How to Study Mathematics

Thinking skills and problem-solving activities are indispensable to every area of our lives. To some extent, we are all problem solvers. The problem solver’s work is mostly a tangle of guesswork, analogy, wishful thinking, observing patterns, and frustration. To become a master problem solver may be as inaccessible as acquiring the skills of a violin virtuoso, but everyone can become a better, more confident problem solver.

Robert Blitzer, Introductory Algebra for College Students, 2nd ed., p. 221 [Upper Saddle River, NJ: Prentice Hall, 1997].

 

Study habits. To master mathematical concepts, all we need is practice, practice, and more practice. A steady pace, practice, and patience will take you further than any shortcuts. You may have to change your study habits, because study habits that were sufficient for success in high school are not good enough for success in university. If you do an exercise for which solutions or answers are provided, do not look at them before giving at least one honest try to find the answer (if there is one). In other words, do not give up easily.

The process of learning from an exercise starts with the right questions:

  • What do I want to know?
  • What do I have to explain?
  • What do I know about this?
  • What is this exercise saying?
  • What is this exercise testing?

Make your own summaries. At the end of each section, make a summary, table, or flow chart of the concepts to be learned. When you look at your summary, you should know what the section is all about, why it is important, and what strategies and skills are needed to solve the problems presented at the end of the section.

Know your textbooks and learning materials. Review all learning materials. Take note of the information provided and learn how to use it. Bookmark the recommended Websites (if any), and go through the material on the links.

Familiarize yourself with all the learning features of your textbook. Most textbooks provide summaries at the end of sections or chapters, quizzes, and review problems. Take note of those you find most useful and do them faithfully. At the university level, not all answers to the exercises in a textbook are provided, because students are expected to develop the ability to know when their work is correct. With learning comes self-confidence, and toward the end of a course, students should feel confident about their learning. More importantly, professionals need to learn to detect their own mistakes and those of others.

Your textbooks are not a collection of assigned problems; you are expected to understand the course material and put it to use to solve problems. Hunting through a section to find a worked out exercise that is similar to the one assigned is totally inappropriate. Solving mathematics problems requires understanding of the concepts discussed. Never read mathematics without pencil and paper in hand. Read the corresponding assigned section before trying the exercises.

You should treat the examples as exercises and try to solve them without the author’s help. Be prepared to check all the details as you read, and above all, read critically. Take nothing for granted. If you do not understand a statement, go back in the section, or to a previous section, to see if you misunderstood something. If you still fail to understand after puzzling over it a while, mark the place, continue reading, and ask your tutor for help on that point.

You will be asked to do exercises as concepts arise—do them as they are assigned, not later; you need to make sure you understand what is presented before going on to the next concept. Do not limit yourself to the assigned exercises in the course. Try to do as many exercises as you can, and do not avoid those that appear to be challenging. The fun in learning is the satisfaction of being able to respond positively to a challenge. The time you have spent learning your materials will stand you in good stead. It might even save you time in the long run.

Expectations. In high school, students are accustomed to the first few weeks of a new course being nothing more than a review of the last course, or even of the last few courses. When a new idea is finally presented, the class spends several days working on it before moving ahead. Students quickly learn that there is no major crisis if they do not follow what is said the first time or miss a class. It is repeated soon (and sometimes ad nauseam).

In university, however, reviews of previous material, if they occur at all, will be brief. From the start you will study something new. Moreover, every subsequent new concept will assume mastery of previous ones and will involve the presentation of new ideas. This is why you must study, definition by definition, topic by topic, steadily and constantly. Mathematics cannot be learned in a hurry. Mastery of the basics is always assumed.

For example, to find the solutions of the equation 5x² + 6 = 7, you would expect your high school teacher to write

5x² + 6 = 7, hence 5x² = 7 – 6 = 1.

This implies x² = 1/5, and we conclude that
x = plus or minus 1 over the square root of 5.

In university, you will find something like

The solutions of the equation 5x² + 6 = 7 are
x = plus or minus 1 over the square root of 5.

No details are given. If you cannot work out the details, review your basics and make sure you understand the algebraic process. Ask for assistance if necessary. Do not leave it for another day. Lack of proficiency on the basics is the major cause of students’ frustration in mathematics courses.

Asking for help. You will receive helpful assistance if you clearly articulate your problems and difficulties. Be prepared to give the exact reference for where your difficulties lie (source, problem number, page, section, etc.) and indicate what you do not understand. If you do not know where to start, ask yourself why. Review definitions, results, and examples for the concepts used in the problem.

Discussing problems and course material with someone can help you to gain insight into difficult concepts. But take warning: others cannot understand the concept for you. It may be tempting to let others do your homework for you, but you will not learn it this way. You must take the time to learn! It is unwise to pretend that you understand when you do not.

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