[yodsanklai]

These problems are examples of questions that can be tackled with the help of mathematical tools. They can show students some applications of mathematics, but IMHO they don't really teach mathematics. Maths have to do with statements and proofs about abstract objects. I'd rather learn about euclidean geometry than trying to figure out the number of piano tuner in manhattan (I'm sure most students will be bored either way).

Actually, this is an endless source of discussion among instructors, not only maths. Should classes be driven by applications in order to motivate students?

[danarmak]

Do applications motivate students? Personally I would expect examples like these to de-motivate, since they make clear that the subject does not, in fact, apply to the students' idea of real life.

If you teach abstract mathematics and make vague promises they will be really useful, some students may believe you. But if you try to demonstrate how they're useful, and the best examples you can come up with are estimating the number of piano tuners in Chicago or the number of air molecules in the room, things students know they won't ever need to do in their lives, they should become less interested.