We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation solver in Mathematica, and analytically constructing solutions from reasoned building blocks. We obtain graphical feedback as soon as possible in each approach and permit "what if" modeling wherever possible. This approach is contrasted with the usual Fourier series development and series solution using boundary value solution strategies. (Contains 9 figures.)