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Hello I want to learn mathematics.

In order to do this I want to get familiar with formulas/theorems by taking one and just analyze it and try to manipulate it to understand it better. I wanted to ask you whether this is a good/efficient idea to become proficient in math.

My theory behind this question is that a theorem reflects several things in mathematics.

  1. It reflects the existence of a category of problems/ideas that is meant to deal with.

  2. It provides a method or part of a method to solve a problem.

  3. It reflects rigorous mathematical expression of the insight of a mathematician. This means that a theorem is the crystallized form of expressing an idea.

  4. Being able to prove or to completely understand the proof leads to enhanced knowledge of methods used in proving mathematical ideas and give a greater clarity in what exactly the use is of the theorem/formula. (I contrast this with solving equations with a formula, which you can do without completely understanding the formula)

If instead this is not a good/efficient idea would you be able to specify why not?

I define efficiency as (energy spent on gaining understanding in mathematics/ total energy spent)

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  • $\begingroup$ The answer to this may be opinion based reflecting the learning style of the students involved. For me learning by proving theorems just doesn't work. $\endgroup$ – Karl May 28 '15 at 8:42
  • $\begingroup$ I do really hope that the answer to this question is not merely opinion based. I hoped to find educators or mathematicians who have experience with this approach, as I want to use it for myself but also teach more basic math to others using this method. Oh, and also my goal is not to pass exams, my goal is to gain a deeper understanding in math. $\endgroup$ – St.Clair Bij May 28 '15 at 8:46
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    $\begingroup$ This is is better suited to the maths educators site. Personally I believe teaching especially basic maths with theorems is a bad idea and a mistake new teachers frequently make. Others will disagree naturally but I'd urge caution with this a approach $\endgroup$ – Karl May 28 '15 at 8:52
  • $\begingroup$ Thank you for the suggestion. In the case of really basic math of course I'd rather have them being able to perform the calculations before going on to understanding theorems. $\endgroup$ – St.Clair Bij May 28 '15 at 8:56
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The reasons you cite are all good ones, and it is indeed the case that learning theorems and their proofs together is an extremely important practice. This perhaps depends upon the mathematics you're doing (certainly true for pure mathematics, perhaps not entirely the case for applications to engineering or other such disciplines), but understanding the proof of a theorem is generally necessary for truly understanding the intuition behind the theorem, the essential ingredients for developing the proper thought-process concerning other similar problems, etc.

That being said, you'll sometimes reach results that are important to simply learn without proof, usually because the proofs are simply much too high-level for your current skills or is just so tediously long that a proof sketch or the like is generally a good substitute. Some examples include, say, Riemann-Roch while studying algebraic geometry at an undergraduate level, Godel's Incompleteness Theorems for logic/set theory at an undergraduate level (though proof sketches are rather easy to give; true verification of all the essential facts is long), Fundamental Theorem of Algebra the first time you're dealing with polynomials (essentially any proof requires the machinery of real or complex analysis), etc.

However, while studying theorems and their proofs is a necessary condition, it is by no means sufficient. One particular thing that I find is extremely important but not necessarily connected to learning theorems and their proofs (though doing this is certainly helpful with regards to it!) is working through problems. While understanding why a result is true is important, the ability to apply those results to derive new ones is at least as important, as it will truly test your understanding of those results and the ideas they present. Related to this is writing carefully your solutions (and not simply just sketches of a solution or only thinking about a solution), as writing your solutions down in a clear manner is useful for making sure your "solution" actually is a solution, as well as developing skills in communicating mathematical ideas.

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  • $\begingroup$ Thank you for a very useful answer. You make a good point about this approach not being capable of being a stand-alone approach to learn math. And indeed pure-math is the first kind of math I want to learn. $\endgroup$ – St.Clair Bij May 28 '15 at 8:54

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