Mostly they teach various theorems and their proofs, so that you can "stand on the shoulders of giants", and they try to train your "intuition" (incommunicable heuristics) by exercise in proving simple theorems.
More advanced math courses teach explicit heuristics (in the form of "proof strategies": general patterns that are often found in proofs), which can be programmed in an automated theorem prover to some extent, but they are not good enough to make up for the lack of human-level intuition.
On the other hand we know excellent heuristics for chess, in the form of position evaluation functions that give an higher score to positions which are more likely to lead to a victory. The chess piece relative value heuristic, for instance, is extremely simple and yet very effective. High-level computer chess programs use much more complicated evaluation functions, but chess piece relative value is usually the dominant component of it.
Ganesalingam and Gowers recently released an ArXiV preprint of their article "A fully automatic problem solver with human-style output" about a theorem-proving program they have written that generates human-readable output. Gowers had blogged about the program in April 2013, after soliciting responses from readers regarding answers generated by the program and by humans (without explicitly revealing that one of the answerers was a machine) in March 2013.
There do exist theorem-proving systems such as Coq. In fact, it was announced about a year ago that the proof of the Feit-Thompson odd-order theorem, which originally appeared in a 255-page journal article in 1963, had been completely verified using Coq. But the Gowers-Ganesalingam program differs in that the output of this program is a human-style proof (for better or worse) whereas Coq uses a formal language syntax (see here for a small part of the proof of the Feit-Thompson theorem using Coq).
The ArXiV preprint contains details of the program and how the authors believe it improves on existing programs. One of the ways they believe they are better than past provers is their ability to more explicitly code for picking promising lines of reasoning.
I'd appreciate comments from LessWrong readers about the specifics of the Ganesalingam-Gowers program, and any broader ramifications of their ability to create this program for predictions related to artificial intelligence.