How to Understand Math: College Student Seeking Guidance

[hapyro]

Hey,

I'm a sophomore in college and I'm on a current path which will lead me to a double major in mathematics and economics. Currently, I'm what people in my school refer to as "great in math." All this means is I get As in the classes I take (finished the calculus series, now taking linear algebra, differential equations, and discrete math). The thing is, although I receive good marks, I still feel like something's missing. Right now, my guess is that I don't really understand the math, I just do it. I'm not sure if anyone understands what I'm talking about, but if so, I'd really appreciate any input in actually understanding math.

Thanks.

 

[Klockan3]

Try to work more with concepts than formulas. When you get a new problem think "How would I go about solving/proving this?", then try for a while without any other help than yourself.

 

[epenguin]

I don't really understand the math, I just do it.

Then maybe you are a genius.

 

[chiro]

Hey,

I'm a sophomore in college and I'm on a current path which will lead me to a double major in mathematics and economics. Currently, I'm what people in my school refer to as "great in math." All this means is I get As in the classes I take (finished the calculus series, now taking linear algebra, differential equations, and discrete math). The thing is, although I receive good marks, I still feel like something's missing. Right now, my guess is that I don't really understand the math, I just do it. I'm not sure if anyone understands what I'm talking about, but if so, I'd really appreciate any input in actually understanding math.

Thanks.

What do you feel that you don't understand? Do you feel that what are doing in mechanical rather than conceptual?

 

[mathwonk]

do you ever do any outside reading? try "what is mathematics?" by courant and robbins. reading works by outstanding mathematicians helps understanding. some suggestions: spivak, apostol, jacobson, courant, bott, euler, gauss, feynman, milnor, sah , shafarevich, lang, mumford, artin, van der waerden, cartan,...

 

[Sankaku]

finished the calculus series, now taking linear algebra, differential equations, and discrete math

I apologize if this is the wrong advice, but have any of your courses been intensively proof-based? Some of the ones you listed can be either computational or theoretical (linear algebra, discrete math). If not, then I recommend finding the first proof-based course that the math-majors at your school take and do that next.

If there aren't any proof-based classes at your school (unusual), then pick up "Understanding Analysis" by Stephen Abbott. It is good for those who have done Calc 1 & 2 but have not done a "Spivak" style course.
https://www.amazon.com/dp/0387950605/?tag=pfamazon01-20

On the other hand, if you are aceing all the proof-based courses, maybe epenguin is right!

 

[ahsanxr]

I know exactly where you're coming from as a first year Math/Physics undergraduate student. I'm pretty much exactly like you, as in who does well in class, can do most problems but still think there's something missing. I made a thread about this earlier too which might be helpful (although most of the replies were replies to certain things which I gave examples of that I didn't understand):

https://www.physicsforums.com/showthread.php?t=462488

Even though I haven't done this I think the general way of truly understanding stuff would be understanding the derivations of everything. Another thing which I've heard is that things become clearer once you go into proof-intensive classes. Which does make sense because I pretty much learned algebra/trig/pre-calc in the same way too (most people do) but when I started doing calculus and matured more I started understanding all that in a much better way (not saying that its a lot but it does make a point). So basically I think once you are "mathematically mature" enough things which don't seem to make sense now would start coming intuitively. At least that's what I think would happen. I would also like to hear more replies to this issue.

 

[Matterwave]

It's all good. As Feynman once said "shut up and calculate".

Although, I'm not sure if he meant that quote for this situation...

 

[jsewell94]

Hapyro, it's interesting that you should post this because I feel the same way in my math classes sometimes. I am currently enrolled in Higher Level Math for the IB Program, and I get A's on all of my tests and my teachers always brags on me..but I don't feel like I truly..understand it. I feel as if I only understand it at a superficial level..or something. It's difficult to explain.

 

[[Quadratic]]

It's all good. As Feynman once said "shut up and calculate".

Although, I'm not sure if he meant that quote for this situation...

Feynman didn't say that. It was David Mermin.

 

[Sankaku]

http://en.wikiquote.org/wiki/David_Mermin

Also, it was meant in relationship to physics. If you want to understand Math you can't "shut up and calculate." In fact, in the higher levels of (pure) mathematics, there is almost no "calculation" anyway.

 

[abiyo]

I second mathwonk's advice. If you have the time, at least learn the motivation for the concepts and how they fit in the over all scheme of things. If you have more time, get a sense of the proof. If you are really dedicated, learn the proof and derivation at heart.

 

[mal4mac]

MathsWonk makes some good suggestions, but my favourite is the recent "Mathematics: A Very Short Introduction by Timothy Gowers", A Fields medallists and Cambridge professor.

He says "I have focused on a more philosophical barrier, separating those who are happy with notions like infinity, the square root of minus one, the twenty sixth dimension, and curved space from those who find them disturbingly paradoxical... It is possible to become comfortable with these ideas...”

His main point is: "One should learn to think abstractly, because then many philosophical difficulties disappear."

More quotes:

"Do numbers exist? Mathematicians can happily ignore this question."

"The abstract method in mathematics takes a similar attitude to mathematical objects: a mathematical object is what it does."

 

[mathwonk]

these attitudes are reflections of a statement by Felix Hausdorff in his Set theory, translation of the 1937 3rd edition, p.28. He dismisses attempts to define cardinal numbers as equivalence classes of sets under bijection, as done by Russell and others.

"This formal explanation says what the cardinal numbers are supposed to do, not what they are. More precise definitions have been attempted but they are unsatisfactory and unnecessary. Relations between cardinal numbers are merely a more convenient way of expressing relations between sets; we must leave the determination of the "essence" of the cardinal number to philosophy."

After reading this I decided I was by temperament a mathematician and not a philosopher.