Studying Mathematics

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.

1) I am not very good at math

It is a myth that some people are inherently unable to learn mathematics. The myth is fostered by an age old misunderstanding caused by a failure of the education system and a societal willingness to accept poor performance in mathematics. Many studies, including a seven year study by Dr. M. Poage and myself, demonstrate beyond a shadow of a doubt that any student can learn mathematics to a 95% mastery level. Stop telling yourself that you cannot learn mathematics. Stop excusing poor performance by claiming a lack of ability in mathematics. Address your poor performance honestly and initiate corrective action.

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:

  • (a) ignorance of prerequisite material,
  • (b) insufficient study time, and
  • (c) inappropriate study methods.

(a) Prerequisites

Mathematics departments go to great efforts to state prerequisites for a course, usually check to determine if a student has mastered all prerequisite material, and are usually insistent that no student enroll in a course without the necessary prerequisites. That obsession with prerequisites results from a clear understanding that a mathematics concept cannot be learned if it depends upon un-mastered prerequisite material.

It is to be expected that occasional prerequisite material will be unknown to the learner. When such 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.

(b) Study time

The amount of time required to learn a particular mathematics topic varies from individual to individual. Mathematics departments across the country are fond of recommending two hours of study time outside of class for every hour of in-class lecture. My personal experience indicates that this is a minimal requirement. Many students need as much as three or four hours of study time outside of class for every hour of in-class instruction. A few individuals will require many more hours of instruction and study in order to succeed. There is a brutal fact embedded in the previous sentences. If you cannot (or do not) devote the amount of time required for you to learn the mathematics presented in College Algebra (or any other math course), you will fail the course.

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 symbol 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.

(c) Study methods

Some students learn better if the instructional stimulus is written while others learn better if the instructional stimulus is audio. There are numerous other individual learner preferences of this nature, none of which will be discussed here.

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.

Learner Activity 1: Memorize definitions

The very first and absolutely essential step to learning a mathematics topic is to memorize the definitions involved. As an example suppose the topic to be learned is the concept of equality of complex numbers. Begin by memorizing:

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.

Learner Activity 2: Study and understand definitions

Decide what the definition says. The above definition states that two complex numbers are equal if and only if their corresponding real components are equal and their corresponding complex components are equal.

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.

Learner Activity 3: Construct questions and answers about each definition

Ask yourself if you know the definition of each mathematical word or symbol used in the new definition. A good way to formulate that question to yourself is: Write the definition of —-.

For example, when studying the definition of equality of complex numbers you should ask yourself to perform the following activities:

  • Write the definition of complex number.
  • Write the definition of real component.
  • Write the definition of complex component.
  • Write the definition of i.
  • Does this define a concept or a mathematical object?
  • What is the value of i2 ?
  • What does the blue = symbol mean?
  • What do the red = symbols mean?

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.

Learner Activity 4: Construct a list of examples and a list of non-examples

Things that satisfy the definition are examples of the creature being defined. Things that do not satisfy the definition are non-examples of the creature being defined. You should construct lists of things that satisfy the definition and things that do not satisfy the definition. For example, when studying the definition of a complex number you might construct the following lists.

At the end of this activity the learner should be able to correctly identify examples and non-examples for the concept being defined.

Learner Activity 5: Evaluate your understanding by attempting some exercises

Use the exercises at the end of the section to evaluate your understanding of the concept. For the above definition turn to exercises like: Find real numbers a and b such that (a – 1) + (b + 3)i = 5 + 8i. Analyze the question by observing the equal sign = refers to equality of complex numbers and therefore the above definition must apply. From that definition you can then conclude that a – 1 = 5 and b + 3 = 8. The observation that solving those two equations for a and b provides an application and a review of one of the properties of equations, helps to reinforce that previously learned concept, illustrates an application of that prevously learned concept, and serves to connect at least two topics.

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.

Learner Activity 6: Study available examples, illustrations, and applications

Go through the text material, website material, and lecture notes to find all examples relevant to this definition. Study each very carefully. To study very carefully means that you must know and understand every word and symbol being used. If an unfamiliar word or symbol is used, you must find and review its meaning in previously learned material. To study very carefully also means that you must be able to provide reasons for each step in the example. Those reasons must always be sound mathematical principles.

Learner Activity 7: Evaluate your understanding by attempting some exercises

The examples, illustrations and applications provided in instructional material are designed to extend your understanding of the concept beyond the very simple basic understanding. In other instances, examples will illustrate how two or more concepts may be used in concert to solve a problem or answer a question. To evaluate how well you have mastered these extensions, you should turn to related exercises at the end of the section.

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 one plus the square root of negative eighteen 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 the square root of negative one , 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.

Learner Activity 8: Periodic Review

Implement a schedule for regularly (weekly) reviewing definitions and their ramifications. When performing your periodic review also try to consider all subsequent uses of the concept.

To facilitate 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.

Learner Activity 9: Study Properties, Procedures, etc

To study any special properties and/or procedures, you should attack each one individually in the same manner that you studied the definitions. Although this is a short statement about studying properties and/or procedures, the process is as involved as the previous eight Learner Activities.

Learner Activity 10: Evaluate your understanding by attempting some exercises

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, conjugate of a complex number, 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.

Learner Activity 11: Understand relations between concepts

Read your notes, text, and website looking for statements that relate concepts to each other. Try to understand how the various concepts fit together to constitute a whole unified structure. Try to answer questions like:

  • What kind of concepts are these ?
    • Are they mathematics objects?
    • Are they mathematics operations?
    • Are they mathematics relations?
    • Are they mathematics procedures?
  • How do the new concepts relate to familiar concepts?
    • Are they similar ?
    • Are they the same?
    • Are they in opposition?
  • How does the new concept extend your ability to use mathematics?

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.

Learner Activity 12: Evaluate your understanding of larger blocks of material

At the end of each section or chapter you should evaluate your understanding of the collection of concepts contained in the section or chapter. For this evaluation, use the more difficult exercises and/or any self-test which might be provided. Test yourself by working old tests and/or quizzes from the website.

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.

 

References and Additional Reading

Advice from Dr. Lynn E. Garner, Chair, Department of Mathematics, Brigham Young University
St. Olaf — Studying Mathematics
Joyce — goal is to learn

 

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