Casting Out Nines

In my upper-level courses — especially the two senior-level math majors courses I teach, Modern Algebra and Topics in Geometry — traditionally I’ve seen timed tests and so forth as being ineffective in assessing the kinds of advanced problem-solving that students in those classes have to do. Mainly the problems are ones in which they have to prove a theorem. It’s hard to do that under a time pressure because it’s a creative endeavor.

So typically I’ve given such problems out as homework, with the instructions that students may work together on understanding the problem and drafting up a sketch of the solution (Polya’s stages 1 and 2) but the main solution itself, as well as any reality-checking, has to be done individually.

This article from the Harvard Crimson from a year ago captures exactly what I wish this process would look like on the students’ level. The article is about Math 55, called “probably the most difficult undergraduate math class in the country”. How do these students handle the homework in this class, which is assigned frequently and hits like a ton of bricks?

Georges Bizet’s Carmen blares from the computer of Menyoung Lee ’10. The boys sit scattered around their gray worktable, their eyes telltale red and fingers sore from countless hours at their laptops, dutifully LaTeXing problem sets. They have been here since 2 p.m. and will work for almost 12 straight hours to complete the problem set due the following day.

As the hours pass, they discuss the problem set. They formalize and write the solutions on their own for academic integrity. Despite the class’s cutthroat stereotype, this session is about community, not competition. [emph. added]

They work hard as a group — they have to — but when it comes time to actually write the solution, they voluntarily break off to work the solution out on their own, because they have a sense of academic integrity. It’s a community, but not a commune. Nobody is taking anybody else’s work and turning it in as their own, because I suppose they have pride in their work and in their abilities. As far as I can tell there are no timed assessments in Math 55 to hold them individually accountable.

I wouldn’t want my Geometry and Algebra classes to be as hard as Math 55, but I’d love it if students would have a solid sense of the correct point when working together on problems needs to stop and individual work needs to begin, and then make that switch from group to individual work as a matter of personal ethics and an understanding of what it means to learn a subject.  And I’d love not to have to shift assessment of problem-solving over to timed tests as a result.

Do students in high school and certain college courses where group work is stressed more and more frequently understand that this point exists?

Another thing about group work and assessment. In some courses, particularly upper-division courses with small enrollments, the same kind of individual accountability I’m looking for can be found through oral presentations, not just timed assessments.

I found this out in the textbook-free quasi-Moore Method abstract algebra course I did this past semester. Students were free to work with each other and consult outside sources on any course task they wished to, but at the end of the day their grade depended on their ability to get up in front of the class (and me) and present their work — answering questions on the particulars, being able to explain the overall strategy of a proof, and defending their work against potential holes. Students who could do this on a regular basis scored highly. Students who couldn’t scored poorly. It worked out.

And I know that the students learned a valuable lesson: You don’t present something unless you know it’s right, otherwise you’ll end up embarrassed. And don’t discount the educational value of potential embarrassment.

One of the things I have learned this semester (which is now officially over, having turned in my last batch of grades this morning) is the following lesson which I am convinced I must implement immediately: Group work has been playing far too great of a role in my student’s grades. From this point forward, assignments which could conceivably be done in groups — not just those that are designated for group work — will count for no more than 10-15% of the grade in my courses.

I like collaborative learning. I think, in fact, that working with other people on math can be not only a highly effective way of doing so but also carries with it a powerful pro-math socialization effect. The best personal friendships that I had during my college + grad school years were those that I formed with my classmates in my various math classes, as we struggled through material that, to us at the time, was really hard. Not only did those friends help me learn, I also associated good times and shared victories over math problems with learning math.

But here’s the deal: At the end of the day, the grade that an individual earns in a class, mine or anybody else’s, has to be an accurate reflection of that individual’s mastery of the material and that individual’s ability to solve problems and think effectively. If were reasonably confident that group effort on problems was translating into individual mastery, I’d be perfectly willing to admit as much group work as students want. But the fact is that this has not been the case.

Case in point: In a recent course, I gave out some pretty difficult advanced problems and instructed students on the usual academic honesty procedures, which boil down to “collaborate if you want but not to the point where you’re no longer doing your own work”. I got back solutions which were eerily similar and all basically correct, and in many cases way out of character for the students handing them in. It was enough to make me suspect a breach of my academic honesty policy, but not enough to make a case. So I simply reproduced the exact same problem on a timed test. And guess what? Whereas before, nearly everybody had a really nice solution — the same really nice solution — this time only one or two people had an idea where to start or even how to correctly parse out the terminology in the problem.

And this has been happening all over, not just in that class — a sort of soft academic dishonesty that nominally stays within bounds. Students work together and hand in work that earns points but does not reflect their understanding of the material. I understand earning good grades is important, but equally important is my ability to identify problem areas and help students grow through them.

So I know what all the digital nativists say about how in the modern workplace, people work collaboratively and it’s a 19th century anachronism to give out timed tests and all that. But you know what? You can’t contribute to a group if you yourself have used the group to feign your own competence. So from here on out, the majority — if not all — of my assessments of students will be done in a timed setting, under conditions that I can set and monitor. For example, in calculus next semester, I’ll assign homework problems and let students work on it all they want in any size group they want. But the grade is going to come from timed quizzes, tests, a midterm, and a final. Some variation on that will also be in place for my two sophomore level courses as well. If you do group work properly, contributing where you can and really working to understand things where you can’t, then it will be no problem to do well on a quiz or test. If not, then the quiz or test will show that up as well.

If that makes me an anachronism, or unhip, or whatnot, then so be it. I’m tired of students not learning the material because they have easy workarounds for doing their own work, and one way or another they will get a good grade in the course if and only if they can show me that they know what they are doing.

This is the final article in the weeklong retrospective series. We’ve seen twelve posts, all on different subjects but also all about the same thing — the stuff I think about regarding teaching, education, math, and technology. It’s been a fun week for me, and I feel re-energized having reminded myself of just what it is this blog is for. Hopefully you’ve enjoyed it too, and thanks for your indulgence.

This last article is a more recent one. I don’t blog about my religious beliefs very often, although I probably would be justified in doing so since my Christian faith is at the core of who I am and why I am doing what I am doing. On this occasion, though, I was moved by my then-pastor’s sermon on Philippians chapter 2 to consider how humility plays itself out in my daily life. Sadly, it doesn’t play out nearly as much as it should. But when you think about it, how often do you hear the word “humility” used in conjunction with “higher education”? That’s even sadder, and it should really not be this way. 

So this last article is both an observation and a challenge for me — for all of us in this business — to approach the call to teach, learn, and serve with selfless enthusiasm. And for me, to blog accordingly. 

Humility and higher education

Originally posted: May 22, 2007

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A couple of weeks ago, our pastor preached a sermon on Philippians 2, the main idea of which is humility. The link gives you the text for the whole chapter (read it!) but verses 3 and 4 give a strong command:

Do nothing from rivalry or conceit, but in humility count others more significant than yourselves. Let each of you look not only to his own interests, but also to the interests of others.

As a Christian whose vocation is to higher education, I find these twin commands to be deeply countercultural, not only to the culture at large but especially to the culture of higher ed. After all, I work in the “ivory tower” — a parallel universe to the rest of the world where the rules of common sense and human behavior seem to take on Bizarro-world-like properties at times.

What would real humility look like, as practiced by a professor actively working in higher education? Taking the verses above at face value, it would seem pretty simple: treat people with a sense of their worth, and value their worth more than you value your own, and look to their interests in addition to your own. This in turn requires that professors think about two big questions. (more…)

Editorial: Here’s the second in a series of reruns retrospectives I’m running all week here at CO9s. This one goes well with the one I posted yesterday. Students learn best in an environment where their curiosity is nurtured and sculpted, and this museum store vignette is a great microcosm of what I wish academia were really like most of the time. I think the message that we have to de-institutionalize learning in order for it to become compelling to young people is an important one, and I hope that I embody that message every day in my work.

The Lesson of the Museum Store

Originally posted: June 23, 2006

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Since writing that post, the 2.5-year old is now almost 4, and she has a 2-year old sister who likes the museum store almost as much. Sadly, this particular museum store in the mall closed over the summer. But there’s still the real thing just a half hour away, and it’s a LOT more dangerous on the family budget.

Last night we packed up and went to the mall for some playtime (the kids’ play area has been a Doodlebug favorite for a long time) and some small-scale shopping. After the Mrs. went off to shop, though, Doodlebug wanted to run around the mall rather than play. We ended up at a newly-arrived store: The Children’s Museum Store, which is evidently a branch of the store found on the bottom floor of the awesome Indianapolis Children’s Museum (consistently ranked as the top children’s museum in the country).

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