Gian-Carlo Rota (1997): Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations: "One of many mistakes of my youth was writing a textbook in ordinary differential equations.... It led me to realize that I had no idea what a differential equation is.... Some of my colleagues have publicly announced that they would rather resign from MIT than lecture in sophomore differential equations. No such threat is available to me, since I am incorrectly labeled as the one member of the department who is supposed to have some expertise in the subject, guilty of writing an elementary textbook still in print. The Administrative Director of the MIT mathematics department, who exercises supreme authority upon the faculty’s teaching, has only to wave a copy of my book at me, while staring at me in silence. At her prompting, I bow and fall into line; I will be the lecturer in the dreaded course for one more year, and I will repeat the mistakes I have been making every year since I first taught differential equations in 1958...
...It is hopeless to expect that I will correct any of my mistakes at this stage of life. To allay my feelings of guilt, I will resort to a ruse. I will present them to you in the attractive literary form of the decalogue. The goofs, gaffes, misunderstandings, and prejudices I am about to list are not exactly hot off the press, and you may find them cloyingly familiar. Why, then, make a public spectacle of them? Well, I myself always find it gratifying to listen to opinions I agree with, and I surmise that you may feel likewise as you listen to my tirade.
Most of the material now taught in an introductory differential equations course is hopelessly obsolete.... The text (any text) will list a number of disconnected tricks that are passed off as useful, such as exact equations, integrating factors, homogeneous differential equations, and similarly preposterous techniques. Since it is rare–to put it gently–to find a differential equation of this kind ever occurring in engineering practice, the exercises provided along with these topics are of limited scope: as a matter of fact, the same sets of exercises have been coming down the pike with little change since Euler.... The sophomore course in differential equations will never be reformed. It will die a natural death, and it will be replaced by several shorter courses that will deal with realistic aspects of differential equations. It is to be hoped that these new courses will be taught by mathematicians rather than by engineers: the budget of any mathematics department is entirely dependent on the number of engineering students enrolled in our elementary courses. Were it not for these courses, which engineers generously defer to mathematicians, our mathematics departments would be doomed to extinction....
Reduce to a minimum the discussion of first order differential equations at the beginning of the course.... None of Boole’s beautiful techniques is of any conceivable use to anyone who deals with differential equations today.... Teaching a subject of which no honest examples can be given is, in my opinion, demoralizing.... teach changes of variables.... forget about existence and uniqueness of solutions.... linear systems with constant coefficients are the meat and potatoes of the course.... stay away from differentials...
#noted
