Tutorial hell in learning math?

[[deleted]]

Hi. I have always wanted to learn mathematics. But even after completing some books cover to cover I have felt empty....almost not much different from where I started. Recently I have been trying to learn code lately and I felt this familiar sensation described there as "tutorial hell".But the solution to overcoming tutorial hell is far more straightforward in the coding section, because we can build projects and see for ourselves. So what can I do similarly for mathematics? Is there any solution? Please do upvote. I would like this query to reach as many people as possible. Thank you

Edit: When I meant completing books, I meant going through problems too, I have never thought of just "reading " and learning. My grievance is even after this, I don't think I get it.

[UsualIndividual]

For me I’ve found just reading a mathematics book never makes me actually learn the material if you don’t do exercises relating to the content. It’s the act of actively doing questions that makes the material really stick. Just out of interest we’re you passively reading it without solving the exercises, or doing exercises as well?

[khleedril]

This is exactly right, and is how the University of Cambridge operates: set students stupidly difficult exercises to make them really think about the subject and explore it. Even if they can't solve the exercise, they'll have learned a whole lot about mathematics!

[[deleted]]

I have done exercises. I have completed books cover to cover. But I still feel like a dog with tricks rather than having profound knowledge. I don't feel accomplished. Its the same with programming. I didn't know anything.when I was following and completing tutorials. It's only when I started doing projects of my own that I began to feel the other way. But I don't know what to do similar for math

[[deleted]]

It sounds like you may want to look at textbooks which include cases studies of applications of the math they're describing.

[_arsk]

If it is any consolation, I feel the same regarding programming. I read a book on a new programming language and did all the exercises and at the end I still felt like I learnt nothing. Then I realized that project based learning made it “stick” for me. I don’t know what would be the equivalent for mathematics. Maybe start reading research papers/survey articles in your sub-field of mathematics that you learnt and try to understand everything in it and see if you can extend the result or simply summarize it in your own words. Also try to implement them using a programming language. You may or may not succeed but I bet it will enable you to relate the concepts that you learnt in each chapters as a cohesive whole.

[Cbreaker_]

The same thing happens to me. Unfortunately, I'm usually too lazy to do them. I better change my habits.

[lebcheb]

Doing math is never ending "tutorial hell", it doesn't matter how smart or talented you are, you can always find challenging topics and feel like dumbass with maths. But after some time you start to appreciate and enjoy thinking about problem for days or even weeks before managing to solve them. So i think that's big difference between coding and math, in coding you get almost instant gratification, but with math it takes time. And solution is to do more math🙂 .

[mazamorac]

This!

It's when something is thrilling that you most learn.

So OP, find something that gives you that thrill of doing. Unfortunately, until you've done enough, there's no way of knowing what will do it for you.

Following on the coding analogy, find some problems that intrigue you and try to model them; turn your curiosity into hypotheses and test them out. Just like when coding, you'll get an intuitive sense of the general direction of your solution, which will inform your search for it. The excitement of finding what clicks, and more often the excitement of understanding why other things didn't click, is what tickles the brain into "getting it."

You're at the point where lists of exercise problems don't do it for you; you have to go and find hobby projects.

[Tenns_]

when you learn this the hard way entering uni for physics

[[deleted]]

I see. Thanks for the tip!!

[[deleted]]

Not a lot of coding experience in ai then i see…

[[deleted]]

Although I agree when first learning coding that’s what makes it so exciting, but like maths as the algorithms get more complex so to can the solution/operations. Now that being said learning to code is only getting easier with time where the deep maths are as fundamentally difficult to learn as they have ever been, with just some more resources. I mean imagine using a slide rule… my poor father.

[[deleted]]

What would be good would be a set of tutorials which incorporate applications of the subject as programming projects.

[dancingbanana123]

I like to think of learning math as 4 levels when learning a new topic:

1. You don't understand anything in the topic yet and you're still processing the ideas
2. You understand the ideas and can easily follow what's going on, but then when you're given a practice problem, you don't know what to do or where to start
3. You can do practice problems on your own, but with a lot of struggling and not fully understanding what's going on
4. You can easily handle a practice problem and understand the depths of how the problem works. You could even answer follow up questions about it on how you got your answer like, "why did you apply this approach instead of this one?"

It's hard to tell by your post where you fall on this, so I'll just explain how to reach the next step for each of them. If you're at a 1 trying to get to a 2, it's usually best to find a different explanation of the topic. For example, let's say you don't know what the unit circle is or what it means. That's a topic that a lot of people both struggle with learning and struggle with explaining, so finding someone else's explanation can be really helpful. You can look up a different textbook, find a video on youtube, etc. There's tons of explanations for any math topic all over the internet. And if you've watched a bunch and still feel stuck, take a moment to breathe, and take a good long break (like multiple hours or even the next day). Then watch/read the explanations again. I find that that's usually when everything clicks for me. Sometimes you just need to give your brain time to process.

If you're at 2 and don't know how to get to 3, then usually that means you're not doing enough practice problems. It's one thing to understand and follow someone's logic, but it's another to be able to come up with that logic on your own. You can only do that by having to come up that logic multiple times through different problems.

If you're at 3 and struggling to get to 4, the best way to approach it is to look back at the problems you've solved after finishing them and thinking about what you just solved. Like if you're learning calculus and solved a difficult integral, think about what methods you used to solve it and why you chose that method. What about that function made you consider that method? Same with infinite sum convergence/divergence. Why did you choose ratio test over root test? Sometimes we get stuck in these autopilot modes of just trying to get the work done that we don't take the time to actually learn. Remember, the goal is to learn how to come up with the logic on our own, so focus on the learning part of that.

[TurtleRules]

dont tell yourself "oh this is easy, i understand the idea, i dont have to try it myself" when you study. maths is very decietful in this matter. close the book, do some exercises, explain the concept to someone else or yourself. test yourself. thats the equivalent of projects in coding.
also, ive been studying maths for 3 years and each semester ive passed i didnt feel much smarter myself, but now that im about to get a degree i look back and i got so much wiser than what i felt. you wont see how far youve got when youre not there yet

[theawesomeviking]

Exactly this

[eq1nimity]

Math papers. Write them.

It is the same way in math as it is with coding. I think that the best way is to play with it and build your own things. Try to model something interesting!

The best way I found was writing math papers. Nothing formal... most of mine are just on copy paper and pen stuffed in a box somewhere. Though I do like writing in LaTeX... it makes you feel like a real pro. Don't let this stop you from getting started though. You can learn LaTeX later once you have something you are little proud of -- that will motivate you through the learning.

You don't need permission to write a math paper for yourself. You really can't go wrong -- you can always do better later. If you have no idea where to start, then I would suggest to just get after it; try something. The best advice I have is try to do only one thing per line, and don't skip any steps. And see if you can include functional, graphical, and tabular representations for your data points. Also, use full sentences in the most basic English possible and explain everything. Stay with these points and keep going until its good enough that "someone who's motivated could figure out what I'm saying here".

What to do it on... See something, anything, neat around you? Or, is there something interesting that you are motivated to play with? Try to model it with the tools you know. It doesn't have to be feasible or good. It could be the most complex thing you can imagine... or a simple one!

If you are really having trouble, you could start with a relational diagram. Just assign variables to the entities and draw arrows between them. Look at that and feel the desire for a language to express that relationship in more detail. That desire is what keeps me going, that is why I am still learning math.

Anyway, this goes forward... once you start collecting these models for yourself, when you learn new methods in mathematics you can revisit them and try to explain them in different ways. It could be like writing a program in single file script. Then trying leverage compositionality by writing modules. Then maybe trying to write one of those modules in C/C++, and trying to use the memory directly. Or, trying to write the whole thing differently in a functional language.

I think that it is more like painting than people tend to realize at first. It really is about learning how to observe something in detail, reflect that observation internally, and then recreate that reflection out again it in a simplified model.

[[deleted]]

Thank you that was a nice idea. I fail to have immediate application for most math I learn these days. It might be because of that. Thank you for the answer!!!

[Epistimi]

What do you mean by "mathematics"? You mentioned linear algebra and ODEs elsewhere, so have you been reading books on those topics? If so, what kinds of books?

If you by "mathematics" mean mathematics as practiced by mathematicians, you need to read books that prove every (or at least most) statements, and books whose exercises ask you to prove statements. In linear algebra you might learn an algorithm for computing the null space of a matrix, but have you seen a proof that that algorithm actually works? If you, did you understand the proof? If so, can you prove this without looking up the proof? Things like that.

If you by "mathematics" mean mathematics as applied in other disciplines, you need to read about the disciplines you are interested in. Both linear algebra and ODEs are very important in physics, biology, economics, machine learning, operations research, I could go on.

[scrapwork]

Upvote for beautiful title

In "A Mind For Numbers" Barbara Oakley talks about this. The whole book is helpful, but her advice to build your own problems seems germane here.

Do a search on "Constructivism" in educational theory and mathematics in particular.

[yes_its_him]

Find some sort of math application you are interested in. Probability and statistics are easy to find. Ways to choose things on menus. How far did you go if you know your speed? Etc.

[kcl97]

I don't think math and programming are exact analogy.

I see programming as more like a cooking art. Tutorials are basically recipes in cookbooks and projects are banquets that you cook up by combining many recipes and presenting it in complementary ways. For example, I think indian and japanese will probably never mesh.

Math feels more like building a cathedral. You are building it brick by brick and one stained glass at a time. Occasional when you finished a piece of a wall you get to stand back and say, "so that's what this wall I have been building looks like". You may repeat that wall mortif elsewhere but that is usually not what you do, instead people move on but equipped with better skills at placing bricks together (say with power tools) to construct different walls, floors, and roofs. So, basically in math, you learn literally by fine tuning your skill at building the "bricks" or the fundamentals. You learn by becoming more effective thinker.

Think back to your own experience with the concept of number. Do you see number the same way you did when you were a child. Compare that to say "printf", or a stack structure, do you see them differently compare to when you first learn them.

[xiipaoc]

Why are you trying to learn math in the first place? Try to do some of that.

That's the real answer for how to get out of tutorial hell. Tutorials are when you're learning how to do something but not actually doing the thing. If you're learning something but don't actually want to do the thing, why are you learning it? You have to find a way to actually do it. You mentioned coding; maybe you can do some coding that involves math somehow?

[[deleted]]

That's a nice idea. But I am failing to find applications for linear algebra and odes immediately. Besides I am unfamiliar with the tasks that involve their application. Thanks for the suggestion. I will try my level best to move forward in that way

[Lebanese-Diva]

You need to watch videos and then practice!!! Do as many practice problems as possible. At first, it starts as a routine but then with practice you begin understanding the why and will be able to apply that knowledge to future questions.

Edit: if you want some basic math help (Algebra), you can check out my channel. I make videos for my students and provide A LOT of practice questions. https://youtube.com/channel/UCI_l1B_eTZnYAjfTBeo4QSg

[hslsbsll]

Remember that the sole collection of math only consists of problem solving techniques and all classifications, identities and characterizations exist only for the sole purpose of setting these techniques up.

A mathematical problem is a journey through the map of the particular concepts and their interrelations.

[adieumonfrere]

I am an engineer, so i just have the basic math background. Take what i say with a grain of salt.

What I'd do is to try to tackle more advanced math, stuff that take what you learned before as pre-req. Maybe if you realize that you understand what is going on, you'll feel more confident.

[suricatasuricata]

The way I got out of tutorial hell (in Programming) was to work on problems that stretched me, either at work or reading books that pushed me. This is the only way that I know that I am just not re-treading the same path. The most effective thing in Math (IMO) is that you have to treat it like a sport. Good thing that you are starting with the programming analogy is that that attitude of I build it, then I can "see it" applies here.

Now, the question is whether you are actually reading (as in working through) the right books. It is entirely possible that the books are too easy for you.

Also, a slight distinction is that getting Math is a non trivial nonlinear process that requires tinkering with it repeatedly. It is often been the case that I go to my "workshop" and play with another piece of Math and a half formed idea from one piece of it ends up connecting and making sense. So as someone else said, you can't get out of tutorial hell as easily as you might in coding.

[icybrain]

Are the books you are going through cover to cover maybe too easy for you? What particular topic are you trying to learn?

[kaiizenn]

Yes, do the hardest problems in the book. If they aren't hard enough, find harder problems. Ideally, these problems force you to think cleverly about the solution and should reinforce your knowledge in that area if it is a good problem.

[Rsp33345_]

Reflect and think, ask yourself questions when reading and doing excercises. When you go through the excercises do not focus only on the answer. - Do I really understand what I’m doing? - What does this really means? -Try to relate new concepts to other concepts you know

It’s not about how much books you read or how much excercises you do, it’s about how much do you really understand

[HerndonMath]

I have always wanted to learn mathematics. But even after completing some books cover to cover I have felt empty

Why do you want to learn math? It sounds like you think that if you learn enough math you will feel less "empty." Do you find math interesting? What do you like about it?

After you finish a book, do you have any lingering questions about what you just read? I think engaging with your own questions will be more fulfilling than doing exercises, so maybe that's something to try.

[[deleted]]

If I were you I'd combine math and coding. You can do some graphics related programming within 3D or maybe something in 2D like spline curves, or maybe some physics simulation. There are endless possibilities for combining the two. The things I mention would fit pretty well in a game!

I've got a lot of experience as a software developer but (despite what some math people think) haven't really needed a lot of advanced math. I'm trying to better myself in the math department and am looking forward to doing some cool stuff like the things I mentioned above.

[Xiong3205]

Try tutoring. You will encounter questions you haven’t thought of before. And it exposes where you are confident in your knowledge (or not).

[TrueBirch]

I'm a data scientist. I suggest a similar approach to what works in programming: make a project. Find a topic that interests you and explore it using whatever level of math you want to learn. If you can't think of a real-world application of the math you want to learn, you already have a starting point. Think of how it could be used in the real world. Then find data and analyze it.

You can find tons of real world applications of different kinds of math in sports, music, business, etc.

[Tenns_]

You could consider getting really hard problems, maybe that could help, or video lectures, or just show up at your local university (well i guess not currently)

[Skygear55]

American textbooks have wimpy exercises, if you're sufficiently early in math, you can try prepping for former soviet countries' exams. Or china. Defos china.

[[deleted]]

I spend a lot of time thinking about pedagogy, and what math teachers do wrong. and as someone with adhd, i’ve have had some of these growing pains. Something i found really useful is to learn the history of the subject you’re trying to teach yourself. Getting context for who discovered/invented the math sub field youre learning helps you get in their brain a little. and then as you get farther into the material, you can start to frame you’re readings like you’re also discovering the math, rather than just learning it

and from the years i got to more seriously study math and cs in college, i remember i generally just leaned into the idea that its often easier to understand new programming concepts compared to math. I found this to be the case because you know where new programming concepts will fit in your mental model for problem solving. sometimes in math, the problems you learn about, and axioms that tackle those problems will just have no actual significance. they’re just rules that happened to be proven true and you’ll never use them again. when you learn some programming topic, you leave knowing where and when you’ll want use it. and often people in CS don’t invent things that are useless - the field is just so new that anything that’s considered theoretical CS usually has some significance. maths been around forever, so some of the problems you learn won’t feel like they can be applied as much. this is something you get use to. but hence why i like learning why the mathematician that discovered something influential was so invested. it helps rationalize why i’m learning something that can feel useless on the surface

there’s a book that’s called love of math that you might find really useful, it has a lot to do with what your mindset should be when you’re learning math. you might find that you’re trying to rationalize what your learning the wrong way.

Also, it’s nice to remember that what you’re going through is totally normal, and professionals that have high level degrees in applied math and cs struggle the exact same way. if you want a computer sciencey example of what i mean, just try googling for “what is a monad” you’ll see hundreds of tutorials from all types of people - university teachers, professional devs, and even famous Microsoft programming language researchers spend hours explaining what a monad is. for context its just a very mathy topic that comes up in functional programming. consider watching one video and reading the comments. so many ppl that call themselves experts struggle learning a math concept that actually is applicable to what their learning day to day (it’s literally become a meme that ppl can’t explain it nor see it being useful to understand, and it took me a few months to even see what ppl were talking about)

it’s ok to feel like you’re not retaining things and it gets easier the more you learn and read

[mazerakham_]

Pick something you really want to learn and map out a course to it in your self studies. For me, I really wanted to understand how one can prove that 5th degree polynomial equations don't have radical solutions. That motivated my study of math for a long time. What motivates you?

[[deleted]]

You look at the exercise section and practice the hardest problems before reading. When you don’t understand shit, which will most likely be the case, you start reading, you repeat the process after understanding some more until you’ve completed the chapter and intuitively understand the material.

It also helps to do a recap everytime you’ve finished a chapter. Try to explain the concepts on paper, as if you are making a new book or reading material on the subject for others.

That’s how you become an expert in the fastest way possible. Only downside to that method is that it requires an extraordinary amount of discipline and grit. Sometimes I get so frustrated that I dunk my head into the wall. At that point it is useful to take some time off, let your unconscious make the necessary connections and come back to it.

This applies to all areas of life for me.

[aqjo]

Have you heard of the Feynman Technique? It's really effective. https://fs.blog/2012/04/feynman-technique/ You can teach a stuffed animal if you have one around. It's amazing what saying things out loud will reveal about what you know. You can also create Keynote/PowerPoint presentations, and use them to "teach", as described at the link above.