Essay | Thoughts on Learning Mathematics
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Learning mathematics requires more than diligently reading books and solving problems: it also requires thought. This is obvious, yet I often forget it while studying.
From following a proof to asking questions
When I read the proof of a theorem, I usually try to understand how its assumptions lead to its conclusion. Once I reach the end of the proof, however, I feel satisfied and put it aside, returning only when an exercise happens to require it.
This is plainly inadequate. The important part of learning mathematics is not merely verifying that a theorem is correct. That matters, of course, but our predecessors have already spent centuries making that road sound. My task is to think further.
Why is this theorem needed? Why was this definition introduced? Why is the proof arranged in this way? These are only the most basic questions.
Mathematics can certainly be tiring, but no rule requires me to finish a book within a month or even a year. It may be better to slow down, examine every aspect of the text carefully, and use exercises to test what I have learned. That, to me, is the path of mathematical study.
The only drawback is that the results are difficult to see at once. Many of the benefits emerge quietly years later and become part of one’s way of thinking.
Preserving interest and rigour
There is no need to be faster or more anxious. We already know that a new mathematical discovery may take a very long time to affect ordinary life. Why, then, do so many people continue to devote themselves to this game of thought?
Because it is interesting, and because it is rigorous. Those two qualities are what matter. Mathematics may be serious, but it should also remain enjoyable.
Before reading a mathematics book, I should arrange the chapters I intend to study and give myself a timetable. The timetable ought to remain flexible and adapt to present circumstances, but it is still useful: without one, study can become too loose to sustain as a habit.
I should then understand each chapter and attempt its exercises. There will certainly be problems I cannot solve, but that is no reason for excessive self-reproach. Exercises exist to test what I have learned and to make me consider why every element has been placed where it is.
Thinking with a pen
Most importantly, while reading and solving problems I should keep asking why.
I should put those questions on paper and think with a pen. Only then can thought take hold instead of remaining a castle in the air.
After that, I can move on to the next chapter and repeat the process.