The Dodecahedron is a character from the book The Phantom Tollbooth by Norton Juster. He lives in the city of Digitopolis at the base of the Mountains of Ignorance. Here is his description from the book (page 145)
 

He was constructed (for that’s really the only way to describe him) of a large assortment of lines and angles connected togehter into one solid many-sided shape — womewhat like a cube that’s had all its corners cut off and then had all its corners cut off again. Each of the edges was neatly labelled with a small letter, and each of the angles with a large one. He wore a handsome beret on top, and peering intently from one of his several surfaces was a very serious face. Perhaps if you look at the picture you’ll know what I mean.

 
Now we can learn a lot from what the Dodecahedron says. Look at this exchange (page 148):

“I’m not very good at problems,” admitted Milo.
 
“What a shame,” sighed the Dodecahedron. “They’re so very useful. Why, did you know that if a beaver two feet long with a tail a foot and a half long can build a dam twelve feet high and six feet wide in two days, all you would need to build the Kariba Dam is a beaher sixty-eight feet long with a fifty-one foot tail?”
 
“Where would you find a beaver as big as that?” grumbled the Humbug as his pencil point snapped.
 
“I’m sure I don’t know,” he replied, “but if you did, you’d certainly know what to do with him.”
 
“That’s absurd,” objected Milo, whose head was spinning from all the numbers and questions.
 
“That may be true,” he acknoledged, “but it’s completely accurate, and as long as the answer is right, who cares if the question is wrong? If you want sense, you’ll have to make it yourself.”

 
To me, this encapsulates a lot about how mathematicians think: to a mathematician, it’s the problem that’s the interesting thing, not the usefulness. In fact, we even define usefulness differently — note how the Dodecahedron uses his beaver example to show the usefulness of problems. Clearly this is not the same as the Humbug’s definition of usefulness. I rather suspect that to the Dodecahedron, it’s useful because it highlights how maths can solve problems — whether the answer is realistic or useful is a side issue.
 
Still, Milo clearly doesn’t get it, but I’m not sure it’s the maths itself that’s the problem — it’s the mathematician: Milo and the Dodecahedron think differently about what is useful. Maybe as teachers, we should help the students understand mathematicians a little more, as opposed to just understanding mathematics.