Word Gems: 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!

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?"