Two classic examples (Cartesian vs polar coordinates, frequency vs time domain) are mentioned in other comments. I would certainly add "looking at values vs lo

Two classic examples (Cartesian vs polar coordinates, frequency vs time domain) are mentioned in other comments. I would certainly add "looking at values vs looking at differences" and "looking at the generator vs looking at what's generated" (which the top level story is an example of). It may take a little squinting, but we can see the common mathematical practice of reducing one problem to an example of another as an instance of this. There are also tweakier math examples where treating something as topology or something makes it easy to solve.