Wordgems - Mathematics: How to Study Math

As a department chair from 1990-1997, I counselled most of the students to do not succeed in our department's engineering majors. Most students who fail academically, do so because of a few critical errors in judgement and in how they approach their studies. As a student (again) myself from 1984-1988, I became an avid student of "how to study" mathematically based subjects. I had the good fortune to learn some of the secrets of studying mathematically based subjects (such as statistics and engineering) and developed a few new techniques of my own. I have shared these methods with students on an individual basis and in numerous workshops and presentations. Students who have used these techniques report excellent results.

How to study "non-mathematical" subjects.

Survey of Available Resources - Much has been written on how to be a good student--but very little on how to be a good math, science or engineering student. On my shelf are twelve books and articles I have collected on how to be a good student or how to study. Only two of them give any special attention to "mathematics". All of the other books are directed to students majoring in non- mathematically based majors.

What Is Different About Studying Mathematically Based Subjects - The study of math (or related subjects such as engineering, science, or statistics) is a process that requires progressive, step-by- step learning of fundamentals or the "order" of things. Engineering goes beyond just learning "order" and further emphasizes the "design" process. The goal of engineering education is to learn, understand, and use laws of science within the discipline of a profession and the needs of society.

Math and science are filled with "ordered" structure. This truth makes them different from the study of humanities and social sciences where philosophy, creativity and inference may be of more concern. Unfortunately most students try to use the same methods in all classes. This approach usually leads to a deficient approach to learning math related subjects. In most cases this deficiency is compensated for by a innate intelligence, added time to the task, or persistently repeated failed courses. In many cases, however, this deficiency is not compensated for and the student fails somewhere along the way.

My belief, based on experience, is that 99% of the students who enter engineering have the brain power to graduate. Those who fail to graduate because of non-outside forces (eg. health, finances, personal problems, etc.) usually do so because of poor strategy, planning, study habits and methods.

An on-line survey is linked below for the purpose of self-assessment and introducing some of the concepts that will be presented later. It is suggested you take the survey and look at your score. The lower the score, the more you may need to re-assess your methods. If your your score is very low, do not be discouraged. Finding out new strategies that allow you to succeed that actually take less of your time can be very invigorating!

FOLLOWING THE IDEAS AND STRATEGIES IN THIS PAPER WILL HELP YOU TO GRADUATE IN A TIMELY MANNER, WITH HIGHER GRADES, LESS WORK, AND MORE FUN.

The following steps should ideally be done before even deciding to go school. School needs to "fit into" your life, not "become" your life at all costs. This concern becomes larger as we get older and more involved with job and family. This step-by-step process can be done on your own or with the help of a parent, spouse, faculty advisor or mentor.

The Big Picture - The Big Picture is really your life. Your studies should fit into your life in such a way that you can sustain yourself for as many years as it takes to graduate. PLAN your week so that you have enough time to study properly and have a BALANCED life. You need to balance your academic, financial (career/work), personal, family, social, and spiritual life. If you are spending more than 60-70 total hours per week on school plus work, then you may be out of balance.

Use a Time Picture - There are 168 hours in a week. Use a "Time Picture" to determine how much time you can devote to school. Then use that information to determine how many classes you can take. Below are the steps for constructing a "Time Picture". Following is an example time picture. Use the blank "Time Picture" provided at the end to go through the steps yourself.

The sample below happens to be for a part-time summer school student taking Chemistry (3 units) and Math (4 units), working two part-time jobs and in a leadership position with an extra-curricular activity.

Analysis: This is an example of a balanced schedule. The student should have plenty of time for both part time jobs, a major extracurricular activity and two math/science classes AND STILL HAVE A BALANCED PERSONAL LIFE. The student should cut back on something if he wants to take more units.

Strategy For Graduation Planning - Follow these basic guidelines and you will minimize your difficulties in completing your program.

Strategy for Learning and Taking Classes - To "properly" learn math, science, or engineering you should plan on spending 2 to 4 hours of outside study for each hour of class. The nature of these courses is that the material is cumulative in two ways:

The methodology explained below is designed to give the student tools to effectively learn the material in a progressive manner rather than by "cramming" (Cramming" is the practice of not keeping up with class assignments and then trying to learn the course material the night before a quiz or test. Some students in non mathematically related courses or curriculums manage to "pass" some of their classes this way. In addition to the obvious possibility of failing a class, the main disadvantage of cramming is that the retention rate for the material is practically zero--which hurts later on). In order to have time to learn progressively the student needs to have a balanced class load because some class types will require more time than others. A "balanced" load is a mixture of types and difficulties of classes. For example, it is not desirable to be taking all engineering science classes or all general education classes.

Try not to take more than one pure math or statistics class at a time. These are not subjects you can cram because you are building and training all the time. Certain engineering classes should not be taken together in the same quarter because they can be very difficult and time consuming. In particular the "engineering sciences" are traditionally very difficult. These classes include statics, dynamics, strength of materials, fluid mechanics, thermodynamics, metallurgy, operations research, etc. In fact, many students find themselves having to repeat some of these classes because a C- or better is required to continue the sequence. Again, try to only take one "engineering science" course each quarter.

Another type of class to take without being hampered by a heavy class load are what I call "critical path" classes. A critical path class is a major class that is embedded in a long series of prerequisites. If you were to fail a critical path class, for example, it could delay your graduation from one quarter to one year, depending on how often it is offered. Critical path classes are often considered very important in your major and you would do well to learn the material thoroughly. Below is a sample classload:

Other considerations: Try to limit the number of classes requiring term papers or quarter projects to three or less. Try to limit the number of labs to three or less--preferably no more than two.

Example: Lets take a look at the Industrial Engineering Curriculum at Cal Poly Pomona. The "Critical Path" is the longest string of consecutive classes that must (or should) be taken in sequence. I would say it is:

MAT 114 Calculus I
MAT 115 Calculus II
MAT 116 Calculus III
MAT 214 Multivariate Calculus I
MAT 215 Multivariate Calculus II
MAT 216 Differential Equations
STA 309 Probability and Statistics
IME 312 Probability and Statisitics for Engineers
IE 311 Math for Engineers
IE 327 Systems Engineering
IE 416 Operations Research I
IE 436 Advanced Production Planning

Strictly speaking, several of these classes could be taken concurrently (eg. STA 309 can be taken concurrent with MAT 215). The problem then becomes taking two heavy math courses at the same time.

Major classes that are engineering science that actually should be taken prior to IE 436 include:

IE 417 Operations Research II - IE 429 System Simulation (on the list of core courses)
ME 214 Statics - ME 215 Dynamics - ME 218 Strength of Materials - ME 219 Strength of Materials
ECE 231/251L Electric Circuits - ECE 333/383L Electrical Controls

All of the above classes should be taken with the Transcription Method (explained later) of note-taking except IE 436 and the ECE classes. (Note: If I were a full-time student I would include the physics classes here and use the transcription method with them as well).

Most of the rest of the core and support courses are technical as well and the Non-transcription Method of note-taking should be used.

Several core and support classes and most of the GE classes can be taken with using the "St. PIE" Method of note-taking.

There are 17 total lab classes (assuming that chemistry and physics classes and labs are taken concurrently). Therefore, it should not be necessary to take more that two lab classes in any one quarter.

Analysis: If I were a full-time, freshman IE student I would try to take the critical path classes close to the order shown (several can be switched) and schedule everything else around the critical path. I would balance the other engineering science and lab classes so that I was not over loaded in any one quarter.

If I were a part-time student, I would map out the critical path, other engineering science courses, and classes with labs so that I never had more than two of them in any one quarter.

If I were a transfer student, I would make sure I had all lower division lab classes and ME classes out of the way early so I would not get caught with scheduling conflicts and difficult schedules later on. I would try be a full-time student (non-working) my last three quarters if at all possible.

Next we discuss "In Class" (Phase II) and "Outside Class" (Phase III) activities. Your ability to be successful here depends on the study habits described below and whether or not you have set yourself up to be successful in planning your time and classes (Phase I activities). The three areas covered below are:

a. Note-taking skills (Phase II & III)
b. Study skills (Phase III)
c. Exam techniques (Phase II)

The Importance of Notes and Note-taking Skills - The focal point of this study technique is the development and use of your notes.
Properly developed notes typically reflect the ideas, concepts, methods, examples, and "do's & don'ts" that your instructor believes to be important and will expect you to know well.
Mastery of your notes will be the best use of your time and is much more efficient than basing your study around the textbook. I recommend three different note taking strategies--depending on the class and/or instructor.

1. Transcription method (for math, statistics, engineering science, and selected critical path classes)

2. Non transcription method (for most technical major classes and technical GE classes)

3. St. PIE method (for humanities, social science, and other non-technical classes)

Each strategy uses the two-column system to start with. What you do after that is what makes the three methods or strategies different. The two-column method is adapted from a notetaking format developed at Cornell. The following illustration shows how 8 x ll, three-hole paper is used for taking notes.

Transcription Method: Most math and engineering textbooks are not easy to follow on their own so you need something additional to study. Preparing proper notes can provide the proper study materials. Take rough notes in class by copying everything down on 8x11 paper with two lines (drawn either freehand or with a straight edge ahead of time) following the Cornell format as best you can (see above). Do not neglect to write down anything because it may be important later (even though not obvious at the time). Then later, outside of class, "transcribe" (recopy) notes using the two-column system. Transcribing is where you learn because it forces you to think things through for yourself-not just follow the instructors thinking.

The two-column system - The two columns are mentioned below. Subsequent notes explain the logic and use of the system. On the last page is a sample page of notes using this system.

Index or Table of Contents Page - Create an Index or Table of Contents by transferring the index headings and page number it is on to a page at the beginning of your notes. This "Table of Contents" for your notes is invaluable when studying and taking open-note exams.

While transcribing you will find gaps in logic or things you do not understand. Missing steps of proofs, errors or incorrect statements, and undefined terms are just a few examples of questions you will find only by using the transcription method. Identify these questions in your notes so you can ask the instructor later. This is done by using Post-it notes. Place the post-it with the question written on it so it sticks out of the page. See your instructor during their office hours and resolve your questions by going through the post-it notes. You will find most instructors will be very happy to help if you use this method.

In math there are two parts to learning:

Color coding - Use color coding of notes to help you study and use your notes efficiently. A suggested use of colored highlighters is:

Pink - Headings
Green - Rules
Yellow - Cautions, things to memorize
Orange - New words or items
Purple (blue) - Examples
Optional Blue (purple) - Old rules and things you should know

Non-Transcription Method - The non-transcription method is the same as the transcription method except that the you do not rewrite the notes. Use the two-column ruled paper and write your original notes as neatly as possible. Then color code and fix up these notes as outlined above. This system works best if you have an instructor that does not go to fast. A speedy instructor forces you to get sloppy sometimes.

St. PIE Method - This method is from the excellent book by Laia Hanau, Play the STUDY GAME for Better Grades, Harper & Row, Fifth Ed. 1972. Briefly stated, use the Cornell Method and write down everything you can in the capture area. When studying you then try to classify the material in your notes as either:

Statement
Proof
Information
Examples.

Thus the acronym St. PIE.

In a humanities or social science class, for example, you would then make all the connections. The instructor does not always "tie things together". They leave that to the student. When you see a statement (postulate, theory, axiom, contention, opinion, premise) for ask yourself where the proof, information, and examples are in your notes. Tie them together with lines, arrows, and notes. When you do this you are predicting the test questions in advance. Aren't many test questions things like: This is an example of ..., Give proof to support ..., etc.

Study Skills - With math and applied mathematics fields it is not enough to simply "know" or "understand" material. You need to know the material well enough to perform quickly-- without hesitation. Therefore, knowing involves DOING AND DOING QUICKLY. Learning the rules well will help you tremendously. Studying consists of studying the rules, trouble spots, and practicing examples. Studying is not paging through the book working problems. Math is a "doing" subject, not a "reading" subject.

Study Steps - Write down the first line of an example problem on a piece of scratch paper, close your book or notes, then work as far as you can without looking. Then start over and repeat the process four or five times until you can do the example QUICKLY al the way through. By the time you have worked through the example repeatedly you have the rules memorized. The repetition in this process is the key to learning. This phase of the study process is where you find the "can't do's" to enter them into column one of your notes.

Now the homework should be the "frosting on the cake" and should only take a few minutes. Contrast this technique to the common practice of digging into homework without proper preparation. The homework may eventually get done after a lot of "page flipping", but the student still does not know the material proficiently and rules are not internalized.

Study Time - How long should you plan on for studying and how should it be used?

HOW TO STUDY MATH AND SCIENCE

PREVIEWING

Before class briefly preview the text material that will be covered in the lecture.

Remember, the purpose of previewing is not to understand the material but to get a general idea of what the lecture will cover. This should not be a very time-consuming process.

NOTE-TAKING

When taking notes in class, listen actively; intend to learn from the lecture.

TEXT READING

If your class lectures provide a good overall structure of the course, you can use your text to clarify and supplement your lecture notes. In order to create a single study source, insert the notes you take from the text into your lecture notes themselves as well as in the margin or the back of the opposite page.

If your text provides the best overall structure of the material, then you can use your lecture notes as the supplementary source. In either case consider the following procedures:

PROBLEM SOLVING

Solving problems is usually the most important aspect of math or science courses. You must, therefore, spend much of your study time either working or studying problems. When working a problem, follow these steps:

During the problem solving process, it is often helpful to say out loud all of the things you are thinking. This verbalization process can help lead you to a solution.

PROBLEM ANALYSIS

After you have worked a problem, analyze it. This can help sharpen your understanding of the problem as well as aid you when working future problems.

TEST PREPARATION

If you have followed an approach to study as suggested in this handout, your preparation for exams should not be overly difficult. Consider these procedures:

TEST TAKING

TEST ANALYSIS

Analyzing returned tests can aid your studying for future tests. Ask yourself the following questions:

IMPORTANT: The knowledge of most math/science courses is cumulative. Many concepts build on previous concepts, and a poor understanding of one concept will likely lead to a poor understanding of future concepts. Consequently, you should seek help early, if you encounter difficulty.

Before I get into the tips for how to study math let me first say that everyone studies differently and there is no one right way to study for a math class. There are a lot of tips in this document and there is a pretty good chance that you will not agree with all of them or find that you can’t do all of them due to time constraints. There is nothing wrong with that. We all study differently and all that anyone can ask of us is that we do the best that we can. It is my intent with these tips to help you do the best that you can given the time that you’ve got to work with.

Also, let me apologize right off the bat if I offend you with some of what I’m going to say in this document. Offending you is not my intent, but in some cases there is simply no other way of saying some of what needs to be said than how I’ve said it here. I have found over the course of time that sometimes in order to get students to understand that they need to do more to be successful in a math I’ve had to be very blunt with my advice. There are portions of this document in which I will continue to be blunt. Again, the point is not to offend but to get some of the people reading this to realize that they need to do more.

Now, I figure that there are two groups of people here reading this document, those that are happy with their grade, but are interested in what I’ve got to say and those that are not happy with their grade and want some ideas on how to improve. Here are a couple of quick comments for each of these groups.

If you have a study routine that you are happy with and you are getting the grade you want from your math class you may find this an interesting read. There is, of course, no reason to change your study habits if you’ve been successful with them in the past. However, you might benefit from a comparison of your study habits to the tips presented here.

If you are not happy with your grade in your math class and you are looking for ways to improve your grade there are a couple of general comments that I need to get out of the way before proceeding with the tips. Most people who are doing poorly in a math class fall into three main categories.

The first category consists of the largest group of students and these are students that just do not have good study habits and/or don’t really understand how to study for a math class. Students in this category should find these tips helpful and while you may not be able to follow all of them hopefully you will be able to follow enough of them to improve your study skills.

The next category is the people who spend hours each day studying and still don’t do well. Most of the people in this category suffer from inefficient study habits and hopefully this set of notes will help you to study more efficiently and not waste time. Also, you will probably find that as your studying gets more efficient you will not need to spend as much time as you once had to.

The final category is those people who simply aren’t spending enough time studying. Students are in this category for a variety of reasons. Some students have job and/or family commitments that prevent them from spending the time needed to be successful in a math class. To be honest there isn’t a whole lot that I can do for you if that is your case other than hopefully you will become a more efficient in your studies after you are through reading this. The vast majority of the students in this category unfortunately, don’t realize that they are in this category. Many don’t realize how much time you need to spend on studying and hopefully reading this document will help you to realize that you do need to study more. Many simply aren’t willing to make the time to study as there are other things in their lives that are more important to them. While that is a decision that you will have to make, realize that eventually you will have to take the time if you want to pass your math course.

Now, with all of that out of the way let’s get into the tips. I’ve tried to break down the hints and advice here into specific areas such as general study tips, doing homework,, studying for exams, etc. However, there are three broad, general areas that all of these tips will fall into.

Math is Not a Spectator Sport

You cannot learn mathematics by just going to class and watching the instructor lecture and work problems. In order to learn mathematics you must be actively involved in the learning process. You’ve got to attend class and pay attention while in class. You’ve got to take a good set of notes. You’ve got to work homework problems, even if the instructor doesn’t assign any. You’ve got to study on a regular schedule, not just the night before exams. In other words you need to be involved in the learning process.

The reality is that most people really need to work to pass a math class, and in general they need to work harder at the math class than they do with their other classes. If all that you’re willing to do is spend a couple of hours studying before each exam then you will find that passing most math classes will be very difficult.

If you aren’t willing to be actively involved in the process of learning mathematics, both inside and outside of the class room, then you will have trouble passing any math class.

Work to Understand the Principles

You can pass a history class by simply memorizing a set of dates, names and events. You will find, however, that in order to pass a math class you will need to do more than just memorize a set of formulas. While there is certainly a fair amount of memorization of formulas in a math class you need to do more. You need to understand how to USE the formulas and that is often far different from just memorizing them.

Some formulas have restrictions on them that you need to know in order to correctly use them. For instance, in order to use the quadratic formula you must have the quadratic in standard form first. You need to remember this or you will often get the wrong answer!

Other formulas are very general and require you to identify the parts in the problem that correspond to parts in the formula. If you don’t understand how the formula works and the principle behind it, it can often be very difficult to use the formula. For example, in a calculus course it’s not terribly difficult to memorize a formula for integration by parts for integrals. However, if you don’t understand how to actually use the formula and identify the appropriate parts of the integral you will find the memorized formula worthless.

Mathematics is Cumulative

You’ve always got to remember that mathematics courses are cumulative. Almost everything you do in a math class will depend on subjects that you’ve previously learned. This goes beyond just knowing the previous sections in your current class to needing to remember material from previous classes.

You will find a college algebra class to be very difficult without the knowledge that you learned in your high school algebra class. You can’t do a calculus class without first taking (and understanding) an Algebra and a Trigonometry class.

So, with these three main ideas in mind let’s proceed with some more specific tips to studying for a math class. Note as well that several of the tips show up in multiple sections since they are either super important tips or simply can fall under several general topics.

These are some general tips that where either important enough to single out or just didn’t seem to fit into any of the other sections.

Getting help when you are in trouble is one of the most important things that you can do in a math class. Here are a couple of things that you can do the get help.

Note that this section contains some general tips on making the most out of your homework. The next section contains tips on actually working homework problems.

In the previous section there were some general tips in regards to homework sets as a whole. Here are some tips to help you actually work the problems. Note that some of the ideas were important enough that they are actually in both sections.

Taking exams is probably one of the most important things that you’ll do in a math class and so it’s important to do the best that you can. Here are some ideas that will help you while you’re taking an exam.

This is probably one of the more important sections here and also one of the most over looked. Learning from your mistakes can only help you.

Physics and Math are Problem-Solving Disciplines. You must learn the underlying principles and connecting themes to solve the problems.

1. Where you sit in the class may be important. Try sitting in the front half of the class.

2. Preview the book before the lecture to see what is in the book. If you know the formulas are in the book you won't have to write them down during the lecture and can listen more attentively.

3. Read the introduction and summary of the relevant chapters and look at the section headings and sub-headings.

6. Make notes of new words, new units of measure, statements of general laws, etc. Study notes and related text material IMMEDIATELY after the class to reinforce your learnings. This should be done within 24 hours.

7. During the lecture, question what is being said continuously whether or not you verbally ask those questions.

9. If note taking leaves no time for thinking in class, copy only the key steps and fill the remaining steps in after class.

1. Examine the information given in the course syllabus carefully before studying.

2. Build up ability to read this kind of material. Do a little bit every day (mind building is like body building).

3. Set aside one hour daily per course in this area and read. When the hour is up STOP. You may have only read 5 pages but you will know what is in them. You can then leave your bigger time blocks for your reading courses.

4. If you do one problem a day for each course you won't be left with 40 problems to do on Sunday night.

5. Go by and see the professor during office hours. Usually they welcome this. When the professor gets to know who you are from the sea of faces in a lecture class, your questions will be looked at as valid and your interest will be noted.

6. Rather than skipping the sample problems in the middle of the chapter, work them. They help in understanding the logic of the chapter.

7. Read the assigned problems before reading the chapter. That way you will know what to focus on. Typically when problems are assigned, it is the concepts in the problem which need to be learned.

8. If you work a problem and get it wrong, it is just as important to know what you are not looking for as well as what you are looking for.

9. Answers in the back of the book may be a bad crutch to use. Often there is more than one way to do a problem and your answer may be just as legitimate as the one in the back of the book.

4. Generate a list of themes, principles, and types of problems you expect to have covered in the test.

5. Review actively. Try to look at all possible ways a principle can be applied.

6. Get as much information about what is important from the professor. Look at the way the professor works through problems.

8. Go into the test thinking you are the Greatest Mathematician or Physicist alive and that you have been called in to solve these problems.

9. Look through the whole test first and do problems you can answer first. Make sure and check the point value of the problems in the test.

10. Don't look at the test as a measure of your ability, "The world has been stumped for hundreds of years on this topic so why should I be able to solve it on the first try?" or "I could have gotten another set of problems which I could have answered: I didn't know these but I know how to do others," or "It's not that I am not cut out for physics, it is that I didn't know these particular problems." If you blow a test go to the professor and ask, "What is it that I'm missing?"