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Back to slope fields again, sorry
Posted:
Aug 21, 1997 5:23 PM


Why we are putting emphasis on slope fields? Unless they are particulaly simple it is sometimes hard to figure out by inspection what function they represent (ie, they can be frustrating to use). Mathematically, they do not seem to convey much in comparison to the phase plane (graphing a function against its derivative). The phase plane for the pendulum, for example, is an ellipse, and gets particularly interesting when you look at the effect of damping or resonance.
Another reason to question the emphasis on slope fields is that we are already teaching students how to reconstruct a family of functions using the graph of a derivative. It seems that this could be sufficient for understanding what the derivative is graphically.
Finally, it will be important later on for students to become familiar with vector fields. Won't slope fields just add to their confusion when they study vector fields? Maybe we should teach vector fields instead of slope fields, since they are a nice calculus application of parametrics as well as vectors, both of which are currently receiving emphasis on the AP. This could be done at a level suitable for HS students.
I've looked at a number of books trying to see if I'm just missing something and almost wonder if our interest in slope fields stems from the fact that they are easy to graph on the calculator (tail wagging dog). Any thoughts on this would be appreciated.



