Developing Mathematical maturity

[Klungo]

I have a question about development of mathematical matury.

First off, I'm a high school senior with almost all my college general education credits due to dual enrollment in Florida. I have been doing independed study since 10th grade and have self-taught everything (computation) from pre-calculus to differential equations which I completed last month. I have credits for calculus 1 and 2 (by AP calculus BC), calculus 3 and am currently taking differential equations by dual enrollment.

As I noticed in these forums, AP calculus, and the calculus 3 and differential equations at college lack any treatment of proof. However, in Florida public universities, usually all the proofs affiliated with these courses at presented as a core course to be taken after the introductory calculus sequence. (Similarly, I'm going to exempt the calculus sequence which shouldn't do much harm since neither regular nor honors calculus presents any proofs here).

So recently, I have been trying to promote my maturity in mathematics. I have gone through proofs of theorems with field axioms and order axioms for real numbers with no trouble. Worked with natural numbers and learned how to use induction. I've done a bit of set theory algebra proofs including union, intersection, and the supremum and infimum of bounded sets. And worked a bit with the proofs using the e-o definition of limits to prove their properties.

Thing is, I usually don't prove the theorems by myself. I usually have to take a peak and the hint will usually get me a bit of the proof done. Once I learn the proof, though, I understand every part of it. (This goes mainly to properties of limits using the e-o definition and supremum/infimum). This is the right procedure to developing mathematical maturity right?

 

[Vanadium 50]

Thing is, I usually don't prove the theorems by myself. I usually have to take a peak and the hint will usually get me a bit of the proof done. Once I learn the proof, though, I understand every part of it. (This goes mainly to properties of limits using the e-o definition and supremum/infimum). This is the right procedure to developing mathematical maturity right?

Nope. That's like trying to improve upper-body strength by watching other people lifting weights. You've got to get it by yourself, and experience struggling with a problem for an extended period of time, trying different things.

 

[Poopsilon]

I'm not sure exactly what you mean by theorems. Generally speaking, I don't think someone working through, say, an advanced calculus text, is expected to be able to prove many of those theorems which are given in the book for the reader to use. What the reader IS expected to prove are the problems at the end of each chapter, which generally don't have solutions in the back of the book.

If you're talking about just, say, a regular calculus text that happens to prove its theorems but where all its problems are computational in nature. Then I think you should use the book for its intended purpose, which is to improve integration skills or whatever.

 

[20Tauri]

Honestly, I think you're already pretty mathematically mature for a high school senior. I'm studying math in college, and I didn't start paying attention to proofs until I took linear algebra. I still have to peek at hints for my proofs sometimes.

If you want to practice writing proofs, taking another look at number theory might be worth your while. There are some fun things to prove if you learn some modular arithmetic, and I am given to understand that number theory is sort of a nice preview of abstract algebra in some ways.

 

[micromass]

Proof writing is a long process that takes some time to develop.

Ideally, here's how a beginner should approach a math book:
1) Read the proofs in the book and be sure to understand every aspect of it.
2) Close the book and try to come up with the proof yourself. (try not to peek unless you really don't find it).
3) Try to make exercises which are pretty close to the proofs in the book (i.e. use the same ideas).
4) Try to make more difficult proofs which challenge your conceptual understanding.

If you're able to read a math book like that, then you're already doing alright.

 

[Klungo]

Whoops. I forgot to mention that by text, i meant Apostol's calculus vol 1. And by theorems, I meant the ones that don't show up as exercises at the end of the text. I usually don't need help on the exercises (that is, not all solutions/proofs appear at the end of the book anyways).