(UNLOCKED May 10 2017; there is no reason for this still to be locked.)
Hunter tells the following story about how hard it is to predict what mistakes a student will make:
Hunter is teaching a section of the university's "calc 2" class. This includes very basic differential equations -- what they are, slope fields and that sort of thing, and how to solve separable ones.
A not-so-uncommon sort of problem given to the students is "verify that the following function satisfies the following differential equation". Despite this being extremely straightforward, students tend to do poorly on these because they're not sure what's going on. Or, commonly, once they know how, they'll solve the differential equation from scratch and verify that they got the expected answer, rather than doing it the easy way. (Hunter thinks it's because they don't quite understand what a differential equation means. My own opinion is that the problem is more fundamental than that, because my experience suggests to me that they would probably do the same with any equation, not just a differential one; but all this is beside the point.)
Anyway. In order to avoid this, Hunter decided to explicitly do a bunch of examples of this in class. He considered several differential equations, and the functions ex, sin(x), and cos(x). For each differential equation, he checked which of the three satisfied it, pointing out such facts as there could be more than one solution, etc. (Wow, that sounds like a really good idea, said Kurt and I upon hearing this. Surely after this the students would understand what to do!)
But that night he got an email from a student asking for help on a specific problem. It was on the web homework, and it was a multiple choice problem; it gave a differential equation, and asked which of the listed functions satisfied it. The student had written to Hunter because he did not understand what the problem was asking -- because the choices weren't ex, sin(x), and cos(x)!
-Harry
Hunter tells the following story about how hard it is to predict what mistakes a student will make:
Hunter is teaching a section of the university's "calc 2" class. This includes very basic differential equations -- what they are, slope fields and that sort of thing, and how to solve separable ones.
A not-so-uncommon sort of problem given to the students is "verify that the following function satisfies the following differential equation". Despite this being extremely straightforward, students tend to do poorly on these because they're not sure what's going on. Or, commonly, once they know how, they'll solve the differential equation from scratch and verify that they got the expected answer, rather than doing it the easy way. (Hunter thinks it's because they don't quite understand what a differential equation means. My own opinion is that the problem is more fundamental than that, because my experience suggests to me that they would probably do the same with any equation, not just a differential one; but all this is beside the point.)
Anyway. In order to avoid this, Hunter decided to explicitly do a bunch of examples of this in class. He considered several differential equations, and the functions ex, sin(x), and cos(x). For each differential equation, he checked which of the three satisfied it, pointing out such facts as there could be more than one solution, etc. (Wow, that sounds like a really good idea, said Kurt and I upon hearing this. Surely after this the students would understand what to do!)
But that night he got an email from a student asking for help on a specific problem. It was on the web homework, and it was a multiple choice problem; it gave a differential equation, and asked which of the listed functions satisfied it. The student had written to Hunter because he did not understand what the problem was asking -- because the choices weren't ex, sin(x), and cos(x)!
-Harry
