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 Think reveals 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, and How Mathematicians Think provides 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 Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.

Developing logical thinking and fundamental mathematical ideas, and using problems that pique students' mathematical curiosity, this work aims to prepare readers for all upper-division mathematics courses and improve their skills in presenting coherent arguments.

This document focuses on how mathematics teaching and learning can be improved by developing more powerful approaches to connect thinking and mathematics. It proposes changing perspectives on what it means to learn and do mathematics and explores how these perspectives can be incorporated into the teaching of secondary school mathematics. Chapter 1 offers a view of mathematics as emerging largely from individual and social activity rather than from textbooks, worksheets, and tradition. The learner is depicted as someone who actively constructs meaning instead of passively receiving it. Chapter 2 considers how a greater emphasis on communication (discussion, debate, recording, and writing) stimulates and uncovers students' learning and thinking and leads to a deeper understanding by both teachers and students. Chapter 3 explores how teachers might encourage greater inquiry and communication in a secondary school class by making minor, but thought-provoking changes in ordinary problems and situations. Finally, chapter 4 gives some practical advice on transforming the mathematics classroom into a place where students are expected not only to absorb and consume mathematics but also to produce and think about it. Contains 17 references and 17 figures. (MKR)

A basic text for young and adult/adolescent literature courses. An exciting new young adult literature text...one that offers a holistic, inclusive approach to incorporating the popular books young adults like to read into a solid English language arts curriculum. A student-centered, whole language emphasis advocates a curriculum that encourages student response to literature and helps develop crucial critical thinking skills. The author suggests numerous approaches to using young adult literature, explores and demonstrates a variety of teaching methods, and discusses an array of literature appropriate for a wide range of students in a number of subject areas, including literature written specifically for children and young adults, popular adult literature, and the classics.