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A comprehensive collection of historical readings in the philosophy of mathematics and a selection of influential contemporary work, this much-needed introduction reveals the rich history of the subject. An Historical Introduction to the Philosophy of Mathematics: A Reader brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject. Presenting historical background essential to understanding contemporary trends and a survey of recent work, An Historical Introduction to the Philosophy of Mathematics: A Reader is required reading for undergraduates and graduate students studying the philosophy of mathematics and an invaluable source book for working researchers.
Raising students' math achievement doesn't mean ripping up your planning book and starting over. In Accessible Mathematics Steven Leinwand (author of Sensible Mathematics) shows how small shifts in the good teaching you already do can make a big difference in student learning. Steve focuses on the crucial issue of classroom instruction. He scours the research and visits highly effective classrooms for practical examples of small adjustments to your teaching that lead to deeper student learning in math. Some of his 10 classroom-tested teaching shifts may surprise you and others will validate your thinking. But all of them will improve your students' performance. Thoroughly practical and ever-aware of the limits of teachers' time, Steve gives you everything you need to put his commonsense ideas to use immediately. His extensive planning advice will help you streamline your teaching to get more from everything you do. Classroom examples from every grade level model teaching language and instructional moves. And his suggestions for professional learning help increase your effectiveness through the power of collaboration. Steven Leinwand shares your priority: raising the mathematical understanding and achievement of every one of your students. Read Accessible Mathematics, try his 10 suggestions in your practice, and discover how minor shifts in your teaching can put student learning into high gear.
In this first modern, critical assessment of the place of mathematics in Berkeley's philosophy and Berkeley's place in the history of mathematics, Douglas M. Jesseph provides a bold reinterpretation of Berkeley's work. Jesseph challenges the prevailing view that Berkeley's mathematical writings are peripheral to his philosophy and argues that mathematics is in fact central to his thought, developing out of his critique of abstraction. Jesseph's argument situates Berkeley's ideas within the larger historical and intellectual context of the Scientific Revolution. Jesseph begins with Berkeley's radical opposition to the received view of mathematics in the philosophy of the late seventeenth and early eighteenth centuries, when mathematics was considered a "science of abstractions." Since this view seriously conflicted with Berkeley's critique of abstract ideas, Jesseph contends that he was forced to come up with a nonabstract philosophy of mathematics. Jesseph examines Berkeley's unique treatments of geometry and arithmetic and his famous critique of the calculus in The Analyst. By putting Berkeley's mathematical writings in the perspective of his larger philosophical project and examining their impact on eighteenth-century British mathematics, Jesseph makes a major contribution to philosophy and to the history and philosophy of science.
John Dee's angel conversations have been an enigmatic facet of Elizabethan England's most famous natural philosopher's life and work. Professor Harkness contextualizes Dee's angel conversations within the natural philosophical, religious and social contexts of his time. She argues that they represent a continuing development of John Dee's earlier concerns and interests. These conversations include discussions of the natural world, the practice of natural philosophy, and the apocalypse.
Boost confidence, reduce anxiety, and spark those "aha" moments for students through effective math tutoring! Filled with research-backed guidance for tutors to ensure students develop the confidence and skills they need, The Math Tutor′s Handbook: Strategies and Tips for Success is the ultimate guide for effective math tutoring. As national data suggest that students are falling behind in mathematics performance, the role of the math tutor is more critical than ever before. The authors bring decades of wisdom and know-how to the tutoring table. Steve Leinwand—a renowned leader in math education—and Caroline Welty—a sought-after tutor who brings current insight into today’s learners—have blended their collective experience into a comprehensive handbook that offers clear guidance for effective tutoring in grades K-12 mathematics. With their conversational style, the authors provide hands-on resources to help diagnose individuals’ mathematical strengths and needs to make sure that each tutoring session carries the necessary impact.. Through a range of activities, checklists, examples, stories, and suggestions, this handbook Focuses on the importance of personalized instruction that no computer program can match Describes how to develop strong, supportive relationships between tutor and student Identifies common stumbling blocks around the big ideas in mathematics across grade levels and suggests how to overcome them Celebrates mistakes as valuable learning opportunities to foster a deeper comprehension of mathematical concepts rather than just getting the right answer Promotes skills needed for the future, such as problem-solving and critical thinking, which are beneficial not only in future math classes but also in various aspects of life and work Let The Math Tutor′s Handbook empower you to create a positive learning environment for your students and help them boost self-confidence, reduce math anxiety, and master essential skills. Whether you are a private tutor or parent, you work in a tutoring center, or you are an in-school K-12 math teacher, coach, or specialist, this is the resource you need for the math tutoring strategies and practices that are foundational to successful learning.
In this book Reale presents Plato and Aristotle. At the center of Reale's interpretation of Plato is the fulcrum of the supersensible, the metaphysical discovery that Plato presented as a result of the Second Voyage. This discovery of the supersensible is, in Reale's view, not only the fundamental phase of ancient thought, but it also constitutes a milestone on the path of western philosophy. Reale presents Plato in three different dimensions: the theoretic, the mystical-religious, and the political. Each of these components takes on meaning from the Second Voyage. In addition, Reale has shown that only in the light of the Unwritten Doctrines handed down through the indirect tradition, do these three components, and the Second Voyage itself, acquire their full meaning, and only in this way is a unitary conception of Plato's thought achieved. The interpretation of Aristotle that Reale proposes depends on his interpretation of Plato. Aristotle read without preconceptions is not the antithesis of Plato. Reale points out that Aristotle was unique among thinkers close to Plato, in being the one who developed, at least in part, his Second Voyage. The systematic-unitary interpretation of Aristotle which Reale has previously supported converges with the new systematic-unitary interpretation of Plato. Certain doctrinal positions which are usually reserved to treatments in monographs will be explored, because only in this way can the two distinctive traits of Aristotle's thought emerge: the way in which he tries to overcome and confirm the Socratic-Platonic positions, and the way in which he formally creates the system of philosophical knowledge.