I work as a private tutor for proof-based math, and I have a lot of students who've spent some time self-studying before coming to me. The comment by godelski matches my experience: the biggest obstacle seems to be the fact that it's hard to learn how to check your own proofs if no one has ever taught you how. I see a lot of variation in how well people have managed to develop that skill on their own.
Having more textbooks with solutions to the exercises would probably help a lot with this, especially if you used the solutions judiciously. I think the fact that this isn't more common sadly has a lot to do with their role in undergraduate teaching: every exercise that has a solution in the back of the book is one that college students can very easily cheat on. I definitely agree that it's frustrating that the product has to be made worse for everyone else just because some people would misuse the better version. Far from the only such case in the world!
Yeah it’s a catch-22. I’m in grad school and occasionally I get to be instructor. When I am I focus far less on tests and more on homeworks and projects (I do ML so it’s well suited for that style). The homeworks are made to be “play around” and the project is to be very self driven (with plenty of help, but they are juniors or seniors so they be fairly self reliant) and to find passion.
The reason I do this is because grades matter so much to students that even if they care to learn material they are incentivized to cheat (and subsequently cheat themselves). I think a lot of academics still don’t get this and are resistant to change (it is a lot of work to create a class but not to much once you worked everything out).
I think this confidence thing is also something that needs to be learned in every subject. Even in CS the compiler, type checking, and even unit tests aren’t enough (though they are extremely useful).
I should also say, one unfortunate thing I find in academic teaching of coding is we often don’t look at code. There’s not enough time. But to me this feels like trying to grade a math proof by looking only at the first and last lines. I think this builds lots of bad habits and over confidence
I totally get that. I was fortunate to get properly educated up through essentially the level of an undergraduate math degree (minus maybe typology), but then continued learning a lot on my own. It’s common to hear that the struggle is part of the learning process. The more I’ve advanced the more I find this to be true. It’s that struggle that makes you pay attention to the small details that are so critical. I was also fortunate to have a professor who would pester me and he later told me he wanted me to be confident in my results, because eventually I would have no one to double check (and he was right). The struggle really helps with this.
I think the main difference between learning provisioning and math is a compiler. To learn either you can only learn by doing. Reading and lectures aren’t enough. What is hard to learn in math is to be the compiler yourself. To be able to verify “programs” (do I even need quotes here?). This is a very powerful tool to add to your tool belt and one I think even helps in programming.
I hope others can add advice here and words of encouragement. The struggle is real, but it is part of the process, for better or for worse.
Do you have any favourite books? like maybe 2 or 3 that you'd grab in the event of ww3 or a zombie apocalypse? I'm always on the look out for good self-learning maths books.
These books are what i would consider the best books for self-study. (I have gone through many textbooks after i went back to school to study statistics.)
• Real Analysis: A Long-Form Mathematics Textbook
• Proofs: A Long-Form Mathematics Textbook
• Mathematical Logic Through Python
• A Course in Calculus and Real Analysis
• A Course in Multivariable Calculus and Analysis
• Differential Equations With Applications and Historical Notes
• Probability and Random Processes (Grimmet)
• Probability Through Problems
• Fifty Challenging Problems in Probability with Solutions
• The Simple and Infinite Joy of Mathematical Statistics
nicf|1 year ago
Having more textbooks with solutions to the exercises would probably help a lot with this, especially if you used the solutions judiciously. I think the fact that this isn't more common sadly has a lot to do with their role in undergraduate teaching: every exercise that has a solution in the back of the book is one that college students can very easily cheat on. I definitely agree that it's frustrating that the product has to be made worse for everyone else just because some people would misuse the better version. Far from the only such case in the world!
godelski|1 year ago
The reason I do this is because grades matter so much to students that even if they care to learn material they are incentivized to cheat (and subsequently cheat themselves). I think a lot of academics still don’t get this and are resistant to change (it is a lot of work to create a class but not to much once you worked everything out).
I think this confidence thing is also something that needs to be learned in every subject. Even in CS the compiler, type checking, and even unit tests aren’t enough (though they are extremely useful).
I should also say, one unfortunate thing I find in academic teaching of coding is we often don’t look at code. There’s not enough time. But to me this feels like trying to grade a math proof by looking only at the first and last lines. I think this builds lots of bad habits and over confidence
godelski|1 year ago
I think the main difference between learning provisioning and math is a compiler. To learn either you can only learn by doing. Reading and lectures aren’t enough. What is hard to learn in math is to be the compiler yourself. To be able to verify “programs” (do I even need quotes here?). This is a very powerful tool to add to your tool belt and one I think even helps in programming.
I hope others can add advice here and words of encouragement. The struggle is real, but it is part of the process, for better or for worse.
globalnode|1 year ago
Al0neStar|1 year ago
• Real Analysis: A Long-Form Mathematics Textbook
• Proofs: A Long-Form Mathematics Textbook
• Mathematical Logic Through Python
• A Course in Calculus and Real Analysis
• A Course in Multivariable Calculus and Analysis
• Differential Equations With Applications and Historical Notes
• Probability and Random Processes (Grimmet)
• Probability Through Problems
• Fifty Challenging Problems in Probability with Solutions
• The Simple and Infinite Joy of Mathematical Statistics
• An Introduction to Error Analysis