Mathematics for High School Teachers- An Advanced Perspective

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9780130449412: Mathematics for High School Teachers- An Advanced Perspective

This book gives readers a comprehensive look at the most important concepts in the mathematics taught in grades 9-12. Real numbers, functions, congruence, similarity, area and volume, trigonometry and more. For high school mathematics teachers, mathematics supervisors, mathematics coordinators, mathematicians, and users of the University of Chicago School Mathematics Project materials for grades 7-12 who want a comprehensive reference book to use throughout their careers or anyone who wants a better understanding of mathematics.

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Mathematics for High School Teachers–An Advanced Perspective is intended as a text for mathematics courses for prospective or experienced secondary school mathematics teachers and all others who wish to examine high school mathematics from a higher point of view.

Preliminary versions of the book have been used in a variety of ways, ranging from junior and senior (capstone) or graduate mathematics courses for pre-service secondary mathematics education majors to graduate professional development courses for teachers. Some courses included both undergraduate and graduate students and practicing teachers with good success.

There is enough material in this book for at least a full year (two semesters) of study under normal conditions, even if only about half of the problems are assigned. With a few exceptions, the chapters are relatively independent and an instructor may choose from them. However, some chapters contain more sophisticated content than others. Here are four possible sequences for a full semester's work:

  • Algebra emphasis: Chapters 1-6
  • Geometry emphasis: Chapters 1, 7-11
  • Introductory emphasis: Chapters 1, 3, 4, 7, 8,10
  • More advanced emphasis: Chapters 1, 2, 5, 6, 9,11.

In each sequence we suggest beginning with Chapter 1 so that students are aware of the features of this book and of some of the differences between it and other mathematics texts they may have used. More information and suggestions in this regard can be found in the Instructor's Notes. Additional instructional resources are also at the web site http://www.prenhall.com/usiskin .

The presentation assumes the student has had at least one year of calculus and a post-calculus mathematics course (such as real analysis, linear algebra, or abstract algebra) in which proofs were required and algebraic structures were discussed. The term "from an advanced standpoint" is taken to mean that the text examines high school mathematical ideas from a perspective appropriate for college mathematics majors, and makes use of the kind of mathematical knowledge and sophistication the student is gaining or has gained in other courses.

Two basic characteristics of Mathematics for High School Teachers–An Advanced Perspective, taken together, distinguish courses taught from this book from many current courses. First, the material is rooted in the core mathematical content and problems of high school mathematics courses before calculus. Specifically, the development emanates from the major concepts found in high school mathematics: numbers, algebra, geometry, and functions. Second, the concepts and problems are treated from a mathematically advanced standpoint, and differ considerably from materials designed for high school students.

The authors feel that the mathematical content in this book lies in an area of mathematics that is of great benefit to all those interested in mathematics at the secondary school level, but is rarely seen by them. Specifically, we have endeavored to include:

  1. analyses of alternate definitions, language, and approaches to mathematical ideas
  2. extensions and generalizations of familiar theorems
  3. discussions of the historical contexts in which concepts arose and have changed over time
  4. applications of the mathematics in a wide range of settings
  5. analyses of common problems of high school mathematics from a deeper mathematical level
  6. demonstrations of alternate ways of approaching problems, including ways with and without calculator and computer technology
  7. connections between ideas that may have been studied separately in different courses
  8. relationships of ideas studied in school to ideas students may encounter in later study.

There are many reasons why we believe a teacher or other person interested in high school mathematics should have this knowledge. Mere are a few. Knowing alternate approaches helps in making decisions regarding curriculum, selection of materials, and lesson plans. Being able to connect, extend, and relate mathematical ideas to each other and to the mathematics a student may take later helps in designing courses and responding to student questions. Having a sense of history and the stories behind the mathematics can make lessons more interesting and engaging for both teacher and student. Encountering the richness of the mathematics that is studied at the high school level helps us to understand why some students are turned on by that mathematics, while others have difficulty with it.

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