Overcoming Math Education Challenges to Pursue Higher Studies

[Vanush]

Hi

I am a third year electrical engineering undergraduate, who would also like to engage in higher study of mathematics. I am in awe of mathematics, having done linear algebra and calculus as part of engineering, plus whatever is necessary for elec engineering (Laplace, DE's, Fourier, complex numbers, LTI systems).

However, I have a problem that is stopping me from taking higher level courses, a problem that I ascribe to the very poor junior and secondary mathematical education in Australia. I freeze whenever I see problem along the lines of "Prove ...". I am sufficient at applying theorems to practical problems, but have difficulty with highly theoretical problems.

How do maths students deal with proofs, definitions and theorems? How do they get it into their head, as well as the 'method' for solving practical problems?

Do you think these books will help me understand on how to master the underlying concepts of mathematics?

https://www.amazon.com/dp/0471135712/?tag=pfamazon01-20
https://www.amazon.com/dp/0691023565/?tag=pfamazon01-20

Can anyone think of other resources to help me think how mathematicians think?

 

[The_Duck]

I don't know anything about those books, but many math departments have classes that effectively serve as introductions to the realm of proof-based math courses. Ask someone in your math department about whether there is a good course to take to introduce you to proofs.

You might find it helpful to think about proofs as just explaining why something is so. It's not supposed to come off the top of your head, you're supposed to play around with things, perhaps for quite some time, until you see why the statement is true. For example, "Prove that the differences between successive squares 0, 1, 4, 9, 16, 25... and so on are the odd numbers 1, 3, 5, 7, 9... and so on." A problem like this is just asking you to play around with the math until you see why this must always be true, and then write that up.

 

[snipez90]

The first book by Zeitz is very difficult. It's not so much a book for learning proofs as it is one that develops exceptional problem solving abilities. For the most part, you probably shouldn't seriously work through this book unless you're already fairly well-versed with basic high school math competitions. However, there is a chapter on calculus that is more or less a brief intro to real analysis, where you learn rigorous proofs behind many of the important concepts in calculus. See if you can borrow this text if this sounds interesting.

The second book is a classic but it also doesn't strictly focus on proofs. It teaches fairly standard problem solving strategies that form the basic toolkit for attacking any type of mathematical problem.

I think probably the best idea is to simply take the first course that most math majors take at your school which involves proofs. Perhaps it is specifically a course introducing one to proofs, but that's not always necessarily the case. The basic techniques for writing a proof are not hard. You simply start with the hypothesis and the definitions and try to tie these to the conclusion, the statement you're trying to prove. Often times, not knowing what the good first step should be is not deficiency in proof-writing knowledge, but rather a lack in problem solving abilities. This is why it's often simpler just to take a higher level math course, learn the methods and techniques by actually reading the proofs, and hone problem solving abilities by working on as many problems as possible.

 

[desertls1]

The statements above are good advice. My wife has an undergrad B.S. in Mechanical Engineering, and now she wants to get a PhD in pure math. So she started by taking a rudimentary introduction to proofs called "principles of mathematics," which serves as the backdrop to taking higher level classes such as real analysis and abstract algebra.

It introduced things like: Logic and proofs, elements of set theory, relations and functions, Cardinality, groups and their quotients, and the field of real numbers. You'll also get the feel of writing proofs using proper notation.

Without this intro level course, you will struggle with proofs in my opinion. My wife was able to do very well in the class with a math background that is quite similar to yours. Once you get that basic class down, you'll be able to self-study.

 

[AlephZero]

If you want to learn "how mathematicians think", you could do worse than reading some books by mathematicians who not only do math, but also write well.

Three to get you started:

G.H.Hardy, "A mathematician's apology" (short, and easy reading)

Hilbert, "Fundamentals of Geometry". Assuming you have a reasonable "practical" grasp of elementary geometry, this is Hilbert's shot at plugging the holes in the logic of Euclid's Elements.

Bertrand Russell, "The Principles of Mathmatics". (NOT Russell and Whitehead's "Principia Mathematica"!) This one will keep teaching you things for the next 30 or 40 years, if you re-read it occasionally.

No apologies for the fact that the Hardy is more than 60 years old, and the other two are just over 100. Unlike science, once maths has been done, it stays done for ever.

 

[Vanush]

The statements above are good advice. My wife has an undergrad B.S. in Mechanical Engineering, and now she wants to get a PhD in pure math. So she started by taking a rudimentary introduction to proofs called "principles of mathematics," which serves as the backdrop to taking higher level classes such as real analysis and abstract algebra.

It introduced things like: Logic and proofs, elements of set theory, relations and functions, Cardinality, groups and their quotients, and the field of real numbers. You'll also get the feel of writing proofs using proper notation.

Without this intro level course, you will struggle with proofs in my opinion. My wife was able to do very well in the class with a math background that is quite similar to yours. Once you get that basic class down, you'll be able to self-study.

Is this the same as 'Discrete Mathematics'?

http://www.maths.usyd.edu.au/u/UG/JM/MATH1004/Resources.html

 

[logarithmic]

Is this the same as 'Discrete Mathematics'?

http://www.maths.usyd.edu.au/u/UG/JM/MATH1004/Resources.html

Yes, discrete maths is where mathematical logic and proofs are introduced. Although since you're from Sydney, how did you manage to get through high school without some experience in proofs, because there are usually a few questions requiring you make a proof in the HSC.

 

[Vanush]

Not really in 2unit mathematics, more so in 3unit and 4unit maths. I should've taken them, but my ego wasnt really good!

 

[╔(σ_σ)╝]

I am also a third year EE student with similar interest what helped me was I took a half semester course on discrete math which introduced me to proofs then I immediately went ahead and took a real analysis class. Doing this I was able to become a lot more competent at read and writing proofs.

The best way I think of learning mathematics is practise it. Pick up or take a analysis book and start reading and doing problems.

I hope what I said is useful to you. :-)

 

[desertls1]

Is this the same as 'Discrete Mathematics'?

http://www.maths.usyd.edu.au/u/UG/JM/MATH1004/Resources.html

No, not based on the syllabus for the link you gave. That hardly scratches the surface of doing real proofs. The course you linked uses some methods to apply to real numbers, etc, but a true class covering an intro to proofs will not have equations to solve.

I'd stick to something more abstract for your first class

 

[jeffasinger]

Just wanted to add that while it might be just one step ahead of where you're at right now, you might want to look at Polya's How to Solve It.

The book tries to teach you how to make conjectures about a problem, and then gives you general suggestions (that are pretty generally applicable) on routes to a proof of your conjecture.

 

[mathwonk]

replace the word "prove" by the words "explain why, with evidence".

 

[Shackleford]

I'm currently a physics major/math minor. I'm swapping those starting next semester. I'll eb taking Abstract Algebra and Intermediate Analysis next Fall. I have virtually zero experience with proofs. I haven't decided what exactly I should do to prepare. Would MATH 1312 prepare me for those two classes? I might buy one of the books mentioned in this thread, too.

http://www.uh.edu/academics/catalog/colleges/nsm/courses/math/

Oh, I'm just on the cusp of my junior/senior year.