Your
mathematics textbook in high school was most probably not a book that you were
required to read. It was primarily a book to provide you with assignments. The explanations
about mathematics came from the teacher who presented examples of the types of
problems assigned as the homework. College textbooks are written to be read by
students and college instructors expect students to read them. In fact, because
the pace of delivering material is faster at the college level, not all the
material a student is expected to master is actually covered in the class
lecture. When a section of the textbook is covered by a college instructor, the
instructor expects students to read the section thoroughly for understanding
and to be responsible for everything in that section, even material that is not
explicitly mentioned in class. Thus, it is important that you read and, in
fact, study the assigned sections of the textbook.
Reading a textbook in
mathematics should be looked at as a slow and thoughtful process. A student
should always be ready with questions when reading the textbook. Good questions
are “why is this true?” or “how does that follow from the previous line?” A
mathematics text should always be read with a pen or pencil and paper at hand
for jotting down notes and working out missing steps in algebra or calculus.
Filling in the missing steps in a textbook example provides good practice
before you tackle the problems at the end of the section. Authors of textbooks
are, general speaking, experienced classroom teachers and only leave out
details which are within the backgrounds of the students for whom the book is
written. So don’t be discouraged by missing details or steps. Instead take them
as an opportunity to check your understanding of the material.
The format of a textbook
in mathematics is really quite standard. Each chapter is made up of sections
and each section usually consists of an amount of material appropriate for one
class discussion. A section begins with an introduction intended to give an
overview of the material, to relate it to previous material, or to link it to
future work. The section will feature important definitions, usually
highlighted in some special way, and introduce notation. An explanation of the
main ideas of the section is then followed by examples and sample problems with
solutions to illustrate the ideas. A section may include other highlighted
items besides definitions. These include algebraic identities, formulas
sometimes derivations, properties, theorems, most usually proofs, and solution
techniques. If your goal is to be successful in mathematics, a standard
procedure to adopt is to understand with precision every highlighted item in a
section of the text, i.e., items which are written in bold print or
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A section usually concludes with an exercise set
containing a collection of problems related to the material covered in the
section. Many of the problems are very similar to those presented in the
examples of the section. A student who has studied the section should have no
trouble working these problems. Even though these problems may be routine it is
important that the student do them to acquire practice in recognizing the
crucial steps to the problem solution. By becoming familiar with a method of
solution through practice a student provides insurance against falling into
avoidable errors on exams and quizzes. (“Avoidable errors” are sometimes called
“silly mistakes” by the students who made them) The higher numbered problems in
an exercise set are usually more difficult. Authors tend to build exercise sets
so that students who work through the whole set starting with the first problem
stand a better chance of solving the later problems than students who skip over
the beginning problems. Often the reason that the later problems are more
difficult for students is that they generally involve not only the material
covered in the section but they also require a working knowledge of material
from previously covered sections. You can solve even the later problems of the
exercise set if you make it a policy to learn for long-term understanding, not
just short-term recall.