*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.

Human papillomavirus (HPV) is a major cause of cervical cancer. Human Papillomavirus Infections in Dermatovenereology pulls together the diverse disciplines of clinical, molecular biological, socio epidemiological, and immunological research to bridge the gap between the clinical aspects and basic biology of HPV. This volume provides a much-needed overview of the scientific and clinical data of HPV and HPV-associated diseases, exploring opinions on current therapies and diagnostic methods. It critically reviews the most frequently used molecular biologic methods, evaluating their potential in HPV detection. Specialists in dermatology, genitourinary medicine, gynecology, urology, as well as pathologists, microbiologists, epidemiologists, and virologists will appreciate this timely examination of the ubiquitous pathogen, HPV.

Eleven years ago the circular DNA of a novel single-stranded virus has been cloned and partially characterized by Nishizawa and Okamoto and their colleagues. According to the initials of the patient from whom the isolate originated, the virus was named TT virus. This name has been subsequently changed by the International Committee on Taxonomy of Viruses (ICTV) into Torque teno virus, permitting the further use of the abbreviation TTV. Although initially suspected to play a role in non A E hepatitis, subsequent studies failed to support this notion.

Within a remarkably short period of time it became clear that TT viruses are widely spread globally, infect a large proportion of all human populations studied thus far and represent an extremely heterogeneous group of viruses, now labelled as Anelloviruses. TT virus-like infections have also been noted in various animal species. The classification of this virus group turns out to be difficult, their DNA contains between 2200 and 3800 nucleotides, related so-called TT-mini-viruses and a substantial proportion of intragenomic recombinants further complicate attempts to combine these viruses into a unifying phylogenetic concept. "

*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.

How do you know if an election is fair? Or if the result truly represents the choice of the people? In Making Democracy Fair students use elementary mathematical methods to explore different kinds of ballots, election decision procedures, and apportionment methods. In the first half of the book, students are introduced to a variety of alternatives to the "winner take all" strategy used in most elections. Determining which strategy is fairest is usually a very difficult question to answer, and many times the strategy chosen determines the winner. In the second part of the book, students investigate different methods of apportionment. How many representatives from each state will there be in the United States House of Representatives? How do countries using a proportional representation decide on the number of representatives from each political party to be seated in their government bodies?

The book is suitable for learners and students from Grade 9 upwards to undergraduate level, and provides firstly an elementary introduction to the fundamental concepts of an implication, equivalence, quantifiers, necessary and sufficient conditions, etc. Thereafter, some basic laws of logic such as Modus Ponens, Contrapositive, Disjunctive Inference, Syllogism, etc. are discussed. Some basic methods of proof such as mathematical induction, direct proof, proof by contradiction, etc. are illustrated with examples from the high school mathematics curriculum. Throughout the book concepts and logical reasoning are discussed with reference to many everyday examples as well as examples from high school algebra, elementary number theory, geometry and trigonometry.