Every mathematics professor has heard "I am not very good at math" or some variant thereof from at least one student in almost every lower level (below calculus) class. That myth is the subject of this essay.
Poor performance in mathematics is almost always the direct result of insufficiently studying the requisite material. Correct this error and you will start doing well in mathematics. The key to addressing this issue is to recognize what causes your study efforts to be insufficient. The three most common causes are:
It is to be expected that occasional prerequisite material will be unknown to the learner. When unknown prerequisite material is identified the learner is expected to interrupt the current instructional activity long enough to learn the necessary prerequisite material. When the amount of such unknown prerequisite material is excessive, the amount of time required becomes excessive and the learner is unable to succeed.
In many instances the amount of required study time is extended because necessary prerequisite material has not been mastered. When prerequisite material has not previously been mastered, the learner must interrupt his/her study of the current topic to study and master that necessary prerequisite material before attempting to proceed. For example when a learner attempts to understand the definition of complex number, if the radical symbol has not previously been mastered, the learner must interrupt the study of this definition while finding and mastering definitions and explanations of square root and principle square root. Obviously that activity requires additional time.
What constitutes appropriate study methods for learning mathematics is less dependent on the individual than on the sequence of activities and desired learning outcomes. I will discuss a necessary sequence of learner activities and the expected learning outcomes from each of these learner activities.
Nationally more than 50% of students in elementary mathematics classes fail mainly because they do not properly study mathematics. National statistics indicate that about 60% of beginning mathematics students in college need to change study habits.
In a handout at St. Olaf College it is observed that when comparing college mathematics courses to high school mathematics courses you will probably find that at the college level;
"there is a greater emphasis on why things happen the way they do as opposed to how things are done. This means your study will require significant amounts of thinking about the material in addition to doing problem sets. There is a BIG difference between just getting assigned problems done and studying the ideas."
Professor Clare Hemenway at The University of Wisconsin - Marathon County points out that reflection is the most important study ingredient for success in mathematics.:
"To be sure a part of studying is doing the homework. But all too often, students complete the assignment and put the book away. It is critical that you take some time to reflect on the work you have just done and that you understand why you solved the problem the way you did and that you understand all the steps involved. Then, write down your reflections!"
"If you simply mimicked the instructor's work and did not bother to understand why you followed a specific process, you will not be able to understand how to transfer this process to similar problems. Also, you will encounter difficulty when you take an exam covering a variety of concepts. Your reflections will probably be one of the first things you look at when it comes time to review for a test." (Hemenway - web reference has disappeared)
As you study try to always remember the advice given by Professor D. Joyce of Clark University.
"Your goal is to learn, not to get answers. ... Try to understand the principles. The particular problem you're working on is of no importance in and of itself; it only helps you to get the concepts. "(Joyce)
The following Learner Activites are used, either consciously or unconsciously, by almost everyone who successfully studies mathematics. It is advisable that you incorporate each of these Learner Activities into your study of mathematics.
Two complex numbers a + bi and c + di are equal, a + bi = c + di,
if and only if a = c and b = d
At the end of this activity the learner should be able to write the definition without errors.
Make note of each symbol and be sure you understand its role in the definition. In the above definition the blue = symbol means equality of complex numbers (the new concept) and the red = symbols mean equality of real numbers ( an old familiar concept). If the old concept is not familiar, you must find and review its meaning in previously learned material. If you are unable to find the appropriate prerequiste material in your notes, turn to the course website or the index in your textbook. Perhaps you need to turn to the Internet where you might use Google, PurpleMath, or some other reliable website.
At the end of this activity the learner should be able to explain each mathematical word and symbol used in the definition.
For example, when studying the definition of equality of complex numbers you should ask yourself to perform the following activities:
You should answer these questions by writing in complete sentences. When writing a definition insist that it be absolutely letter perfect. When any of us studies a new topic we are not knowledgeable enough to rephrase a definition.
At the end of this activity the learner should be able to write a precise definition for each mathematical word and symbol used in the definition. The learner should be able to intelligently discuss the use of each word and symbol in this definition.
At the end of this activity the learner should be able to correctly identify examples and non-examples for the concept being defined.
If you are unable to answer the question, you must conclude that you have not mastered the definition of equality of complex numbers and you must work through the first four activities again. More carefully redoing the first four activities may clear up the difficulty but it may be necessary to redo them with some assistance from an instructor.
Comment:Observe that working the problems should occur rather late in the study process. Significant study should occur before working problems.
Mathematics is not learned by working problems. Mathematics is learned in order to solve problems. The primary purpose for exercises in a well written mathematics textbook is to provide a means of self evaluation.
For example, when studying the definition of complex number, at this stage you might try exercises like: “Write the complex number 4i + i2 in standard form.” or “Write the complex number standard form.” If you can correctly perform those activities and provide sound mathematical reasons for your process, then you should probably conclude that you have mastered the definition of complex number, the definition of , and the definition of the principle square root of a negative real number. If you cannot answer those questions or if you cannot provide reasons for your work, then you should reexamine the definitions, examples, illustrations, and any other explanations, including help from your instructor, you have at your disposal.
For periodic review you might consider a set of flash cards containing each of the definitions in the course. Another method which has been successfully used by many persons (myself included) is to maintain a notebook containing each definition ever encountered in mathematics. Regularly read a part of this notebook. For example, devote 15 minutes each day to reading in this notebook.
To evaluate how well you have mastered the additional properties and procedures, you should turn to related exercises at the end of the section.
For example, after mastering the concepts of norm of a complex number, and conjugate of a complex number, and multiplication of complex numbers and after studying the procedure for converting a division problem to a multiplication problem, you should turn to some division problems at the end of the section. If you can perform the divisions correctly and if you can provide reasons for each step in your work, then you should conclude that you are ready to move on to the next topic. If you cannot correctly perform the divisions or if you cannot provide correct and complete deductive reasoning for your work, then additional study is required.
At this stage of your development you should be able to write intelligently about the topics.
For example you should be able to write a paragraph or two about the similarities and/or differences of division of fractions and division of complex numbers. You should be able to write each definition, concept, property, and procedure without any prompting.
At this stage of your development you should be able to present a correct lecture on the topics of the section and/or chapter. You should also be able to construct tests and quizzes which test all levels of knowledge of the topics.