Metal, Mantle, DX12, GLnext, etc are The Next Big Thing™ and I wanted to get some practice in with them. A texture mapping demo won't light anybody's world on fire, but it serves as a good unit test.
Alright, fair enough, but just fyi I can tell you that you won't get much experience with the new APIs by doing these small demos though. I have hands-on experience with more than one of those APIs you listed here, and while I can't go into specifics I can tell you that doing these very basic things will barely require you to learn any new concepts at all within these new frameworks.
The whole concept of shading or texture mapping are not getting changed or replaced. As a matter of fact, you could probably even use your existing DX11 shaders on a couple of these new platforms without having to make any changes to them at all. These new APIs are mostly about the bigger picture and the optimizations which can be made by reducing driver overhead and state changes and by giving the developer more control over his/her application. The only way to really master those concepts is by having a more fleshed out demo in which you can actually play around with and see the effects of these changes.