Notation and Beginners

You’re interested in learning a new mathematical topic. After reading some popular articles that speak in broad terms about the subject, you are energized to go on a deeper dive and learn the details. You open up a textbook, pencil in hand and ready to expand your horizons. You start going through the introductory chapter, motivated with the dream of working through the whole textbook. You make a little progress, but you find yourself pausing…

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Discounting

I am very good at finding the bad aspects of any given idea or thing. I wouldn’t call myself cynical. Rather, I have a tendency to find ways to justify not doing something. This has to do with my reluctance to try new things, and I suspect I’m not the only one who does this. Imagine the following scenario. You’re presented with an amazing opportunity, but there is something you have to do. Maybe you…

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Control of the Future

It’s tempting as a student to think that nothing is in your control. We often feel like our whole lives revolve around the whims of professors and their decisions for assignments and tests. It can be easy to retreat into “reactive” mode, making sure that everything which is thrown at you gets done. When we operate like this, we tend to be exhausted, since we can never look further than a week. The issue with…

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Slick Versus Pedagogical Proofs

In mathematics, there’s almost always an opportunity to make proofs more concise. For example, when you first learn a concept, it might take a while to prove a result using the definitions that were developed. The reason it’s longer to do is because the definitions require you to spell out ideas explicitly. As a result, you might get to the end of the proof, but it takes a bunch of little intermediate steps to do…

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