# Math can't be all memorizing! How do you get to the real, more creative side of math?

#### MissSilvy

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.

#### slider142

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

You're just being prepared to have certain rote processes accessible as second nature instead of through slower active logic. This will be useful for simple rote applications later on. As such, you have not really seen what mathematics is yet. Try checking out the book https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20 for a glimpse of most of the mathematics that mathematicians study.
For a whimsical look at mathematics (and politics), try https://www.amazon.com/dp/0554354128/?tag=pfamazon01-20, or https://www.amazon.com/dp/073820675X/?tag=pfamazon01-20.
Also, before you get bogged down by "What is Mathematics?" one of the books that really inspired my study of the pure sciences as a child was "One, Two, Three, ...Infinity" by George Gamow.

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#### maze

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

I think most mathematicians wouldn't call memorization and solving tons of trivial problems "mathematics". They would call it doing mindless computations, and hate it tremendously like you. It is unfortunate that the system is set up this way at the lower levels, but that is another story.

Look for courses intended for math majors, and ones that emphasize proofs. If you post a list of course descriptions, we could probably be of more help in this.

By the way, you should beware of physicists who want to teach you math in a physics course. It will be taught very superficially and without rigor, with focus on computing things rather than understanding what you're doing. In other words, it sounds to me like you wouldn't like it (I'm the same way). You should take actual math courses taught by the mathematics professors whenever possible.

#### Unknot

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

You have to put some lower-division classes behind you (yes, those horrible classes aimed at engineering students) and take upper-division courses. Then you will realise math has nothing to do with speed, computational ability, or memorization.

I do know some mathematicians who enjoy computing. But that's not an advantage though.

#### Bob S

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

Wait until you get to integral calculus in college. The first problem I encountered was a glass (truncated cone), tilted at an angle of 45 degrees, and filled with water. How much water is in the glass? At least we could bring integral tables.

#### csprof2000

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

MissSilvy:

I agree with whoever said the stuff about the physics classes. It's best if you learn all the stuff they are going to tell you from a mathematician as well... in physics, things just "are" however they need to be... and you should know how they need to be since, well, that's the only way it makes sense, see?

Just try to make the best of the lower-level math courses. Like somebody said, it's just training you the basics, like teaching words to an infant from flash cards. Learning vocabulary isn't any fun, but once you know enough words, you can write whatever you can imagine. Math is sort of the same way.

And yes, nobody likes actually computing things by hand. That's why we have machines to do that. Generally, if a machine can do it fast, cheap, and easy, then humans will find it fairly boring. As a CS major in undergrad, I often found that it was easy to keep myself interested in the more trivial math by figuring out how to make computers do the sorts of calculations I found trivial. You don't really know how to do something until you can tell a computer how to do it. [/CS-related rant]

#### Klockan3

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

I do know some mathematicians who enjoy computing. But that's not an advantage though.
It actually is an advantage, the faster you can compute the more iterations of thoughts you can process in any give time and thus the faster you can learn new concepts.

If you struggle with the basics it will go really slow, and yet even slower the higher you get. And by computation I am talking all sorts of operations, not just the basic +-*/.

#### arildno

Homework Helper
Gold Member
Dearly Missed
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

MissSilvy:

If you were to become a great and creative soccer player, do you think you could achieve that without repetition and rote learning?

What about training to become a violinist? Doesn't that require boring, repetitive sessions until you have mastered the elementary steps?

And so on..

Maths is no different. You need to flap your wings repeatedly before you are ready to fly.

Besides, try to UNDERSTAND the examples, why they have to be true, rather than merely memorizing them.

#### CRGreathouse

Homework Helper
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

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.

#### Klockan3

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

MissSilvy:
Besides, try to UNDERSTAND the examples, why they have to be true, rather than merely memorizing them.
This, once you start getting a bit higher it goes much faster to understand everything than to memorize everything.

However most students have just learned that to do math you memorize formulas since it was the easiest way in things like highschool, but that mentality will really put a strain on you later on.

#### matt grime

Homework Helper
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

It actually is an advantage, the faster you can compute the more iterations of thoughts you can process in any give time and thus the faster you can learn new concepts.
At what level are you talking about? I would hope you think you're not making assertions about anything higher than high-school mathematics. The idiot savant can do computations. The mathematician understands why one does those calculations in the first place.

What next? Asserting that knowing pi to 100 decimal places makes you a mathematician?

#### Tac-Tics

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

Unfortunately, being a highly technical subject, math requires a large vocabulary and has a very rich foundation of concepts.

There is a lot of room for creativity in math, but not so much in the classroom. One of the most creative and difficult of all math activities is simply COMING UP with a good math problem. There's a tricky balance to it! If you come up with a problem that's too restrictive, you will find there are no solutions to it. If you come up with a problem that's too easy, you will find there is an obvious solution to it. But a good mathematician can come up with simply problems whose solutions are very hard (but yet possible) to find.

A "problem" can be just about anything that requires logical thinking and has a provable solution set. Find a way to color any map in 4 colors. Find a way to arrange these tiles to satisfy these conditions. Figure out optimal strategies for a card game.

I think one of the hard parts of learning math is simply figuring out the motivation behind the theory. Things like real numbers, square roots, and derivatives are just abstractions. Much (if not most) of mathematics has a concrete example that served as an inspiration which was later generalized.

#### Klockan3

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

At what level are you talking about? I would hope you think you're not making assertions about anything higher than high-school mathematics. The idiot savant can do computations. The mathematician understands why one does those calculations in the first place.

What next? Asserting that knowing pi to 100 decimal places makes you a mathematician?
Way to misinterpret my post, you are obviously just talking about the ability to add numbers together or multiply them together which people stopped doing in early middle school, I am talking about algebraic operations in general which should have been obvious from my post.

A person who can do most algebraic operations in his head faster than someone else can speak will easily be able to follow with what the other person is saying, while someone who have to sit down and do them slowly will be lost somewhere on the way and thus not understand.

Its like learning a language, even if you know thousands of French words if you can't conjure up their meaning on a whim you will not be able to understand someone who speaks it. And I hope you understand how much it helps in math to understand what people say...

#### matt grime

Homework Helper
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

I didn't misinterpret your post at all, Klockan3. What are 'algebraic operations' in general? (You can't be that vague and accuse people of misunderstanding you.)

Understanding is what counts. The ability to work out, say, the fundamental group of some topological space on the fly is nice but hardly going to be very important. Better that you know and understand the van kampen theorem and amalgamated products. Yes, that will involve the memorization of definitions, and you're never going to get very far from that in mathematics. And yes, the best way to learn definitions is through re-inforcement.

#### maze

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

Yes, that will involve the memorization of definitions, and you're never going to get very far from that in mathematics. And yes, the best way to learn definitions is through re-inforcement.
Hopefully you should have seen enough examples and developed your intuition such that it is obvious why definitions are set up the way they are and no memorization is necessary. I had a prof who once said, "all theorems are trivial, once you understand the right examples".

Now maybe you need to do some memorization at first as an intermediary step (like building the scaffolding when you put up a building), but eventually the memorization should be replaced with firm understanding.

#### lurflurf

Homework Helper
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

^
-Some results are of frequently usefull and lengthly to derive, so it would be of use to memorize some results that you could derive. "I do not recall that result, but I should be able to construct it given fifty hours and a hundred shets of paper."
-Conventions and names are not always obvious so they need to be memorized to avoid confusion. "I understant calculus well, but I do not know of this derivative of which you speak. Give me the deffinition and I will be able to follow along."
-Understanding surely reduces the memorization required, but a greater benifit is that it is much easier to memorize things you understand. It is easier to understand things you have memorized.
-Memorization and understanding are complementary not in opposition.

#### matt grime

Homework Helper
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

Hopefully you should have seen enough examples and developed your intuition such that it is obvious why definitions are set up the way they are and no memorization is necessary.
And therefore you had to memorize the examples, or do enough of them to learn what was going on and memorize the ideas by re-inforcement.

If you think you won't be memorizing definitions (by which we really mean the names given to the definitions) then you're fooling no one but yourself. For instance what are the differences between these terms in group theory:

almost simple
semi-simple
quasi-simple
virtually simple

There are many situations where it is necessary to memorize definitions in mathematics, understanding what is going on in them is entirely different. Sadly, sometimes one has to memorize too much. Does anyone really think that having to remember the difference between an T1 and a T1.5 space is particularly good for the mind or soul?

I had a prof who once said, "all theorems are trivial, once you understand the right examples".
He was talking about the proofs of the theorems.

#### MissSilvy

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

Oops, I didn't mean to start a debate! Nevertheless, it's informative to see what each of you thinks about various ways of learning math :)

Perhaps I panicked too soon. I just assumed that since most of what we're dealing with now is memorization and acquiring tools that it would be that way forever. I didn't mean to say that I think memorization itself sucks or doing book exercises is a waste of time, because (as some of you have said) you need know the vocabulary before you can claim you're fluent. I was a little afraid that because the emphasis isn't on creativity just yet that when I actually get to that point, I would be handicapped because I've never done it before.

I do appreciate the emphasis on understanding and I know it's important (learning without understanding makes quite a fragile memory in any case). Would a better question be what else could I be doing in addition to my classes in order to improve my mathematical fluency? Thank you for all the responses!

#### maze

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

This is going nowhere. I'm not going to convince you nor are you going to convince me. But anyways I must make 1 correction:

He was talking about the proofs of the theorems.
I know for a fact that he was actually talking about the theorems themselves (not the proofs).

#### arildno

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Gold Member
Dearly Missed
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

I was a little afraid that because the emphasis isn't on creativity just yet that when I actually get to that point, I would be handicapped because I've never done it before.
Well, before the fledgling have tried to fly, it never flied, did it?

But in most cases, the transition works just fine, as long as the fledgling's muscles have been properly trained.

You shouldn't think of drill and memorization as plugging techniques that will prevent creativity from bursting forth.

Rather, what such techniques do is
a) Hone your skills in explicit problem-solving

b) By automatizing certain elementary techniques, your brain won't be distracted by the sub-problems these are meant to solve, so that you may use your full intellect at the new task at hand.
Thus, memorization OPENS the gate for creativity to flow in, it doesn't shut it.
But to open that gate is an, admittedly, boring feat on its own..

#### matt grime

Homework Helper
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

I know for a fact that he was actually talking about the theorems themselves (not the proofs).
Fair enough: triviality usually refers to a proof rather than a statement of a theorem; I can't recall anyone calling a statement of a theorem trivial.

Like it or not mathematics does involve lots of memorization at all levels, just like any subject. You can't get away from that. I found that writing out lecture notes 5 times was the best way to get things in my head.

You have to remember the names attached to definitions. I agree that what the definition encapsulates ought to be clear from examples that lead you to think of that as the correct definition. Sadly not all teaching can be done that way: some times the definition necessarily comes first for expediency, since we can't just let people get on and discover things at their own rate.

Passing back to the T1 stuff, it should be clear that there should be a name for what we call a Hausdorff space. Remembering that we attach the name Hausdorff to is not necessarily easy: names are not always chosen to be suggestive. And you have to remember conventions: some people require that Hausdorff is part of the definition of a compact space, and would use the term quasi-compact instead.

The best way to learn is to do. At the stage the OP is at now, that means lots of worked examples (how many depends on the person). Later on, it is not necessarily even possible for there to be such a thing as an example.

There is another quote I like, along the lines of:

"Definitions are hard, that makes theorems easy."

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#### DukeofDuke

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

Take a course called "linear algebra". NOT algebra, linear algebra.
Trust me it wont be just memorization. You WILL learn to derive those theorems or you'll get hurt.

#### g_edgar

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

I have seen the other end of this problem. Mathematics courses for pre-med, pre-law, business, architecture, etc. Give them problems where using memorized formulas will give the solution, they are content. Include a problem where you have to think about what is going on and analyze it .... practically rebellion!

#### DavidSnider

Gold Member
Re: Math can't be all memorizing! How do you get to the real, more creative side of m

It's just not that important for most people to know "why" in mathematics.

Think of how many people who enjoy and play music without knowing the first thing about music theory.

#### DukeofDuke

Re: Math can't be all memorizing! How do you get to the real, more creative side of m

I have seen the other end of this problem. Mathematics courses for pre-med, pre-law, business, architecture, etc. Give them problems where using memorized formulas will give the solution, they are content. Include a problem where you have to think about what is going on and analyze it .... practically rebellion!
yup, and its even worse in the premed physics classes. Even the engineers do this, though its not as bad.

Makes me sad, because with rudimentary math you can get away with computation, but with physics you really can't! Physics, you have to pay attention to what the universe is telling you, if you just write down formulas you miss all the fun!

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