Shift, paradigm, shift!

This morning was painful.

I had the uncomfortable task of watching 30 of the students in my morning Calc I section puzzle their way through a difficult (but fair, I feel) exam, their last "mid-term" exam of the semester and a traditionally hard one (on applications of the derivative). This awful duty was the last straw.

The penultimate one came yesterday: I was called upon to help adjudicate a case of "academic dishonesty" (to use the lovely euphemism) in a colleague's class. This colleague wanted to know if the students in question had indeed "cheated" (to stop beating around the bush). After cursory (and then more in-depth) inspection, I agreed that yes, indeed they had.

However, to me the incident said more about the culture of the academy than it did about the "dishonesty" in which the students were involved. More specifically, the students were clearly guilty of "cheating," but to me the more crucial issue concerned why it is they felt the need to "cheat" in the first place.

To "cheat" requires that there be a "game," and that it matter that people follow the rules of that game, and further that it matter that in order to succeed at the game one must "do better" than anyone else playing the game. This was definitely the case in this particular course: the students had been given a (very) high-stakes exam, which to them was more than an assessment instrument; it was moreover one of a very small few means of receiving feedback on the degree to which they were mastering the concepts of the course the exam was given in. To them, "cheating" on the exam was a natural response, given the way in which they've been acculturated to consider the exam a must-win game in which success is measured by high marks.

The point I'm getting at is the following: the more we as educators eliminate, or to the greatest extent possible downplay, the competitive aspect of education (high-stakes testing, rigid and number-driven grading schemata, individualistic learning paradigms, etc.), the less likely we are to find our students "gaming" the system by engaging in "cheating." In some regards, "cheating" will cease to exist, as it simply will have been defined away.

As I hinted above, most "cheating" (I truly believe) is undertaken as an act of desperation, a means of coping with failure as measured by receipt of lower-than-average academic grades. "Cheating" is a means of striving to succeed within a system which provides extrinsic rewards for optimal performance rather than intrinsic rewards for authentic mastery and authorship. I cannot but believe that the vast majority of students would welcome an academic system in which the goal is not to earn high marks but rather to learn, and that students accustomed to this system would see no need to game the system by "cheating." I'm not so naive as to suppose that every student will respond well: there will always be those so acculturated by thirteen-plus years of a largely competitive educational system that it's in their blood to fight tooth and nail for every last percentage point that might tip them from a B+ to an A-...but I believe that even those who are very comfortable with this traditional system will abandon it if given the chance to do so.

All of this gets me back to this morning, and the thoughts I had as I watched my Calc I students (even the strongest of them) wiggle and squirm in completing their in-class exam, stressing out not over whether or not they'd really learned the concepts we've been working on together for the past few weeks, but rather over the grades that will unmercifully adorn their papers when they get them back tomorrow morning.

My thoughts can be boiled down into two short words: no more.

No more in-class exams. Ever. I'm through with them. The one I gave this morning (and will give again in a couple of hours) will be my last.

For quite some time now I've not given in-class exams in my upper-level courses, feeling that little meaningful could be asked on such exams, aside from requiring students to parrot already-proven theorems or give short answers to requests for definitions. In these courses, in-class exams offer none of the opportunity offered by take-home exams to ask authentically engaging and probative questions, and therefore I've found them pointless time-sinks.

For quite some time now I've resisted the banishment of in-class exams from my first-year courses, thinking, I suppose, as I've heard some of my colleagues to think: "there are certain computational techniques the students will have to learn to perform, and to perform quickly." True, but even the most straightforward computational skills will be as well developed (and much more readily understood) if performed in the service of completing the more meaningful, authentically engaging exercises included on take-home exams. This is as true in Precalc or Calc I as it is in Abstract Algebra I or Topology. Even in lower-level courses take-home exams offer a much more meaningful sort of assessment, and even if those exams are still meant as individual exercises and not collaborative ones (of the sort with which I've been experimenting in Linear Algebra this semester) they go much further than do in-class exams in encouraging a culture of collaborative engagement, simply by downplaying high-stakes individualistic assessment.

Keep in mind that I'm not boo-hissing exams entirely: exams offer students a means of reflecting on the ideas they've learned for the past ___ weeks, and if properly responded to they're fantastic tools for giving students feedback. Well-designed and well-delivered exams give students a healthy way of furthering their learning and assimilating their knowledge as they think critically about it. Exams are here to stay.

In-class exams, however, for me will soon be a thing of the past.

I've already responded to a couple of concerns I anticipate colleagues (and even some students...very bright ones, in fact) might have about this decision of mine, but let me respond to a couple more hypotheticals.

Colleague/student: "If you de-emphasize high-stakes individualistic exercises like in-class exams, some students will game the system and prop themselves up on others' work without really learning anything themselves."

Me: "No matter how you set the system up, students are always going to find a way to game it. Gaming the system I propose means cheating themselves out of learning. Gaming the system as it currently stands involves engaging in a behavior that's viewed as sociopathic but is really little more than a symptom of deeper systemic problems. I find the former course far less pernicious. Sure, there'll always be students who frankly don't give a shit, but we're not going to serve them well in any system, and with a system more conducive to authentic learning, they might just pick up a thing or two along the way."

Colleague/student: "You can tilt at as many windmills as you'd like to in your own little class, but you've got to assign grades at the semester's end, anyway. Won't the system you propose result in massive grade inflation?"

Me: "My response to this concern is twofold. First, in courses in which I've begun doing away almost entirely with individualistic activities (begun in Foundations, Topology, and Abstract Algebra I and II in previous semesters, and taken to the extreme in my Linear course this semester), I've seen little noticeable change in the final grades for the courses. This was true even in Topology, in which students had unlimited opportunity to revise and resubmit all work, with no constraint on collaboration. I simply don't see evidence for grade inflation. Second, even if there were grade inflation, so what? It would be incredibly difficult to determine whether students' grades were made higher because they were bracing themselves on each other ("gaming" the system in the sense expected by my first colleague above), or simply because they'd actually managed to more richly and more fully understand the ideas addressed by the course. That is, maybe the grades are higher for a reason: the students are actually, for the first time in their lives, getting it."

I truly feel this way.

Moreover, I truly feel that I've become proficient enough as an educator that my courses offer the sort of rich collaborative learning environment wherein in-class exams no longer serve a meaningful purpose. They're simply anathema to my teaching philosophy, and they're counterproductive to my goal of establishing a safe, stressless, and supportive setting in which we can all learn from one another in robust and authentic ways. From here on out, they're gone.

Before I leave, let me pass my apologies on to the students currently enrolled in my Calc I course: I'm sorry that you had to suffer through the last iteration of this practice. I know how hard the topics you're being tested on are, and I'll keep that in mind as I respond to the exams tonight. I'll respond to them not with an eye toward giving you a grade, but with an eye toward letting you know how well you're learning. I hope you'll receive them back from me with the same thoughts in mind.

One last note: I should point the interested reader in the direction of Alfie Kohn's No contest: the case against competition, about which I've blogged before, and which over the years has probably proven to be the single book which has exerted the greatest influence on me as a teacher. It should be required reading for all educators, at every level.

Desperados

Yesterday was a long day, and after working pretty much nonstop (class prep to research meeting to class to research to grading to another class to more class prep to another meeting to home to grade and grade and grade) from about 6:30 a.m. until 8:30 or so p.m., nothing could have made it seem longer than discovering at that latter hour that a couple of students had cheated on yesterday's Calc I exam.

Ugh.

Given that I've not much time to write right now (class beckons with a gently arching arm), I'll merely reference an older post which, written in a time of much more leisure, says all I feel like saying right now.

On the positive side, I've found from informal conversations that a number of my Calc I students continuing on to Calc II with me in Spring 2010 are open to the idea of portfolios and other outcome-based assessment methods.

Reading day

One course down, two to go.

Abstract Algebra II is now wrapped up, all grades successfully submitted by noon today. Overall the students did very well, which is hardly a surprise since the class had a number of our best and brightest and most senior students, five of whom are leaving us after commencement in a few weeks.

So far the questions from students in the other two courses (both sets of students are dealing with take-home exams) have come in a steady trickle.

Yup, two take-home exams.

I was a bit reluctant to make the Calc I exam a take-home exam, after the debacle from Fall 2007, but I decided that what I perceive to be the benefits of granting the students a take-home far outweigh the negative aspects, including the risk that one or more students might again decide to cheat.

1. By affording the students time for meaningful reflection on the concepts learned in class, take-home exams offer a further formative learning opportunity rather than simply a summative assessment of a student's ultimate performance.

2. By eliminating the largely artificial "high-pressure, high-stakes" environment of an in-class test, take-home exams are more apt to measure more clearly students' understanding of (or at least ability to recover and synthesize deep ideas involving) the course's subject matter than simply aptitude in test-taking.

3. As hinted in the previous point, take-home exams more closely approximate "real-world" settings in which students will eventually find their skills tested, in which they will generally have access to resources (books, notes, mentors, and, yes, the outlawed-even-on-take-home-exams colleagues!) in order to better to meet the challenges with which they're faced.

We'll see how it turns out. I did give a little bit of a lecture when I distributed the exam sheets a few days ago, an uncharacteristically stern admonition at the outset, hoping to instill in the students the gravity of the trust I'm placing in them.

I've received one student's exam so far, but I've not had a chance to look over it. We'll see.

For now, I'm working away, one day at a time.

Tomorrow: donuts and derivatives, and a meeting about writing assessment. Oh boy!

A cathartic post

It was one of those days yesterday.

One of which?

One of those.

As I write this, the dawn is creeping up in the east, grasping at the leafless trees with Homerically rosy fingers. Pink clouds are scudding on a periwinkle sea. It looks like it'll be a beautiful day, and the prettiness tempers the ugliness of the purely visceral righteous indignation I felt last night.

Yesterday at 5:00 p.m. was due my Calcsters' take-home final, their final obligation to me this semester. Their duties done, it was the last I'll see of most of them until next year.

I had some wonderful parting moments as I said my farewells for the semester. Many of these people will be back for more in Calc II, and I look forward to it. I'm very open about the fact that Calc II is my favorite class to teach. (Although now that I've got 280 down pretty pat, I'm going to miss it next semester!)

The partings were rendered bittersweet by a revelation made last night as I began grading the Calc I final exams.

Six of my students cheated off of one another.

Exams from three pairs of people (two pairs of folks from the first section, and a third pair straddling the sections) show incontrovertible signs of collaboration.

This hurts me.

It hurts me because it insults my intelligence (did they think I wouldn't notice?), and because it takes the respect I've shown them all semester (in treating them like responsible adults all semester long, in trusting them with a take-home exam in the first place when many of my colleagues would likely say "you can't trust freshmen with a take-home exam"), the respect I work ceaselessly all semester to build, and it throws it back in my face.

Did you think I wouldn't notice?

News flash, folks: I'm a pretty smart guy, and I've been playing this game for a decade now. It ain't hard to recognize when someone's worked with someone else: there'll be identical use of uncharacteristic phrases, identical idiosyncratic errors, identical stray notational boo-boos. What the cheatee writes, the cheater will often copy. If the cheater wants to try to cover her or his tracks, she or he will often modify a line or two here and there, but if said cheater has no idea what the cheatee means by a particular line or computation, the result can be a wholly incongruous modification. (I saw a couple of doozies last night, changes that rendered the corresponding sentences ungrammatical.)

Did you think I wouldn't care?

In class, I'm a nice guy.

I'm a nice guy in class mostly because I'm just a nice guy in general. I could never be a hard-ass. It's just not who I am.

But I'm also a nice guy in class because I truly believe that in showing you kindness and understanding, and in showing you respect as adults (young adults yet, to be sure, but adults nonetheless), you will return the favor.

And for the most part, you do.

Of the thousand-odd students I've had over the past ten years or so, only a very, very small handful have bobbled the trust I've placed in their hands. By and large, I receive little but respect, gratitude, and positivity from my students. Even those who don't like me so much generally agree that I treat them fairly, that I treat them like sane, rational, adult human beings capable of managing their affairs with me with all due maturity.

So, believe it or not, it really does hurt me when a few folks pull this kind of crap.

I ask again, did you think I wouldn't care?

"Do they realize how much you care?" Maggie asked last night as I was ranting and raving about the six students who'd clearly gotten a little undue help from one another. "Do they know how much work you put into their classes? How much you care about how they do?"

Yeah, they do. I know they do.

Not a semester goes by without many students remarking (to me directly and anonymously on their evals) on the effort I put into class, the consideration I show them, the amount of work that goes into making sure they'll have a quality experience in my courses.

They do know that. I still believe that.

Why, then? Why cheat?

These aren't bad people I'm talking about. I like them all. They're smart, they're funny, they're kind. They're good people.

But, at least momentarily, they're also desperate, and they're inconsiderate.

Desperate times call for desperate measures.

For the most part, these folks are first-year college students. As such, though they may be sharp as tacks and quick as whips, they're also pretty piss-poor at managing their schedules. A year from now most of them will look back at their first year and realize just how much time they were wasting simply because they hadn't yet learned how to organize themselves and get their shit together to get it all done. They feel harried and hurried. They're continually, but artificially, pressed for time. They feel they don't have enough time to finish everything they need to, mostly because they've squandered their time through mismanagement. So when time's run out and they've yet to get it done, they'll resort to desperate tactics.

A couple of disclaimers are in order here.

First, I'm not claiming that you're not working hard. In fact, I'm sure you're working your tails off, I've seen evidence of your hard work all semester in my class. I'm rather claiming that you're working inefficiently; you've not yet learned how to write effectively and efficiently, you've not yet learned how to streamline your use of your supporting resources (textbooks, internet, outside assistance), you've not yet learned how to prioritize.

These things will get easier for you. By the time you graduate, you'll get better at doing these things, mostly because you'll have to: almost without exception, every one of you will look back on your first year of college as the easiest year of your college career. Hands down. To make it from here, you're going to have to get better at managing your time.

Nidra, a student from the first class I ever taught at UNCA, stopped by my office yesterday morning. She'd just finished off her thermodynamics final, and was on her way to go study for another exam. She was in the neighborhood and thought she'd say hello. We talked for a bit, and I made some cute remark along the lines of, "ah, freshmen!" and she agreed. She's sufficiently far away from her own freshman experience to look back on herself at that time and laugh.

Note to current freshmen: you'll laugh, too. You'll have your chance. The next few years will be hard as nails, but you'll love them, and you'll sincerely wonder how you ever thought your first year of college was difficult.

My other disclaimer? There are a few of you (and I'd like to think you know who you are) who truly do have heinous, untameable schedules. Some of you work forty hours a week, and/or have small children, and/or have ridiculously long commutes to campus nearly every day of the week. Some of you have had major personal crises that have unexpectedly affected your performance in your classes this semester. Some of you can gather no more than three or four measly hours of sleep each night, night after night after night. I know who you are, and I thank you for the hard work you've given me this semester; it touches me that you value our class enough to make it such a high priority.

But you know what? Almost without exception, these are not the people who cheat. Why this is, I don't know. Perhaps respect is born of their work ethic, perhaps they simply don't enough time to orchestrate the dishonest maneuvers cheating would require. For one reason or another, these folks, who at the semester's busy end are likely the most desperate people in the class, aren't typically the ones who cheat.

As I said above, the cheaters are also being inconsiderate.

I'm using the term in its somewhat literal sense: they're simply not considering the effects their actions will have on others. I cannot believe that they would act as they have if they'd thought about how strongly it would effect me.

Or so I think.

And so it goes.

So where do we go from here?

To those of you who cheated on the final exam through undisguiseable collaboration: you will be receiving a zero on the exam. If you feel this penalty is a bit harsh, please refer to Section 2.1 of the Student Handbook, dealing with "academic honesty": I would be well within my rights if I were to fail you for the course and report you to the Student Affairs Office.

And so it goes.

So.

So yesterday wasn't all bad, in spite of those late-evening grading revelations (and no, I'm not yet done grading. I didn't want my choler to color the remainder of my job, so I've put it off until after I'll have written this cathartic post, at which time I'll set to the task once more with renewed alacrity!), and in spite of a somewhat disappointing mid-morning meeting on grant-related stuff about which it would be inappropriate to say more (actually, I lie: the meeting was a productive and pleasant one, I just wish we'd had it three months ago!), I had a good day yesterday.

I shared a group hug with Quincy, Olivia, and Davina, three of my tip-top 280 students, in the Math Lab. I have had a blast working with those three, truly with everyone in the class, this semester. I've got one more meeting with them, on Monday.

Early in the morning Sieglinde came by to drop off her Calc final. She was touched by the e-mail I'd sent her letting her know how happy it makes me that she's happy she's chosen a Math major. She'll be a great student. We met for a half hour or so on Thursday, at which time I gave her a crash course on graceful graph labelings, enough to get her started over break in thinking about the gracefulness of lobsters and other closely-related graphs.

I gave a similar problem to Trixie yesterday afternoon. On Thursday Maggie and I had lunch with five of the folks who'd helped with Super Saturday this semester, in order to show my gratitude for their efforts. Trixie was among them. Half-(but only half-)jokingly she told me she was going to miss not having any math to do over break, so I promised her I could give her something to think about. "Not calculus," I said. "Real math." She agreed to consider it, so I spent an hour or so yesterday morning typing up much of the information I'd given to Sieglinde in person the day before, indicating a related problem Trixie could work on, dealing with asterisks instead of lobsters.

As she left I put the notes in Trixie's hands. "I'm excited," she said, "I don't know why."

"Because, one, you like math," I told her, "whether you admit it or not. And two, you're good at it, whether you know it or not. And three, you, maybe not over break, maybe not by the end of this coming semester, but likely within the next year or two, if you get started now and work hard at this stuff, you, Trixie Muddleston, have a chance to discover new, original mathematical results."

"That's exciting," she said.

"I know! And that's why you're excited!"

I've got her, I hope. I've got her hooked.

After all, if I care about it, so will they.

If I'm excited, they'll be excited too.

To all beginning teachers out there: be conscious, confident, and competent, to be sure, but above all else, be passionate, and show your students that you truly care about what you do. There's no better way to do this than to get them involved in your own efforts, to give them a spot on the team. The sooner this is done, the better for them.

"But freshmen?" the traditionalists cry. "Some of them can barely add, and they butcher the Product Rule, can't expand a binomial to save their mothers' lives! What business have they got working on research?"

Here's something I've noticed: many of the students who did well at math in high school and who in their first year of college elect to pursue a Math major truly have no idea what math is really about. They're toeing up to the starting line without even knowing how to run the race.

How do they get themselves into this? Many of them believe math is little more than an extended elaboration of the almost entirely mechanical classes they went through in high school. (True quote from a first-year Math major, earlier this semester: "I'm going to be a math major. I love derivatives!") They're wholly unaware that doing math well takes more than attention to detail and carefulness in computations, that in addition it demands creativity, originality, analytical perspicacity. They're unaware that math is an art as much as it is a science, that at heart most active math researchers are no less artists than are poets, painters, and concert violinists. Though they may know that math has to do with physics and engineering, and perhaps even chemistry, they're unaware that math shares intimate bonds with biology, sociology, economics, and philosophy. They're unable to see past the numbers and plusses and minuses, to see the abstract patterns and structures that underlie it all.

Sometimes by the time they've finished off a year or two of a college mathematics curriculum (and have thereby learned a bit more about what math is about), they've discovered they're in over their heads, they don't particularly like what they're doing, they're not very good at it, but if they want to finish their degree in time, there's no time to change majors.

They've no means to see past the numbers, for the numbers are, for the most part, all they've ever known. How on Earth can you convey to a first-year college student the depth and breadth of mathematics without letting them involve themselves in its enterprise? And the sooner, the better: they'll thereby learn early on what math is about, and they'll grow stronger at it.

I'd like to put together a one-credit course titled something along the lines of "Mathematics: a Survey," whose content would comprise an overview of the mathematics discipline: what are the major branches of mathematics, how did they evolve, how do they relate to one another, how do they touch on other fields of study? It would introduce the idea of mathematics as an extended exercise in abstraction and pattern recognition, of deduction and induction. It would also showcase some of the beauty of mathematics, and so might make an effective article of propaganda. It would have no prereqs, and would be recommended to anyone majoring in or thinking about majoring in Math. "Take this course," one could tell one's students, "and it'll help you decide if math is really for you."

Anyway, I've been writing this for two and a half hours now. I'd best be getting back to grading. I'll check in later, perhaps, and follow up on some of those loose ends I keep meaning to tie up (Newton v. Leibniz, math poetry, and so forth).

For now, take care. To all of my students, even those desperate few: thank you for your work, thank you for your efforts. I look forward to working with you further.