4 ms·

When you hit upon a proof like that, you know it. If you're struggling and need feedback, then your solution is probably more complicated than the elegant one

by dkarl 8y ago

When you hit upon a proof like that, you know it. If you're struggling and need feedback, then your solution is probably more complicated than the elegant one in the book, and the question you're asking is, okay, my solution isn't great, but is it even a solution? There's a reason these books declare some level of "mathematical maturity" as a prerequisite. By the time you get to certain content you have to either be comfortable checking a proof (maybe set it aside for a few days like you would any other writing) or have human help.

jeff76 8y ago
Think you missed my entire point of including solutions for the beginner, i.e., someone that lacks math maturity and the reasoning of how easily such a person could fool themselves into thinking their thought process is valid.
- dkarl 8y ago
  Someone who is a beginner will be working on easy material with a textbook that has very mild expectations of "mathematical maturity." Textbooks mostly (with a few intentional exceptions) calibrate the mathematical maturity they require according to how much work a student will have done in order to be ready to tackle the book's subject matter. Some people might think they can be ready to do problem sets in an advanced math textbook while skipping the work that would have given them the maturity it requires, but they are kidding themselves. They can't advance on the spectator track and then make a horizontal move over to actually doing the math. There's nothing wrong with people passively reading the material if they enjoy it, but if they want to tackle the problems, they are going to have to work through the earlier material by solving problems as well, and if they do that, they'll find that their sophistication increases along with the demands on it.
  - jeff76 8y ago
    The way you were arguing makes it sound like you are against beginners having access to solutions, or that you completely misunderstood my initial argument. I am not arguing about anything other than that. If you want to address that in particular sure. I am not saying anything about people skipping material and getting bitten by it later.
    - dkarl 8y ago
      Even the first textbooks that start introducing abstract math and developing mathematical maturity in beginning college students often (usually?) have problem sets without solutions, and I'm defending that. I think it's better than providing solutions, even for people working alone. Of course it isn't ideal to work alone; it's helpful to have other people to point out errors in your proofs and to present their own proofs to you. I just don't think providing solutions helps at all in that way. What gives you the best approximation of that experience is spending a long time working on problems. Sometimes you'll realize you've proved something false and have to debug your own proof, so you'll be collaborating with yourself. Solutions will only make it harder for you to force yourself to have those experiences.

by dkarl 8y ago

jeff76 8y ago

Think you missed my entire point of including solutions for the beginner, i.e., someone that lacks math maturity and the reasoning of how easily such a person could fool themselves into thinking their thought process is valid.

dkarl 8y ago
Someone who is a beginner will be working on easy material with a textbook that has very mild expectations of "mathematical maturity." Textbooks mostly (with a few intentional exceptions) calibrate the mathematical maturity they require according to how much work a student will have done in order to be ready to tackle the book's subject matter. Some people might think they can be ready to do problem sets in an advanced math textbook while skipping the work that would have given them the maturity it requires, but they are kidding themselves. They can't advance on the spectator track and then make a horizontal move over to actually doing the math. There's nothing wrong with people passively reading the material if they enjoy it, but if they want to tackle the problems, they are going to have to work through the earlier material by solving problems as well, and if they do that, they'll find that their sophistication increases along with the demands on it.
- jeff76 8y ago
  The way you were arguing makes it sound like you are against beginners having access to solutions, or that you completely misunderstood my initial argument. I am not arguing about anything other than that. If you want to address that in particular sure. I am not saying anything about people skipping material and getting bitten by it later.
  - dkarl 8y ago
    Even the first textbooks that start introducing abstract math and developing mathematical maturity in beginning college students often (usually?) have problem sets without solutions, and I'm defending that. I think it's better than providing solutions, even for people working alone. Of course it isn't ideal to work alone; it's helpful to have other people to point out errors in your proofs and to present their own proofs to you. I just don't think providing solutions helps at all in that way. What gives you the best approximation of that experience is spending a long time working on problems. Sometimes you'll realize you've proved something false and have to debug your own proof, so you'll be collaborating with yourself. Solutions will only make it harder for you to force yourself to have those experiences.