Finding the Creative Side of Math: Advice for Undergrads

[arildno]

Practice IS the most effective way to memorize something.

There was nothing in OP to suggest that we were to limit our attention to some particular memorization techniques, and not others.

Furthermore, as matt grime has repeatedly said in this thread (and he happens to be a professional mathematician), definition is one of those fields where memorization is absolutely necessary, and of the greatest of help.



If, when reading a text and some sequence is said to converge pointwise to some function f, well, if it has been years since you read maths and have simply forgotten (or never learned properly) the definition of "pointwise convergence", then you are stuck, and should not proceed further in the text.


This is one other field besides practice where memorization is of crucial importance.

 

[arildno]

Actually this debate is quite interesting. I was referring more to the standard 'here is a formula, now use it on these twenty problems and memorize it for the test'-type thinking. .

I'm glad you are still with us, MissSilvy!

What you refer to, is typical low-grade math exercise, the value of which cannot be underestimated, and the difficulty of which for the average school kid is also gravely underestimated.

Many kids DO have problems of
a) Putting the right numbers into the right "slots" of the formula
and
b) Perform the arithmetical operations upon that in the correct order, getting the correct results.

You certainly cannot progress into maths until this has become completely trivial, and dependent upon the particular individual, that can take quite a lot of practice.

 

[anonperson]

I agree with csprof. If you think you understand something, try programming it. You can't fool a computer! Often that has helped me to understand the fine details of something I thought I understood before.

I concur. I'm not much of a mathematician or a programmer but when I taught my students basic Turtle programming you should have seen how the kids took to the math afterwards. Their teacher couldn't keep up with all of their questions about math after they had their programming class. These were third graders chomping at the bit for complicated algebra!

 

[camilus]

get into the theory of numbers, that's where math is the most interesting and beautiful.

 

[Elucidus]

I am not sure where the original poster is as far as coursework is concerned, but these are courses that I have found that contain substantial material that challenges the creative side of doing mathematics:

Logic
Set Theory
Combinatorics
Elementary Probability (not Statistics)
Introductory Number Theory
Dsicrete Mathematics
Linear Algebra
Elementary Abstract Algebra
Integral Calculus (to some extent, including series)
Any undergrad problem seminar

Some of these courses are accessable with just an intermediate algebra/precalculus background.

I'd also suggest looking into books on math puzzles or anything by R. Smullyan. For those interested in proofs I'd suggest "How to Prove It" by Velleman (I think). I have been particularly enchanted by the encompassing theory of Categories (William Lawvere has a great book on the subject - "Conceptual Mathematics: A First Introduction to Categories").

I can't agree more with the sentiment that one should take serious math courses instructed by serious mathematicians. I've had too many undergrad courses taught by those who weren't really steeped in the material (including an abominable Logic course taught by a philosophy major who had trouble with derivations by contrapositive). Being a mathematics professor obviously shades my opinion, but the best courses I've had were taught by animated, engaging, and knowledgeable instructors.

I hope this information is helpful.

--Elucidus

 

[27Thousand]

I'm not sure if I'm the only undergrad who has ever felt this way but I hope this is the right forum to ask for advice in.

In math classes, generally we were given a bunch of very silly, trivial problems and made to solve the same exact thing fifty times (what's the root of (x^2)+3? Now what's the root of 3(x^2)+4? And on and on and on). Math is taught by 'memorize this stuff and stick it here and watch your signs'.

Historically, I've done well in these sorts of classes but sometimes I can't help but feel that there's a whole side of math that I'm missing. I know methodology and memorizing are important but it's taught to the exclusion of anything else. It's like being stuck in a gray room but there's a peephole in the wall that shows something more more exciting and creative than memorizing example problems. This simply can't be all there is to math. I talked to my adviser and his only advice was 'wait until you get to the proofs class, it gets a lot more 'fun' then'. The textbook isn't any help either, since it has the same tired problems and definitions.

My question, if I'm making any sense at all, is how can I get to this other side of math? I enjoy finding things out seeing patterns (that's why I'm majoring in physics) but I want to do the same in math. I know it's possible, but I just don't know how to get there. Any advice would be vastly appreciated and sorry for all the questions. Thank you.

I'm like you, I don't like rote memorization either. It's always more fulfilling to grasp the concept of how something really works. Just like you can use sticky tape to grab a piece of paper, you can grasp the concept by doing whatever you can to turn details into concepts, or the process of concept-ualizing.

I always try to do my best to visualize in my mind why/how the equations work, why something may specifically be in the denominator, etc, etc. Then I think about what I worked out to myself when doing the practice exercises, so the grasped concept is strengthened. Then relating to what you already know/outside knowledge helps.

Although this sounds elementary, some of the Complete Idiot's Guides and for Dummies have ways of visualizing.

 

[pbandjay]

I never force myself to "memorize" math. Rather, I strive for understanding of proofs of theorems and such...

I have a professor who once said: "People assume: I am a mathematician, therefore I solve numerical problems... This is equivalent to: I am a novelist, therefore I write words."

 

[lurflurf]

^Well good on you. So do you just remember all the theorems, proofs, and definitions or do you just not worry about the ones you forget. I know if I did not "memorize" I would not remember trigonometric identities, integrals, all those variations on compact, which types of rings are named after which people, 7 times seventeen, or many other things.

 

[DukeofDuke]

^Well good on you. So do you just remember all the theorems, proofs, and definitions or do you just not worry about the ones you forget. I know if I did not "memorize" I would not remember trigonometric identities, integrals, all those variations on compact, which types of rings are named after which people, 7 times seventeen, or many other things.

Well, mainly for the little theorems they are pretty obviously true, its just a matter of remembering whether or not its true by theorem and whether or not you can say it. For the bigger ones, there's a difference between straight memorizing, or understanding and then remembering. I try to do the latter, where you create the concept in your mind and develop the theorem out of that, rather than just trying to remember the words. But yeah, some things are unavoidable, you have to straight up memorize names. But things like integrals are so second nature now its not really memorizing. And I only remember two trig identities, takes me all of 12 seconds to derive all the others.