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So far (calculus 1), the math concepts that I learned have been pretty easy, and the greater difficulty is usually in solving certain problems. However, as you go higher and higher do the concepts become more and more difficult to grasp in relation to doing the actual problems they are used for? Basically I am asking does the ratio of difficulty of $\frac {lesson}{problems}$ increase with harder classes? For example if I take a very advanced class, will it be more difficult for me to understand the concepts, or to solve the problems from that class after I've understood the concepts? (I know you can make very hard problems even with the easiest concepts, but I'm referring to the textbook type) Thanks.

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closed as primarily opinion-based by Zev Chonoles, Amzoti, Pedro Tamaroff, Danny Cheuk, Potato Jul 8 '13 at 6:27

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

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The general trend that I'm noticing is that the higher the level the math, the more abstract it becomes. Also, you begin to ask different questions - instead of "what is the answer" you ask "why is the solution like this," as an example. –  kvmu Jul 8 '13 at 3:46
    
Check out some of these texts: Algebra, Thomas W. Hungerford, Topics in Algebra, 2nd Edition , I. N. Herstein , Abstract Algebra, 3rd Edition, David S. Dummit , Richard M. Foote. What do you think? –  Amzoti Jul 8 '13 at 3:49
    
I am wondering why are you asking this question? I mean are you wondering about how more advanced math courses are like that you are planning to take afterwards? I don't know if I can answer the exact question you asked, but I noticed in the more advanced math courses, you are required to do a less amount of homework problems since they are a bit harder. Compared with calculus you are usually asked to do many problems but more are computational. –  user77404 Jul 8 '13 at 3:55
    
@user77404 yes you were correct I am planning to major in math. I've updated my post, but what I was referring to is this: if I take a very advanced class, will it be more difficult for me to understand the concepts, or to solve the problems from that class after I've understood the concepts? –  Ovi Jul 8 '13 at 3:58
    
@Ovi: In my experience everything gets harder, and I think it should. I have found that it is more difficult to grasp the concepts and more difficult to solve problems. Grasping concepts and being able to solve problems are probably the same thing in higher mathematics. –  Eric Kightley Jul 8 '13 at 4:08

3 Answers 3

The question is based on a distinction between understanding concepts and being able to solve problems about those concepts. I'm not sure I see a real distinction here. An "understanding" of concepts that leaves one unable to solve problems is probably an illusory understanding.

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Higher mathematics is easy ... in the same way that neurosurgery, world-class ballet and winning Wimbledon are. All you have to do is show a (small) amount of talent and then be prepared to dedicate a large chunk of your life to becoming among the best in the world at it. About 10,000 hours of solid practice is generally considered to be sufficient.

A tad less humorously, mathematics is both a language and a mode of thought.

Considered in this way, developing fluency in mathematics is not dissimilar to developing fluency in a foreign language. Starting with a few words and basic concepts, you expand your knowledge of vocabulary and conversation until you suddenly find yourself "thinking" in the foreign language rather than your native language. Just as there are concepts in German that do not translate well to English (a problem English solves by sucking the German words into the lexicon) and even more such concepts in Vietnamese, because of the greater cultural gap; there are concepts in mathematics that only make sense when thinking in mathematics - they have no simple analogy that can be expressed in English (or Vietnamese).

The important thing is to make sure that you understand the current concepts before moving on to the next. I can distinctly remember the wistful longing, after my first year of University, to go back and redo the final high school maths exam that was such a struggle with my now greatly expanded knowledge base. It was like after being forced to use a hammer on a bolt, I had had revealed to me an entire toolkit of spanners and sockets!

Mathematics is also like a foreign language in that if you don't use it it goes away. I know I studied metric spaces and I must have been pretty good at it because there is a Credit mark next to it on my academic transcript but I would not now have the first clue about how to recognize much less solve a problem that uses them.

If you follow a properly structured course of mathematics and follow the specific branches that interest you; you should find that the concepts and your fluency build upon each other. I would say that if you wanted to be a mountaineer, you would be ill-advised to start with Everest but that if Everest is your goal then by starting at the local indoor rock climbing wall and dedicating the time to practice, and moving on (and up) there is no reason that you can't get to Everest.

Now, professional mathematicians start building ladders when they get to the top of the mountain and strike off into the blue!

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Since this question is basically asking for an opinion, I will give mine although I believe it is true for most people.

I have found that as I go "higher and higher", the concepts get more difficult to grasp and the problems also get more difficult. I believe this is true in just about every field though.

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I expected that the concepts get more difficult to grasp, but for example if I take a very advanced class, will it be more difficult for me to understand the concepts, or to solve the problems from that class after I've understood the concepts? –  Ovi Jul 8 '13 at 3:56
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It is mixed. A good many problems in beginning "abstract" courses are easy to solve if you have a good clear understanding of the definitions, and a modest supply of concrete examples. –  André Nicolas Jul 8 '13 at 4:08
    
@AndréNicolas Ok thanks –  Ovi Jul 8 '13 at 4:14

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