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?

I can’t say that my students are at that level (but they’re only freshmen in high school). I think they understand that the group work is to help with their individual understanding. They know they will be individually assessed on problem solving. My hope is that by the time they are juniors and seniors they’ll fully understand the group process.

The issue with Math 55 is that, since it attracts only the best and the brightest, and (as far I can tell) is not required of any student, the students in that classroom probably all

wantto be there andwantto learn as much mathematics as they can. Therefore they will not cut corners, because they know they are cheating their future selves. The students in a more ordinary class, however, may not necessarily want to be there; they may know that they will not need the material in the future and are not particularly invested in learning it, just in getting a good grade. Therefore some of them will cut corners, since their motivation is a mark on a transcript, not the learning it ostensibly represents.Another way to put it is: The behavior you see in the students in Math 55 is the result of a teacher or teachers, somewhere along the way, getting students inside the true enjoyment and challenge of mathematics. They don’t just spontaneously start acting like that. I’d like to ask them all who taught them how to do real work in math. Do you think they learned in high school?

I don’t think this has to do with math specifically. The behavior I’m referring to — working hard and contributing to a group process but then switching to all-individual work when called for — doesn’t appear to me to be confined to mathematics courses, even hard ones. It seems more like jsut the product of growing up knowing what’s right and wrong, and with having a strong work ethic and pride in your work. I don’t think you have to be a math genius, or even a math student, to have that behavior.

I think that kind of thing comes from parents and friends. Teachers merely reinforce it and make it content-specific.

Yes. We (or at least I do and I’m generalizing about my peer group) understand. It’s too bad that in high school our group projects are not as challenging or we’re stuck with a useless group with only one or two people doing all of the work. It’s to the point where I want to cry when I hear ‘group work’ and ‘we’re picking the group’ in the same sentence.

I don’t think it translates well out of mathematics… but in my high school, and specifically in the freshman and sophomore math classes (another teacher and myself) we actively cultivate this.

Jonathan