100 Commonly Asked Questions in Math Class: Answers that Promote Mathematical Understanding, Grades 6–12

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Alfred S. Posamentier, William Farber, Terri L. Germain-Williams, Elaine Paris, Bernd Thaller & Ingmar Lehmann

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    About the Authors

    Alfred S. Posamentier is Dean of the School of Education and Professor of Mathematics Education at Mercy College, New York. Previously, he held these same positions for 40 years at The City College of The City University of New York. He began his professional career as a high school mathematics teacher for six years in New York City. He is the author and coauthor of more than 55 mathematics books for teachers, students, and the general readership. He is also a frequent commentator in newspapers on topics relating to education.

    Dr. Posamentier has extended his reputation in mathematics education to Europe. He has been visiting professor at several European universities in Austria, England, Germany, and Poland, and at the University of Vienna, he was Fulbright Professor in 1990.

    William Farber is an Associate Professor of Mathematics Education at Mercy College in New York. He presently serves as Director of the Graduate Level Clinically Rich Teacher Preparation Pilot Program, which was awarded to the School of Education of Mercy College to establish an ongoing partnership with participating high-need schools in the Yonkers School District. His professional experiences include serving as K–12 Mathematics Specialist for the New York City Department of Education's Department of Mathematics, Director of the Dr. Charlotte K. Frank Mathematics Education Center at The City College of New York, and a teacher of secondary school mathematics in the Bronx. He is the author and coordinator of many grant awards involving innovative professional development programs in mathematics education.

    Terri L. Germain-Williams is an Assistant Professor of Mathematics Education at Mercy College. Prior to accepting the position with Mercy College Graduate School of Education, Germain-Williams worked as an Achievement Manager with the New York City Department of Education, supporting more than 25 K–12 schools in the areas of instruction, strategic planning, professional development, federal and state data and accountability, scheduling/programming, and student services. She has taught Grades 8–12 mathematics and served the New York City Department of Education as a high school Assistant Principal.

    Elaine Paris holds BS and MA degrees in mathematics from Brooklyn College, CUNY, and a doctorate in mathematics education from Teachers College, Columbia University.

    She began her teaching career at Erasmus Hall High School in Brooklyn as a math teacher. Later, Dr. Paris began teaching mathematics and computer programming at Mercy College. She coauthored the first computer manual used at Mercy College and published several math and computer texts and articles.

    She has been a professor in the Department of Mathematics and Computer Information Sciences for more than 30 years, Assistant Chair and Chair, as well as the Director of the U.S. Department of Education federal grant McNair Post-Baccalaureate Scholar Program. Dr. Paris recently retired and holds the rank of Professor Emerita.

    Bernd Thaller is a mathematical physicist and Associate Professor at the University of Graz, Austria, who has made contributions to quantum mechanical spectral and scattering theory. He has written several books on quantum mechanics, introducing and applying new visualization methods for quantum wave functions. In recent years, his focus has shifted toward educational math. He was coordinator of a European Community project for teaching math in secondary high schools between 2005 and 2008. In 2008, he founded and is currently the head of a regional educational competence center for mathematics and geometry, whose main task is the coordination of activities of the math teacher training institutions in the region.

    Ingmar Lehmann is retired from the mathematics faculty at Humboldt University in Berlin. For many years, he led the Berlin Mathematics Student Society for gifted secondary school students, an organization with which he is still closely engaged today.

    He is the author of numerous mathematics texts in Germany and the coauthor with Alfred S. Posamentier of some other books, including The Secrets of Triangles, The Glorious Golden Ratio, Magnificent Mistakes in Mathematics, and The Fabulous Fibonacci Numbers.

    Introduction

    It is well known that students will pose questions to teachers about a variety of topics. Oftentimes, these topics are related to what is being taught. Yet on occasion, students who hear about things mathematical from family, friends, or the media will not hesitate to ask their math teacher to explain something that may have been confusing to them. Most of the time, teachers can respond to these questions in proper fashion. However, there may be times when the question asked is not at ready reference for the teacher. We hope that this book will provide a resource for teachers who need a quick response to a variety of questions relating to school mathematics.

    In general, mathematics education practitioners and researchers agree that teachers need to present mathematics as a motivating subject to their students—one that promotes and fosters thinking, and, at times, flexible and unrestricted inquiry. In fact, when explaining the Process Standard “Reasoning and Proof,” the National Council of Teachers of Mathematics states thus: “The ability to reason systematically and carefully [in mathematics] develops when students are encouraged to make conjectures, are given time to search for evidence to prove or disprove them, and are expected to explain and justify their ideas.” Clearly, this statement indicates a national imperative in mathematics education for teachers to emphasize the importance of process as well as mathematical thinking in their classes.

    At the same time, students want—and need—to know the answers to their mathematics questions without the teacher telling them to think through the problem. In other words, although process in solving mathematical problems and conceptual understanding of mathematical topics are both important, students need to learn procedural methods that will help them solve problems efficiently and effectively.

    In this book, the authors identify common mathematics content questions that students, almost exclusively at the secondary level, often ask in class. We then follow each question with clear and concise answers that are aligned with the Common Core State Standards. Although the question-answer dyads themselves will span the middle school and high school grade levels, some questions might be introduced in grade levels prior to middle school. Accordingly, secondary-level students frequently ask questions that appear to be late elementary in terms of content. Because of their seemingly elementary-level nature, teachers often dismiss such questions. Given the emphasis on secondary-school student mathematics questions, our focus is to tailor seemingly elementary-level questions toward secondary mathematics courses.

    This book will help to prepare the novice mathematics teacher or serve as enrichment for the more experienced mathematics teacher in anticipating common content-related mathematics questions that students of all ages and grade levels will undoubtedly ask during the school year. In addition, the authors strive to provide efficient answers that encourage flexibility in ways of solving various mathematics problems. The topic of this book is of increasing importance so that teachers of mathematics answer students’ questions effectively and anticipate what students will ask so that the presentation of answers is clear and concise. Of equal importance is the book's emphasis on answers to mathematical questions as they relate to efficient test-taking strategies in different assessments that students will need to take, especially at the secondary level.

    Each chapter provides a detailed list of common mathematics questions along with their answers in the most comprehensive way possible. This book focuses on practical application in mathematics and highlights ways that teachers can engage and motivate students. At the same time, the practical nature of the book is supported by research in mathematics education and human development.


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