How to study and learn math

          Learning takes effort and time. 

The Basics

Attend class. Most serious students attend every class.

Learn during class. Don’t just take notes – pay attention and learn during class.

Do the homework. Good students take the commitment seriously. "Practice makes perfect." Not doing a homework on time is a warning signal that you are getting behind.

Beyond the Basics

Review very soon after studying. It is hard to get things you “learned” into long-term memory. Immediate review (10-15 minutes of review right after studying something) is far more effective than review a day or more later. Research shows that, with immediate review, 83% of class discussion is retained 9 weeks later. Waiting even one day drops retention to only 14%!
          Forming a memory takes about 17 seconds of close attention. If you don’t linger on a thought for 17 seconds you won’t remember it.

Do the homework by remembering how, not by looking up how as you go along.
          You don’t know math if you don’t remember it. Learning requires remembering.
          This has important implications for how you should do homework. Doing homework by looking up how to do it as you go along often fails to result in learning because it fails to emphasize memory. Getting homework “done” is not the same as learning how to do it!
          That is why you should try to do the homework without looking at the text or solution manual.

                  Study all the material first (concentrate on key points for at least 17 seconds each) and then
                  Do the homework by remembering how.
                  If there is a problem you don’t remember how to do, look in your text or notes to find out how, but with the intent to remember how. Doing without remembering is not learning.

Review after 15-30 minutes. When you find out how to do something (say, a homework problem), there is a good chance you will forget how to do it almost immediately after you finish. Reinforce your learning before you forget by reviewing it after 20 minutes of other work. For example, if there is a homework problem you have to look up how to do, look it up and do it, but resolve that you will come back to that topic. Then continue with the rest of the homework and 20 minutes later return to that topic and try a similar problem to reinforce your learning before it fades. Immediate review really works for moving thoughts to long-term memory.

“Seeing how” often doesn’t work.  To learn, you must be engaged with the work. Many students watch in class, nod, and think "I can do that." Many copy the solutions manual or a friend’s work and think "I see how it's done. I can do that." These thoughts are often wrong. Don't kid yourself. The proof of your ability to do it is in your doing it by remembering how.
Serious Students

•         Figure it out! If something in the text does not make sense right away, take the time to figure it out. Read it again with your brain in gear. Study the text's example again. Go over it, carefully, it until it makes sense.
Asking your instructor to explain something is sometimes necessary, but not as often as students seem to think. If you read the text and work on an example with the intent to figure it out, you probably can. Then your learning will be better and longer lasting than if you ask someone else to do it. Plus, in the process, you will be learning how to learn!  You will find you are getting better and better at understanding what you read. The effort pays dividends, not just for today's lesson, but for all future reading. Believe it!

                  Devote substantial time to learning (not just doing homework) outside of class.
At a university you are expected to work two hours outside of class for every class hour. Two hours may seem like a lot of work, but that is what it takes – even if you finish the assigned homework in less time. Study in addition to doing homework. Do the work and reap the benefits. School is a serious job for which the pay is your education.

To learn math more efficiently you must learn to read math. You learn to read by reading.

                  Read your text.
              Read with two goals: 1) to learn the current material
                                        2) to practice reading (to learn to read more fluently). The ideas of mathematics are best expressed in written symbols (not aloud in English). By learning to read you learn how mathematics really works.
                  Think of the book as a lecture you can follow at your own pace.
                  Read with pencil, paper, and calculator. Do the calculations with your own calculator. Reproduce the graphs on your calculator. Try to fill in missing steps. Take notes. All this is designed to encourage you to think while you are reading. And, it works!  (Reading math is much, much slower than reading a novel.)
                  Pay special attention to theorems. Theorems are a mathematical way of summarizing general methods that apply to many examples. They tell you what to do and when to do it.

           If you find reading your text difficult, you can blame the text, or take responsibility and recognize the strong signal that you are dangerously weak at an important skill – reading comprehension.  The harder you find reading, the more you need to work at it. Don't kid yourself. Not reading is a sign that you are not comfortable with an important mathematical skill. No one else will take time to teach you to read – you’ve got to do it yourself.
Reflection.  Research has shown that learning has a passive component. For example, while you are asleep tonight your brain will categorize and file things you "learned" today. However, you will remember far less tomorrow if you get only five hours of sleep. Get enough sleep.
          Right after class you can "mull over" the lecture and later find you remember it much better than if you proceed straight to a different sequence of thoughts. (This is related to the “review after 20 minutes” idea.) Driving home, if the car radio is off, you may find your thoughts returning to the lecture. This sort of repetition is very valuable.
          Do not walk out of class and immediately put on your earphones and join the world of entertainment. Right after class is the very best time to review the material, even if only in your mind as you walk across campus, and move it to long-term memory.

Success.  At MSU, about 30 percent of the students are "over traditional age." Many retake math courses they did poorly in years ago, and do very well the second time around. What is the explanation?
          Attitude. They want to learn it this time. They do the work. They pay attention. They have learned how to put their brains "in gear."   

          "Deferred gratification" is a reward to be received later. Many of the returning "over traditional age" students have learned the hard way that it takes serious effort to develop skills for which employers pay well. Put that effort in now. Learn to appreciate your developing skills. Enjoy your education. Enjoy the process, and you can excel.