What is the most efficient way to gain mathematical insight?

[k4ff3]

Hi everyone.

Little bit of my background:
Im currently studying mathematics in the third year at university (or maybe you call it college in your country), and I have two more years left. I feel that I have affinity for physics and maths, and it is something I really want to master. My grades in these subjects is good - but my way (the easiest way ... ) to obtain good grades has been to practice problem solving techniques, rather than obtaining the deep mathematical insight I am wanting. As of now it's difficult for me to apply my current mathematical skills to new subjects independently, meaning that if I am facing problems i have not seen before, there is a great possibility that i will manage not solve it. I want that kind of insight that allows me to apply my math skills to any problem faced!

My question is therefore: How do I obtain this kind of mathematical knowledge? Do you have any books to recommend? How should i use the books? How should i work with the topics? I am interested in all topics, and its up to you to recommend in what order those topics should be studied. I am willing to lay down some hard work.. Both general tips, and specific study-tips (like write down the teorems on a sheet as you read) is of great help.

Any reflections on this matter is truly appreciated.

Thank you.

-k4ff3

 

[twofish-quant]

Don't worry about efficient. Improving your problem solving skills is a highly inefficient process. There are some tricks to help you, but a lot of it involves just getting use to staring at a problem for long periods of time. My understanding of the cognitive science research on grandmaster chess players is that they get to where they are by learning large numbers of patterns, and this seems to make sense with what I've seen with mathematicians and physicists. A good physicist can solve a problem because it's similar to one that they've seen before, but they've seen thousands of problems.

The other thing that helps is to not think in terms of having or not having mathematical insight, but in terms of improving skills. If you study problem solving for several years, you will *still* be stumped, but you'll be stumped by a much harder set of problems.

There are three books that I've found useful

"Problem Solving Through Problems. Loren Larson"
"Problem-Solving Strategies (Problem Books in Mathematics)" by Arthur Engel
"The Art and Craft of Problem Solving" by Paul Zeitz

 

[hellbike]

I always though that being mathematician is about finding new patterns, and not master usage of those already invented. This is what engineering is about.

 

[Phivar]

Whenever you learn something new or when you start a new "chapter" in your book, take some time to focus on understanding the core principles of the formulas/theorems etc.
When you have fully understood the principles behind a formula you will much easier know when to use it, and be able to apply it in unfamiliar problems.

That is what I tend to do, and it has worked quite well. Of course you can't expect to understand everything in for example a very complex formula, but try to get an insight in the basic principles and why numbers or symbols are placed where they are.

 

[k4ff3]

Whenever you learn something new or when you start a new "chapter" in your book, take some time to focus on understanding the core principles of the formulas/theorems etc.
When you have fully understood the principles behind a formula you will much easier know when to use it, and be able to apply it in unfamiliar problems.

That is what I tend to do, and it has worked quite well. Of course you can't expect to understand everything in for example a very complex formula, but try to get an insight in the basic principles and why numbers or symbols are placed where they are.

Good point, but what is the most efficient way to understand these theorems and formulas? (In your opinion of course, it's impossible to state something truly general)

 

[Phivar]

Good point, but what is the most efficient way to understand these theorems and formulas? (In your opinion of course, it's impossible to state something truly general)

I often just stare at stuff until I understand it, if that's a way to put it. Even though math and physics can be kinda abstract sometimes, I just try to imagine the concepts and put them into situations. Take your time, look at examples in the book and do not be in a hurry!

 

[eof]

I always though that being mathematician is about finding new patterns, and not master usage of those already invented. This is what engineering is about.

It's more about seeing old patterns in a new context. I've heard the phrase "there's a finite number of good ideas" quite a few times. Even the Fields medal is often given to someone for building a bridge between two areas of math, so that old and well understood concepts from one field can be used in another field. Another way of saying that is that you notice that two things are essentially the same.

 

[eof]

I often just stare at stuff until I understand it, if that's a way to put it. Even though math and physics can be kinda abstract sometimes, I just try to imagine the concepts and put them into situations. Take your time, look at examples in the book and do not be in a hurry!

Go tell that to a first year grad student and he'll laugh. :)

 

[Phivar]

Go tell that to a first year grad student and he'll laugh. :)

Hehe, that may be true since I'm still in high school :P So far it has worked well, but I know I'll probably have to do a lot more work in higher education.

 

[Sankaku]

Good point, but what is the most efficient way to understand these theorems and formulas? (In your opinion of course, it's impossible to state something truly general)

Personally, I try to get as many different sources as I can. I find multiple textbooks. I try to find and watch opencourseware lectures (if they exist for the topic). Quite often the second book or video will suddenly make sense and then it will seem obvious...

Sometimes I try to read ahead in the subject, even if I don't totally understand it. The material I am working on now has more depth if I know where it will be used later.

 

[Phivar]

A good teach can do wonders. I hope you got a good one ;)

 

[k4ff3]

Thank you all for responding.

I would rather not rely on a teacher or something external. I am searching for the ultimate way of self studying. Interested in tips on especially good books, good working habit techniques, and the best way to approach mathematics (and physics) when doing both problems and reading. The more spesific tips, the better! :)

Any more opinions on the matter is greatly appreciated (=

 

[eof]

Thank you all for responding.

I would rather not rely on a teacher or something external. I am searching for the ultimate way of self studying. Interested in tips on especially good books, good working habit techniques, and the best way to approach mathematics (and physics) when doing both problems and reading. The more spesific tips, the better! :)

Any more opinions on the matter is greatly appreciated (=

Here is kind of how I usually self-study something. Try to prove every theorem without reading the proof first. If you can't get anywhere in 15 minutes, then look at how the proof starts. Rinse and repeat. Make notes of ideas you had, because they can often be as valuable as the stuff in your text.

When done try to summarize the proof down into steps that give you enough information, so that you can fill out the details. After you've done this, write down the theorem and this summary of the proof in your own notes. Put your notes aside and try prove the theorem without looking at your notes. If you can't do it, look at your notes and try over with a blank paper.

When writing the notes, the idea is to cram as much stuff into one page on your notes as you can. The next day before you go forward, skim over your notes for the past few days trying to fill in the proofs for stuff that you don't find completely trivial. The point of having short notes is that it's faster to skim over what you've done. When something that you previously thought was a step worth mentioning in the proof starts to become completely obvious, delete the step from your written notes (use LaTeX) and or the whole lemma or theorem.

This is essentially how I study. If I don't do something similar, I will not remember anything I've read a few weeks later. It takes time, but you'll remember it. I keep looking at my notes constantly and often copy paste important theorems and proofs into a flash cards program called Mnemosyne (you can google it online).

It takes time, but I've come to the conclusion that it's better to learn the foundational material really well than trying to expose yourself to lot of stuff. When you do research on a problem, it's hard to find a substitute for a lack of good foundations, but the lack of having seen some stuff that might help you is easily mitigated by talking about your problem to others.

 

[k4ff3]

Thank you for a great answer! How long have you done this, and how well has it worked? I am also wondering in what year of study you are in?

 

[eof]

Thank you for a great answer! How long have you done this, and how well has it worked? I am also wondering in what year of study you are in?

I've been doing it for a few years and I'm a 1st year Ph.D. student in math at an Ivy school, so it has worked pretty well if grad school admissions is any indication.

As a grad student the course load is too hard, so it seems that I can't continue doing this the way I used to, but in undergrad it was doable and I was able to keep up with the pace. However, I'm probably going to go over the stuff we covered this fall during Christmas break by using the same method and similarly next summer for stuff we cover next spring.

 

[k4ff3]

That's impressive. I wish you all the best!

How much work have you laid down in your studies on average? It seems that your die hard system must come with the price of no spare-time. Is that true?

 

[eof]

Short answer: lots. Meaning I go to the gym or for a run each morning for an hour together with a friend of mine, spend one evening a week with friends, watch maybe a movie or two per week before bed, but otherwise I spend pretty much all my time awake studying.

 

[k4ff3]

And for how long have you kept up the tempo?