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The popularity of books like Visual Group Theory and Visual Complex Analysis validates the importance of pictures and visualization for complex subjects. Unfortunately, I'm not aware of similar books for other common subjects like linear algebra or real analysis. I hope there will be. For now though, what can students do about this and how can they help themselves?

Maybe this could be a list of suggestions, like the Group Explorer software from Nathan Carter, Visual Group Theory's author?

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A few suggestions for things that might improve the situation:

(1) Buy the books that are well-illustrated. This will indicate to authors and publishers that illustrations are important. Another very well illustrated one is Trefethen's book on approximation. See also this question.

(2) Learn to use the graphical tools in packages like Matlab and Mathematica. Good for situations where you have objects described by known equations.

(3) Get proficient with some free-hand drawing software package, preferably a 3D one. Good for those (more common) situations where you don't have any equations.

(4) Learn to draw by hand. Personally, I think drawing by hand is best for learning and conceptualizing; using a computer typically interrupts the thought process (for me, anyway). Take a drafting class (if you can find one these days) or an art class. There is an entire chapter explaining how to draw good diagrams in this book by Koenderink . You can see some examples on its cover. Koenderink's advice is especially interesting (to me, at least) because he knows a lot about human vision/perception. Simply making your drawings "accurate" or "correct" is not really the point.

(5) If some lecturer tells you that "pictures have no place in mathematics because they do not constitute proof", walk out of the room.

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    $\begingroup$ Excellent answer! Thanks! $\endgroup$ – Pamela Lee Mar 9 '14 at 8:46

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