Using a sample of 69,919 red giants from the SDSS-III/APOGEE Data Release 12, we measure the distribution of stars in the [α/Fe] versus [Fe/H] plane and the metallicity distribution functions (MDFs) across an unprecedented volume of the Milky Way disk, with radius 3 < R < 15 kpc and height | z| \lt 2 kpc. Stars in the inner disk (R < 5 kpc) lie along a single track in [α/Fe] versus [Fe/H], starting with α-enhanced, metal-poor stars and ending at [α/Fe] ∼ 0 and [Fe/H] ∼ +0.4. At larger radii we find two distinct sequences in [α/Fe] versus [Fe/H] space, with a roughly solar-α sequence that spans a decade in metallicity and a high-α sequence that merges with the low-α sequence at super-solar [Fe/H]. The location of the high-α sequence is nearly constant across the disk however, there are very few high-α stars at R > 11 kpc. The peak of the midplane MDF shifts to lower metallicity at larger R, reflecting the Galactic metallicity gradient. Most strikingly, the shape of the midplane MDF changes systematically with radius, from a negatively skewed distribution at 3 < R < 7 kpc, to a roughly Gaussian distribution at the solar annulus, to a positively skewed shape in the outer Galaxy. For stars with | z| \gt 1 kpc or [α/Fe] > 0.18, the MDF shows little dependence on R. The positive skewness of the outer-disk MDF may be a signature of radial migration; we show that blurring of stellar populations by orbital eccentricities is not enough to explain the reversal of MDF shape, but a simple model of radial migration can do so.