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    $\begingroup$ As a counterpoint, I never understood Jordan normal form until I learned that it was a special case of the classification of finitely generated modules over a PID. In other words, my difficulty with Jordan normal form came from teaching this application of representation theory before the subject! $\endgroup$ Commented May 28, 2015 at 22:14
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    $\begingroup$ Both of your points are true. It is a good idea to bring up the Jordan normal form before the theory of modules over a PID, but it is not at all necessary to teach its proof and the algorithm before the general case of a PID. $\endgroup$ Commented May 29, 2015 at 0:12
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    $\begingroup$ Well, I think most good teaching is either motivated theory or theoretically sound applications, because these two things should almost never live without each other. $\endgroup$ Commented Oct 20, 2016 at 20:38