[yshklarov]

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.

[bick_nyers]

I disagree. I have to learn by committing to muscle memory. I always say I'm a slow learner, but a fast thinker.

Yes, I can read/hear a concept and understand it in abstract and visualize it pretty easily, but it will leave my brain just as easily as it entered. Just the way my memory works.

Solved problems speeds up that muscle memory learning process significantly as opposed to going line by line and attempting to generate your own problems/solutions. In addition, you can solve a problem correctly, but not have the correct prose, solution manuals can help there as well.

Edit: Honestly the biggest thing about solving problems is that it gives a sense of progression and a dopamine-reward loop that most people just don't get from reading one line at a time. That being said, good problems and good solutions can be time consuming to generate, so it makes total sense to me that the PhD-level textbooks don't follow that format.

[cultofmetatron]

> Solved problems speeds up that muscle memory learning process significantly as opposed to going line by line

completely agree. I would recommend checking out mathacademy.com. I've been using it the last two months to raise my skill levels in math and its been a great investment of my time. It gives you a short lesson on the topic and then just gives you problem sets to burn though. what I like about it is that you don't think about your learning objectives. I just log in every day and do the problem sets and lessons until I feel like I did enough for the day. repeat everyday and you'll just naturally find yourself improving.

[fivestones]

The website for math academy looks almost too good to be true. Can you describe more of your experience using it?

[gmays]

It’s been great for me. As of today I crossed 324 days of using it straight. I wrote about my experience here: https://gmays.com/math

And yes, I’m a total fanboy. I’ve also known the founder for over a decade and he’s been working on it for most of that time. Math Academy came out of him helping his son learn math even before that.

I haven’t found anything close to teaching math than this. It’s legit.

[cultofmetatron]

Been using it for 3 months now. When I started, I took their placement test. its been 20 years since I had to do any difficult math so I pretty much bombed haha. It recommended their foundations 1 which is for adults returning to math after a long time.

in terms of difficulty, I found the problems to be somewhat easy. you see a set of modules on the right and your progress on the left. you click on a module to start and it gives you a written tutorial on how to solve a problem. Then you go though several of them. If you complete it with less than 3 wrong, it gives you the xp and youc an continue on another module.

if you do get 4 wrong, it stops the module and you go with the next one. usually a day or so later, it throws the lesson at you again. I've had this happen twice. usually its a sign I've been on the platform for too long for the day and my brain just can't process math anymore. Over time, it introduces more lessons and harder topics along with periodic review modules to test stuff I haven't looked at in awhile.

I'm a few days away from completing course 1 and its been a much needed review of a lot of topics I remember from algebra 2 back in highschool. One of my biggest weaknesses back then was factoring polynomials and I think mathacademy explains the process 1000x better than any of my highschool teachers did. on certain topics like factoring, i'd say I'm much stronger now than when I was in highschool.

The key is to make time for it. I view it like going to the gym. every day I set aside an hour minimum and just crank through them till I either get tired or have some other obligation I have to take care of.

Is it worth it? so far Id say its probably one of the best roi activities I've partaken in. it really does automate the process of learning math. I love that the feedback is rapid so the time between I attempt the problem and can learn from my mistakes have let me progress much faster than I could have in a classroom. you just have to make sure use it consistently. have a notebook and a calulator next to you so there's no distractions and just crank thoguh problems.

TLDR: its worth the $50 every month. As long as you're consitent with it, You WILL learn math with this program.

[BossingAround]

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.

[enriquto]

> spending 4 weeks on like 5 pages of a 10-page chapter. But is that the best use of your time?

Look at it as the best use of paper :)

Many math books are dense. They don't bullshit you around. Spending several hours on a single page is the normal usage.

[moomin]

Yeah, and for better or for worse, it is the best arrangement. I remember ploughing through Alan Baker’s number theory book. You have to sit with a pencil and paper and convince yourself of half the steps, but you sure as hell know the material afterwards. And you do need the skills you gain by doing this.

[082349872349872]

> And you do need the skills you gain by [engaging].

The slogan I've heard is: "Mathematics is not a spectator sport."

[moomin]

I haven’t heard that one before but it’s on the nose. I have heard Euclid’s much older “There is no royal road to geometry”.

[dunefox]

> because making solutions is a lot of work!

Imagine how much more work it is to try to solve these exercises then. Exponentially more than creating solutions for someone who already mastered this topic. This tells me a lot about how a teacher thinks of students.

[yshklarov]

> This tells me a lot about how a teacher thinks of students.

This is quite an uncharitable perspective!

[dunefox]

It comes from experience. I had several professors who did not even release the script for a course, much less the problems and never solutions - everything had to be written down by hand by students while you were supposed to listen, understand, and ask questions. If you couldn't read something or were too slow, you could not study this part.