not a well-planned exam, since young minds are fragile.

when i taught for the first time as a graduate student, i made an error in arithmetic.

it took me 10 minutes to solve a quiz i wrote, so i figured that my students could do it in 20.

how wrong i was; half the papers had blanks in one or more parts of the last question (there were only two problems).

since then, i used a factor of three to gauge time.

now it appears that i should change that.
the average score is about 57%.

for my classes last week, the midterm i wrote took about 16-17 minutes for me to solve. it was supposed to be a 50-minute exam. i even chose variants of the practice problems that were listed in the course schedule and examples i did in class.

after grading it, i think i have to use a factor of four now.

sometimes you can tell that a student has some understanding, but panicks due to a lack of time. on many papers i see good work scratched out, and oversimplified, incorrect work takes its place.

the more i think about it, the more crucial it is to have an exam that students can do in the time allotted, and some with a few minutes to spare.

only two students in a class of 62 finished early [1], and in another class of 48, nobody finished early. if your best students need the entire exam period, then this is a bad sign ..!

there is another reason: give a student an exam that (s)he cannot finish in the time allotted. that only reinforces the suspicion that yes, i am bad at math. then, depressingly, (s)he just stops trying.

when the exam is curved, the instructor knows that as long as everyone has done comparatively badly, the majority of student grades will go unscathed.

but some students never realise this. they only consider their own performance, and in this american culture, it seems that "if you can't get the correct answer quickly, then you aren't good at what you're doing."

i don't believe in that, myself -- mistakes are a natural part of learning -- but i'm only reporting what i observe.

i haven't returned the exam yet. already, though, some students have scheduled appointments with me, in efforts to determine "what they are doing wrong."

---

there's also something unusual about our syllabus, at least to me. at this university, students learn about the definite integral and methods of integration at the end of calculus i, which is confusing to me.

subsequently, one assumes that students know how to integrate at the start of Calculus II. in two lectures, we are supposed to cover all of the usual methods of integration -- substitution, parts, trigonometry, partial fractions!

this sounds to me like a recipe for disaster. there are too many schools who do not cover integration in calc i -- michigan, for one -- and there are always transfer students.

each method, from my own habits, takes its own lecture to learn. there's barely enough time to run through the characteristic examples of each of them, in 3 lectures!

other topics of the calc ii syllabus strike me as odd:

1. methods of integration, numerical methods, improper integrals
   
2. applications to geometry (areas, volumes, arclength) and to physics
   
3. differential equations, including second-order linear homogeneous ODE ?!?
   
4. power series and the like;
   
5. three-dimensional geometry ?!?
   

this is .. a lot. moreover, the topics are less cohesive than i'd like. it's good to learn geometry, but wouldn't it be more fitting to fit it into multivariable calculus (calc iii)? [2]

this is an unfounded suspicion, but i suspect that there was some politicking, which led to this syllabus.

perhaps the school of engineering complained that their students don't know how to solve ODE early enough in their training, and demanded an accelerated program.

if this is true, then they never accounted for the fact that it actually takes time to learn anything of substance. heck, it took the great minds of newton and leibniz to invent calculus. moreover, leibniz was interested in computational engines, and wanted to make calculus accessible for applications.

put another way, engineers: if you think that calculus is easy, it's because leibniz designed its implementation that way.

take, for example, the standard problem of showing that



using only the definition of a derivative. you actually have to know how to use the binomial theorem to understand this fact!

even putting n=3 causes trouble to most students. judging from how much trouble students have with using (not proving) even the quadratic formula, be glad that leibniz had an applied mind!

all i know is: any decent calculus instructor would have known that integration is hard enough for students, and it would be folly to demand more of them otherwise, and more quickly.

in retrospect, i should have ignored the syllabus and incorporated basic methods of integration questions on the exam. i should have realised what my students would find difficult.

most of the problems, instead, used integration at one step, such as improper integrals, hydrostatic pressure and force, ODE ..

[sighs]

why do i get the feeling that the next two weeks will be full of instructional "damage control?"

[1] "early" means handing in the exam before the last 5-10 minutes. there will always be some students that will try to leave a few minutes early, if only to make it to their next class early or simply because they can't stand looking at the exam any more.

as a result, you cannot trust these flighty students to gauge how hard the exam was.

[2] having taught calc 3 here, last year, i can assure you that these geometric topics are assumed and there is no budgeted time to review them.