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I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was not the best. (I am always too lazy, and not that smart enough)

However time has changed. Now I am dealing with abstract ideas of math, and I realized that I am really depending on visualization. For example, when compactness was first introduced to me, I have no idea about it because my instructor only wrote the definition, so did Rudin. I had trouble understanding the finite subcover of infinite cover because I was thinking "Why not always making the cover be the one covering everything? It is finite." Then I searched a Youtube video and saw a professor drawing circles (open cover) and I was like "Aha! Now I understood it!" I had a period not knowing what cosets are until I realized that I can think of partitions. Till now, after the idea has been introduced for 2 months, I still think of partitions first.

The problem is that very often before I realize what a concept "looks like", I have a hard time understanding the definition/theorem. Often some visualization understanding of a concept come up a period later after it is introduced. During this period, the instructor will go on the study of this concept, and I will be completely clueless. I probably will see why some properties or theorem introduced are true by directly looking at criteria in the definition, say. However I will soon forget without understanding, and such understanding usually is a visualized one.

I don't know whether my method is good. This method somehow looks reasonable but somehow it is like the way to deal with elementary math, but not high-level math. Please help. I've searched some similar topics on this site, many of them are closed because they are too broad or there are too many answers. I hope this one is good enough. I tried my best to make it not too broad. Also please pardon for my bad English, which is my second language.

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    $\begingroup$ Nicely phrased question. I cannot even imagine solving a non-trivial question without visualization as a guide. $\endgroup$ – André Nicolas Apr 20 '15 at 22:25
  • $\begingroup$ Visualising, and drawing in particular, is an important guiding tool for many an argument. But be careful, as it might lead you astray (don't use your intuition about how the topology of the plane works to make false assumptions in general topology, for instance). $\endgroup$ – Arthur Apr 20 '15 at 23:00
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    $\begingroup$ @Arthur In topology I usually consider topologies of a set containing 3 elements, those familiar examples like $\Bbb R^N$, lower limit topology, etc, or imagine some candidates that seems to fit the question and looks like topologies, in order to try to enrich different kinds of topological spaces that the question applies. Otherwise often I have no first impression on the statement of the question. True sometimes I got false assumptions, but disproof of this false assumption is not so trivial, even if I was not misled by intuition. $\endgroup$ – MonkeyKing Apr 20 '15 at 23:13
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    $\begingroup$ This is a very valid question, but unfortunately it can only attract opinion-based answers (who can tell for anyone else how they do maths?). You may profit from discussing this in chat. $\endgroup$ – Lord_Farin Apr 25 '15 at 16:40
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As Lord Farin indicated in the comments, the answer to your title question is heavily subjective, depending on personal preference.

Ultimately this depends more on your learning style. There is no right or wrong way to learn mathematics, so long as you ultimately reach the same truth.

If learning by visuals helps you understand the concepts behind a problem then by all means use it! They are indeed a useful way of putting together letters and numbers in a more physically meaningful way.

On the other hand, while visuals provide helpful guides to understanding what is going on, they do not always constitute mathematical proof. (Though in some geometric cases, such as the Pythagorean theorem they work just as well.) That is to say, do not use images or drawings on your average mathematics examination without proper context or related work.

For instance, if you are taking out a ruler and a pencil to figure out the slope of the tangent to (2,4) on the graph $y=x^2$, you are doing something incorrectly. Drawing the graph may be helpful in contextualizing the algebra, and even help check your answer, but in this case, it is not, strictly speaking, rigorous, an attribute mathematicians pride.

As our friends at Cognitive Sciences StackExchange can tell you, every person has a unique way of learning things:

Learning Types

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  • $\begingroup$ I don't agree that there is no right or wrong way to learn mathematics. the only right way is by definition the one which allows discovering the mathematics alone without help, as humanity did since ~2000 years. furthermore, it is obvious that listing definitions and theorems without thinking to the underlying concepts is exactly what computers can do: nearly nothing as they aren't able yet to discover the Pythagore theorem without help. hence concepts are central in mathematics and reasoning in term of concepts (farther than just the definitions) is the only right way to learn mathematics. $\endgroup$ – reuns Feb 9 '16 at 0:05
  • $\begingroup$ On that level yes I would hazard to agree with you. But the way that you reason and phrase concepts can be done in many different ways. Visual cues is but one more way of doing that. $\endgroup$ – KR136 Feb 9 '16 at 0:13

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