TITLE: Understanding the intuition behind math
QUESTION [61 upvotes]: I'm currently a Calculus III student. I enjoy math a lot, but only when I understand its beauty and meaning. However, so many times I have no idea what it is I am learning about, althought I am still able to solve problems pertaining to those topics. I'm memorizing the sounds this language makes, but I dont understand them. I think a big reason why most children and teenagers really dread math over any other subject is because the only thing that is taught to them is equations and numbers, and to them is not being explained its significance. For if people really knew what it all meant, then they'd realize it's probably the most important think they should ever study. Even in college, at this relatively high level of math, they still do not preach its meaning, but rather scare you into cramming pages after pages of material so you can pass the next exam. When you have passed the exam, then it is safe for you to forget the material you just absorbed. This is the reason I often find myself bored of studying my current topics. For some things, I see the intuition behind it, and those are the things that keep me interested in calculus, but its often so hard to come up with a good meaning of what I'm learning all by myself.
It took mankind hundreds and thousands of years to come to where we are with math, so I dont expect to understand its true meaning in an hour, but I'd really like to. The school curriculum, here in America at least, doesnt teach meaning or utility. This is the reason so many youngsters always ask "when will we ever need to use this stuff?"-what a naive question this I learned to be. So I guess what I'm trying to say is that I've grown bored of this material, as we often cram 2 full sections of a topic in one day. Its impossible to keep up with its meaning, but if I am to survive this course along with more advanced courses to come, I must be able to understand its meaning. My textbook doesn't help me with this issue though- no math book can really teach intuition, but they dont even attempt to. They barely even go into the history of the current topic, and thats one of my favorite parts-I like to read about how a normal man came up with this theory that revolutionized the world. It makes math feel human to me, and that I too can understand it like men before me. 
Most of the question answerers on this site are mathematicians, if not at least one in training. This means you have come to where you are by understanding what you have learned in the past. How have you done this? How have you been able to connect all the pieces? What are some good resources that will help in what I hope to do? How do I stop being a robot, and actually connect with what I'm learning?

REPLY [6 votes]: I'm in my third year of a maths bachelor (in Italy), and the situation is the same over here. We are now dealing with topological vector fields and advanced measure theory and I don't have a clue of what's going on, but this is not a new issue. 
In my first year I found very helpful betterexplained.com. Unfortunately, it is quite deficient on advanced topics, but fortunately, 3blue1brown came to life recently to fill this gap, and it is a source of astounding material.
Ultimately, I feel like the source of the issue is that there's too much teaching on too many topics.
I am huge believer that self-learning is the most effective way of learning. However, there's just no space for it in modern curricula. After 4-6 hours of morning lesson, there's just no way I will be sharp enough in the afternoon to come up with my intuitions, my solutions, my understanding. I will just look at how many notes I took in the morning, be scared, and start racing through it understanding as little as needed to understand the following topic. It is really sad to be said, but that's the way it is. 
I would love to be faced with a problem and be told to seek the solution (which is what I do in my free time, on topics unrelated to university classes). At university level, solutions are likely to be very difficult, and it's unlikely that anyone will really find it, but that does not matter. If you have been struggling with a problem, and you really know what you are trying to address, what caused you trouble, ecc, you will welcome the lesson in a different way. And it's not just a matter of "I'm more interested in it now that I have struggled with it", it is also a matter of efficacy. When I see the solution to a problem I really have worked on for a week, it will sink in me.
But building this way of doing in a math curriculum would necessarily imply that way less topics would be covered, because each of them would take more time, and there would also be less lesson hours. Would that be bad? I don't really think that you are a mathematician only if you know what a Hilbert space is, or what Riesz's theorem states, or what Baire's lemma says. I believe math is a matter of how you face a problem, not a set of notions. Anyway, I really don't see the point of doing moving on to another topic if the previous one hasn't thoroughly been understood. After all, even from a working perspective, knowing advanced analysis is unlikely to benefit you (unless you do research), whereas having true intuitions about basic calculus topics could be really helpful.
We are not in the 1800s anymore, there are a lot of other ways to gather knowledge other than university. This is not to say that we don't need universities anymore, but that it is a huge opportunity! It means we can stop giving students notions, thinking that otherwise it will be difficult for them to obtain them, and start cultivating the math mindset in them, which will allow to go through any theory/notion in the future, whenever it will be needed.
This does also mean that a student cannot have to go through 3 courses simultaneously. How is one supposed to deeply get into a subject, if he has to juggle through topological vector fields, probability, singular homology and linear programming? There's just no way: as soon as one stops to really consider a subject for a couple of days, he's gonna be left so much behind on everything else that he's gonna be crying!
All of this to say: we need to teach less stuff. There's no other way as far as I can see. Teach less, let students do more.