Checking out what is on the other side of the fence
As I wander the internet looking for colleagues who are doing modeling in differential equations I stumble upon lots, I mean lots, of peripheral or related materials. One type of material is what I would call review material or prep materials for cognate courses. Often, these are notes on fundamental review materials for what differential equations or calculus material will be needed for the course under discussion, say in engineering or physics.
One recent find was Basic differential equations (for before the beginning of class) authored by Paulo F Bedaque of the Department of Physics at the University of Maryland. PROF Bedaque offers an informal, indeed chatty three page overview of differential equations. In looking over this material or other sets of materials from other disciplines one gets an idea of what is important to colleagues. Incidentally, this can be accomplished over coffee or a beer, at lunch or in the office, just by talking during a visit. Perhaps just a visit to determine "what do you want your students to know from our differential equations course?"
As an example PROF Bedaque has highlighted in a box (we all know physics texts highlight important stuff in boxes!) the following, "A differential equation involving up to the n-th derivative has a n parameter family of solutions and we need to know n additional conditions to specify one unique solution." So he is telling his students you will need additional conditions to fully determine that unique solution to the differential equation before you. This is something we tell our students early and often.
At one point he says of solutions for differential equations, "Where do these solutions come from ? It does not matter ! You can think that the muses whispered the solutions in my ears. What matters is that, given these solutions, you can verify for yourself that they are, in fact, solutions." What he is telling his students is that they do not have to know how to produce solutions, but rather to confirm that a function IS a solution of the differential equation, and I suspect see the form and relate it to the physics in question posed by the differential equations.
In reading the lines (and in between the lines) of what colleagues say to our (and their) students about differential equations we can learn much about what we might offer our students, what directions and approaches we might suggest to them, and what we might emphasize for the downstream client discipline which is forcing them to take a differential equations course.
Listen to the other discipline muses is the lesson here.
Incidentally, we can learn from our mathematics colleagues and "steal" their advice to their students. For example in a Hand Out on MATH 274 Differential Equations we find,
"The purpose of written homework and activities in this course is to develop skills in understanding and communicating mathematics. It is not to give you busy work. Don’t think of your homework paper as a certificate proving that you have done the assignment. Think of the homework as an exercise in learning, and then reporting what you have learned. There is a lot of truth in the statement, “If you can’t explain it, you don’t understand it.” Communicate with the reader. Don’t write to the instructor (who already knows how to do the problems), but explain your solutions as if working with someone who needs help, perhaps a classmate who has been sick. Start at the beginning and be clear, logical, and complete.
"The ultimate test of what you write is this: Can someone learn from your paper? Easily? Remember, the reader will see only what you wrote, not what you meant to say. So all of your work must all be there, and be accurate. Make you paper reader friendly."
This is good advice for any student and written well enough that it might be worth repeating in your syllabus.