Understanding math - what does it mean?

[ltjrpliskin]

I know lots of people say that when you are learning something it is better to understand it rather than just blindly try to memorise it. This might be a silly thread, maybe I'm thinking too hard about it but I would really like some clarification.

So I'm wondering, what does it really mean to understand something?
If I do understand something, how would I know?
What are all the necessary conditions that one must satisfy to understand something?
How would YOU go about the learning process in order to gain understanding of a certain topic?

Any input will be greatly appreciated! :)

 

[DiracPool]

So I'm wondering, what does it really mean to understand something?

At its very basic level, to understand something is the ability to construct a model of it, either physically or mentally. For example, "understanding" the operation of adding 3 plus 7 requires that you grab 3 members of one species of object and then manually combine those objects with another 7 members from the same species. Now you have 10 objects all of the same species. You don't have to manually do this, although you might want to start that way, you can visualize the operation in your head. Either way, you have constructed a model of the operation and this act demonstrates that you "understand" that operation.

The counterexample is a situation whereby you were given a table of numbers added together and their solutions and were told to memorize them. Say one of these entries were 3+7=10. Now, on a test, you were asked what 3+7 was, and you'd say 10, because you memorized it from the list. However, you have no idea why 3+7=10, you just remembered it from the table. In this case you do not understand what this is because you didn't arrive at and, for our arguments sake, were unable to arrive at the solution through constructing a model of it.

If I do understand something, how would I know?
What are all the necessary conditions that one must satisfy to understand something?
How would YOU go about the learning process in order to gain understanding of a certain topic?

See above

Any input will be greatly appreciated! :)

No problem, hope that helped

 

[Floid]

An example that might illustrate the point:

The derivative of sin(x) = cos(x) and the derivative of cos(x) = -sin(x)

It is easy to just memorize these two equalities, but why does d/dx cos(x) = -sin(x)? If you understand that the first derivative is the rate of change you can sketch out cos(x) and sin(x) and see why it is -sin(x) and not sin(x). You could even visually "prove" the solutions.

I can never remember which one has the negative sign so on the few occasions I actually need to perform such a derivative I find myself mentally plotting what they look like and figuring out which derivative has the negative sign. That might be as deep as my mathematical understanding extends but maybe it shows the difference.

 

[mathwonk]

I try to strive for ability to: motivate it, calculate with it, apply it, generate examples of it, derive/prove it, and explain the concept to others. I often work on this by writing it up in detail - hence all the notes on my web site. It also helps to lecture on it to an audience, even of one person.

 

[AlephZero]

I would say "understanding" is what you need to answer the sort of questions we got in our BS Maths finals exam:

A "special vector space" is a vector space with the following properties [details omitted]. Prove some interesting theorems about special vector spaces.
(Time allocation: 1 hour of a 3 hour exam).

But It appears the current US system has been dumbed down a bit compared with that!

 

[ltjrpliskin]

No problem, hope that helped

You certainly did, thank you! I should be able to construct proofs myself without memorising them line by line.

I try to strive for ability to: motivate it, calculate with it, apply it, generate examples of it, derive/prove it, and explain the concept to others. I often work on this by writing it up in detail - hence all the notes on my web site. It also helps to lecture on it to an audience, even of one person.

I like this idea of writing notes in lots of detail. If I am not able to explain a certain detail clearly, then I have not understood this part properly. Thanks mathwonk!

I would say "understanding" is what you need to answer the sort of questions we got in our BS Maths finals exam:

A "special vector space" is a vector space with the following properties [details omitted]. Prove some interesting theorems about special vector spaces.
(Time allocation: 1 hour of a 3 hour exam).

I guess I won't get better at solving "unseen" problems unless I practice a lot of problems, so this is also key to understanding. Thanks AlephZero!

 

[mal4mac]

At its very basic level, to understand something is the ability to construct a model of it...

Surely it's more basic to actually experience something. If you say "The grass is green" I understand what you're saying 'cause I've experienced grass & green *directly*. I don't need to model grass or green! You can visualise, or actually see, adding one spoon to a collection of two spoons and seeing that there are three spoons... again that's direct. But start with a collection of 341 spoons and 456 spoons and there's no "instant" way of seeing that you get 797 spoons... so you use standard addition rules to get to 797, and you can be said to understand the process ("model") of addition, but the understanding is not as strong as actually seeing "1 + 2 = 3". Your understanding of the harder addition might be "beefed up" with a proof, or a demonstration, but nothing beats direct experience.

 

[Robert1986]

But It appears the current US system has been dumbed down a bit compared with that!

Why in the world would you think this??As a matter of fact, the exam you describe seems a lot like ordinary exams I took. Perhaps the one you describe is slightly more involved than an ordinary midterm exam, but it seems about right for a final exam.

 

[WannabeNewton]

Why in the world would you think this??

It would depend on the university of course but I can certainly attest to what Aleph alluded to. At Cornell, I took the honors linear algebra class and it was insultingly easy. It honestly felt like a waste of time; I learned nothing theoretically rigorous from that class. A month spent on Roman would have been infinitely more productive.

On the other hand, if you take a look at honors analysis at UChicago or math 55 at Harvard, you will see what is quite possibly some of the most difficult material for a freshman math major at any university. It is brilliantly hardcore stuff. In a similar spirit, physics 8.022 at MIT which is the Purcell based electrodynamics class for freshman is also quite a difficult class.

 

[jedishrfu]

Understanding comes when you think you know the topic and then try to teach it to someone else. My example was in teaching the F=ma to some teachers for a job interview.

In preparing for it, I developed some simple examples like deciding whether or how to stop a truck rolling downhill ( a real-life example that happened in front of our house when my sister-in-law forgot to set the brake and leave it in gear on my brother-in-laws brand new stick shift Mazda truck. It started to roll and a neighbor stopped it with a brick before it had rolled too far.

It was then that I learned that applying a force to the truck to stop it changes its velocity that is first slows it down meaning if you try to stay your ground you'll get run over first. ( A good darwin test for physics grads). It doesn't just stop it.

Anyway I passed the interview only to be shot down for failing to have the requisite 18 grad credit hours in physics (I had 17) for community college teaching.