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*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.
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. "
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.
*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.
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