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Handbook of Mathematical Cognition
How does the brain represent number and make mathematical calculations? What underlies the development of numerical and mathematical abilities? What factors affect the learning of numerical concepts and skills? What are the biological bases of number knowledge? Do humans and other animals share similar numerical representations and processes? What underlies numerical and mathematical disabilities and disorders, and what is the prognosis for rehabilitation? These questions are the domain of mathematical cognition, the field of research concerned with the cognitive and neurological processes that underlie numerical and mathematical abilities. The Handbook of Mathematical Cognition is a collection of 27 essays by leading researchers that provides a comprehensive review of this important research field.
eBook
Willful ignorance : the mismeasure of uncertainty
\"An original account of willful ignorance and how this principle relates to modern probability and statistical methods Through a series of colorful stories about great thinkers and the problems they chose to solve, the author traces the historical evolution of probability and explains how statistical methods have helped to propel scientific research. However, the past success of statistics has depended on vast, deliberate simplifications amounting to willful ignorance, and this very success now threatens future advances in medicine, the social sciences, and other fields. Limitations of existing methods result in frequent reversals of scientific findings and recommendations, to the consternation of both scientists and the lay public.Willful Ignorance: The Mismeasure of Uncertainty exposes the fallacy of regarding probability as the full measure of our uncertainty. The book explains how statistical methodology, though enormously productive and influential over the past century, is approaching a crisis. The deep and troubling divide between qualitative and quantitative modes of research, and between research and practice, are reflections of this underlying problem. The author outlines a path toward the re-engineering of data analysis to help close these gaps and accelerate scientific discovery. Willful Ignorance: The Mismeasure of Uncertainty presents essential information and novel ideas that should be of interest to anyone concerned about the future of scientific research. The book is especially pertinent for professionals in statistics and related fields, including practicing and research clinicians, biomedical and social science researchers, business leaders, and policy-makers\"-- Provided by publisher.
Book
How mathematicians think
To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics,How Mathematicians Thinkreveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results.
Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure.
The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, andHow Mathematicians Thinkprovides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a \"final\" scientific theory?
Ultimately,How Mathematicians Thinkshows that the nature of mathematical thinking can teach us a great deal about the human condition itself.
eBook
A Mathematics Course for Political and Social Research
Political science and sociology increasingly rely on
mathematical modeling and sophisticated data analysis, and many
graduate programs in these fields now require students to take a
\"math camp\" or a semester-long or yearlong course to acquire the
necessary skills. The problem is that most available textbooks are
written for mathematics or economics majors, and fail to convey to
students of political science and sociology the reasons for
learning often-abstract mathematical concepts. A Mathematics
Course for Political and Social Research fills this gap,
providing both a primer for math novices and a handy reference for
seasoned researchers.
The book begins with the fundamental building blocks of
mathematics and basic algebra, then goes on to cover essential
subjects such as calculus in one and more than one variable,
including optimization, constrained optimization, and implicit
functions; linear algebra, including Markov chains and
eigenvectors; and probability. It describes the intermediate steps
most other textbooks leave out, features numerous exercises
throughout, and grounds all concepts by illustrating their use and
importance in political science and sociology.
Uniquely designed for students and researchers in political
science and sociology
Uses examples from political science and sociology
Features \"Why Do I Care?\" sections that explain why concepts
are useful to practicing political scientists and sociologists
Includes numerous exercises
Complete online solutions manual (available only to
professors)
Selected solutions available online to students
eBook
Individuelle Lernunterstützung Im Mathematikunterricht
Long description: In der Diskussion um die Gestaltung von Lehr-Lern-Prozessen in der Schule finden aktuelle Erkenntnisse neuro-wissenschaftlicher Forschung immer mehr Gehör. Die vorliegende Arbeit setzt aus der immer genaueren Kenntnis der Vorgänge im Gehirn erwachsende Folgerungen mit aktuellen Ansätzen der Schulpädagogik und der Mathematikdidaktik in Beziehung und leitet Forderungen an einen zeitgemäßen Mathematikunterricht ab. Dabei wird deutlich, dass Rückmeldungen der Lehrkraft an die Lernenden eine enorm wichtige Hilfestellung im Lern-prozess sein können – wenn sie bestimmte Kriterien erfüllen. So werden, ausgehend von theoretischen Konzeptionen und empirischen Befunden, verschiedene praxistaugliche Feedbackmethoden entwickelt. Dazu zählt beispielsweise eine optimierte Auswertung der Ergebnisse von Leistungserhebungen mit dem Ziel einer sinnvolleren und effektiveren Nutzung der daraus gewonnenen Erkenntnisse für den weiteren Lernprozess. Alle Methoden werden im Unterricht erprobt, ihre Auswirkungen evaluiert und mögliche Weiterentwicklungen und Variationsmöglichkeiten thematisiert.
eBook