How does Compton scattering demonstrate particle over wave behavior? Why is Compton scattering thought to demonstrate light's behavior as a particle over that as a wave. I'm interested in the thoughts at the time of Compton, but also how it contradicts current theory of light-waves. 
The closest I've found to an answer to my question are statements such as this: "[Compton's] data were inconsistent with the wave theory, which predicted that the scattered light would spread in all directions, like water waves radiating from a stone dropped in a pond."
Is this statement true, and is it saying that in Compton's time, wave theory predicted that a single light-wave would scatter in all directions...as in Huygens' "spherical" waves? Or am I not understanding the statement correctly?
 A: I think this is an interesting question. Unfortunately, many hasty sketches of the history of physics, as they are taught, tend to draw somewhat biased conclusions for the sole purpose of avoiding delving into these types of questions (some people consider it to be a waste of time apparently).
As far as I can tell, the classical scattering theory at the time was essentially Thomson scattering theory which is basically an elastic scattering limit of the Compton scattering.
Since it is based on the assumption of an elastic scattering, it cannot explain what is observed for Compton scattering where, if anything, the frequency of the light is observed to change.
Now, to enter into a proper inelastic model one must try to figure out where goes the energy of the wave and propose an actual model for it, especially when using classical wave theory; and that's the difficult part.
In looking for references recently, I came across this paper by Raman (the same gentleman as for the Raman effect) who concludes his paper in a retrospectively surprising way by saying:

Incidentally, it becomes evident why the argument of the "triangle of
  momenta" by which Compton obtained his formulae gives the same result
  for the change of wavelength as the classical theory of the emission of
  spherical secondary waves by the electron.

Although there is no doubt that QED does a great job describing light-matter interaction better than any theory has done so far, it is thus safer to refrain ourselves from setting sharp demarcation lines with historical experiments allegedly signing the death warrant of classical theories; it is always much more complicated than we think.
