A wave-particle dual nature was discovered and publicized in the early debate about whether light was composed of particles or wave. Evidence for the description of light-as-waves was well established at the turn of the century when the photoelectric effect introduced firm evidence of a light-as-particle nature. This dual nature was found to also be characteristic of electrons. Electron particle nature properties were well documented when the DeBroglie hypothesis, and subsequent experiments by Davisson and Germer, established the wave nature of the electron.
This movie helps students to better distinguish when nano-things behave as particles and when they behave as waves. The link below connects to an exercise on these concepts.