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I wish more textbooks, especially free resources like in the link, would be better about providing more solutions. A book with a lack of solutions tends to crea

I wish more textbooks, especially free resources like in the link, would be better about providing more solutions. A book with a lack of solutions tends to create a circular problem for me. Knowing whether my solution is correct or not is dependent on how well I truly understand the concepts. However, if I truly understood the concepts, then I wouldn't need to solve the problem in the first place. How is one supposed to learn without feedback?

j7ake 2y ago
These days chatgpt can fill the gap surprisingly well for many questions.
- yshklarov 2y ago
  I wouldn't count on it -- LLMs make lots of errors in reasoning, and errors in solutions are very frustrating to most math students.
- BossingAround 2y ago
  ChatGPT can help with very simple problems that people have already solved online. That said, typically, you can search for the actual solution online without the ChatGPT's possible hallucinations. A better use of LLM is to input your reasoning and ask if it is correct, but even then, you can't rely on the output. Probably the best is to ask on the math side of stack overflow and rely on the kindness of a stranger.
- phforms 2y ago
  Without any solutions or even problems to begin with and without having a teacher by your side, I found that LLMs can be a good-enough learning assistant. Of course, you cannot rely on their answers and they sometimes lead you astray, but what works for me is to: - throughout the text, use the LLM to ask questions, generate problems (I haven’t tried this yet, so it might not work very well) and give you hints on how to close the gaps in your understanding - reduce a given problem to the simplest possible case and ask how to solve that (if you want to check if your solution seems plausible) or if it can give you hints on how to approach the problem (when you are stuck) - vary the problem description to see if different approaches are suggested - critically ask questions about the (steps to the) solution or any provided hints (does something seem odd or logically inconsistent?) - other LLMs sometimes give you better answers (or contradict each other), so, if possible, try asking different LLMs the same questions Having this “conversation” with the LLM where you need to critically check everything it says has the added benefit of being more actively engaged with the material as opposed to simply looking up the solution. It may be a huge waste of time if you’re not careful, but I think if you use it intelligently to guide your own thinking, it can be very helpful.
yshklarov 2y ago
Not providing solutions is quite common in math textbooks, in part because professors (including the author!) want to be able to assign problems from the textbook to their class, and in part because making solutions is a lot of work! Outside of a classroom setting, the way you learn from a textbook without external feedback is by engaging more actively with the material. Treat each statement in the main text as an informal exercise. Each time you come across a proposition -- whether it's a formal theorem statement or a claim in the body of the exposition -- try proving or otherwise justifying it to yourself before reading on. Take a look at Theorems 2.3.1 and 2.3.2 -- they are very similar. Once you have absorbed the proof of 2.3.1, you can attempt 2.3.2 on your own. If you can't finish the proof, you can read a couple of sentences from the included proof for "hints"... or, if you do finish a proof, you can compare it to the proof in the text. If you read actively enough, you can learn the material quite well without doing any problems. Many people will claim that you need to do formal problems in order to learn math, but this is untrue. Many math textbooks at the higher level don't include formal exercises or problems at all, and people learn from them just fine. Admittedly, reading mathematics is a skill in its own right, and you shouldn't expect it to come easily right away. Of course, the best thing is to have a one-on-one teacher, but few of us are so lucky.
- BossingAround 2y ago
  Honestly, this seems needlessly painful to me. Of course, you can be scanning each sentence for a proposition, then pause, and try to reason it out, thus spending 4 weeks on like 5 pages of a 10-page chapter. But is that the best use of your time? The bigger problem is that not every thing that the author says is within your level of reasoning. Some very simple things can be exceedingly hard to prove, and you, as a learner, don't know which is which. That's why there are the problems at the end of the chapter, which are designed for the level that you should have attained by the end of the chapter. Without solutions though, you have no way to check your understanding, and you are forced to try and squeeze every little problem from the text. Not having solutions is simply not suitable for a self-learner. It is sufficient for a class settings, where you can ask the professor if your solution is correct. To me, a good compromise is to provide solutions to every odd- (or even-) numbered problem. Thus, the self learners have at least half of the problems within their reach, and t he teachers can assign the other half of the problems.
humanfromearth9 2y ago
I think that each may book should at least have all problem solutions at least, and most problem resolutions as well, leaving some room problem to be solved without giving the resolution, for training. Anything short of that can only be used as a reference book by a teacher, because teachers need to confirm that the resolution is indeed complete and right. Without that, as a engineering and economics student (decades ago), my resolutions would often have been incomplete and lack some details and specific cases.
wjholden 2y ago
By solving the problem in more than one way! This should be very achievable in a field like discrete math. First solve the problem by hand, then model the program in something like Mathematica or OR-Tools and see if you can get the same solution. This would work even better in a lower level maths like Algebra and Calculus. Just use Mathematica's Solve[] function for many problems and you'll know if you were right or wrong. One can also use this approach in an algorithms class. Write your own naïve program to trivially but reliably solve test cases, then compare your more sophisticated algorithms to these results. Alternatively, use a reference implementation from some other library. For example, compare your own graph algorithm solutions to what Neo4j returns.
- justin66 2y ago
  Or save yourself hundreds of hours and find a textbook with solutions.
oscarlevin 2y ago
Author here. For what it's worth, deciding what percentage of the exercises should have solutions is a continual challenge. I chose to use a lot of interactive exercises (thanks to PreTeXt these are easy to embed in the text) for which students can enter their answer and get feedback on whether they are correct. That works well for computational problems. For proof-based or otherwise theoretical problems, I tired to provide enough examples with full solutions and a few exercises with them as well, while still giving those who would like to have a more authentic open-ended problem without solution that opportunity too. And of course, I want other professors to find the book useful for courses they teach, and providing problems without solutions that can be graded for credit is also important. Anyway, hope you find the resource helpful.
- dunefox 2y ago
  Exercises without detailed solutions are useless. If I don't have direct access to a tutor I need solutions, otherwise there's no point - I can never check whether I'm right, I'm stuck, or I'm making mistakes that are invisible to me. This invalidates the whole point of a textbook, it's your entry to a field without access to a teacher.
  - oscarlevin 2y ago
    They may be useless to you (although I think even seeing examples of types of questions that can be asked serves a purpose), but the book was not written only for you. I think it is reasonable for books to have features that are useful for students in courses that assign a grade, as well as other features that are useful for students using it for self study. It is also worth pointing out, that at least for math textbooks, there is never an expectation that a student should solve all the exercises. If you solve those that do have a solution, then you will hopefully still have a good experience and learn lots.
    - soul_grafitti 2y ago
      I've been using textbooks to learn things for over 50 years and during my development writing code many (most?) of those times was not in the context of a classroom but in the context of needing to understand something so I could write code to do it. For folks in this situation having answers, especially to some complex problems, is a huge benefit in terms of gaining understanding and saving time. I don't think I'm alone here when I say I frequently gain understanding of things by writing code to model them. I know I understand something when I can write it in code. So after ingesting the lesson about a topic I'll see if I can write code to do what I just learned. I have spent a lot of hours stuck at a juncture of not being entirely sure my code, which works on the simple examples, is also working in the complex ones. The difference in results may not be evident. Am I troubleshooting my understanding or my code? I need a known result to actually know. This is one place being able to copy something from your textbook, for example, paste it into ChatGPT and ask questions is a huge benefit. ChatGippity doesn't always get it right but usually the interchange is much more useful than being stuck.

j7ake 2y ago

These days chatgpt can fill the gap surprisingly well for many questions.

yshklarov 2y ago
I wouldn't count on it -- LLMs make lots of errors in reasoning, and errors in solutions are very frustrating to most math students.
BossingAround 2y ago
ChatGPT can help with very simple problems that people have already solved online. That said, typically, you can search for the actual solution online without the ChatGPT's possible hallucinations. A better use of LLM is to input your reasoning and ask if it is correct, but even then, you can't rely on the output. Probably the best is to ask on the math side of stack overflow and rely on the kindness of a stranger.
phforms 2y ago
Without any solutions or even problems to begin with and without having a teacher by your side, I found that LLMs can be a good-enough learning assistant. Of course, you cannot rely on their answers and they sometimes lead you astray, but what works for me is to: - throughout the text, use the LLM to ask questions, generate problems (I haven’t tried this yet, so it might not work very well) and give you hints on how to close the gaps in your understanding - reduce a given problem to the simplest possible case and ask how to solve that (if you want to check if your solution seems plausible) or if it can give you hints on how to approach the problem (when you are stuck) - vary the problem description to see if different approaches are suggested - critically ask questions about the (steps to the) solution or any provided hints (does something seem odd or logically inconsistent?) - other LLMs sometimes give you better answers (or contradict each other), so, if possible, try asking different LLMs the same questions Having this “conversation” with the LLM where you need to critically check everything it says has the added benefit of being more actively engaged with the material as opposed to simply looking up the solution. It may be a huge waste of time if you’re not careful, but I think if you use it intelligently to guide your own thinking, it can be very helpful.

yshklarov 2y ago

Not providing solutions is quite common in math textbooks, in part because professors (including the author!) want to be able to assign problems from the textbook to their class, and in part because making solutions is a lot of work! Outside of a classroom setting, the way you learn from a textbook without external feedback is by engaging more actively with the material. Treat each statement in the main text as an informal exercise. Each time you come across a proposition -- whether it's a formal theorem statement or a claim in the body of the exposition -- try proving or otherwise justifying it to yourself before reading on. Take a look at Theorems 2.3.1 and 2.3.2 -- they are very similar. Once you have absorbed the proof of 2.3.1, you can attempt 2.3.2 on your own. If you can't finish the proof, you can read a couple of sentences from the included proof for "hints"... or, if you do finish a proof, you can compare it to the proof in the text. If you read actively enough, you can learn the material quite well without doing any problems. Many people will claim that you need to do formal problems in order to learn math, but this is untrue. Many math textbooks at the higher level don't include formal exercises or problems at all, and people learn from them just fine. Admittedly, reading mathematics is a skill in its own right, and you shouldn't expect it to come easily right away. Of course, the best thing is to have a one-on-one teacher, but few of us are so lucky.

BossingAround 2y ago
Honestly, this seems needlessly painful to me. Of course, you can be scanning each sentence for a proposition, then pause, and try to reason it out, thus spending 4 weeks on like 5 pages of a 10-page chapter. But is that the best use of your time? The bigger problem is that not every thing that the author says is within your level of reasoning. Some very simple things can be exceedingly hard to prove, and you, as a learner, don't know which is which. That's why there are the problems at the end of the chapter, which are designed for the level that you should have attained by the end of the chapter. Without solutions though, you have no way to check your understanding, and you are forced to try and squeeze every little problem from the text. Not having solutions is simply not suitable for a self-learner. It is sufficient for a class settings, where you can ask the professor if your solution is correct. To me, a good compromise is to provide solutions to every odd- (or even-) numbered problem. Thus, the self learners have at least half of the problems within their reach, and t he teachers can assign the other half of the problems.

humanfromearth9 2y ago

I think that each may book should at least have all problem solutions at least, and most problem resolutions as well, leaving some room problem to be solved without giving the resolution, for training. Anything short of that can only be used as a reference book by a teacher, because teachers need to confirm that the resolution is indeed complete and right. Without that, as a engineering and economics student (decades ago), my resolutions would often have been incomplete and lack some details and specific cases.

wjholden 2y ago

By solving the problem in more than one way! This should be very achievable in a field like discrete math. First solve the problem by hand, then model the program in something like Mathematica or OR-Tools and see if you can get the same solution. This would work even better in a lower level maths like Algebra and Calculus. Just use Mathematica's Solve[] function for many problems and you'll know if you were right or wrong. One can also use this approach in an algorithms class. Write your own naïve program to trivially but reliably solve test cases, then compare your more sophisticated algorithms to these results. Alternatively, use a reference implementation from some other library. For example, compare your own graph algorithm solutions to what Neo4j returns.

justin66 2y ago
Or save yourself hundreds of hours and find a textbook with solutions.

oscarlevin 2y ago

Author here. For what it's worth, deciding what percentage of the exercises should have solutions is a continual challenge. I chose to use a lot of interactive exercises (thanks to PreTeXt these are easy to embed in the text) for which students can enter their answer and get feedback on whether they are correct. That works well for computational problems. For proof-based or otherwise theoretical problems, I tired to provide enough examples with full solutions and a few exercises with them as well, while still giving those who would like to have a more authentic open-ended problem without solution that opportunity too. And of course, I want other professors to find the book useful for courses they teach, and providing problems without solutions that can be graded for credit is also important. Anyway, hope you find the resource helpful.

dunefox 2y ago
Exercises without detailed solutions are useless. If I don't have direct access to a tutor I need solutions, otherwise there's no point - I can never check whether I'm right, I'm stuck, or I'm making mistakes that are invisible to me. This invalidates the whole point of a textbook, it's your entry to a field without access to a teacher.
- oscarlevin 2y ago
  They may be useless to you (although I think even seeing examples of types of questions that can be asked serves a purpose), but the book was not written only for you. I think it is reasonable for books to have features that are useful for students in courses that assign a grade, as well as other features that are useful for students using it for self study. It is also worth pointing out, that at least for math textbooks, there is never an expectation that a student should solve all the exercises. If you solve those that do have a solution, then you will hopefully still have a good experience and learn lots.
  - soul_grafitti 2y ago
    I've been using textbooks to learn things for over 50 years and during my development writing code many (most?) of those times was not in the context of a classroom but in the context of needing to understand something so I could write code to do it. For folks in this situation having answers, especially to some complex problems, is a huge benefit in terms of gaining understanding and saving time. I don't think I'm alone here when I say I frequently gain understanding of things by writing code to model them. I know I understand something when I can write it in code. So after ingesting the lesson about a topic I'll see if I can write code to do what I just learned. I have spent a lot of hours stuck at a juncture of not being entirely sure my code, which works on the simple examples, is also working in the complex ones. The difference in results may not be evident. Am I troubleshooting my understanding or my code? I need a known result to actually know. This is one place being able to copy something from your textbook, for example, paste it into ChatGPT and ask questions is a huge benefit. ChatGippity doesn't always get it right but usually the interchange is much more useful than being stuck.