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I have recently been engaging more and more with my creative side by drawing, writing, and (trying to) playing piano. I have come to see mathematics as much more of a creative field of study than I have previously and it got me thinking...Is there any research into creativity or creative aspects of mathematics? Specifically, dealing with the teaching and learning of math, as we are teachers after all :)

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    $\begingroup$ One of my professors one day just stopped in the middle of lecture and stated along the lines of "Math is more of an art than science. There is an art to constructing beautiful proofs." $\endgroup$ – Chris C Mar 22 '15 at 16:10
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Yes, there is a growing literature at the nexus of mathematics education and creativity. The main name to know is Bharath Sriraman (google scholar) though the classic pieces to read for mathematical creativity are the book by Hadamard (see the answer of Lucas Virgili) as well as the chapter Mathematical Creation by Poincare that served to inspire the aforementioned work.

You may also wish to check out several earlier questions (to which I provided some sort of answer) that mention mathematical creativity; these include the following trio:

  1. MESE 4093: Moving From Rote Learning To Creative Thinking

  2. MESE 2236: Jason Padgett's “Struck by Genius”

  3. MESE 630: How can mathematics educators encourage innovation and creativity?

The above links and their many (many) references should give you a good start, and may lead you to refine your question into something more specific. I should also note that there is a literature on creativity and teaching that is not necessarily connected to mathematics, but which is in many cases (potentially) applicable! To this end, I think R. Keith Sawyer (google scholar) is producing some interesting work; see, e.g., his book Structure and Improvisation in Creative Teaching.

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    $\begingroup$ Sriraman: "social interaction, imagery, heuristics, intuition, and proof were the common characteristics of mathematical creativity." Thanks for this reference! $\endgroup$ – Joseph O'Rourke Mar 21 '15 at 22:58
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I believe you would like Jacques Hadamard, The Mathematician's Mind, which has changed the name since I've read it (it was called The Psychology of Invention in the Mathematical Field).

From the synopsis:

It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity.

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If you are looking for information about relation math and creativity, you should take in mind something about intuition. This topic is very important in the field of mathematics education. For it, you may read Fischbein's book.

Fischbein, E. (1987). Intuition in science and mathematics: An educational approach (Vol. 5). Springer Science & Business Media.

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Decided on elaborating my comment.

You might find the article The experience of mathematical beauty and its neural correlates (Front. Hum. Neurosci., 13 February 2014 | doi: 10.3389/fnhum.2014.00068) interesting. They found that some mathematics (such as $e^{i \pi} +1 = 0$) activates similar parts of the brain for mathematicians that viewing a work of art or other sources does. Thus there is an art to creating proofs, equations, and statements in aesthetically pleasing ways.

The BBC has a nice article on it: Mathematics: Why the brain sees maths as beauty.

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    $\begingroup$ My recollection from when I read this study was that it wasn't clear, to me, that the mathematicians were experiencing the mathematics underlying the presented formulae, or whether it was a matter of the representations (so that Euler's identity looks more appealing than a Ramanujan morass based more on its compactness/parsimony than because of the mathematics it encodes). $\endgroup$ – Benjamin Dickman Mar 22 '15 at 18:17
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    $\begingroup$ That is a good distinction. I would say it, at least, shows that we can strive for an aesthetically pleasing formulation of the mathematics. It is always good to go back over a proof looking for refinements. $\endgroup$ – Chris C Mar 23 '15 at 2:33

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