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Intermediate Algebra, An Individualized Approach - Textbook
Preface
Intermediate 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 year of Algebra as it is
normally taught in high school. Intermediate Algebra primarily involves an expansion
and extension of the topics of an Elementary Algebra course. Most of the titles
of the chapters of this book will be quite familiar to the student who has successfully
completed an earlier Algebra course, but each topic will be presented at a higher
level of development. For example, in an Elementary Algebra course the study of
exponents is limited to integers without any direct relationship being shown between
exponents and radical expressions. Here in Intermediate Algebra the study of exponents
is extended to rational number exponents with much attention being devoted to
exponents and their relationships to radical expressions.
The spiraling effect on learning is clearly evident in this book. After Chapter
1, each chapter in this book reviews some portion of the work of Elementary Algebra
and then extends the concepts to new number systems or more sophisticated problems
requiring more techniques for their solutions. In other words, the spiral effect
begins each new topic at its level of prior knowledge and then lifts the knowledge
of the topic to a new, higher level.
Chapter 1 reviews the vocabulary of Elementary Algebra while teaching the learner
that words and their meanings will be of great importance to success in this course.
The learner is instructed that the acquisition of vocabulary has been directly
linked to expertise in the subject itself. That is, knowing the language of Intermediate
Algebra is a necessary step to successfully learning the concepts and skills described
by that language. A good vocabulary goes beyond the goals of effective communication
between the learner and the instruction. A good vocabulary enables the learner
to efficiently store new information, retrieve it more effectively, and, most
importantly, assimilate it into his/her own knowledge base where it may become
the source of abilities which are far beyond the scope of this course. Chapter
1 stresses the importance of the meanings of words in a mathematics course.
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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