From R. G. Collingwood, The Principles of Art (with thanks to David Pierce, who recommended me the book):
This is not because (as Oscar Wilde said, with his curious talent for just missing a truth and then giving himself a prize for hitting it) ‘all art is quite useless’, for it is not; a work of art may very well amuse, instruct, puzzle, exhort, and so forth, without ceasing to be art, and in this ways it may be very useful indeed. It is because, as Oscar Wilde perhaps meant to say, what makes it art is not the same as what makes it useful.
Of course, G. H. Hardy’s famous saying immediately crosses mind:
The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’.
Now, rephrasing of Collingwood’s maxim for mathematics is obvious:
What makes it mathematics is not the same as what makes it useful.
A paper by Paul Andrews The curricular importance of mathematics: a comparison of English and Hungarian teachers’
espoused beliefs, J. Curriculum Studies 39 no. 3 (2007), 317–338, shows that English school teachers of of mathematics emphasise that mathematics is useful:
Their number work is extremely important and that’s just fundamental. … If they can’t use numbers then how can they go on to use anything else? And how can they see how things happen? I think just putting everything into a practicalsense so that they know they can measure dimensions. They know, if they’re going to put up wall-paper or whatever, that they can estimate how much they’re going to need. Estimate, being able to estimate, is important. How much is it going to cost me, roughly, that sort of thing.
But Hungarian teachers put emphasis on mathematics for its own sake, emphasising, for example, problem solving aspects of mathematics:
I think it is very important to make children realize that mathematics is actually solving problems. … So I find it really important to help them collect what they know in connection with the problem, what conditions they know, what knowledge they have, and how they can order it to create a solution. … You can’t be sure they always get to the solution, but it’s still good if they can find a way where, or how, they can look for a solution. The other important thing is that … children have the right to make mistakes. They have to learn that when solving a problem, if they make an error, it doesn’t matter. It’s a part of solving the problem … and goes with any kind of problem-solving, particularly in mathematics. And if they get this experience in mathematics, it helps them get through problems in life too. … It’s very important for children to understand that a problem is solved when they have … checked it and are sure that there are no other solutions; and if there are other solutions, then they have found them all.