Understanding the intuition behind math

Question

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?

Answer 1

I like this question a lot and I think that it's an important one. So here goes a (necessarily incomplete) attempt at answering such a broad and personal question.

First, "motivation" and "understanding for the essence" can mean very different things. There is of course physical motivation and intuition, and that probably applies most immediately to the Calculus III course that you are talking about. E.g. for the concept of derivatives of vector valued functions, you can think of the vector valued function of time that gives the position of an object as a vector. Of course, its derivative with respect to time will be the velocity (also a vector, since it described the speed and the direction of the movement) and the second derivative will be the acceleration. A good course in such an applicable subject will not just ask question like "compute the derivative of such and such a function", but will actually confront the student with real life examples.

But there is also intuition for less physical and more platonic concepts, such as that of a group, or of a prime number. Again, examples help. Also, you should always try to ask yourself the question "Could I have invented this?". If you see a new definition, ask yourself "What concrete problem might have prompted someone to define such a thing?". If you see a new result, ask yourself "Why was this to be expected, why would it be at least a reasonable conjecture?". Then try to convert your intuition into a proof. When you see a proof, ask yourself "Why is this a natural approach to try? Could I have proven this?". I agree with you that knowing the historical development can be very helpful in this and you should invest time in researching it.

I would like to contradict you in your assertion that intuition, motivation and historical context are black magic secrets that mathematicians acquire and then keep to themselves. It is true of some books and some teachers. So, you just have to find the right books. For that, you could ask for a specific recommendation here, including the area you want to learn and the books you have looked at, together with the reason you found them deficient. Of course, you can also ask specific "intuition" type questions.

To learn to appreciate mathematics, it is important to think about mathematics in your "spare time". Go out into nature and think about what your lecturer just told you in the last lecture. Or just think about whatever you find interesting. Then come back home with specific questions and look them up or ask them here.

Finally, something that I preach my students all the time is that they should develop a critical approach to what they are taught: if I give them a definition, they should try to come up with as many examples as possible. If a state a theorem of the type "A implies B", they should go home and find an example that "B does not necessarily imply A". If they do find such an example, they should ask themselves what additional hypotheses they need to impose to get the converse. If they don't, they should come back to me and ask me "but you haven't told us the whole story. What about the converse?".

In short, don't expect your lecturers to tell you everything you need to know. You should expect to have to think, to investigate yourself, to ask questions, and, above all, to think about mathematics because you can't help it, rather than because you are told to. This is not something, most people are born with, it's something that you have to cultivate.

Answer 2

I can offer a few things that help me gain an intuition for new material, in addition to the more mathematically-related advice given by others:

Talk about the mathematics, either with other students, or with your professor during office hours. I've noticed that while lots of times lectures can be dry and fast-paced, visiting after hours and asking about the concepts underlying the material can be very rewarding. Talking with other students in the class can also expose the meaning as long as you find a group of other students who are also interested in learning the "why" behind the "how." Ask each other questions ("what do you get out of this theorem?") and challenge the material.

Find books in the subject you are learning that are a bridge between the typical undergraduate text book and the more advanced (graduate level) books and read them as a supplement to the course. For instance, Tom Apostol's Calculus (Vol. 1 & 2) is a great companion book for the calculus series, and will answer many of the mathematical "why" questions that the typical undergrad books leave out (plus, you'll be a leg up when you get to advanced calculus!).

Lastly, read lots of mathematics books that aren't textbooks. I started learning about mathematics through books like Euclid's Window, Fermat's Enigma, and Prime Obsession. Journey Through Genius was also a good one, as it runs through the motivation behind several important results in mathematics. Since then I have switched to reading biographies and autobiographies to get to know the mathematicians who have come before me, how they thought, how they lived, and how their lives are all intertwined (some good ones: The Man who Love Only Numbers (Paul Erdos), The Man who Knew Infinity (Ramanujan), I Want to be a Mathematician (Paul Halmos)). In my experience, learning this side of mathematics has been invaluable for my understanding of the discipline in general.

Answer 3

Great teachers, great books, and (more recently) some great bloggers like Terry Tao and Tim Gowers. There are, in fact, some books that do a great job at giving context, rationale and intuition; Silverman and Tate's "Rational Points on Elliptic Curves" is one example, and some of it should be accessible to a motivated high-schooler. You should definitely read "Mathematics: A Very Short Introduction" by Gowers, and maybe browse the Princeton Companion to Mathematics from time to time.

Also, study physics.

Answer 4

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.

Answer 5

I just read this paper: On proof and progress in mathematics, it could help you.