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The issue is here: "I have no problem solving most of algebra or calculus problems, but if you asked me to explain what I'm doing I wouldn't be able to tell you

by m11a 6y ago

The issue is here: "I have no problem solving most of algebra or calculus problems, but if you asked me to explain what I'm doing I wouldn't be able to tell you, at least not using mathematical constructs/words." This is slightly unclear; there are two possible problems here from what I can see, and they're totally different problems to have. Either you cannot do maths, and you're only good at solving repetitive problems of the same genre from school textbooks. Or you can do maths, and you struggle to follow the generally accepted set of notation. If your problem is the first, you need to explore a wider range of mathematical problems, which will fundamentally require you to think, apply your skills to the problem in various new ways. Maths is a subject where learning to think and learning to solve a specific set of problems are two very different things. Good resources for this are the Art of Problem Solving, and various Olympiad questions. You can also just try to apply your mathematical knowledge to real life examples. If your problem is the second, I don't know a solution, but this is a far better problem to have than the first. I'd rather have the problem solving ability and use my own set of notation that nobody else understands, than have less problem solving ability and understand the notation. But, in short, that'll just come down to practice, seeing the general notation, and somehow forcing yourself to use it. Alternatively, just try to define your own set of notation at the start of your solutions. It may depend on what kind of professor you have as to how many credits you'll get, though. A good professor will not penalise you that much for making up your own notation, if your solutions are correct, but may try to encourage you to use standard notation (as this is required for any kind of collaborative mathematics).

hyperpallium2 6y ago
The problems in Spivak are pretty good I think - they'll show if you can do maths or not. Some also have answers (and an answer book for the rest), so there's examples of explaining what was done, for harder ones. The explanations are sometimes too concise, but at least it gives some guide for how you can explain your answers. At a much easier level, Khan Academy videos mostly are explaining how a specific problem was solved - another resource.
- swiley 6y ago
  Meh. Everyone pushes their favorite calculus book but I think better is to just try to push yourself to gain an intuition for your subject. Try to visualize what it is the text is saying, try to apply the problem etc. When I took calculus there were three things that really helped me with this: 1) A study partner that's competitive. when you have an intuition you're more likely to get the correct answer first because you can check what's going on quickly without having to recalculate things. Also when they get stuck you'll be expected to explain how you figured it out. Studying with someone like this will force you to grow an intuition. 2) Writing programs that use what you've learned. At least for me this helped a lot. I learned to program before I learned any interesting math so being able to rewrite what are often implicit ideas in an explicit language can really help you think things through. Like many people here I gained an intuition for integration early on by writing physics engines for games. 3) Just meditate on the ideas.
meken 6y ago
I suffered from problem 1. I was able to steamroll my way through calculus II. Then I hit a brick wall in calculus III and came to the harsh realization that I had simply rote memorized everything up to that point. I resolved to actually understand the material, so I took the rather drastic step of going through all of khan academy, starting with arithmetic, all the way through calculus. This was easier back in 2010, when all khan academy videos were on a single page. Luckily, the internet archive can help us out https://web.archive.org/web/20100206074742/http://www.khanacademy.org/ https://web.archive.org/web/20100206074742/http://www.khanac... By the time I finished, I had built up a good mathematical intuition. This was actually one of the most satisfying intellectual journies I have gone on. As a shout out to circa-2010 khan academy, it’s still so beautiful to me that all the videos are there, on one page. You can see them all, unlike the layout of the current website, where you can easily go down rabbit holes and lose your sense of progression.

by m11a 6y ago

hyperpallium2 6y ago

The problems in Spivak are pretty good I think - they'll show if you can do maths or not. Some also have answers (and an answer book for the rest), so there's examples of explaining what was done, for harder ones. The explanations are sometimes too concise, but at least it gives some guide for how you can explain your answers. At a much easier level, Khan Academy videos mostly are explaining how a specific problem was solved - another resource.

swiley 6y ago
Meh. Everyone pushes their favorite calculus book but I think better is to just try to push yourself to gain an intuition for your subject. Try to visualize what it is the text is saying, try to apply the problem etc. When I took calculus there were three things that really helped me with this: 1) A study partner that's competitive. when you have an intuition you're more likely to get the correct answer first because you can check what's going on quickly without having to recalculate things. Also when they get stuck you'll be expected to explain how you figured it out. Studying with someone like this will force you to grow an intuition. 2) Writing programs that use what you've learned. At least for me this helped a lot. I learned to program before I learned any interesting math so being able to rewrite what are often implicit ideas in an explicit language can really help you think things through. Like many people here I gained an intuition for integration early on by writing physics engines for games. 3) Just meditate on the ideas.

meken 6y ago

I suffered from problem 1. I was able to steamroll my way through calculus II. Then I hit a brick wall in calculus III and came to the harsh realization that I had simply rote memorized everything up to that point. I resolved to actually understand the material, so I took the rather drastic step of going through all of khan academy, starting with arithmetic, all the way through calculus. This was easier back in 2010, when all khan academy videos were on a single page. Luckily, the internet archive can help us out https://web.archive.org/web/20100206074742/http://www.khanacademy.org/ https://web.archive.org/web/20100206074742/http://www.khanac... By the time I finished, I had built up a good mathematical intuition. This was actually one of the most satisfying intellectual journies I have gone on. As a shout out to circa-2010 khan academy, it’s still so beautiful to me that all the videos are there, on one page. You can see them all, unlike the layout of the current website, where you can easily go down rabbit holes and lose your sense of progression.