Is wave-particle duality a real concept or a pedagogical tool? In a less opinion-based way: what are particle properties and wave properties of a particle (that we speak of particle properties of particles nicely encompasses the problem).
Judging by the recurrent themes in the questions about the basics of quantum mechanics in this community, many difficulties arise from not being able to distinguish the particle properties from the classical properties (there is less confusion about waves, perhaps because people new to QM rarely have much intuition from wave theory).
Let me give two examples:
We say that, if the size of slits is comparable to the de Broglie wave length, the electron behaves as a wave. In the other limit, if the slits are very wide, it will behave as a particle. This is the expression of the duality.
However, if we were to approach this situation rigorously, we would treat the electron as a wave: solve the Schrödinger equation and demonstrate that for wide slits the coarse-grained probability distribution is particle like. Particle-like behavior here refers to a classical trajectory, and is an approximation to "true" wave-like behavior.
Many questions arise from naively associating photons with particles, having momentum and trajectory. In reality, what makes the photons particle-like is their countability.
Thus, I would say that the main particle-like property is the countability, while the wave-like properties are due to the mode structure described by a partial differential equation (Schrödinger, Dirac, Maxwell, etc.) On the other hand, having simultaneously measured velocity and position is not a particle-like, but a classical particle property. Am I still missing something?