Guide to being a successful math student

As a successful student of mathematics, I have encountered a few insights, general strategies, and bits of advice which I personally found helpful during my first ten years of studying mathematics. As an active researcher in mathematics, these lessons continue to guide me. As an instructor of others in the study of mathematics, I have collected these on this webpage to share with my students, current and future, as well as anyone else who reads them.

Learning Mathematics

The first step in being a successful math student is knowing how to study. If this sounds obvious, let me say that it really isn't. I didn't really learn how to study until my first semester of graduate school, and then it came as a complete epiphany, motivated by desperation. Both in high school and as an undergraduate I put very little effort into school, nothing more than the minimum to get by. It wasn't that I wasn't interested in learning, far from it; rather, I had a perception of "studying" as a kind of cheating, like cramming for a test. I had always gotten by simply by sitting in class, absorbing lectures, and not really thinking too much about it.

Graduate-level math classes changed all that. Sitting and hoping to absorb material does not work with graduate math classes. The situation forced me to learn how to study effectively. Once I learned how to study effectively, I realized I could have had a much higher GPA as an undergrad than I ended up with, had I known how and had the motivation to study mathematics effectively. To be fair to my high school and undergraduate teachers, the blame for this rests squarely on my shoulders; though ultimately, it only seemed to take material that was sufficiently difficult to make me see the light.

By "studying mathematics" I mean learning mathematical ideas and techniques. I do not mean "cramming in preparation for a test," nor am I referring to techniques for memorization. Memorization is a poor learning technique at best; it no more helps you learn mathematics than buying a Japanese dictionary enables you to speak Japanese.

So, how does one study mathematics? The answer is both simple and elegant: read the text very carefully, one sentence at a time. Do not go on to the next sentence until you have fully understood the current one. It helps to take notes, writing a summary of the ideas you're reading or seeing in a lecture. Then, once you've finished reading or once the lecture is over, take a blank sheet of paper (or a stack of them :) and write an explanation of the ideas as though you're writing a lecture. It's not important whether you actually give the lecture to anyone; writing and rephrasing the ideas forces you to clarify the ideas in your own mind. It is also extremely useful for exposing gaps in your understanding, which you can then remedy by rereading the relevant portions of the text and/or asking questions.

Depending on the level of the mathematics text and your own level of understanding, this technique may always not be necessary in its full form; however, learning mathematics requires the patience to take small steps and make certain that you understand each step before continuing on. Many small steps add up to a large result.

Taking Mathematics Tests

A well-written math test is designed to measure your understanding of the ideas and techniques you've learned in class. This typically involves working problems in lower-level courses and proving theorems in higher-level courses. Ideally, if you have a firm grasp of the material and are comfortable with your understanding, you can start by stating the problem and continue writing each step explicitly, proceeding until you've solved the problem or proved the result.

Of course, you may not immediately see how to complete or, possibly, even how to start a given problem. This can be indicative of not having learned the material effectively, but it isn't necessarily. Even when you know the material well, it can be hard to see how to start a problem, especially in a test situation where you're already feeling under pressure from limited time.

In general, the best cure for test anxiety is a healthy dose of confidence in your understanding of the material, which comes from effective studying and doing practice problems. So many people have test anxiety specifically in mathematics classes that the condition has its own name, "math anxiety." I'm sure this is a result of the popular perception of mathematics as prohibitively difficult, but it is really quite backwards. Mathematics is unique among human activities in the there is no uncertainty about the results of mathematics. If you are careful at each step in a mathematical solution or proof, then your final result will be every bit as certain as what you started with. Be patient and careful, and you have every reason to be 100% confident in your results.

That said, what if you don't see how to start? First and foremost, you should not consider cheating or trying to fool your math professor. Aside from the obvious point of dishonesty, trying to fool a math professor with a bogus answer is unlikely to work, and cheating is likely to have serious repercussions, ranging from getting a much worse grade than you would have if you had honestly tried, up to to academic suspension or worse. Every math professor was once a student, and we are familiar with the techniques students use to cheat, having learned them they same way current students do.

Further, answers listed without work are not worth many points, since the point of a math problem is not getting the actual answer but showing that you understand the method of getting the answer. When two people who sit together turn in the same incorrect work, it is hard to avoid drawing the conclusion that cheating is going on, and this both a disappointing and angering situation for an instructor to be in. Academic dishonesty is a serious matter, and if you are caught cheating there are serious consequences, which can include expulsion. In any case, cheating yourself out of the education that you are paying for does not make sense.

OK, so suppose you don't know how to start a problem. What should you do? First, don't leave the problem blank, because that guarantees that you get no credit for it. Write something down, but not just anything; read the problem carefully and write out as clearly as you can what the problem is asking for. That is, rephrase the problem in your own words. Very often, in the course of explaining what the problem is looking for, you will see how to use the information you've been given to answer the question.

If you still don't see how to proceed, write down something that's true. Start by summarizing what you know. Read the problem again and see if there's anything more you can say based on what the problem says. If you're still stuck, then move on to the next problem and come back when you've completed the problems you know how to do. There's no shame in admitting to not seeing how to proceed, and such an admission is far less insulting to your instructor than trying to fake an answer or cheating. Further, a correct beginning to a problem is worth more partial credit than a "complete" but bogus answer.

Avoiding Common Errors

In every area of mathematics, especially in the first courses on algebra and calculus, there are certain common mistakes that students tend to make, at least until they've lost points on tests, quizzes or homework several times and learn the hard way. The following are just a few of the most common mistakes I see when grading my students' work. Each of these mistakes can indicate simple carelessness or can be indicative of a deeper problem, a misunderstanding of how the rules of mathematics work. You can avoid making these common mistakes by first understanding why they are mistakes, and then being careful with each step you do when working problems.

Common Algebra Mistakes

  • forgetting to distribute
  • distributing an exponent
  • canceling terms instead of factors
  • misunderstanding fractions
  • misunderstanding negative and fractional exponents

Common Calculus Mistakes

  • forgetting to use the product, quotient and chain rules
  • attempting to distribute or cancel trig functions or logarithms
  • forgetting to add C when solving indefinite integrals
  • misusing notation, e.g. leaving off the "dx" in an integral or forgetting to remove the integral symbol after integrating
  • using the old limit values with the new variable after substitution in a definite intgeral
  • fogetting to change the differential when substituting
  • confusing sequences with series
In my calculus classes, I often assign quiz problems designed to make my students aware of these and other common errors and encourage critical reading and thinking. These problems consist of a normal problem and a complete but incorrect answer. The student's job is then to identify the error or errors and to give a correct solution. It is extremely important to get into the habit of reading carefully and critically!

Copyright © 2002-2008 Sam Nelson