Spring Break

For many people, this is the time of year that students look forward to a week's worth of debauchery and drowning whatever little information they've gained over the year up to this point.  This is also the time when I  openly admire my colleagues in the liberal arts who enjoy assigning large writing assignments over the extended break.  There is untold joy to be found by taking a walk through the campus library in the middle of the break and see the harried and frustrated students surrounded by books or typing away on their laptops.

For math teachers, the situation is a little more delicate, in my opinion.  Sure, you could assign a project over the break.  However, having to deal with a lecture hall's worth of students who can barely compose a text message, much less organize a project, does not sound in the slightest bit appealing to me.  It works better in higher level coursework, but I avoid it for freshman classes.

There is a temptation to give a large exam the day before spring break.  However, there will be all too many people hounding you with excuses to miss the exam, some with "legitimate" reasons that would not be worth proving false.  You do not want to have to deal with makeup exams for half the class, especially if the makeup will happen after spring break, screwing up the timing of everything.

On the other hand, you might want to give an exam the Monday after spring break, claiming that students can "study over the break," knowing full well that almost none of them will do any such thing.  While it won't be as big of an issue, there will still be plenty of people doing their best to weasel out of the test.  While these can often be deflected, it's not worth the trouble when there are better alternatives.

The best thing to do was discovered by a peer who stumbled on it accidentally.  (The fool was actually trying to be reasonable toward his students, and it went disastrously.)  If you want to utterly brutalize your class, while making it seem like you're being reasonable, schedule an exam for the Friday after they get back from spring break.  Then you get the advantage of them having forgotten almost everything over the break, but since it's several days after they get back, they get this sense of security that they will be able to study.  While a few students can handle it, most will not make up for everything they've lost over the break.  The average on that exam will be gloriously abysmal, and they will have no excuse at all for doing poorly.

To add a little bit of sugar on top, I suggest you also have a quiz on the Friday immediately before spring break.  If it's a quiz but not a test, you can declare that there is no way to make it up, and so you can still penalize the large number of students who will be gone.  The lazy and distracted students will fall into both traps, and will come back just about in time to have to face the final drop day and decide what they will do with the rest of their lives.

These little things can even make teachers that are otherwise popular with students twist the knife and make anyone but the best students suffer.  And ultimately, that is what we all want.

Know-it-alls

You've seen them in classes before.  You know the ones I mean.  The students that are desperate to show off their knowledge not only of the class they're in, but of life in general.  I'm not talking about the (moderately) intelligent student who tends to talk more than others, if only because they're sick of sitting in a silent room waiting for anyone to answer the teacher's question so the class can move on.  To some extent I can sympathize with this student - plagued with the feeling of being surrounded by a sea of brain-dead husks.  Yet I don't sympathize with them enough to not enjoy the look of impatience and desperation on their faces.

No, I'm talking about the student who wants the teacher and the entire class to know how smart they are, and impress them with their insight and knowledge.  The sad thing is they tend to be oblivious to the fact that they themselves are the only ones enjoying their performance.  I've seen one student who would pause after every question to look around the room with this self-important smile on his face, eager for everyone to recognize his cleverness.  Yet only he seemed to be unable to hear pockets of the room groan that the teacher had answered this exact question not moments before in the course of lecture.

Even better is the student who tries to start up a conversation with the professor in the middle of a crowded lecture hall on a topic with only the most tenuous connection to the subject at hand.  I've heard of one particular freshman statistics class.  The lecture was supposed to be about probability and basic counting rule, but it ground to a halt when one particular student determined that the discussion should shift toward cardinality and the various kinds of infinity, and then into calculus and analysis and other topics wholly unrelated to the class (at least at that level).

If you think this sort of student gets on my bad side, drives me up the wall, or fills me with untold rage, you're looking at it from the wrong point of view.  Make no mistake, this student will never win points with me through these tactics.  I have delighted in turning down requests for directed study or letters of recommendation, glorying in the looks of dejection on their faces as I tell these students that they were a disruption in my class and would get no favors from me.  But even this is secondary to the misery they impart on their fellow classmates.

For one, the class is annoyed as the student wastes their time with wandering topics irrelevant to anything on the syllabus.  Even better, since class is on a tight schedule, if not everything can be covered in lecture, I simply assume that the students can learn it on their own time.  So even if it's not covered in class, that does not exempt the students from having to learn it for the exam.  It's the misery that keeps magnifying as one student strives to stroke their own ego.

And so this semester I was happy to see one of these know-it-alls show up in my second semester calculus class.  They wanted to talk about vectors when we talked about solids and surfaces of revolution.  At the merest mention of infinite series, this student couldn't help but bring up Fourier series and open up that whole can of worms.  The class was clearly turning on him, and he was blindly forging ahead with his self-importance.  But then one day after he wasted a good ten minutes of class, I saw that imp of a girl pull him aside after class.  After that day, he sat next to her, almost entirely quiet.  It appeared they pass notes back and forth - her way of appeasing his need without disrupting class.

So now she is not only deflecting each effort for me to crush her mind and spirit, but she is working to improve the morale and atmosphere of the class in general.  Unacceptable.

Going backwards

I am learning again to enjoy teaching integral calculus.  Second semester calculus, at least the portion of it relating to integration techniques, is a wonderful garden of delights, filled with hemlock, nightshade, and oleander.  Yet while it is glorious to see the expressions on the students' faces when you first show them volume and surface of revolution, and the formulas therein, I'm thinking of simpler pleasures.

If I had a nickel for every time a student in my class differentiated instead of integrated, or integrated by parts or substitution incorrectly, I could fund a month-long tropical vacation.  And it's sad, really, that it gives them so much trouble.  After all, they spent the better part of a semester practicing every single permutation and contortion of derivatives.  The integral is posed as an antiderivative, integration by substitution is just the chain rule in reverse, and integration by parts is nothing more than the product rule backwards.  So with all the work they've done taking it one direction, you'd think it wouldn't be so hard to just reverse the process.

And yet, I suppose I really shouldn't be surprised.  Don't think I mean this from the point of view that the reversal of many mathematical processes is highly nontrivial (which it is most certainly not, as evidenced by the wealth of research currently being done in various flavors of inverse problems).  No, instead I'm talking about the fact that these children have been stymied by simple reversals as long as they've been doing the most basic arithmetic.

Ask almost anyone who teaches math what sort of mistakes crop up most frequently, and you can almost guarantee you will hear variations of things involving negatives, fractions, and division.  Negatives and subtraction, arise as the inverse of addition.  Fractions and division similarly arise as the inverse of multiplication.  The panic rises whenever students have to do anything with logarithms, the inverse functions for exponentials.  They can add, multiply, and exponentiate all day long.  But ask them to back up and go the other way, and they look at you like you've told them to grow wings on a dog.

Over and over these students stumble whenever asked to do anything in reverse, no matter how little or big, no matter how much preparation and coaxing they are given.  It's sad, as it points to their lack of even having a single dimension of motion in their thinking.  They can only go forwards, never backwards, and heaven help you if you ask them to look left or right.

But that's okay, I will continue to enjoy pulling them forward, shoving them backwards, pushing them left and right, and delighting in their confusion as all the symbols and ideas swirl around them.  Other teachers can spend all their time and energy trying to "reach" the students and have them understand.  I know they are wasting their time, and I will instead focus my energies in better, and more entertaining, directions.

Spring Semester

Against my better judgment, I have actually volunteered to teach Calculus II this semester.  Sure, part of the reason is that one particularly annoying student from Calculus I did far too well last semester, and I want to make sure her ego and her spirit, is crushed this semester.  But beyond that, I have decided to embrace this freshman class.

Why is that?  Well, second semester calculus has a bit of a reputation among students as being the hardest math class in the calculus sequence, if not the curriculum.  I suspect this is partly to do with the fact that many of the students taking this class are in the less technical BS degree programs, possibly even education, where most of the bottom of the barrel will have committed their lives.  These poor souls will never know, or even probably have the ability to comprehend, the beauty in boundary value problems, measure theory, or the structure of rings and varieties.

Beyond the intrinsic mental and curriculum finality of this course, I know there are portions of it that, to the students, seem out of place among all the mechanical computation and memorization of the rest of the sequence.  I speak, of course, of sequences and series.  The understanding of sequences and series is absolutely essential to the entire subject of analysis, especially the underpinnings of calculus itself.  Yet I know the students do their best to resist paying attention during the section on Riemann sums.  I know they'll probably leave this class still trying to pretend that a capital sigma can be treated as a letter "E."  But I will enjoy the look of suffering on their faces as they come to terms with the fact that they will face at least one exam consisting almost entirely of sequences and series, with almost nothing coming out to a solid, concrete answer.

Well, maybe I'll throw them a bone and give them a geometric series.  Maybe even a Taylor polynomial.  One with varying parameters and odd boundary behavior.

A Beacon of Inspiration

Clearly I have been busy and distracted lately.  The transition between semesters will always do that.  But this is not what I'm here to talk about today.  No, today I want to share a story I heard recently.  A story of a person who may be the most wonderful math teacher ever.  Many people will fail to see it that way, so I will make sure to keep everything as anonymous as possible, even though this happened years ago.

There was this university.  It was a small university, with an even smaller math department.  Many students there majored in engineering, but few majored in mathematics.  There wasn't even a masters degree available in math.  As a result, the math classes were plagued by students who were determined to waste their time insisting on practical applications to every problem.  The faculty in the department, naturally, were not terribly sympathetic toward these creatures.

This department did something very interesting.  When they posted their courses for an upcoming semester, they didn't list professor names with any of the classes.  Why?  Well, there was this one particular professor who had a bit of a reputation.  How bad of a reputation?

Imagine you have signed up for a vector calculus class.  There are about forty people signed up for the class at the outset.  The professor walks in, and he has a very distinct look to him.  He walks to the front of the class and looks out over the room.  Half the class stands up, gathers their things, and walks out the room.  This professor then turns and writes his name on the board.  Half the class again gets up and walks out, leaving about ten or so to sit through the lecture.  The next time the class meets, the attendance will have halved yet again, leaving about five students left who had no choice but to take this section of this class.  By the end of the semester, perhaps one or two of the remaining students will have passed the class.  All told, a successful semester.

And what was his big crime?  Proof-based tests that allow no partial credit and no homework.  Cry me a river.  Those engineering students don't deserve their degrees if they can't think logically for a few hours a week.

I must admit, this is the sort of reputation that most teachers can only dream of having.  The reputation of instilling such fear and hate in the student body that the entire department will adjust how they list courses to try to dupe students into taking your classes.

My only criticism is that this technique is a bit too harsh.  If you chase off all your students in the first day, then that leaves altogether too few to torment over the course of the semester.  The best suffering is that held until after the drop date has passed.