Mathematics for High School Teachers- An Advanced Perspective [Paperback]

Zalman Usiskin (Author), Anthony L. Peressini (Author), Elena Marchisotto (Author), Dick Stanley (Author)

Book Description

ISBN-10: 0130449415 | ISBN-13: 978-0130449412 | Publication Date: December 5, 2002 | Edition: 1

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.

Editorial Reviews

From the Back Cover

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.

Excerpt. © Reprinted by permission. All rights reserved.

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.

Product Details

• Paperback: 596 pages
• Publisher: Prentice Hall; 1 edition (December 5, 2002)
• Language: English
• ISBN-10: 0130449415
• ISBN-13: 978-0130449412
• Product Dimensions: 10.1 x 8 x 1.1 inches
• Shipping Weight: 2.8 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #156,925 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

25 of 25 people found the following review helpful:

5.0 out of 5 stars Sui Generis, July 14, 2003

By 

A. Ali "Harkonnen" (Minneapolis, MN USA) - See all my reviews
(VINE VOICE)    (REAL NAME)   

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This review is from: Mathematics for High School Teachers- An Advanced Perspective (Paperback)

This book is one of a kind. It affords an integrated perspective of traditional high school mathematics, making explicit the intimate relationships between arithmetic, algebra, and geometry. Additionally, it indicates and suggests lines of development that are pursued in undergraduate courses. Both purposes - showing the unity of the subject, and indicating further development - are accomplished by placing traditional high school topics in a broader conceptual and historical perspective.

The book is divided into two parts; the first, titled "Algebra and Analysis with Connections to Geometry", deals with numbers, functions, equations, polynomials, and number systems. The second, titled "Geometry with Connections to Algebra and Analysis", deals with congurence, symmetry, similarity, area annd volume, axiomatics, and trigonometry.

To give some idea of coverage, the second chapter (on real and complex numbers) discusses irrational numbers, a proof of the irrationality of e, the nested intervals property of the reals, countable and uncountable sets, and the diagonal proof of the uncountability of the reals. The chapter on equations briefly discusses cubic and quartic equations and states the unsolvability of the general quintic; the names of Gauss, Ruffini and Galois are mentioned. The chapter on integers and polynomials discusses induction, recursive definitions, simple diophantine equations and the fundamental theorem of arithmetic. It also indicates the analogies between the integers and the set of polynomials (both are integral domains). The chapter on number system structures discusses modular arithmetic, the Chinese remainder theorem, and gives examples of number fields other than the real and complex number systems (e.g. quadratic fields, and finite fields).

The projects at the end of each chapter extend the material covered in a natural way, and are challenging. To give some stray examples, the coordinatisation of the Riemann sphere, the Cardano-Tartaglia method for solving cubic equations, Fermat's last theorem for n = 4, constructible numbers, and the impossibility of squaring the circle and doubling the cube.

The chapter bibliographies are annotated, up-to-date, and list excellent books for further study.

I have a few criticisms. The first is that surjective functions are not discussed, and in this connection the Schroder-Bernstein theorem does not get mentioned or proved. A second and more serious criticism is the slender coverage of analytic geometry. Only five or six pages are devoted to this. As a consequence, the authors cannot discuss the rich field of algebraic curves in particular, and algebraic geometry in general. There is also no mention of projective transformations (i.e. projective geometry) or continuous transformations (i.e. topology). Finally, there is no mention of Klein's Erlanger program.

These quibbles aside, the book is well-conceived and well-written. It can join Courant and Robbins' "What is Mathematics", and Stillwell's "Mathematics and its History" as a book that gives a bird's eye perspective of (part of) the discipline.

Professors teaching undergrad courses would want this book on their shelves; it shows some of the connections between high school material and the relatively abstract courses taught at college (e.g. Galois theory, group theory, algebraic number theory, and real and complex analysis). Undergrad students might want this book for the same reasons. High school teachers who want a bird's eye perspective of high school mathematics from a sophisticated point of view might also want a copy; suggested lines of development can be used as enrichment topics.

11 of 17 people found the following review helpful:

5.0 out of 5 stars good source, August 17, 2006

By 

Physicsmind "Physicswish" (CA United States) - See all my reviews

This review is from: Mathematics for High School Teachers- An Advanced Perspective (Paperback)

This is a good resource book. However, I think it would be a good idea to learn from other countries that are more successful in math education. Students from Singapore, Korea, China, Russia, and Hungary all are better than our students ON AVERAGE!!! Generally, of course!! For example, the Russian have many books that are available for those who do not have access to great teachers. Books are important source for self learning. In the high school mathematics, for example, among many of the books available, I will mention one: --High School Mathematics by Yakovlev, ISBN 5030010114. Over 800 pages comprehensive yet concise and lucid!! Take a look and you will seee. They say nothing more than just taking a student to understand math--itself. THEY DON'T SAY MATH IS FUN AND INTERESTING, WHY, BECAUSE...nonsense...! The joy of math comes from the history, the moment when a great problem is solved, the men and women who struggled with it...