How to Study:  A Plan

There are many reasons for a student’s shortcomings in mathematics.  One of the most serious is the lack of the right kind of study.  Proper application leads to the assimilation of the concepts and principles of mathematics rather than merely to obtaining “the correct answer” to each problem in the homework assignment.

Learning is a process of acquiring facts or gaining experience in dealing with certain situations.  One method of learning is by constant repetition.  In order to avoid a part of this monotonous task, the following provides a list of devices for learning and retaining more easily. 

1.      WHEN TO STUDY

        A. The after-class study period. As soon as possible after each class meeting, review carefully the new ideas and problems that you acquired from the class discussion.  Crystallize your thinking on these points by writing or saying aloud the gist of the principles.  This should require only 5 or 10 minutes and could be done during a free period or even while walking to the next class.

Reorganize your notes if any parts are not legible or understandable.  Underline the things that were stressed in class.

        B. *The main study period.*   Study the new assignment at your first opportunity which can be the “after-class study period” (i.e., do the homework).  Try not to postpone it until the evening before the next class meeting.  Many daylight hours are wasted in social amenities and small talk.  All study need not be done at night. 

        C. !!The before-class study period!!  Allot 5 or 10 minutes just before class to review the basic concepts & homework that you learned in the main study period.  This should include new definitions, formulas, theorems, homework, and a careful recital of the unpardonable mistakes that you have made recently and that you have sworn not to repeat. 

       D. The review periods.  In order not to forget half of what you have learned, plan to have review periods

(1.)every week

(2.)before each regularly-scheduled examination, and

(3.)before the final examination

2.      HOW TO STUDY NEW MATERIAL

        A. Begin with mental calisthenics.  With your textbook and notebook closed, work the problems of the previous lesson or group of lessons.  Compare your statements with those in your notes and/or homework.

        B. Review briefly the general concepts of the previous lesson.  The new assignment is usually much easier if the preceding one has been understood or, better yet, mastered.

        C. Read and WRITE as you study the new material.  In studying a new article or topic in the textbook, read the author’s statements slowly and carefully.  Reading aloud may be more effective than reading silently.  Eventually, you want to progress to the point where you can make notes as you read, but for the time being, read to gain the first exposure.  At the next class meeting, ask the instructor to explain the statements that you do not fully understand.  Or you could make a list of these difficulties and take them to his or her office for further clarification.  Be specific in your questions.  Show him/her, the paper containing your solution to that tough problem – the one for which you got the wrong answer – and the one that you never could complete.  He/she can then diagnose your troubles because he/she can identify your errors – of commission and omission.  Don’t say to your instructor, “I don’t know anything at all about this material.” 

       D. The original problems (the homework assignment) should be worked (either mentally or on paper) without referring to the text.  If, in working the unsolved original problems, you continually refer back to the illustrative examples in the book, you will most likely fail to assimilate the principles involved.  You will be working problems without learning mathematical concepts.  As a consequence, on the examination, you will probably do poorly because the textbook will not be available for reference.

3.      HOW TO ATTACK THE STATED PROBLEMS

Before attempting any of the original problems, you should study the illustrative examples and/or your notes.  In attacking a stated problem:

        A. Read the entire problem carefully to get the overall picture.

        B. Read the problem again, noting the things that are given and those that are required.

        C. If the problem suggests a figure, draw one and label the given quantities.

       D. If there is only one unknown, choose a letter (or symbol) to represent it.  If there are several unknowns, represent one unknown by a letter and try to express the other unknowns in terms of this letter.  If you are unable to do so, choose other letters for the remaining unknowns.

        E. Read the problem again and translate each stated condition into an equation, i.e., find two quantities that are equal.  In general, the number of equations must equal the number of unknowns.

        F. Solve the system of equations.

       G. Check the solutions against the verbal statements in the problem.  Reject any solutions that do not satisfy these stated conditions.

4.      HOW TO REVIEW FOR AN EXAMINATION

A test in mathematics usually consists of

                                                                                                                                                       (1.)      Solving problems,

                                                                                                                                                       (2.)      Stating definitions and theorems (including formulas), and

                                                                                                                                                       (3.)      Proving theorems and deriving formulas.

The “percentage breakdown” of the exam into these three categories depends upon the individual instructor and the course.  As the student progresses toward higher mathematics, he will get fewer questions of type (1) and more of type (3).

   Don’t stay up late the night before an exam!  Get a normal amount of rest.  Do most of your reviewing at least 12 hours before the test is to be given.  Rework at least one problem of each type of problem that has caused you trouble.   It would be most helpful if, sometime earlier, you had distinguished the assigned problems that you could not work by yourself in your homework.  These are the problems that will cause the most trouble on the exam.  Study them thoroughly.  Don’t spend much time on the things that you were able to get without assistance.  Common sense says that you will probably be able to get them again. 

   **Try to foresee what the instructor will ask.**  His/her performance in class (the amount of time he/she spends on various situations and types of problems, together with his/her attitude, announced, or implied), should enable you to identify the most important items.  The exam will probably consist of all the important topics plus some of the secondary items.  The number of exam questions covering a given topic could be proportional to the number of lessons spent on the topic.  If item A was studied for three lessons and if item B was covered in only one lesson, then one might expect more questions on item A than on item B, but this is not always the case.

   Save each test paper that your instructor has graded.  Here is evidence (black on white, with tinges of red) showing:

(a.)  The teacher’s concept of questions that cover this material, and

(b.) Your shortcomings.

Rework each question that you missed. Some instructors will want these corrections to be handed in.  You will profit by filing away your quiz/test paper with corrections for use in reviewing for the final exam.

"To the Student," Elementary & Intermediate Algebra by Carson, Ellyn, & Jordan, pp. xiii-xviii, 2005