Teaching and playing the game of chess has often been advocated as a form of mental training.
Benjamin Franklin, in his article The Morals of Chess (1750), advocated such a view:
"The Game of Chess is not merely an idle amusement; several very valuable qualities of the mind, useful in the course of human life, are to be acquired and strengthened by it, so as to become habits ready on all occasions; for life is a kind of Chess, in which we have often points to gain, and competitors or adversaries to contend with, and in which there is a vast variety of good and ill events, that are, in some degree, the effect of prudence, or the want of it. By playing at Chess then, we may learn: 1st, Foresight, which looks a little into futurity, and considers the consequences that may attend an action ... 2nd, Circumspection, which surveys the whole Chess-board, or scene of action: - the relation of the several Pieces, and their situations; ... 3rd, Caution, not to make our moves too hastily...."
The U.S. Chess Center in Washington, D.C., teaches chess to children, especially those in the inner city, "as a means of improving their academic and social skills."
There are a number of experiments that suggest that learning and playing chess does, indeed, aid the mind in certain ways. The U.S. Chess Federation (USCF) chess research bibliography contains a collection of many such experimental results.
Benjamin Franklin: The Morals of Chess
U.S. Chess Center
USCF Chess Research Bibliography