How to effectively self-study mathematics

[Eppy]

What are some methods for studying textbooks like Euclid's Elements where there aren't a set of problems in each section and a list of answers in the back?

I've played around with things like copying a proof directly into a notebook, and I imagine once I have more theorems and postulates and things in my head, I can start trying to prove things on my own and then compare to Euclid's proof. I've worked through several proofs before individually, but since I'm working on doing things in order, I'd appreciate some input so my time is spent more efficiently and I get more out of the process.

A broader approach to my question may be: How are classes that focus on logic and proof taught? Specifically, studying of already-proven results.

 

[JasonRox]

The book Euclid's Elements wasn't meant to be taught. More of a thing to read and enjoy I guess.

You can't expect questions out of that because that's like expecting questions out of a journal. If you want questions regarding geometry, simply get a geometry text. There is plenty out there, and that is how it is done in schools, most or all the of the time.

I find the best way to learn mathematics is to discuss it with others, and ask yourselves questions. In contrary to what most people think, I believe mathematics is a social subject. Something that should be discussed amongst students.

Unfortunately, even as a student, I have not found a student willing to engage in conversations regarding mathematics... not even MATH MAJORS!

 

[Tide]

Besides proofs written up in your textbook or elsewhere, try as many exercises as you can. Also, don't underestimate the power of exploration and doing it on your own! For starters, you may want to try to prove things you're already familiar with and which may even seem obvious or trivial (such as proving that all the angles in an equilateral triangle are congruent). Challenge yourself with exercises like independenly deriving Heron's formula for the area of a triangle or coming up with 2 or 3 proofs of Pythagoras' theorem or finding the area of a regular pentagon - without using trigonometry. When you start to get the hang of it, pick out some of Euclid's theorems and prove them yourself before reading how he did them. Postulate your own theorem's and prove - or disprove them!

 

[waht]

I find that copying proofs straight from the book works good for me, even if I don't understand what's it mean.

 

[fourier jr]

you'd probably learn the stuff better if you only looked at the definitions & the statements of the theorems. it would be better still if you didn't even look at the statements of the theorems, but could make them up, or discover them yourself.

one reason i like willard's topology book so much is because i can see a pattern emerging when a new kind of space is defined. the first theorem he proves after defining something concerns subspaces (under what conditions does a subspace of a certain kind of space have the same property), continuous images, products, and sometimes quotients. you might have to add in certain conditions like for the image of a certain kind of space to fit some definition the continuous image has to be open also, etc etc. it's like that for every kind of space (compact, paracompact, hausdorff, regular, etc etc), so you get to the point where you know what kind of theorems are coming up (re: subspaces, continuous images, products), and you can just focus on the more complicated theorems. if only every textbook could be written that well...

 

[NewScientist]

I loved "Fundamentals of Number Theory" it began very simply and presumed you knew next to nothing and built up a good picture of a complex world

 

[Eppy]

Thank you for the responses, even these few have been very helpful.

I didn't make it clear, but it's worth saying that I have already taken geometry and am comfortable with my ability to do things like plug and chug the equation for the volume of a sphere, so that's not an issue. I'm interested more in geometry as a deductive science. It was part of the class, but there was not enough of it for me.

 

[amcavoy]

I find that copying proofs straight from the book works good for me, even if I don't understand what's it mean.

Looking at as many as you can works well. I haven't had analysis yet, and I don't know many proofs, but I certainly feel more confident after just observing others prove certain theorems.

 

[Cexy]

In my experience, the best way to teach yourself mathematics is to do it yourself. A good way to approach learning something is as follows:

1) Look at the theorem.

2) Really look at the theorem, and try to figure out what everything in it means. How do the different parts fit together?

3) Try the theorem on a simple example (e.g. with small integers, or in a low number of dimensions) and try to find counterexamples. Convince yourself that it really is true.

4) Try to prove the theorem generally. Have a good go at this, until you really get stuck.

5) Look up the proof of the theorem given in the text, up until you reach something that you hadn't thought of yet. Maybe this extra hint is what you need to prove the theorem!

6) Try to prove the theorem using your new knowledge. Keep going until you get stuck again.

7) Go to step 5.

Good luck! :)

 

[Deep water]

I believe different approach should be taken by different people to learn Math as we all have different ideas. As I studied biography of great mathematicians I found their learning approach are different. Gauss, Erdos, Jacobi, Abel, Galois, Ramanujan, Neumann, Riemann, Euler all are great but they learned Math in way different from one another. It becomes visible when we read their comments, anecdotes etc.

 

[snipez90]

Well just because they developed the successful ability to do mathematics doesn't mean that different approaches will each lead to success. For instance I strongly disagree with what someone said earlier (and judging by the date of the original thread, WAY earlier) about the helpfulness of copying proofs without necessarily understanding them. There are good and bad approaches in math, and this is no different in any other subject. I mean sure you can go throw crap on a famous painting and call it abstract art, but many people will disagree.

 

[Deep water]

I just wanted to say that different people learn in different way.

 

[Deep water]

One of my friend can not understand class. Whatever he does is reflection of his self study (All of his teachers say he is the most brilliant student they have ever taught). He learned geometry from a book similar to Euclid's Element by himself. He says about self study, "to wander in the path of knowledge with no bound". He says to self study geometry one should draw diagram as running through proof.

 

[indialpc]

I realize this post was a while ago, but I don't think any of the people who replied have studied Euclid's Elements before...

I understand that you're doing it to learn logic more so than to learn the actual math.
Studying Euclid is really difficult because it's not something that people normally study. It's not something you know how to study at first.
Generally it helps to learn the Propositions with someone. You will be at a chalk board, white board, or something of the sort. Have the other person read the annunciation, then read the steps to you. It's helpful to clearly write what information you are given, then write what you are trying to prove. After you have these down, your reader will read the steps to you as you draw what is being described. Write and number each step as you do them, even if they seem obvious. The reader should watch and make sure you don't miss anything in terms of the drawing and of the written steps. Only continue when you truly understand what Euclid is saying, and what has been proven and why. You'll know when you understand it; you'll definitely have an "ah-ha" moment!
Do this until the proposition is finished and you understand it entirely. After that, erase it, then try to present the proposition to your reader without looking at the book. If you can present it, then you know that you understand it and can prove it. This is definitely the best way to learn Euclid's Elements; I've honestly never heard of another way that works well.
Studying Euclid is really useful for seeing how logic works and being able to follow or carry an argument through. Most people think it's just about math.