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by zrobotics
1282 days ago
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One thing though, doing exercises (or problems, whatever you want to call them) is extremely helpful for actually learning math. For your matrix example, I had a very excellent teacher for linear algebra and one of the most helpful things they did was spend the first few weeks of the course teaching manual calculation and actually forcing us to do the manual calculations. At the time this felt like a total waste of time, since after all that is what computers are for, right? However, actually doing the calculations helped greatly to reinforce my memory for matrix manipulation and what is actually occurring. Yes, proofs are a powerful part of mathematics, but they aren't at all the only goal. For someone like a machinist, proofs seem entirely impractical and like a waste of effort. My high school geometry class was heavily proof based, and most of the people in my class couldn't understand why we were doing these exercises. As a result, they didn't actually learn many of the helpful tricks from intro geometry that can help with things like machining or carpentry. In fact, I'd talked to someone who was in that class that now runs a cabinetmaking shop and they mentioned feeling cheated in high school due to our geometry class. I thought it was very well taught, but I ended up doing a BS in math. I was surprised that we ended up with very different views of the same class >10 years later. Plus, even after MS level math courses, I can appreciate the need to do exercises. I look at them the same way as practicing playing an instrument. Sure, you can intellectually understand everything about a performance, but the only way to learn is through practice. I haven't had to touch 3d calc in a hot minute, but an upcoming side project will need that. First thing I am doing is (legally acquiring) some undergrad textbooks and solution guides and working through problems before comparing results. Yes, I can do proofs, but going through a worked example and comparing results is an excellent way of learning. |
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Ultimately, they are.