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© 2016,In 1936, when he was just twentyfour years old, Alan Turing wrote a remarkable paper in which he outlined the theory of computation, laying out the ideas that underlie all modern computers. This groundbreaking and powerful theory now forms the basis of computer science. In Turing's Vision , Chris Bernhardt explains the theory, Turing's most important contribution, for the general reader. Bernhardt argues that the strength of Turing's theory is its simplicity, and that, explained in a straightforward manner, it is eminently understandable by the nonspecialist. As Marvin Minsky writes, "The sheer simplicity of the theory's foundation and extraordinary short path from this foundation to its logical and surprising conclusions give the theory a mathematical beauty that alone guarantees it a permanent place in computer theory." Bernhardt begins with the foundation and systematically builds to the surprising conclusions. He also views Turing's theory in the context of mathematical history, other views of computation (including those of Alonzo Church), Turing's later work, and the birth of the modern computer. In the paper, "On Computable Numbers, with an Application to the Entscheidungsproblem ," Turing thinks carefully about how humans perform computation, breaking it down into a sequence of steps, and then constructs theoretical machines capable of performing each step. Turing wanted to show that there were problems that were beyond any computer's ability to solve; in particular, he wanted to find a decision problem that he could prove was undecidable. To explain Turing's ideas, Bernhardt examines three wellknown decision problems to explore the concept of undecidability; investigates theoretical computing machines, including Turing machines; explains universal machines; and proves that certain problems are undecidable, including Turing's problem concerning computable numbers.

© 2016,The Math Myth expands on Andrew Hacker's scrutiny of some widely held assumptions: that mathematics broadens our minds; that mastery of arcane concepts  cosine, logarithms, the area of a sphere  will be needed for most jobs; that the Common Core's single format should be required of every student. He worries that a frenzied emphasis on STEM (science, technology, engineering, and mathematics) is diverting resources from other pursuits and subverting the spirit of the country.

© 2016,Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematicsbut, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twentyfirstcentury viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving wellknown theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

© 2015,Following hot on the heels of The Imitation Game , this is the first modern biography of Alan Turing by a member of the familyAlan's nephew, Sir Dermot Turing Alan Turing was an extraordinary man who crammed into a life of only 42 years the careers of mathematician, codebreaker, computer scientist, and biologist. He is widely regarded as a war hero grossly mistreated by his unappreciative country and it has become hard to disentangle the real man from the story. It is easy to cast him as a misfit, the stereotypical professor. But actually Alan Turing was never a professor, and his nickname "Prof" was given by his codebreaking friends at Bletchley Park. Now, Alan Turing's nephew, Dermot Turing, has taken a fresh look at the influences on Alan Turing's life and creativity, and the later creation of a legend. Dermot's vibrant and entertaining approach to the life and work of a true genius makes this a fascinating read. This unique family perspective features insights from secret documents only recently released to the UK National Archives and other sources not tapped by previous biographers, looks into the truth behind Alan's conviction for gross indecency, and includes previously unpublished photographs from the Turing family album.

© 2015,When Critical Multiculturalism Meets Mathematics details the development and outcomes of a teacher professional development project that merged multiculturalism and mathematics. In six compact chapters the authors describe the impetus for their multiyear project and present rich case studies of nine teacher participants. The cases stand alone as compelling reading, yet Marshall et al. extend beyond their distinctiveness to explain the statistical data related to the project s broader impact. Emphasizing both qualitative and quantitative findings makes this book ideal for novice researchers interested in mixed method study. Likewise, the authors unveil the anatomy and a few complexities of conducting research in the real world contexts of schools including participant recruitment and resolution of unanticipated matters that can arise within research teams. A unique twist in the final chapter is Marshall et al. s critique of their own missteps as researchers, which are used skillfully and unobtrusively to proffer tips for future studies. They conclude by theorizing affirmed intersectionality, identified as the critical element that facilitated teachers recognition and acceptance of the compatibility between the study s two components."