Teacher Proof

‘Tom Bennett is the voice of the modern teacher.’ - Stephen Drew, Senior Vice-Principal, Passmores Academy, UK, featured on Channel 4’s Educating Essex Do the findings from educational science ever really improve the day-to-day practice of classroom teachers? Education is awash with theories about how pupils best learn and teachers best teach, most often propped up with the inevitable research that ‘proves’ the case in point. But what can teachers do to find the proof within the pudding, and how can this actually help them on wet Wednesday afternoon?. Drawing from a wide range of recent and popular education theories and strategies, Tom Bennett highlights how much of what we think we know in schools hasn’t been ‘proven’ in any meaningful sense at all. He inspires teachers to decide for themselves what good and bad education really is, empowering them as professionals and raising their confidence in the classroom and the staffroom alike. Readers are encouraged to question and reflect on issues such as: the most common ideas in modern education and where these ideas were born the crisis in research right now how research is commissioned and used by the people who make policy in the UK and beyond the provenance of education research: who instigates it, who writes it, and how to spot when a claim is based on evidence and when it isn’t the different way that data can be analysed what happens to the research conclusions once they escape the laboratory. Controversial, erudite and yet unremittingly entertaining, Tom includes practical suggestions for the classroom throughout. This book will be an ally to every teacher who’s been handed an instruction on a platter and been told, ‘the research proves it.’

‘Tom Bennett is the voice of the modern teacher.’ - Stephen Drew, Senior Vice-Principal, Passmores Academy, UK, featured on Channel 4’s Educating Essex Do the findings from educational science ever really improve the day-to-day practice of classroom teachers? Education is awash with theories about how pupils best learn and teachers best teach, most often propped up with the inevitable research that ‘proves’ the case in point. But what can teachers do to find the proof within the pudding, and how can this actually help them on wet Wednesday afternoon?. Drawing from a wide range of recent and popular education theories and strategies, Tom Bennett highlights how much of what we think we know in schools hasn’t been ‘proven’ in any meaningful sense at all. He inspires teachers to decide for themselves what good and bad education really is, empowering them as professionals and raising their confidence in the classroom and the staffroom alike. Readers are encouraged to question and reflect on issues such as: the most common ideas in modern education and where these ideas were born the crisis in research right now how research is commissioned and used by the people who make policy in the UK and beyond the provenance of education research: who instigates it, who writes it, and how to spot when a claim is based on evidence and when it isn’t the different way that data can be analysed what happens to the research conclusions once they escape the laboratory. Controversial, erudite and yet unremittingly entertaining, Tom includes practical suggestions for the classroom throughout. This book will be an ally to every teacher who’s been handed an instruction on a platter and been told, ‘the research proves it.’

*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.

This book presents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning. Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new mathematical results are unaffected by the digital era. The reality is quite different. Digital technologies have transformed how mathematicians work together, how proof is taught in schools and universities, and even the nature of proof itself. Checking billions of cases in extremely large but finite sets, impossible a few decades ago, has now become a standard method of proof. Distributed proving, by teams of mathematicians working independently on sections of a problem, has become very much easier as digital communication facilitates the sharing and comparison of results. Proof assistants and dynamic proof environments have influenced the verification or refutation of conjectures, and ultimately how and why proof is taught in schools. And techniques from computer science for checking the validity of programs are being used to verify mathematical proofs. Chapters in this book include not only research reports and case studies, but also theoretical essays, reviews of the state of the art in selected areas, and historical studies. The authors are experts in the field.

What should citizens know, value, and be able to do in preparation for life and work in the 21st century? In The Teaching of Science: 21st-Century Perspectives, renowned educator Rodger Bybee provides the perfect opportunity for science teachers, administrators, curriculum developers, and science teacher educators to reflect on this question. He encourages readers to think about why they teach science and what is important to teach.

Focuses on essential knowledge for teachers about proof and the process of proving. It is organised around five big ideas, supported by multiple smaller, interconnected ideas-essential understandings. Taking you beyond a simple introduction to proof and the activities involved in proving, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students...and teachers.

ANDY HARGREAVES Department of Teacher Education, Curriculum and Instruction Lynch School of Education, Boston College, MA, U.S.A. ANN LIEBERMAN Carnegie Foundation for the Advancement of Teaching, Stanford, CA, U.S.A. MICHAEL FULLAN Ontario Institute for Studies in Education, University of Toronto, Canada DAVID HOPKINS Department for Education and Skills, London, U.K. This set of four volumes on Educational Change brings together evidence and insights on educational change issues from leading writers and researchers in the field from across the world. Many of these writers, whose chapters have been specially written for these books, have been investigating, helping initiate and implementing educational change, for most or all of their lengthy careers. Others are working on the cutting edge of theory and practice in educational change, taking the field in new or even more challenging directions. And some are more skeptical about the literature of educational change and the assumptions on which it rests. They help us to approach projects of understanding or initiating educational change more deeply, reflectively and realistically. Educational change and reform have rarely had so much prominence within public policy, in so many different places. Educational change is ubiquitous. It figures large in Presidential and Prime Ministerial speeches. It is at or near the top of many National policy agendas. Everywhere, educational change is not only a policy priority but also major public news. Yet action to bring about educational change usually exceeds people's understanding of how to do so effectively.

The study of curriculum, beginning in the early 20th century, first served the areas of school administration and teaching and was used to design and develop programs of study. The field subsequently expanded and drew upon disciplines from the arts, humanities, and social sciences to examine larger educational forces and their effects upon the individual, society, and conceptions of knowledge. Curriculum studies now embraces an array of academic scholarship in relation to personal and institutional needs and interests while it also focuses upon a diverse and complex dynamic among educational experiences, practices, settings, actions, and theories. The Encyclopedia of Curriculum Studies provides a comprehensive introduction to the academic field of curriculum studies for the scholar, student, teacher, and administrator. This two-volume set serves to inform and to introduce terms, events, documents, biographies, and concepts to assist the reader in understanding aspects of this rapidly changing, expansive, and contested field of study. Key Features Displays different perspectives by having authors contribute independent essays on the nature and future of curriculum studies Presents a unique and in-depth treatment of the Twenty-Sixth Yearbook of the National Society for the Study of Education (NSSE), a 1927 publication that has taken on legendary dimensions for the field of curriculum studies Contains bibliographic entries which feature specific publications by curriculum leaders that helped to define the field Helps readers to learn unfamiliar terms and concepts, to become more comfortable with specialized phrases, and to understand the many significant and perplexing concepts and questions that characterize the field Key Themes Biography and Prosopography Concepts and Terms Content Descriptions Influences on Curriculum Studies Inquiry and Research Nature of Curriculum Studies Organizations, Schools, and Projects Publications Theoretical Perspectives Types of Curricula The Encyclopedia of Curriculum Studies offers the careful reader a surprisingly revealing depiction of the conventions, mores, and accepted research and writing practices of the field of curriculum studies as it continues to expand and change. Availability in print and electronic formats provides students with convenient, easy access, wherever they may be.

The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to English-speaking students. This book is an elaborate version of the course on Conjecture and Proof. It gives miniature introductions to various areas of mathematics by presenting some interesting and important, but easily accessible results and methods. The text contains complete proofs of deep results such as the transcendence of $e$, the Banach-Tarski paradox and the existence of Borel sets of arbitrary (finite) class. One of the purposes is to demonstrate how far one can get from the first principles in just a couple of steps. Prerequisites are kept to a minimum, and any introductory calculus course provides the necessary background for understanding the book. Exercises are included for the benefit of students. However, this book should prove fascinating for any mathematically literate reader.

Research on teaching and learning proof and proving has expanded in recent decades. This reflects the growth of mathematics education research in general, but also an increased emphasis on proof in mathematics education.