How to Read your Mathematics Textbook Properly

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

                                                               

     placed in a box.

 

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.