The buzzing B

And nonplussed

The end of each semester is a time for reflection and renewal. The school term is over, the new term has yet to begin, and the days are free for contemplation, consideration, and ... complaining students. You can always count on the student who learned the “squeaky wheel” adage better than he learned the subject matter. He imagines that his grade is negotiable and fails to note that no negotiating is actually occurring. It can take weeks for the spate of wheedling communiqués to peter out.

If ever there was a time to consider a grading scheme where if the majority of your exams are A's including the final you get an A. My dad said he got a math teacher to bump him up a grade by doing a card trick.  Are you game?

Family legends and Rudyard Kipling notwithstanding (“If you can make one heap of all your winnings and risk it on one turn of pitch-and-toss”), skill at sleight of hand does not translate into grade points in my class. Sorry about that. He moved on to plan B:

My grandmother would give an A if you got an A on the final but maybe she gave harder finals or something.  

Not to criticize the young man's sainted grandmother, with whom I should never be confused, the old girl was offering her students the ancient “sucker bet” routine. I've seen it often enough before. It's deadly.

When an instructor tells a class at the beginning of the semester that grades will be based on either an overall average or the final exam score, whichever is best, a significant minority of the students immediately falls into the trap: I just need to do well on the final. That'll be enough! Of course it never is. Most such students begin slacking off in that class in order to concentrate on other courses or activities. More immediate concerns take over because I just need to do well on the final. They dig the hole deep, taking comfort in the thought that a single Olympian jump at the end will permit them to escape their subterranean situation, even as they neglect the exercises that would make the feat feasible (and, more to the point, unnecessary, because they would be earning the points that would put them into a position to pass without a miraculously redemptive performance on the final exam).

I never offer my students the sucker bet. My student was undeterred.

Just curious now; did anyone else get two A's on exams and an A on the final and still not get an A?  Is there anyone I can commiserate with or is this an anomaly?

Although misery may love company, privacy considerations intervened. I answered him:

Yes, there were two other students, so it wasn’t exactly an anomaly. It was a matter of getting relatively low A’s that were counterbalanced by lower grades, preventing the composite score from being in the A range. You’d be welcome to commiserate with them, but privacy concerns forbid me from sharing their names. —Z

My student kept harping on his “majority” argument and insisted on ignoring the relative strength of his scores. The semester grade was a weighted average of six scores: one for homework and quizzes, four chapter tests, and one final exam. The composite score was computed thus:

Comp = 0.15*HQ + 0.70*E_average + 0.15*Final

My diligent correspondent had a low A for HQ, a high C for E_average, and a low A for Final. His Comp result was 83.4. That's not A territory. Interestingly, he kept his focus on the exams and ignored the HQ result. Thus his argument was, in effect, three A's on five exams should work out to an A in the class. But here are the exam scores:

Exam 1: 90, Exam 2: 80, Exam 3: 95, Exam 4: 54, Final: 92

That's right: He outright flunked Exam 4. It's really tough to be an A student when you flunk an exam, especially that severely. This never figured into his arguments, for obvious reasons. He fussed over the weights. The final wasn't worth enough! Sorry, but short of going the “sucker bet” route it could hardly ever be worth enough to suit his purposes. Besides, I had already sweetened the pot by building some bonus points into the final exam's grading scheme, giving a perfect paper a value of 105 instead of a mere 100. In reality, his 92 on the final was 87.6%. I had already cut everyone as much slack as I intended to.

He had one more card up his sleeve:

Is there nothing that can be done... a test I can take to challenge?

Lord have mercy! Can you imagine? I tried to be nice:

No, there isn’t anything. If you think about it a little bit, you’ll realize for yourself there couldn’t be any after-the-fact exam that students could take to tweak their grades. Otherwise the college would spend the first several weeks of summer vacation giving the special exams to students who were unsatisfied with the outcome of the semester. Six of your classmates who earned B’s did better than you; ten did more poorly. You earned an unambiguous mid-range B in the class, a good solid grade. —Z

It did not satisfy him. I received one more lengthy message in which he noted his regular attendance, active participation, his “majority” of A's, and his work ethic. “It seems that for one reason or another, I end up coming up short somehow.”

The main reason, as best as I can tell, is that you're a B student whose grades range across the spectrum from A to F. It's not mysterious.

I'm sure you're tired of this by now.

Quite.

One page at a time

The importance of packaging

He seemed smart enough, but he was an extremely unreliable student. He confided to me that he was under treatment for an anxiety condition, but his therapy had clearly not resolved his problem. Still, he started off the semester by powering through the lessons and it looked like he would be all right in the long run.

But looks can be deceiving. His scores on the exams eroded steadily throughout the term, and the erosion was eating away at his chances of passing the class. We met during office hours. We conferred after class. He e-mailed me questions, which I tried to answer promptly. Nothing worked. On top of all that, anxiety feeds on itself, so his emotional condition was not improving.

Of course, we tried that old stand-by of extending his time and letting him linger over the exams, but there was no significant pay-off. The situation was desperate—and so was he.

Despite decades of teaching, I was slow to recognize the significance of an anomaly in my student's performance. Although his exams were increasingly disastrous, his quiz scores remained persistently decent, hovering between B's and C's. It was nearly too late when inspiration finally struck me.

“We're going to do something different on the last chapter test,” I told him.

My announcement did not please him. He mistrusted change. However, he was docile enough and desperate enough to cooperate with whatever I wanted to try.

“I will dole out this exam to you one page at a time,” I continued. “You won't get page two until you return page one to me. If there's time at the end of your extended period, you can ask for individual pages back, but only one page will be on your desk at a time.”

His eyes widened. “It'll be like quizzes!” he said. “A series of quizzes! I can do quizzes!”

“Yes, you can,” I agreed.

On exam day, we followed the one-page-at-a-time protocol rigorously. He never had multiple sheets of paper simultaneously on his desk. When I graded his exam, his score soared into the nineties. I was both astonished and gratified. It had worked ever so much better than I had dared hope.

We did it again on the final exam. He broke discipline this time and filched old pages from my desk when he came up for new pages. I noticed that he sometimes had two or three pages on his desk. He did a lot of flipping between them, making clear to me for the first time what he had been doing on the exams when he had crashed and burned. The one-page approach permitted him to stay focused. Under its restrictions he couldn't compulsively jump back and forth between problems, aborting each solution before he finished. I began to monitor him more closely when he came up to swap pages.

Although he never had the entire final exam on his desk at one time, his occasionally divided attention cost him and his final exam result was not superior. Nevertheless, it was good enough to secure him a passing grade for the course, much to his (and my) relief. He learned a new way to manage his difficulty in expressing his knowledge.

And, obviously, I learned something, too.

The template student

When thinking is too much trouble

My exams seldom contain surprises, but my students' answers do. Since I'm a firm believer in keeping track of student progress with frequent quizzes, I telegraph my punches. Students have plenty of opportunity to discern what facts, techniques, procedures, and calculations I deem the most important. (They also—most of them—learn the importance of regular attendance so as not to miss these pre-exam rehearsals.)

Of course, some students take it too far. These are the students who have had the unfortunate educational experience of intensely patterned teaching to “the test.” These are also the students who badger me for “practice tests” in advance of each exam. What do they want? Problems that are exactly like the ones they'll encounter on the exam.(I presume I'm permitted to change the numbers a little bit.)

What surprised me most this past school year was the discovery of this tendency among my calculus students. I was used to seeing it in my lower-level classes like algebra, but in multivariate calculus? An example of the behavior of the template-driven student will suffice. You'll see the problem, even if the terms are mysterious.

On a quiz I asked multiple questions about the gradient of a function of two variables. In part (a) I asked them to compute the gradient and evaluate it at a given point. In part (b) I asked them to use the gradient to compute the directional derivative in a given direction. In part (c) I asked them to calculate the greatest possible value of the directional derivative. In part (d) I asked them to find the direction in which the greatest possible directional derivative would occur. Pretty standard stuff.

On an exam I asked my students to (a) compute the gradient of a function of two variables and evaluate it at a given point. No problem. In part (b) I asked them how large the directional derivative could be? Several students were thrown for a loss. They wanted to compute a specific directional derivative, but instead I was asking them for its maximum possible value. There was no way they could do what they wanted to do because I had not provided a direction, so they made one up. They had memorized the pattern in the quiz and insisted on replicating it exactly on the exam. Since I had, in effect, swapped (b) and (c), they were deeply perplexed and forged ahead with the moves they had learned by rote.

Embarrassing! It wasn't a very large number of students, but I had been hoping they had been weaned away from this tendency by the time they arrived in the calculus III class. I learned otherwise.

Brain damage!

Explaining students

Two of the greatest minds in pedagogy recently came together to ponder some of the profoundest educational conundrums of the era. Or, to put it more prosaically, I called up a former student of mine for a chat. I, of course, hold a prestigious tenured professorship at a California community college. He is a lecturer in English at an out-of-state university. No, we are not universally recognized as the leading experts in our respective fields, but we figure that's mostly the fault of other people. Whenever we talk, we quickly reach agreement in our perspectives and opinions. It immediately follows that the many deficiencies in modern education must stem mostly from a failure to sufficiently adopt our preferred policies and emulate our instructional practices. Just listen to us! It's really quite a pity that so straightforward a solution to so many problems continues to languish unrecognized.

However, we should accept the need for a modicum of caution. There is an unfortunate gap in our grasp of the educational enterprise. Upon comparing notes, PiD and I have come to the unhappy realization that our immense intellects have yet to figure out what makes our students go. (Or not go.) It's perplexing!

For example, I told my entire algebra class that our mastery of the quadratic formula meant that we would never again face a quadratic equation for which “no solution” was a satisfactory answer. Solutions would always exist, whether rational, irrational, or complex. Always! Yet on the next exam several of my students labeled some of the quadratic equations as “prime” and solemnly wrote “no solution” in the answer blank.

PiD advised his English composition class that rewrites were a fundamental component of composition and that course grades would rely much more on their diligence in rewriting and improving their essays than on generating sparkling first drafts. As the academic term progressed, several students asked him how to get better grades. “Have you submitted rewrites of all of your essays?” he asked. “We didn't know we had to do that!” they told him. “But the due dates for rewrites are on the syllabus and I send out e-mail reminders as the dates approach!” “Yeah, but we can't know those unless we look at the syllabus or check our e-mail. You should have told us in class.” “But I did tell you in class!” “Well, maybe. But did you check that we were in class that day? I'm too busy to come to class every day, you know.”

A student wrote me a note in response to a problem on the calculus final exam. The problem asked, “What is the area inside the circle r = 3 and outside the cardioid r = 2(1 – cos θ)?” My student wrote, “Failed because I forgot the eq. for a circle in polar coordinates.” Um. Did you notice? The problem said, “the circle r = 3”?

The great minds of the age crumple in defeat.

Perhaps I did not sufficiently recognize a learning opportunity of my own from decades ago, back when my intellect was forming and a famous educator was trying to teach me some vital lessons.  William H. Cosby, Jr., Ed.D. (1976, UMass, Amherst), discovered these clues during highly personalized field research on his own family. Perhaps our students, like Cosby's children, are led into irrational, disturbing behavior by brain damage. It would explain so much!

If only I had paid more attention back then.

The twelfth hour

And even the eleventh hour is too late!

May flowers are supposed to follow April showers. Well, an even more reliable sequence is the arrival of plaintive e-mails after the publication of semester grades. Here's one from a student who missed a passing grade by half a percentage point. Yes, sometimes I round up an average score of 69.5 and assign a student a C, but only when that average includes a solid passing score on the comprehensive final. This poor student, however, did not pass the final.

Professor is there any way possible to retake the final or to roll up  the .5 of the grade? The last half of the semester were very difficult for me, I have encountered family hardships and fell ill. I know it is not an excuse but it would mean the world to me as so much depends on it. I am truly sorry to put you in such a situation. I'm just lost and I'm pleading for some help. I'm just hoping you can assist me in my situation. Again thank you very much for your attention, I am glad to have a professor who is willing to lend a helping hand.

It's a poignant missive, but much too late. We do strive to provide reasonable accommodate for hardship cases, including deadline extensions or makeup exams when illnesses or other emergencies intrude into a student's academic program. We can't, however, do a darned thing about it after the fact. And, in his case, there had been no hint of any difficulties during the semester itself. Only now, when it was too late. I replied to his message.

It's an intriguing idea, but absolutely forbidden by the college rules. No mitigation or revision is permitted after semester grades are assigned. It’s a strict but necessary regulation. Otherwise we would face a constant onslaught of students seeking to improve their grades by post-semester work.

I tried to let him down gently, explaining that his best option was to retake the class immediately, thereby taking advantage of what he had learned while he still retained it. The summer session sections of the course were already fully booked, but I contacted one of the summer instructors and explained why my student was in need of last-minute enrollment in summer session. He consented to add my student, going above the enrollment limit, so I passed the word along.

Dr R has promised to let you into his summer session class. Be sure to show up on Day One!

My student, whose fall enrollment at the University of California appears to be contingent on his passing this class before he gets there, was relieved to learn he had an option to fulfill the requirements of his UC transfer agreement:

Thousand thanks Professor you just made my day, thank you so much!!! Again thank you !!!

A happy ending? That remains to be seen. But he's got a shot.

The Ritualists

A new strain of tardiness

The old pattern was very familiar, especially since I tend to give my students lots of short quizzes, often at the beginning of a class period: A student arrives late, sees a quiz in progress, and leaps into action, yanking a pencil out of the old book bag, snatching a quiz off the table in the front of the room, and scribbling quickly in a desperate attempt to catch up. That's the old pattern and it's not a surprising one.

Lately, however, I've seen several instances of a new pattern that is, frankly, utterly bewildering. In over thirty years of teaching, I had never seen this behavior until the last few semesters. A few of my tardy students have an unprecedented sang froid. They arrive late, see a quiz under way, and then progress casually to their desks. They never rush up to the front of the room to pick up a quiz. Their leisurely saunter gives me plenty of time to stroll over and hand them one. (Service with a smile!)

This new breed of tardy student is calm and generally unruffled, except sometimes a small moue telegraphs the unspoken thought, “Oh, here we go again!” The serene latecomer positions the water bottle or energy drink or Starbuck cup on a corner of the desk,  carefully tucks away the cell phone or iPod, peels off the coat and rolls it up to tuck in the book cage under the desk, rummages about in the book bag for a pencil or pen (sometimes deliberating over his or her choice of several writing implements—mustn't pick the wrong one!), digs out a calculator and places it precisely in the corner opposite the beverage (whether or not the quiz requires number-crunching), and then finally (as if in surprise) takes note of the quiz sitting atop the desk and begins to ponder it.

This settling-in ritual, in its various versions, eats up at least two minutes, sometimes three. Sometimes there is a lengthy interlude with the smartphone, scrolling through messages and tweets received in the interval between breaking eye contact with the screen upon arriving at the classroom door and arriving at the desk, occasionally extended by the imperative of replying to urgent missives. I imagine most of them are in the vein of

L8 agin
prof :(
lol

Strangely enough, the explanation does not appear to be the simple one: Such casually late students are the class's losers, doomed to fail, and have fatalistically accepted their fate. Nope. That describes very few of them. My unruffled tardies are mostly C students mired in mediocrity. Perhaps they've figured out that they're doing enough to survive and it would be too much trouble to put in the work necessary to rise to the B level. I really don't know.

One thing, however, has not changed. After arriving ten minutes late and getting only five minutes to work on a fifteen-minute quiz, many chronically tardy students are quick (for a change) to complain: “I didn't have enough time!”

“Yes, you did,” I explain. “You just chose to use most of it for something else.”

Brain pain

Lesson unlearned

My students were not happy with me and they weren't keeping it a secret. After a unit on scientific notation, I gave them a quiz containing a question they deemed terribly unfair:

The mass of a proton is 1.7 × 10^–27 kilograms. What is the total mass of 7.2 × 10³³ protons? (Write your answer in scientific notation and include the units.)

I was told, with exquisite care and patronizing precision, that it was wrong of me not to tell them which arithmetic operation was expected. Addition? Multiplication? Subtraction? Division? How dared I give them numbers without specific instructions!

With professional patience, I waited out their lengthy complaints. Then, without saying a word, I turned back to the chalkboard and wrote out a brand-new problem:

The mass of a nickel is 5 grams. What is the total mass of 6 nickels?

With frowns still on their faces, they blurted out, “Thirty grams!”

Another long silence as I waited for their reactions. The faces went neutral. One brave soul ventured a comment: “Were we supposed to know that?”

“Sure,” I replied. “All of you know that you multiply to solve problems like this. You just yelled out the answer to the nickel problem because it was so easy. What I'm trying to get across is that numbers written in scientific notation are still just numbers. You work with them just like you work with other numbers. You're letting your minds shut down because they look different, but you actually already know what to do.”

A smug expression is bad pedagogy, so I maintained a mild and neutral mien. I was quietly satisfied that I had gotten an important point across. My self-congratulation was just a little premature. (You'd think I would know better by now.)

A student in the back row grunted in dissatisfaction and posed a question in an irritated tone: “So on the next exam are you going to tell us what to do with the numbers?”

My spirits fell a notch.

“What do you think?” I asked.

I hope indeed that they do.

Self-diagnosis

This ought to hurt a little

It usually happens right after the first exam of the semester. Somewhere between one third to two thirds of the class is disappointed with the results. Nearly everyone expected an A or a B. Many are surprised they earned a C, D, or even an F. I give an assignment:

Send me an e-mail message by noon on Wednesday that contains two things:

1. A description of what you think went wrong on your exam and why you didn't score better.
2. A description of what you plan to do to deal with the problem(s) described in #1 and how you're going to do it.

Except for the few fatalists who signed up for the class in full expectation of miserable failure (why are they even there?), the students tend to take the assignment seriously. Interesting and often thoughtful responses come in:

I think part of my reason for scoring so low was I didn't thoroughly double check my work. I missed a good amount of points by not double checking that I knew answers to, or just wrote down wrong but still knew the right answer to.

An excellent observation. We have a long class period and I allow my students the entire time for the exam. They have ample opportunity to review and check their answers. Getting out the door before the class period is over is not a good priority.

I skipped the last two sections of the chapter so I didnt get enough practice with the word problems. I also made a lot of simple mistakes that could have been avoided if I would have checked my work carefully.

Self-knowledge. A beautiful thing. The next step is to actually do something about it.

I didn't push myself hard enough to finish all chapter homework. which would of help me master solving linear equations, inequalities and problem solving ect.

Indubitably. Hardly anyone succeeds without practice, and that's what the homework is for. Thanks for noticing.

I think my problem for the test was I didn't study enough.

Yes, I do recommend the practice of studying. Do please give it a try.

1. I think what was wrong with the exam and why I got such a low score was the fact that I barely got any sleep. 2. Study more, go to bed early, and be more prepared. Also I should try to understand certain problems more

Okay, I think you raise some good points. People do better on exams when they are well rested. However, I sense an element of denial. Of the last five class days, you missed three. See the problem?

Plus or minus

Rather missing the point 

One of my favorite negative reviews on RateMyProfessors.com is the following:

I don't understand why people say he is a good instructor. Many students in his class struggle to get a good grade. yes he is clear but his tests are extremely difficult. And expect a ton of repetitive homework assignments.

Let's deconstruct my student's complaint piece by piece:

Many students in his class struggle to get a good grade.

Yes? You mean they don't get good grades automatically? The student in question was enrolled in a calculus class. Such classes are notorious for easy grades, right? Yeah, right. More to the point: In a typical college class you can expect a distribution of grades, most of which are C's. Not what I would call “a good grade.” Good grades are A's and B's, earned only by those students who put in the effort.

[E]xpect a ton of repetitive homework assignments.

I checked. The syllabus contained homework assignments for each section with, typically, 12 to 20 problems. There were 33 sections that we covered, so students were expected to solve approximately 500 problems over the course of a 16-week semester, or a little over 30 exercises per week. (My bleeding heart weeps for them.)

Funny thing: There is a remarkably high correlation between doing the homework and getting one of those good grades. There were thirty students in the class. I note that only one student in the top half of homework performance was not earning an A or a B (and that one student was pulling a solid C). Of the fifteen students in the bottom half of homework performance, only four had “good” grades (three B's and one A [there's one in every crowd]). Conclusion: Do the work, get a good grade.

[H]is tests are extremely difficult.

Evidently not the case for those who work at it by doing the “repetitive” assignments. (Average scores were actually in the low eighties.)

yes he is clear

Thank you very much. Clarity is something I strive for and I am pleased that you noticed.

I don't understand why people say he is a good instructor.

Indeed you don't.

Must be present to win

A cry for help

One of my students—let's call him “Dick”—sent me a distressed e-mail. He was not doing well in class and was hoping for some wise words of guidance from his teacher. His semi-coherent message ran thus:

hey Dr.Z dick here,
hey i wanted to run over a little bit of questions, 1.please tell me if there is anything you can pinpoint from my work that i can work on to develope the grasp of this sections.i do not want to fail and sometimes i feel i can grasp it then sometimes i fail it.i do not want to fail this class i meet with tutors every week twice and home tutors and i can do decent but cannot prove my worth on every other test.im using the dropin ctr efficiently...any help you can recommend i do not want to lose my financial aid as it is viable to my continued succession.i can retake the course next semester as a retry but do not want to receive a W as it may discontinue my aid as well..
dick,

I often reply immediately to such messages, both to reassure the student and to prevent them from getting lost in the in-box maelstrom. Students benefit most from timely feedback. This time, though, I sat on my hands and just stared. And stared. And walked away from the computer.

Dick was in class the next day. I asked him to see me at the end of the period. He dutifully approached me as his classmates filed out of the room.

“I got your message, Dick, but I have to say I'm puzzled. Isn't it obvious what you need to do?”

“Huh? I'm trying everything I can, Dr. Z!”

“Even attending class? You routinely miss one class session per week and you often skip two. I'm less impressed about the frequency with which you meet with tutors if you don't attend actual class sessions.”

“Well, uh, sometimes I can't make it.”

“So it seems. But if you can't attend class, you can't reasonably expect to pass it. And where is the work you're doing with your tutors? I didn't see any homework from you for the last two chapters. So far, in fact, you've missed about thirty percent of the homework and quizzes. You'd barely be passing if you got perfect scores on the remaining seventy percent, but you're nowhere close to that.”

Dick had nothing to say, but he was nice enough to look embarrassed.

“Dick, I was astonished by your message, especially since it should be perfectly obvious that you desperately need to come to class and pay attention to the lessons. You can't skip out on a third of our sessions and survive. Few students could get away with that. I need to see you in class, on time, every day for the rest of the semester. That's my advice.”

He nodded his head. He even showed up the next day. Two days in a row. That's good! I wait to see if he makes it to three, which has occurred before—but rarely.

One thing sticks in my mind, though. Dick was clearly surprised—startled, even—at my advice. The notion of actually coming to class regularly had never occurred to him.

Self-validation

Oops! ... I did it again

It was an accident.

I gave my students a take-home quiz, due at the beginning of our next class period. This doesn't happen too often, but it's a nice opportunity for them to score maximum points by working together and carefully comparing notes before submitting their results. With a few exceptions (the handful of students who prefer to keep their work as secret as possible), my students spring at the chance to cooperate and rack up the points.

This time was no exception. However, one student e-mailed me with a concern. “Abe” had transportation issues and was afraid he might be late to class or even miss it entirely. As a precaution, he had scanned his solution to the quiz and attached the image to his message. I wrote back to put him at ease, confirming my receipt of his work, and wishing him good luck in making it to class the next day.

As it turned out, Abe was in class that next morning and handed in the original version of his quiz. I slipped it into my binder along with all of the others. Like the absent-minded professor I am, I quite forgot that I had printed out his scan and already had that in my quiz folder. During my grading session that afternoon, I inadvertently graded Abe's quiz twice, marking up both the original and the scan.

I noticed my oversight while sorting the quizzes into alphabetical order for purposes of entering the scores in my gradebook. I placed the two versions of Abe's quiz side by side and discovered that they were still identical: My red-ink marks on the two quizzes were identically placed, the corrections were a perfect match, and both quizzes bore the exact same score.

Naturally I was pleased. Consistent grading is one of the most important factors in treating students equitably. Here I had evidence that my correction process was rigorously—even rigidly—consistent. I have achieved the gold standard in the potentially capricious and subject process of grading!

Either that, or I'm a robot.

Too cool for school

No royal road to algebra

Although its hours have been trimmed by the current state budget crisis, my college's Tutoring Center continues to serve as a lifeline for many of our students. Each semester, therefore, I make a point of ensuring that my math students are aware of the facility's existence. I don't just tell them, I show them. Thus it was once again that, during the second week of the semester, I gathered up the entire class and took them on a “field trip.”

It puzzled my students when I announced it, of course. I told them to leave their books and papers behind in the classroom, which I would lock behind them. We would take a few minutes to stroll down the sidewalk to the Tutoring Center, which was only a couple of buildings over. Short field trip. Once I mentioned where we were going, some students nodded their heads in comprehension, grasping my purpose. Other students, however, had a different reaction.

One came up to me, backpack in hand, clearly ready to make a break for it.

“Is this required?” he inquired.

“We're all going to the Tutoring Center,” I said, in oblique response.

“Yeah, but do we have to? Is it an assignment?” He was nothing if not persistent.

“We're all going to the Tutoring Center and we'll be back in a few minutes to start on the next topic,” I said, demonstrating a charming obtuseness.

I don't think my student was charmed. He got to the point.

“Does this affect our grade? Are we getting participation points?” he asked.

I looked right at him, allowing my surprise to show.

“‘Participation points’? In a college class?”

He fell silent but unrepentant. He wanted points if he was going to go to the Tutoring Center with his classmates. It was finally obvious I wasn't giving any. He trailed along behind the rest of the group and I expected him to lag increasingly until he took a “wrong turn” and vanished toward the parking lot.  I was thus mildly surprised and pleased to see that instead he stuck it out and hung at the periphery of the group as I introduced them to the instructional assistant who managed the math tutors in the Center and walked everyone over to the area where drop-in tutoring occurred. Now that my students had been physically present in the facility and had met the key personnel, I figured it was much more likely that they would feel comfortable about returning to it when they needed help.

We returned to our classroom and launched into the lesson for the second half of the class period. The point-grubbing student sat quietly in the back, apparently ruing his decision not to skip out. At least I assume so, since in the next few days he developed a habit of nonattendance or early departure. When our first exam came along, he achieved the class's low score, missing a D by several points. (In fact, his score in the thirties might reasonably be characterized as an F-minus-minus.) He had never come to my office hours and he had never darkened the door of the Tutoring Center again.

I guess he really needed those participation points.

But gay sex is icky in my head!

Whiny-ass bigots 

 “Serena Locksley” was a classmate of mine in graduate school a dozen years ago when we were both enrolled in a doctoral program. Another thing we had in common was our day jobs as teachers, although she was dealing with high school and I had the advantage of dealing with (ostensibly) adult college students. President Obama's mild-mannered and rather halting endorsement of equal marriage rights for all Americans—and the apoplectic reaction of the religious right—reminded me of Serena's serene response to a related controversy in her secondary-school classroom.

Her students were doing a unit on human rights in their senior social studies curriculum. The amicable consensus that human rights are a good thing was beginning to unravel as students began to draw lines in the sand. Sure, it was wrong to discriminate on the basis of race or religion, but sexual orientation? Trying to avoid crossing the lines of politically correct terminology, the dissenters made the point that “queers”—oops! uh, gays—were different from “normal”—oops! uh, most—people.

“But, Mrs. Locksley, what they do isn't natural!”

Years of experience had made Serena all but unflappable.

“What they do isn't ‘natural’?” she replied. “If it's inborn, how can it be unnatural? It is your argument that some people don't know their own sexual impulses?”

Students on both sides of the gay-rights aisle were writhing in agony, praying for the clock to run out on the day's excruciatingly sensitive topic. One student took a stab at making an irrefutable argument:

“Mrs. L, I don't care if people want to be gay, but I don't like it when they make a spectacle of themselves! That's not fair to the rest of us!”

Serena probed for more information.

“You mean, like prancing around in gay pride parades?”

Several students nodded their heads. One went further:

“Or hold hands in public!”

“You find it offensive when people hold hands in public?” asked Serena.

“Well, not when straight people do it. But when two guys hold hands, that's like flaunting it in your face. Then you can't help thinking about the stuff they do, and that's gross!”

“You have to think about what they do? You mean, besides holding hands?” asked Serena.

The student hesitated.

“Yeah ... cause, like, you can't help it. And it's icky!”

Serena let the moment stretch out for several seconds, but the students remained anxiously quiet.

“That's an interesting reaction,” said Serena. “So what about when a man and a woman hold hands? That doesn't force you to think about what they ‘do’? All of you call me ‘Mrs. Locksley’ or ‘Mrs. L,’ meaning that all of you know that I'm married. That doesn't force you to think about what my husband and I do together?”

“Ewwwww! Mrs. Locksley!”

Here endeth the lesson.

Fill in the blanks

Template tests

I was flummoxed. Under normal circumstances, algebra students abandon the complete-the-square technique for solving quadratic equations as soon as they meet the quadratic formula. It is by a significant margin the least-favored of the solution techniques, trailing badly after formula and factoring.

Why, therefore, were so many of my students diligently completing the square when they didn't have to? Even worse, they were doing it on an exam problem, when time is at a premium. Worst of all, they were completing the square to solve a quadratic equation where its use was clearly contraindicated! I was at a loss.

As you may know, the solution of the quadratic equation is the great pinnacle and climax of your traditional introductory algebra class. The end of the semester wraps up with the astonishing revelation that one can now solve any quadratic equation. No exceptions! Such universality is rare, and I try to engender a little appreciation in my students for so powerful a conclusion, the big finish of Algebra 1.

Of course, I also try to get them to approach quadratic equations thoughtfully and methodically. First of all, does the equation factor easily? Then go for it! Is it (or does it appear to be) prime? Then one can apply the never-failing quadratic formula or—in certain specific cases—resort to completing the square. The specific case, naturally, is one in which the quadratic polynomial in question is monic (has a lead coefficient of one) and possesses a first-degree coefficient that is even (making it easy to take half of it and square the result, as required for completing the square).

Otherwise, don't even think of completing the square.

The problem that was puzzling me was monic, all right, but its middle term had an odd coefficient, making it a quite unsuitable candidate for square completion. Why, then, did so many of my students plow right in and start juggling fractions and slogging through more and more complicated expressions? They didn't know and couldn't tell me why they had done it.

The reason finally came to light while I was paging through my collection of quiz keys. I paused to consider the quiz containing the combined-work problem (or “joint effort”—computing the time a job takes if two or more people pitch in and you know how long it takes each person to do the job alone). This was exactly the kind of problem that had caused so much square-completion grief on the exam.

I noticed that I had solved the resulting quadratic equation on the quiz's solution key by completing the square. The polynomial had been monic with an even linear coefficient, so completing the square gave a quick and easy solution ...

... and my students had learned the lesson that combined-work problems are solved by completing the square! After all, the teacher had demonstrated this in a quiz solution key that he had posted on the course website. Did he not constantly encourage them to emulate his example? Follow his lead? Write solutions like he did? Indeed! Indubitably!

Damn.

They learned a lesson I wasn't teaching. They had studied my solution to a particular combined-work problem and then followed it slavishly when next they encountered a problem of the same type—even though the resulting quadratic equation had different characteristics and argued for a different solution technique.

I failed to banish the template problem. My fault!

You know what a “template problem” is, don't you? I'm sure you do. Lots of books are full of them. It occurs when a section of the text presents a carefully worked-out problem in Example 1, you turn to the homework section, and Exercises 1 through n follow the prompt “See Example 1.” And then all of the problems are exactly like Example 1 except that the numbers got tweaked a little. Or maybe Example 1 was a word problem about Sally and Exercise 1 is about Sam. Trivial changes. You can copy the solution of Example 1 as a template and go through filling in the old numbers with the new numbers.

Hardly any thought necessary.

I don't want to be too harsh. Routine drill problems are useful for building basic skills. They are, however, too bland for a steady diet and do not do much (if anything) for building conceptual understanding. Students, however, often prize them for their dull predictability and lack of challenge. They even ask for more, as when they beg for a “practice test” before a big exam. The most favored practice tests are those full of templates for the real thing. Woe betide the instructor who gives in to the pleas for a practice test and then changes the problems too much in the actual exam! Students will feel betrayed.

I refuse to give practice tests. I decline to channel my students' attention too narrowly to specific kinds of problems solved in specific kinds of ways. I want them to consider each problem independently, with a minimum of prompting, examining their knowledge of solution tools and picking the most appropriate one to apply.

The complete-the-square affair demonstrates, I'm afraid, that I have discouraged template thinking less than I had hoped. Perhaps I should ask my colleagues how they avoid it and then do exactly what they do....

Professor Google

Live and interactive!

This is not a rant about students relying on things like Google and Wikipedia for their schoolwork. It's not that at all. It is instead a rumination about the tendency of some students—at least one in particular—to rely on an information-at-your-fingertips approach to education. The student I have in mind appears to have lost—to a remarkable degree—the trait of self-reliance. Let's call this student “Shawn.” He took my algebra class. He took it twice, having flunked it the first time. In both semesters I was treated to Shawn's pop-up arm, which waved for my attention at the least provocation. As a highly interactive instructor, I welcome student questions and want them to feel that their queries are welcome. Shawn, however, was trying my patience (along with that of his classmates).

It took me a while to figure out what was going on. Ever in the moment, Shawn had abandoned thinking. If something—anything—gave him pause, his arm shot up. He wanted immediate clarification from the instructor in preference to actually thinking about it himself. Somewhere along the line he had discovered that the path of least resistance involved asking the professor.

Back in the old days before the World Wide Web and Google, I sometimes fussed for hours (or even days), wracking my brain trying to recover some odd bit of information. I'd pull books off the shelf and page through them. Did I read it in this one? Should I dig out the encyclopedia? Did a friend mention it to me in conversation? Should I try calling him? (Which one?) As you may have heard, we call these the “good old days.” I confess, however, that the Internet and Google have become two of my most cherished friends. Facts and factoids are at my fingertips and I discover or rediscover things with alacrity and pleasure.

Shawn doesn't have Google in the classroom. He has me: Professor Instant Gratification. (I had a lesson to learn as much as Shawn did.)

The starkest example of Shawn's abandonment of thinking arose during a lesson on nonlinear systems of equations. I had created an elementary introductory problem involving a parabola and a line, writing their equations on the board and telling the students we were going to discover the points where their graphs crossed each other. The computations were straightforward (amazing, isn't it, how simple the answers are for Example 1?). Then I told the students we were going to examine the plausibility of our solutions by graphing the two curves and considering their appearance.

I plotted two points (the axis intercepts!) that satisfied the linear equation and sketched the line. Shawn was keeping his peace and presumably keeping pace. The quadratic equation was a little more challenging to graph, but it was still pretty elementary, so I created a short table of values to find some points that would help us sketch the parabola. In rapid succession, I plugged in x = 0, x = 1, and x = 2.

Up went Shawn's hand.

“Yes, Shawn?”

“How did you compute those values?”

“The values in the table?”

“Yeah. Where did those come from?”

Mind you, we had done the parabola to death in the previous chapter. We had graphed vertical parabolas and horizontal parabolas. We had found their axis intercepts with the quadratic formula (if necessary). Several old quizzes and the most recent exam had featured the parabola most prominently. All of the students, including Shawn, had had multiple exposures to the mundane task of plotting a parabola. He had had even more examples of plugging in conveniently chosen numbers to evaluate algebra expressions for graphing. Shawn's question was extraordinarily lame.

“Shawn, I want you to think about that.”

“What?”

“I want you to think about that. I'm finding points that lie on the graph of a parabola, picking x's and computing y's. You've done that yourself, right?”

“Lots of times.”

“Okay. Do it now.”

Long pause.

“Oh,” he said.

“Right,” I said, hoping that he was indeed right in what he was thinking.

It was not easy, but “think about it” became a standard response to many of Shawn's too-quick questions. By degrees, he eventually became more self-reliant instead of instantaneously asking the professor. He never learned to postpone gratification to quite the degree I would have liked, but he got a little better. He stopped asking questions without thinking and I stopped answering so automatically.

And Shawn's classmates got more rest for their eye-rolling muscles.

It figures

Or perhaps it doesn't

I'm still disappointed when it occurs, but I'm no longer surprised. Sometimes, such as when I give an exam on the last day before spring break, I send out a grade update via e-mail so that my students don't have to wait till school resumes to find out their status in the class. My report, which pops up in student e-mail, presents the latest grade distribution in descending order. The closer to the top you find your secret student ID number, the better off you are.

I also provide the weighted components that go into computing each semester score (and grade): homework, quizzes, and exams. I present averages rather than individual scores, and therein lies the rub. Students write back when they receive the grade report and ask, “What was my score on Exam 5?”

Let us consider this. What does the student have in hand?

The student has his average exam score: the grades on Exams 1 through 5 all added together and divided by 5. The student has his old exams, numbers 1 through 4.

How on earth is an algebra student supposed to figure out the unknown value of his score on Exam 5? It is a puzzlement, is it not? If only they had a better teacher, perhaps they could do it for themselves, but I'm afraid the classroom door is a portal to real life, beyond which nothing in the classroom has any relevance. It's not as though the stuff I teach them can actually be used for anything! (Not even for classroom-related applications!)

I recently responded to an inquiry from a student who was earning a B:

Didn’t you realize you could have computed it yourself? You have your average exam score from the grade distribution I sent out. Multiply your average exam score by 5 and then subtract your scores from Exams 1 through 4. What’s left is your Exam 5 score.

He gave me a cheery reply:

I should of know but thanks I'll make sure I put that in my notes.

I'm thinking of forwarding that to his English teacher.

Dates in conflict

Personal priorities and education

“Rob” was a good student, so I was unaccustomed to having him linger after class to ask questions. To make things even more unusual, Rob insisted on waiting till everyone else was gone from the room. Evidently he did not want any classmates to overhear his question. Rather tentatively, he voiced his query:

“Is it okay if I take our next exam a day early? On Thursday instead of Friday?”

Oh, no problem. I can answer that question easily: No.

I was, however, nice enough to give Rob a reason:

“Sorry, that's not possible. The exam doesn't even exist until the night before. I don't finish writing it till we've had our review session.”

In fact, I sometimes tweak an exam after I find out what students have questions about. If we spend a lot of time on a particular topic, I want to be sure that the topic is not neglected on the exam. For example, if my students want to work especially intensely on mixture problems, then I'm more likely to choose a problem of that type to include among the various application exercises on the exam. (And—I admit it—I don't worry too much about the likelihood that this slightly disadvantages the students who choose to skip the review session.)

Furthermore, I dislike the very notion of giving exams early. The student who gets special treatment by getting the exam in advance is obviously subject to the temptation to leak information to friends in the class. The less opportunity for that, the better.

Rob seemed to be surprised by my answer. Perhaps he had had instructors in the past who wrote all of their exams at the beginning of the semester. Not me. I'm much more adaptive than that. Or less farseeing.

Rob was also dismayed by my answer.

“It's not really my fault, Dr. Z. I got tickets to a concert before the semester started, and I wasn't expecting to enroll in a class that met on Fridays.”

“Nevertheless, that's what you did, Rob. The conflict is of your own making. You have a conflict between our exam date and the concert date. I don't have a good way to accommodate that and it's not my problem to resolve it anyway.”

“Well, could I take the exam later? Would that be okay?”

“Sorry, Rob. Think about it. I post the solution key on-line right after the exam. I'd have to withhold it from your classmates all weekend and leave them in suspense so that you could be accommodated. They wouldn't get the answers until Monday, until after you've taken the exam.”

Rob hesitated a moment.

“Oh, I was meaning to tell you. I'm going to be out on Monday, too.”

The conversation ended soon after that.

I've half a mind

Bad teacher!

When it's early in the semester, I tend to cut my students a little more slack. Of course, I expect them to pay attention when I explain why I take off points for some calculations that manage to produce correct answers. For example, how many minutes does it take you to travel 12 miles at 18 miles per hour? Here's what one student told me:

Yeah. Well, I'm really not happy with that. Sorry, but 12/18 is simply not equal to 40. Equality is supposed to be a transitive property, folks! Of course, this could be redeemed with the appropriate use of unit conversion:

This I like. Careful use of units is a powerful way to keep one's calculations in order and to make sense of the results. Full marks! But then you get the woefully calculator-dependent student who presents this travesty:

Heck, you can keep your puny old leap-seconds! My students can conjure up a dozen seconds out of the thin air of feckless rounding. This is a particular gripe of mine. You actually need to grab for a calculator to compute two-thirds of sixty? Good grief!

Thoughtless calculations like these were sprinkled throughout the early semester quizzes and exams. But the pièce de résistance came in a different problem. One that had nothing to do with rounding. I gave my students (gave them, mind you) some volume formulas. All of the most popular shapes were there: cone, cylinder, sphere, box (ahem! Sorry. I mean rectangular parallelepiped, of course). The formulas were actually written out on the assignment sheet. I then asked my students to use the formulas to compute the volumes of some specified shapes. One of the shapes was a hemisphere.

Sure enough, several students decided the formula for a sphere was the best match they could make, computed the result, and ended up with an answer that was two times too big. Arrggh! Naturally, I took off points for that mistake. One of my students waxed indignant when he got his paper back and issued a two-part complaint: (a) I had not given them the formula for the volume of a hemisphere and (b) I had not done an example in class where we had to divide a result by 2 to get the correct answer.

I offered a plea of “no contest” to both charges. They were irrelevant. I patiently explained: “I have higher expectations of my students than merely plugging mindlessly into formulas. I want my students to think about what they're doing. This is not just a plug-in and grind class. Sorry.”

But not very.

A seraphic school seminar

Guardian angel

John Vasconcellos was a well-known figure around the State Capitol. A big teddy bear of a man, his rumpled figure had all the debonair flair of an unmade bed. He briefly achieved national fame when his “self-esteem” initiative drew mocking attention from the Doonesbury comic strip. John himself, however, was unfazed, even if his more substantive contributions to the state of California passed unremarked.

Anyone who serves in a California community college tends to associate the name of John Vasconcellos with his landmark education reform bill, AB 1725, which in 1988 rewrote the sections of the state education code dealing with our schools. One legacy of that legislation is a greater emphasis on professional development for faculty members. On most community college campuses, professional development opportunities are embodied in various seminars and training programs, especially on “flex days” when faculty assemble in the absence of students to rack up their required hours. The flex days, how ever many there are, are ordinarily scheduled at the beginning of each semester. We hear talks, participate in meetings, attend panel discussions, enroll in training sessions, or watch subject-specific demonstrations.

Some flex sessions are great. Most are okay. A few have been dreadful enough to be entertaining. (I recall one in which a colleague quipped—but was it a quip?—that he was thinking of killing himself and several people in the room offered to help. Now that is supportive!) In other words, flex is like any other activity, with its ups and downs, successes and failures. In general, though, we all give it the good old college try and make the most of it.

However, sometimes you run into professional development opportunities that strain credulity just a teensy-tiny little bit. In looking at the flex program books posted on various California community college websites, I have encountered seminars that strike me as, well, odd. Do teachers really need an introduction to “qigong breathing techniques”? I suppose it could be lumped in with those other activities involving movement and health activities, although yoga and various stretching routines seem to be more popular options. No doubt the “Happy Fanny” workshop announced at one school is one of those feel-good PE-type sessions—especially with that Middle Eastern dance component.

But qigong and fannies cannot compete with my favorite among all of the spring sessions I perused. The angel seminar wins it going away:

An Inquiry into the Existence of Angels

There are many who claim that any lingering belief in angels is merely the residue of imaginary wishful thinking. There are others who hold that angels (wings, halos, harps) literally exist. How is one to reconcile such contradictory beliefs? In this session, you will discover how C.G. Jung’s theory of synchronicity provides a vehicle for the exploration and possible reconciliation of this question. Rather than echoing the skeptic who says angels cannot exist or the religious enthusiast who affirms their immanence, this study asserts that by expanding our understanding of both synchronicity and angels, we might be able to resolve the conflict.

It may well be that you are having an uncharitable reaction to the description of this 90-minute program, indicating that you are one of those anti-angel skeptics. If so, how close-minded of you! Are you not open to the possibility of a synchronicitic reconciliation of (A) angels don't exist and (B) sure they do! (Synthesis: Angels maybe exist!)

I confess that I am one of those cynics who has been known to remark that a good course in probability is the best cure for folks who cannot stop seeing significance in random occurrences and coincidences. Still, I must admit that it behooves one to examine carefully the credentials of the seminar leader. Perhaps there might be some substance here:

The faciliator earned her Ph.D. in Philosophy and Religion from the Philosophy, Cosmology, and Consciousness Program at California Institute for Integral Studies.

Whoa! “Cosmology”? (Of course, angels are indeed reputed to hang out in the heavens.) What exactly is this peculiar doctoral program? Here's the on-line description:

Philosophy, Cosmology, and Consciousness (PCC) graduate programs in San Francisco are dedicated to re-imagining the human species as a mutually enhancing member of the Earth community.

They attract intellectually engaged individuals who are in varying degrees dismayed by what they see happening in industrial societies and who are striving to find meaningful ways to develop their gifts to serve the future of the world.

We support those called to meet the Earth community's unprecedented evolutionary challenge by offering students a challenging and supportive learning community in which to find their voice and vision as leaders.

Please return to the links on the upper left of the screen to explore the PCC mission, faculty, curriculum (including our Integral Ecology track), current students, alumni and community, as well as how to apply to the program.

Okay! That's clear enough, isn't it? Well, I don't know about you, but my doubts are completely assuaged. Perhaps I should write the angel-seminar school and suggest a topic for a follow-up seminar next year. I hear that business about the number of angels that can dance on the head of a pin is still outstanding.

A grade goeth before a fall

And it's all my fault

While I doubt it registers with my students, I am at pains every semester to explain to them that they earn grades. I do not merely give them. Unfortunately, the students who most need to hear this message seem to be the least likely to retain it.

I recently taught an algebra class in an accelerated format. Students were warned at the outset of the course's brisk pace and the need to work diligently to stay abreast. The faint-hearted quickly folded their tents and stole away. The braver students stuck it out to the end—a bitter end for a few of them. Overall, though, the success rate was over 80 percent. I was happy that so many of my students passed the class.

One of the students was less than enamored with her “success.” Yes, she passed the class, but she passed it with only a C after having spent most of the semester at the B level. She had spectacularly flunked the comprehensive final (earning fewer than half the possible points on it) and her average plummeted. I declined to award a B to a student who couldn't even earn a D on the final exam. She called me up to complain at the injustice of the result.

Her particular complaint focused on what she perceived as the inequity of students getting a C grade with composite semester scores of 68.5 while she was being denied a B despite a composite score of 78.5. Why did I “round up” the scores near the C-D boundary but not hers at the B-C boundary?

Several factors influenced my decision. First of all, the C-D boundary is basically academic life versus death. A grade of D forces you to repeat the course for credit. I give very close scrutiny to the scores of all students teetering on the precipice of the C-D divide. Furthermore, the three students in question had all beaten my complainant by several points on the final (and the weakest of the three was in the enviable “hammock” position). Unlike my former B student, they had not used the final exam to demonstrate utter confusion and lack of subject-matter retention (a consideration of some significance in a prerequisite course like algebra).

Then, of course, there's the other tiny factor: Among the students with passing grades, the student in question had one of the lowest participation rates in the quizzes that I used throughout the semester to gauge my students' progress. To be fair, it was not chronic absences that caused her to miss so many quizzes (although her attendance did suffer near the end of the semester). No, it was her refusal to submit her paper to me when I collected them, even when I made a point of asking her directly. “No,” she'd say. “It's no good.” Brimming over with sweet reason, I would explain, “Five points on a ten-point quiz may be a little embarrassing, but five points in the grade book is significantly better than zero points!” She'd shove the crumpled quiz into her binder and resolutely refuse: “No, I don't want you to look at it. It's no good.”

In the end, she withheld or missed over twenty percent of her quizzes. A series of truly bad decisions. Those points were not there to reinforce her against a bad result on the final exam, which turned out to be a significant matter in the end. I guess her problems weren't just in algebra.