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.

Back to the basics

I'm feeling a bit less stressed-out than I was this afternoon when I put together that last post. A good run always helps me out.

While handing back exams in both my morning and afternoon Calc I sections today I brought up the idea of using portfolios as a means of assessing student learning in mathematics courses. This idea was couched cozily inside of a conversation about the shock of receiving a "bad" grade on an exam (as some of the students no doubt experienced today). "I hate having to grade y'all," I told them. "I'm more and more opposed to grading in general, and to the simplistic distillation that goes into assigning a single letter grade to such a Gestalt as the sum-total of a student's learning activities throughout an entire semester."

There were many nods of agreement when I described how I'd like to be able to supply them with all of the same feedback I give them already...without the numerical rankings, the stigma-making marks that say "she's more highly-ranked than he is."

"I'm not going to do it this semester, since it wouldn't be fair to any of us, you all or me, to change the system midway through. But I'm seriously thinking about it for future semesters."

More nods of agreement. I'm convinced that students are not against this.

But if I were to move to portfolios, the first question would be, What goes into those portfolios? Clearly students would be asked to submit materials of various sorts that purport to demonstrate mastery of course learning goals. Ultimately, then, the question becomes twofold: What are the learning goals of the course? and What course activities (projects, exams, written assignments, homework assignments, etc.) would be sufficiently rich to demonstrate clear mastery of the learning goals selected?

As I reminded my 280 students today (quite forcefully, I hope), when you've got no idea what to do, you go back to the basics. In 280 in particular and in mathematics in general that usually means you'll want to take a long, hard look at the definitions. In course design, it means you'll want to take a long, hard look at the reason you want the students taking your class in the first place.

My current learning goals for Calc I (as stated in this semester's syllabus) are as follows:

1. Be able to explain to a peer the concepts of limit, continuity, and derivative.

2. Demonstrate how basic problems in physics, engineering, chemistry, and other fields can be couched in math terms using mathematical models.

3. Be able to follow confidently the course of a simple proof.

4. Be able to perform and properly interpret derivatives.

5. Demonstrate (through informed question-asking) a healthy skepticism regarding mathematical and scientific arguments.

6. Demonstrate how to approach a (not necessarily mathematical) problem effectively by breaking it down into smaller problems, arguing by analogy, and applying other basic problem-solving techniques.

I think it's clear that mastery of some of these would be very difficult to assess using "traditional" assessment instruments. While (1) and (4) could be got at with a well-designed traditional test, assessing (2), (3) and (6) would require a more robust (and likely highly nonstandard) project of some sort, and (5) would require something extremely atypical...maybe a dialogue of some sort, or some other "creative analytic practice" (to use Laurel Richardson's term).

Of course, the above learning goals are merely my own...I'd love to see what students could come up with for learning goals of their own. Maybe I should ask them? Yes, I think I shall.

Clearly there's a lot of thinking left to do, on many persons' parts.

For now, I'm off to eat dinner. I hope that if you read this, you'll reflect on it for a moment or more and offer me a few thoughts of your own in the comments section.

(Re)start your engines...

All righty, then.

Tomorrow we recommence, revving up for the straightaway dash to the end of the semester.

This is as good a time as any to take stock of where we are in the semester, content-wise. Accordingly, I'm going to ask folks in each of my three classes to spend around half of their respective class periods tomorrow in reviewing what we've done so far: what have we learned? What techniques have we developed? How does it all fit together?

I've been doing a good deal of reading on pedagogy over the break, from the text for this semester's Learning Circle, Maryellen Weimer's Learner-centered teaching: five key changes to practice (Jossey-Bass, San Francisco, 2002), and Alife Kohn's No contest: the case against competition (Houghton-Mifflin Company, Boston, 1986). The latter does not deal strictly with pedagogical theory, but I came to it through Weimer's text, and I've found its insights useful in designing new classroom concepts.

A digest of ideas:

1. "Our classrooms are now rule-bound economies that set the parameters and conditions for virtually everything that happens there" (Weimer, p. 96; emphasis mine). A page later: "our classrooms are now token economies where nobody does anything if there are not some points proffered" (p. 97, again my emphasis). This economic image is an oft-used and apt metaphor for the give-and-take between the student and the professor, and I've come across it in one text after another. Surely some such variety of exchange is inherent in whatever classroom structure one could imagine, but my question is: must the classroom economy always be a capitalist one?

Given the research that Kohn lays out (suggesting that competition in the classroom and elsewhere is generally detrimental to both group and individual achievement), doesn't it make more sense that the classroom economy be one in which cooperative values serve as the "gold standard" for the course's currency? To carry the metaphor one step further, what if we redesign the economy so that it takes on a more "communist" hue?

For instance, I can envision, in a sufficiently small course (no more than, say 7 or 8 students), an untimed, class exam. Either in lieu of or in addition to a stand-alone individual exam, the entire class would be asked to complete a few problems as a unit, the professor sitting by as an observer and as a "clarifier," roles she or he typically already plays in proctoring an ordinary final exam. All students participate in generating solutions, offering ideas, helping to synthesize ideas already put forth. At the outset of the exercise, a single student could be chosen as a scribe in order to create a single solution to the problems presented, and perhaps no solution could be submitted which had not been "ratified" by every person present.

Yes, yes: there are problems with this idea. For instance, there would almost inevitably be "slackers," those who would get the same grade as everyone else without having participated at all, whether out of lack of knowledge or out of shyness. The more outgoing students would also have a tendency to monopolize the discussion.

A compromise between this innovation and the "traditional" exam format might look something like Weimer's study group exams, presented on pages 89-90 of her text. I think Weimer may have turned me off of this idea with her heavy-handed treatment of the "best" students who chose not to participate in the group exam (p. 90).

2. An idea transversing both Chapters 2 and 5 of Weimer ("The balance of power" and "The responsibility for learning") is the following: grant the students the opportunity at the semester's outset to, within reason, decide the distribution of point values for various types of assignments. This student-led distribution could occur on the first day of class, students breaking into small groups to meet one another and discuss the pros and cons of weighting this sort of assignment that much, and so forth. After giving each small group the change to come up with some rough guidelines, the class could be reconvened as a whole, and ideas shared. A consensus can then be approached: how much will this be worth? Once point values are arrived at, we'd record the result and all stick to the deal.

Obviously there should be some initial parameters outside of which the students would not be allowed to deviate. For instance, in Calc I class, I would ask that each of homework, quizzes, projects, and exams count for some percentage of the class's points, and I would likely set some minimum values (HW must be worth at least 10%, quizzes at least 10%, and so forth). But from there, the students would be on their own. I'd even let them throw in extra requirements, like attendance, if they saw fit to include them.

This arrangement has the benefit of providing students a chance to take control of the grading system to some extent, and thus while it gives them greater power (and less excuse for complaining should they not keep up!), it also invests them with commensurate responsibility.

3. Through Kohn's text I've found some interesting tidbits on pedagogical competition, from other sources: Morton Deutsch, in Education and distributive justice: a social-psychological perspective, Yale University Press, New Haven, 1985, writes: "If educational measurement is not mainly in the form of a contest, why are students often asked to reveal their knowledge and skills in carefully regulated test situations designed to be as uniform as possible in time, atmosphere and conditions for all students?" (p. 394, from Note 48, Chapter 2 of Kohn). Good question. As a fairly non-competitive soul myself, I hate in-class exams and see little purpose to them in the long run. It was this line, in part, that made me think up the class exam scheme in (2) above.

Also, Kohn says on one of the works of the brothers David and Roger Johnson ("The socialization and achievement crisis: are cooperative learning experiences the solution?," Applied Social Psychology Annual 4, L. Bickman ed., Sage, Beverly Hills, 1983): "In fact, even the widely held assumption that 'students learn more or better in homogeneous groups...is simply not true.' A review of hundreds of studies fails to support this assumption even with respect to higher-level students" (Note 28, Chapter 3 of Kohn). There's some ammo for the folks who take flak for "making the smart students work with the dumber ones."

All in all, I'm enjoying both books. Weimer, though I'm not always agreeing with her and I find her tone a bit condescending at times, has given me a good deal of practical ideas, while Kohn's work has been a great fount of references to other authors who purport to prove claims I've heard bandied about before but have never been able to track to the source.