This volume is intended for mathematicians and mathematics instructors who want to enhance the learning and achievement of students in their undergraduate mathematics courses. The chapters in this volume are based on research that closely examines the cognitive and social complexities of how learners build mathematical ideas. Academic mathematicians spend much of their time teaching, preparing to teach, or thinking about how to improve their teaching. Their conversations with colleagues and personal reflections often raise difficult questions about teaching and student learning. All the while the undergraduate mathematics education research community produces theories, models, curricula, and learning materials that speak to the questions mathematics instructors ask. To date, however, few vehicles have been available to assist instructors in using this research knowledge to better understand students' conceptual growth and to facilitate their reflection on teaching practice. This volume is intended to bring some of the knowledge created by mathematics education researchers to the attention and service of mathematics instructors. The 23 chapters in the volume are divided into two sections. In Part 1, “Student Thinking,” the chapters describe perspectives and findings derived from investigations about how people learn central ideas in the undergraduate mathematics curriculum. Part 2, “Cross-Cutting Themes,” contains chapters that focus on the teaching of mathematics and various ways to frame issues that are inextricably related to the art of teaching. In the table of contents, for each article there is a brief annotation describing what issues the article addresses.
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