How to learn mathematics
The asterisk method
Alan U. Kennington

This is an excerpt from my differential geometry book (work in progress).
How to learn mathematics.
The asterisk learning method focuses attention on difficult points which need it most.

This is the asterisk method of learning which I discovered in 1975. It worked!

  1. Find somewhere quiet. Turn off the radio (and your portable digital music player).
    A library reading room is best if it is a quiet library reading room.
    Preferably have a large table or desk to work on.
  2. Open the book or lecture notes which you wish to study.
  3. Start copying the relevant part of the book or the lecture notes to a (paper) note-pad by hand.
    If you are studying a book, you should be writing a summary or paraphrasing what you are reading.
    If you are studying lecture notes, you should copy everything and add explanations where required.
  4. Whenever you copy something, ask yourself if you really understand it completely.
    In other words, you must understand every word in every sentence.
    As long as you are completely comfortable with what you are copying, keep going.
  5. If you read something which is difficult to understand, stop and think about it until you understand it clearly.
    If a mathematical expression is unclear, determine which set or class each term in the expression belongs to.
    All sub-expressions in a complex expression must belong to some set or class.
    Every operator and function in an expression must act only on variables in the domain of definition of the operator or function.
    Draw diagrams of the relations between concepts.
    Draw diagrams of everything!
  6. If you find something that you really can't understand after a long time, copy it to your notebook, but put an asterisk in the margin.
    This means that you have copied something that you did not understand.
  7. While you continue copying, keep going back to the lines which are marked with an asterisk to see if you can understand them.
    If you find an explanation later, you can erase the asterisk.
  8. When you have finished copying enough material for one sitting, look over your notes to see if you can understand the lines which still have an asterisk.
    If you have no asterisks, that means that you have understood everything.
    So you can progress to the next section or chapter of the text.
  9. If you still have one or more asterisks left in your notes after a day or more, you should keep trying to understand the lines with the asterisks.
    Whenever you get some spare time and energy, just look at the lines with asterisks on them.
    These are the lines that need your attention most.
  10. If you discuss your work with other people, especially with teachers or tutors, show them your notes and the lines with the asterisks.
    Try to get them to explain these lines to you.
If you keep working like this, you will find that your study becomes very efficient.
This is because you do not waste your time studying things which you have already understood.

I used to notice that I would spend most of my time reading the things which I did understand.
To learn efficiently, it is necessary to focus on the difficult things which you do not understand.
That's why it is so important to mark the incomprehensible lines with an asterisk.

Copying material by hand is important because this forces the ideas to go through the mind.
The mind is on the path between the eyes and the hands.
So when you copy something, it must go through your mind!

It is also important to develop an awareness of whether you do or do not really understand something.
It is important to remove the mental blind spots which hide the difficult things from the conscious mind.

When copying material, it is important to determine which is the first word where it becomes difficult.
Read a difficult sentence until you find the first incomprehensible word.
Focus on that word.
If a mathematical expression is too difficult to understand, read each symbol one by one and ask yourself if you really know what each symbol means.
Make sure you know which set or space each symbol belongs to.
Look up the meaning of any symbol which is not clear.

First learn. Then understand.
Insight requires ideas to be uploaded to the mind first.
It is often said that learning is most effective when the student has insight.
This assertion is sometimes given as a reason to avoid rote learning, because insight must precede the acquisition of ideas.
But the best way to get insight into the properties and relations of ideas is to first upload them into the mind and then make connections between them.
For example, a child learning the five times table will easily notice the redundancy during or after rote learning.
The best strategy is to learn first, then understand more deeply.
Connections can only be made between ideas when they are in the mind to be connected.


Japanese (日本語) translation: 数学の勉強の仕方 – アスタリスク学習法.


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