What is the best way to develop Mathematical intuition?

Question

I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the intuition to do so.

Answer 1

I can say that a big part of mathematical intuition comes from experience. That being said there is no "best way". I like to think about experience like this:

You're stuck in thick, dense jungle like the amazon rain forest. You have a machete, and you're chopping down plants struggling to find a way out. You walk around headless, knowing not even of a path to the nearest village.

However now say you are flying in a helicopter with a reasonable view around. Now not only can you see where you would be in the jungle, but can also see where the nearest village is. You can see the terrain, map out a path, avoid crossing big streams, etc.

This is the analogy: In the first situation you are stuck with how to move, in the next one with a lot of experience you can see many connections and the way out.

Now we come to your next problem. You say that when going through exercises they ask for proofs. What I would recommend if you have never ever seen a mathematical proof is to go through an article like this one.

Of course it is not possible to grasp these techniques at once. You should try out a few suggested exercises, look at how things are done first, look at how other people do things. Why do they think like this? Slowly after a while, your mind will get used to it.

I am willing to edit my answer and improve on it if you provide more detail on the specifics of your difficulties.

Answer 2

To learn how to prove something you want to do something like this:

First pick something at an adequate level matching your current level of skill.

Then gather a bunch of exercises that ask for proofs together with answers. But don't look at the answers. Sit down and try to solve them. If you get stuck, look at the first line of the answer and try from there. If that still doesn't get you anywhere, look at the next line, and so on.

Repeat until you have gained some confidence. This will teach you how to prove some of the things you come across.

I also recommend reading "Thinking Mathematically". This will help you change your thinking so that you will get stuck less often in general.

Hope this helps.

Answer 3

"I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the intuition to do so."

Honestly, I think you should not build a theory about this. You simply don't know how to solve certain problems. That's it. Don't interpret it that there is something fundamentally wrong with you. Don't conclude that you don't have intuition. Not all exercises are that simple.

Also, intuition comes from experience. The best way is experience. What kind of experience? Well, it depends on what you want. Do you want to be able to solve problems in your book? To become a mathematician? To become a math teacher?

Whatever you do, DON'T read http://zimmer.csufresno.edu/~larryc/proofs/proofs.html suggested in another post. This will do more harm than good. I read this very article recently and I was speechless. In my humble opinion, this guy clearly contributed to ruining math education in this country, even though he probably meant no harm. You don't need to read anything special to understand proofs. Unfortunately, many people in this country want to sell you books about proofs and so they present it as a special skill. Mathematics is proofs. If you have a course without proofs and it is called mathematics (and it is beyond high school), you are paying your money for nothing, period. Read Paul Lockhart's article on education, A Mathematician's Lament: http://www.maa.org/devlin/LockhartsLament.pdf

If you still don't understand why I don't like this article by Larry Cusick, let me quote from the introduction: (http://zimmer.csufresno.edu/~larryc/proofs/proofs.introduction.html)

"The basic structure of a proof is easy: it is just a series of statements, each one being either An assumption or A conclusion, clearly following from an assumption or previously proved result."

---Who needs to be taught the basic structure of a proof? Most people who don't understand proofs will only get more confused. Unless you study math logic, you don't need to analyze the structure of a proof. Furthermore, in many cases it is easier to understand a proof than to dissect it into assumptions and conclusions. It's like a second grader who knows that 5*7=35, but doesn't know "what is the product if the multipliers are 5 and 7".

Continuing with the article. "And that is all. Occasionally there will be the clarifying remark, but this is just for the reader and has no logical bearing on the structure of the proof. " ---Well, this piece of information is not very useful. It is probably more confusing than the actual "clarifying remark" it is warning you about. Anyway, I am not really sure, what it means. It says "...just for the reader". Does it suggest that the proof itself is not for the reader?

"A well written proof will flow. That is, the reader should feel as though they are being taken on a ride that takes them directly and inevitably to the desired conclusion without any distractions about irrelevant details. Each step should be clear or at least clearly justified. A good proof is easy to follow." ---This is almost a joke. I particularly like "...without any distractions about irrelevant details". Forgive me my sarcasm.

"When you are finished with a proof, apply the above simple test to every sentence: is it clearly (a) an assumption or (b) a justified conclusion? If the sentence fails the test, maybe it doesn't belong in the proof." ---No, when you are finished with a proof, just see if it makes sense. There is no need to apply any tests. This is just another trick to make people afraid of proofs. Furthermore, this "test" is wrong. There may be justified conclusions that don't belong to the proof. On the other hand, a statement may belong to the proof, but it may be unjustified, then it doesn't pass this test.

Answer 4

Seems like you're at a basic level. I would recommend reading carefully the proofs from some book you like. You need to get used to the language, to the ordering of the arguments, to notation... You may need support from your teacher, and that's OK. Then moving to the exercises should be easier. But you should try really hard -- thats hard work, and takes some time. If you fail to do some exercise, do your best to learn from it, and ask yourself why weren't you able to do it, ask if the solution is "natural". I believe that, as you get used to the arguments, proving theorems by yourself will come naturally.

Good luck :)