This is the first in a series of posts about reading actively.

Learning to read your textbook actively is perhaps the single most important study habit for success in the mathematics major.  Recently, I spoke with a math major who had transferred from another major.  He read his math textbooks in linear order with a highlighter in hand, in a distraction-reduced environment.  He copied out key vocabulary and facts onto a study sheet, neatly colour-coded.  But he was finding his reading entirely ineffective as a study tool.  What was he doing wrong?

The key is one single principle:  in mathematics, we learn by doing.  An illustrative example is given by Professor Dancealot, whom you can find on YouTube teaching foxtrot by monotone lecture.  How do you best learn to dance?  You dance.  Professor Dancealot is doing no one a favour by droning on about foot position, when he should have his students up in the aisles, trying it out.

Mathematics is no different.  Mathematics is like dance, or language learning.  The only effective way to develop mathematical skills is to practice.  The key to active reading is to learn how to get yourself practicing, not just reading.  This practice is far from automatic — it will not happen as a side effect of pronouncing the textbook words in your head, or even copying them on another piece of paper.  It is a skill in itself.

Perhaps mathematics is better illustrated by the metaphor of yoga, which asks you to assume complicated body positions.  Mathematics asks you to assume complicated mind positions.  As you read the instruction, “Place your feet shoulder width apart,” for a full understanding, you should get up and assume the position indicated before reading the next sentence.

Mathematical texts are no different.  If your text says, “Let S be any non-empty set,” you should create such a set, or a range of possible such sets, in your mind.  This is not automatic!  You must consciously do it.  How do you do it?  You would do the following:  recall the definition of a set for yourself (say it out loud from your own understanding, in your own words); recall the definition of non-empty (out loud again, from your understanding); give a few examples of sets which are non-empty (yes, out loud); and give a few examples of sets which fail to be non-empty (once again, out loud).

Ok, you can whisper.  But the point is not to let yourself get away with anything less than a full, honest, explanation of your understanding as you go along.  If you cannot produce it, you should go back and study the relevant material before proceeding.  The best test of understanding is your ability to explain.  Don’t let yourself off the hook!  It is easy to feel honest and say, “I remember what that means,” but it is a step further to force yourself to give the definition out loud — and this step is the key to strengthening your neuronal connections.

Let us give another example.  In your text, you may find the sentence, “Notice that 3 divides A.”  This is an instruction to perform a mental move!  Remind yourself what A is (out loud), what the formal definition of “divides” is (out loud, from your understanding, in your own words), and then verify, using the definition, that 3 divides A.   In this way, you have yourself performed the yoga move of logically verifying that 3 divides A.  Only then can you move to the next sentence.

And so you continue, sentence by sentence, through the text.  It is a slow process, but by engaging actively, you are growing and strengthening neuronal connections as you read.  This is your most valuable study time.