for universities to embrace lifelong learning through a social compact with government, employers, and learners themselves. Drawing on the latest developments and debates around generative AI, Robot-Proof is a blueprint for the university as a force for human reinvention in an era of technological change—an era in which we must constantly renegotiate the shifting boundaries between artificial intelligence and the capacities that remain uniquely human.
Lance Rips describes a unified theory of natural deductive reasoning and fashions a working model of deduction, with strong experimental support, that is capable of playing a central role in mental life.
THE STORY: On the eve of her twenty-fifth birthday, Catherine, a troubled young woman, has spent years caring for her brilliant but unstable father, a famous mathematician. Now, following his death, she must deal with her own volatile emotions; the
*THIS BOOK IS AVAILABLE AS OPEN ACCESS BOOK ON SPRINGERLINK* One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving. The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.
