Why are we still teaching Newtonian physics if it's "false"? Why are we teaching an outdated theory when the math in general relativity isn't that hard? I really don't see a reason why we should teach a highly oversimplified theory in our schools. For me, it's like teaching that atoms are small balls that cannot be divided any further.
 A: Two reasons.
First, I strongly recommend Asimov's essay Relativity of Wrong that explains very concisely and clearly why questions such as this one are more wrong than Newtonian physics.
Second, the concepts of Newtonian physics are necessary for explaining pretty much anything in physics, including general relativity and quantum mechanics. These explanations usually go like this: "Take Newtonian physics and slightly modify this parameter (for GR) or that parameter (for QM)".
A: Classical mechanics are very important for everyday physics. For the energy scales, relative velocity differences, and mass scales that we experience is our everyday lives, Newtonian physics provide us with an extremely valuable tool of predicting outcomes of events. In other words Newtonian physics are an accurate enough approximation to the more precise theory, special relativity.
Lets not forget that Newtonian physics is accurate enough to take us to the moon!
Other than that some conservation laws from Newtonian physics carry on to the rest of physics, for example conservation of energy and momentum. A young physicist would therefore need a lot of experience with applying these laws, and the best way to do this is with the more 'intuitive' Newtonian physics. 
A: Because you dont need general relativistic (tensors ,differential geometry etc) calculation to send a rocket to the moon.
A: 
(1) We shall never have the right to affirm categorically of any one of the principles of the mechanical and physical theory, that it is true. (2) We are not allowed to affirm of any one of the principles on which the mechanical and physical theory rests, that it is false, so long as there has been no discovery of phenomena that disagree with the consequences of the deduction of which this principle constitutes one of the premises.
What I have just said applies particularly to the principle of inertia [or, in this case, the principles of GR]. The physicist has not the right no say it is certainly true; but still less has he the right to say it is false, since we have so far met with no phenomenon (if we leave out of consideration the circumstances in which the free will of man intervenes) that compels us to construe a physical theory from which this principle would be excluded.

—Pierre Duhem, "Note on the Validity of the Principles of Inertia and Conservation of Energy"
