In Physics, one is often misled in thinking that the mathematical model of a system is part of or is  that system itself. Think of expressions commonly used in physics  like « point» particle, motion «on the line»  «smooth» observables, wave function, and even «going to infinity» , without forgetting perplexing phrases like «classical world»  versus «quantum world» ... On the other hand, when a mathematical model becomes really inoperative  in regard with correct predictions, one is forced to replace it with a new one. It is precisely what happened with the emergence of quantum physics. Classical models were (progressively) superseded by quantum ones through quantization  prescriptions.  These procedures appear often as ad hoc recipes. I will describe  well defined quantizations,  based on integral calculus and  Weyl-Heisenberg symmetry. They are described in simple terms through one of the most basic examples of mechanics. Starting from  probability distribution(s) on the Euclidean plane viewed as the phase space for the motion of a point particle on the line, i.e., its classical model, I will show how to build corresponding quantum model(s) and associated probabilities (e.g. Husimi) or quasi-probabilities (e.g. Wigner) distributions. I will highlight  the regularizing role of such procedures  with  the familiar example of the  motion of a particle with a variable mass and submitted to a step potential. 

