LEARNING MATHEMATICS

• Studying to learn is different from studying for grades. Popular study techniques often fail to produce adequate long-term learning.

Most students spend a majority of their study time on four activities:

1. highlight the text
2. read through the examples
3. try the exercises
4. copy solutions from the answer book

These activities involve no real commitment. Passive activities such as highlighting, reading, and copying have little long-term benefits, and trying the exercises before copying solutions is only minimally active. This kind of study leads to the common complaint, "I 'understood' everything, but I didn't do well on the test."

• People learn mathematics best by doing mathematics and then reflecting on what they have done.

The words "doing" and "reflecting" imply activity on the part of the learner, rather than passivity.

DOING MATHEMATICS

"Doing" mathematics means reading and working problems actively.

• Active reading is done as much with the hand as the eyes. Neatly list key definitions, ideas, and results for each topic. Stop frequently to work out details and to rework text examples.



• Writing up a few exercises neatly is just as important as doing a lot of exercises.



• When you don't understand something, work to frame a specific question. Then seek help.

Active reading and careful writing of exercise solutions takes a long time, but it is what teachers do when they prepare to teach a course for the first time, and it is a crucial learning activity.

REFLECTING ON WHAT HAS BEEN DONE

• After learning a body of material, prepare a careful summary, as though you were going to teach the material to others.

We never really know that we understand something until we have successfully written it down or explained it to others. Often we think we understand an idea, but we find our written or spoken explanation inadequate. The search for a better way to say something can lead to a better understanding.