Elementary Algebra, An Individualized Approach - Textbook Preface

Elementary Algebra, An Individualized Approach is designed to aid the student in learning its material.

The text possesses strong teaching qualities that virtually assure the conscientious student of steady, confident progress. At the same time, the text’s interactive processes assure the student of continuous opportunities to get help where and when it is needed. Solid mathematics, quality instruction, and personal assistance are the hallmark of this series of texts.

The text is designed in a format that constantly creates an interaction between the student, the mathematics to be learned, and an experienced teacher either in a classroom or in cyberspace. The majority of the book is a series of questions/answers that achieves a true dialogue with its student reader. Most of the time, the dialogue teaches the mathematics effectively, and students learn easily without benefit of outside assistance. Sometimes, however, the dialogue may be insufficient; it is in such cases where the student can find extra, personal assistance through conversation with a friend or teacher. These helpful conversations can occur in person, but also are available with a cyberspace instructor who is always as close as the nearest computer.

The quality of the mathematical content and instructional use of this text have been thoroughly documented in a wide range of situations and with great numbers of students. Specifically, there are no units or chapters where common difficulties can be expected. When a student has a problem understanding the material, the difficulties will be unique to that student. Something of importance was overlooked, misunderstood, etc. That is the reason for providing personal assistance in those instances and getting the student back into the flow of the book’s dialogue. The personal assistance will both answer the questions of individual students and give guidance in the best ways to study mathematics.

Because of the design of this book, the student can expect other dramatic improvements in the study and learning of mathematics. One of those advantages is the student’s ability to vary the pace of instruction to match their ability to learn. At times, a student needs to work slowly through the instruction and carefully ponder each new concept. At other times, the student grasps the material quickly and can accelerate their learning rapidly through the material. As a result, most students will move through this material more quickly than in a regular class situation.

Another great advantage of this book is the fact that students have an opportunity to learn to ask intelligent, specific questions about the mathematics in a context that encourages and does not threaten the student. Study skill deficiencies can be corrected within the context of these materials while maintaining traditional levels of achievement and rigor.

The content of this book is comparable to a second semester of Algebra as it is normally taught in high school. Chapter 1 reviews the algebras of the counting numbers, integers, and rational numbers. Chapters 2-3 extend the skills of beginning algebra to equations involving two variables. Chapter 4 reviews the use of counting numbers as exponents and then teaches integers as exponents. Chapters 5-7 involve polynomials. In Chapter 5 polynomials are added, subtracted, multiplied, and divided. Chapter 6 is devoted to factoring polynomials. Chapter 7 teaches the skills of polynomial fractions. Chapter 8 reviews the skills of solving linear equations and then extends those skills by attacking equations which involve polynomials. The last three chapters of the book introduce the irrational numbers. Chapter 9 solves quadratic equations when the solutions are rational numbers and hints at the need for other (irrational) numbers. Chapter 10 introduces the irrational numbers and teaches all the skills for simplifying expressions with radicals. Chapter 11 solves equations with irrational or rational solutions.

The spiraling effect on learning is less obvious in this book than in Beginning Algebra, but nevertheless is present throughout; many topics are introduced at one point and later extended. Each chapter ends with an Application Unit teaching word problem-solving processes. Unlike most text presentations of word problems, the student is taught a process for breaking down the information of a word problem and developing methods for extending these processes to the more difficult problems which will be encountered later.

The book contains many opportunities for the student to make wise decisions about the level of understanding achieved. Each chapter begins with an Objectives Test that illustrates, by example, all of the concepts and skills which are to be presented. Each problem is accompanied by a designation showing where it will be taught in the chapter. Each chapter is broken into units in a way that divides the content into topics of such length that it gives the student a way of organizing their own studying to make it most effective. Most units are designed to be completed in a single study session and, for best results, should be handled in that way. The unit ends with a Feedback which is an opportunity for the student to test the level of understanding achieved. It is strongly suggested that the Feedback be taken at the beginning of the next study session so that some time intervenes between the learning and assessment. The chapter ends with a Mastery Test that is similar to the Objectives Test. When a student can do all the problems correctly on a Mastery Test, it indicates a complete understanding of all the material in the chapter. Again, each problem is accompanied by a designation showing where it was taught in the chapter. Answers for all problems in the Tests and Feedbacks are included at the back of the book. Good learning can occur only when the student is aware of what is known and what still needs to be learned.

We are greatly indebted to thousands of teachers and hundreds of thousands of students who have successfully used our previous materials and given us the benefit of their own experiences and ideas. The strength of this new book is a result of that special kind of study of the work and challenges we face in teaching mathematics to students — some in desperate need for overcoming past experiences that were markedly unsuccessful and have interfered with motivations to try again.

Robert H. Alwin
Robert D. Hackworth

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