Irrational fear of wasting time thinking

I believe that many mathematicians, including myself, have often sought to make exciting mathematical discoveries during our childhood, only to realise they had long been known. These experiences tell us (and rightfully so) that we could learn a lot more by consulting literature, in addition to thinking on our own.

Yet taken to the extreme, it instills a fear of thinking for yourself. The fear prompts us to read endlessly until we’ve found the answer to our questions instead of doing it ourselves. Perhaps the most wasteful of all is reading without thinking. It strongly reminds me of the quote

學而不思則罔，思而不學則殆。

Learning without thought is labor lost; thought without learning is perilous. - Confucius

This symptom is especially perilous in algebra, where almost every theorem could be generalised.

Specifically to maths, perhaps one should replace thinking with doing. It’s all too easy to think that you understand a piece of mathematics only to struggle in the details and realise you’ve understood nothing at all.

The value of having problem sheets marked

Increasingly, getting my problem sheets marked at this stage of my education seems to miss the mark. Back in my first year, it held tremendous importance because I had genuine misconceptions that would go unoticed unless corrected. Now that I’m further along, I often already know or have a sense of where I’ve gone wrong. Unless the marker invests considerable time and effort, the feedback doesn’t provide much value.

This situation really prompts me to question the true value of having graduate students be markers. In spite of that, good teaching is still very valuable. I learn the most whenever the lecturer / tutor goes off the script and talks about something they are passionate about. I strongly feel that personal teaching would be the value propisition of universities going forward.

Even more applied maths

A year ago, I said I wouldn’t be taking any more applied maths. I couldn’t be more wrong now. Despite initially fearing electromagentism would contain too many calculations, the book by Griffiths on the topic proved to be excellent. It’s quickly become my favourite mathematics expository textbook I’ve come across, perhaps with the exception of Dummit & Foote for abstract algbera.

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