When asked to explain the Galilean non-invariance of classical electromagnetism on the basis of pre-relativistic considerations alone, students--and sometimes their teachers too--may face an impasse. Indeed, they often argue that a pre-relativistic physicist could most obviously have provided the explanation "at a glance", on the basis of the presence of a parameter "c" with the dimensions of a velocity in Maxwell's equations, being well aware of the fact that "any" velocity is non-invariant in Galilean relativity. This "obvious" answer, however popular, is not correct due to the actual observer-invariance of the Maxwell parameter "c" in pre-relativistic physics too. A pre-relativistic physicist would therefore have needed a different explanation. Playing the role of this physicist, we pedagogically show how a proof of the Galilean non-invariance of classical electromagnetism can be obtained, resting on simple pre-relativistic considerations alone. (Contains 8 footnotes.)