Doing math, but without actually understanding it

[inception7]

Hi,

I'm taking calculus and I was wondering if problem-solving and comprehension go necessarily hand-to-hand. For example, I often do not understand long and extensive texts and simply try to narrow it down to understanding the concepts intuitively. I have so far managed to do well in math, even despite not relying on formal definitions. I also have probably forgotten a few stuff from my pre-calculus class (definitions or textual stuff) but I retain the concepts.

For example, sometimes I just know how it's done, but I don't know how to formulate things into words or a coherent set of logical phrases.

All this makes me extremely anxious, because I feel like I'm not really understanding anything at all, even though I manage to pull of grades between 80-95%...which is pretty weird!

How about you guys, do you share some similarities with my situation?

 

[Angry Citizen]

I'm taking calculus and I was wondering if problem-solving and comprehension go necessarily hand-to-hand.

Nah. I'm terrible at computation, but I'm pretty confident I understand the material and where it comes from. I'm sort of the complete opposite of you, though. I learn through definitions and derivations.

 

[chiro]

Hi,

I'm taking calculus and I was wondering if problem-solving and comprehension go necessarily hand-to-hand. For example, I often do not understand long and extensive texts and simply try to narrow it down to understanding the concepts intuitively. I have so far managed to do well in math, even despite not relying on formal definitions. I also have probably forgotten a few stuff from my pre-calculus class (definitions or textual stuff) but I retain the concepts.

For example, sometimes I just know how it's done, but I don't know how to formulate things into words or a coherent set of logical phrases.

All this makes me extremely anxious, because I feel like I'm not really understanding anything at all, even though I manage to pull of grades between 80-95%...which is pretty weird!

How about you guys, do you share some similarities with my situation?

Hey inception7.

True understanding is not usually immediate, especially for something like math.

It usually takes the right environment, a bit of work, and continuous energy expenditure whether that's through continual thought and/or actual physical work on something (in mathematics its usually through research projects or other similar projects).

Let me propose the following situation to you:

Consider the difference between a scientist and say your average person.

One of the major differences between the scientist and the average person is not so much the intelligence aspect, but the intention of trying to comprehend something and putting in the time to study and think about something.

We all have the ability to look at the world whether its a mathematical representation or a flower. Most of us look at these things and say "Ohh that looks nice" and move on. Some of us decide to allocate large chunks of our time to think about and make sense of said things in the hope that we may gain at least some understanding.

Also mathematics is a language. Like any language you have to "tune in" to the language before you are able to juggle many words, phrases and paragraphs in your head to make real sense of what you are hearing and reading in order to comprehend and listen.

So in light of that, I wouldn't feel bad about not being able to comprehend something immediately. If you focus more energy and make use of what is out there (information, opinions, perspectives and so on) I think you have every chance to gain more understanding.

 

[Stephen Tashi]

Hi,All this makes me extremely anxious, because I feel like I'm not really understanding anything at all, even though I manage to pull of grades between 80-95%...which is pretty weird!

Your anxiety is justified if you intend to study any mathematics that requires understanding it to the level of being able to understand and write proofs. Often people who "do good in math" find out that they don't do so good when they hit that type of material.

Having done a little teaching, I don't think it is unusual to encounter students who are good at a intuitive type of pattern recognition and skilled at "mechanical" types of symbolic calculation. Some people with such talents also have the talent of comprehending the technicalities of higher math and being able to reason clearly about them, others don't. This may have something to do with skills, but it also has to do with a person's outlook on life. Some people have intense curiosity and a fierce drive to understanding things precisely, others are happy not to worry exact details.

 

[Newtime]

Paul Halmos on studying math:

"Don't just read it; fight it! Ask your own question, look for your own examples, dicover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"

 

[Ben Zina]

I personally know how you feel. I've managed to get a B+ in a vector calculus class and I only have a vague idea of what vector calculus even is.

Now I don't know if that's my fault or simply the way the class was taught (mostly cook book formulas and theory, no proofs).

 

[mathwonk]

understanding just sort of bursts upon me some times, but only after a lot of hard work. it is sort of like the saying that the harder you work the luckier you get. i.e. to understand, you have to try to understand.

 

[homeomorphic]

If you can understand the concepts, you have the most important aspect taken care of. Doing proofs just follows from that, plus a little ingenuity. And sometimes more detailed concepts. "Formal" definitions can usually just be thought of as formalizations or abstractions of intuitive concepts. If you really understand the concepts, remembering the formal definitions should be easy.

If you understand the concept of a derivative well enough, then you should be able to use that understanding to write down the formal definition.

What's not good is if you don't know why you are doing what you are doing at all and are just parroting what other people have told you. There can be exceptions to the rule. Mathematicians will usually skip some complicated and unenlightening proofs to save time, and maybe come back to them later, if it's interesting enough or try to find a better proof.

 

[nonequilibrium]

Ramanujan was one of the most brilliant mathematicians who ever lived (creative-wise, productive-wise), but couldn't justify his creative outbursts in any formal way. He could conjectury infinite series formulas for pi from the back of his hand, but couldn't derive them if his life depended on it.

Three disclaimers:
- I'm exaggerating
- I could be the victim of a common misconception about Ramanujan
- If you're not an ingenious Indian, it is probably less tolerated

 

[Windowmaker]

Sometimes understanding the concepts make more sense after I've seen other sort of math. I am a non-traditional student and i spent some time in the military. I wasnt the greatest at math in high school, but with some age, i became really good at it. My understanding improved as i got older. Weird uh? Anyways, I think it depends really. If you are an aspiring engineer, math or physics major, you should understand the concepts very well. But if your a major that doesn't require mathematics in the field, such as history or business ( not saying business analyst don't need math, just not above calculus) get through the class the best you can. If your an engineer, math or physics major, understanding the mathematical concepts is much more important.