Compton effect: interacting particles or interacting waves

Traditional textbook explanations of the Compton effect treat the photon electron interaction as a particle collision. This explanation is a pedagogical disaster, implying that sometimes interactions are particle-like whereas quantum mechanics always demands that they be wave-like; a photon wavefunction evolves according to a wave equation until its collapse at measurement. If this is so why then does the classical radiation wave equation fail to predict the Compton effect? We address these issues and propose a clearer explanation.