This course calls for two distinct kinds of teamwork. First, the basic sort: we should all see ourselves as part of the same journey of learning together. From the respect we accord each other in preparing for, showing up to, and participating in class discussions, to the problem sets and everything in between, we need to be on the same page and each others’ unconditional allies. Second, there is a team teaching this course. We will be eating our own dog food, so to speak. Third, you will be working in small teams on major projects, namely the “great theorem” presentations, as well as the “show and tell” sessions.

What We Owe Each Other

In terms of general teamwork, we are here to support and motivate one another in going farther and digging deeper. This is harder than it might seem, but the first step really is simply to realize that this is really what we’re here to do.

• We are not here together to compete for a place as the smartest in the class.
• We are not here to prove that we’re good enough to be here.
• We are not here because classes are supposed to meet in person.
• We are not here because participation is 25% of the grade.
• We are here for each other, to make each other smarter and to be part of something together.

Not only does this describe why we have a responsibility to be prepared and present, but also gives us something to go back to when we get into the messy details of how to be together and do math together. Most of our social interactions concerning this subject revolve around intimidation. Even when we don’t mean to, even when we try to help each other, there’s so much baggage to overcome and so little to model good behavior on. I’ve been working directly on this for a decade and yet I know, here at the beginning of another semester, it will be a lot of work to earn your trust and to be able to come in here and work to get the wrong answers, to be able to ask questions, about things that are confusing, to do this without worrying that you’re asking a “dumb question”, or that you don’t deserve to be here with us.

• We have learned that others are likely to get the answer and so we can save embarrassment by not getting our pencils out.
• We have learned that the way to succeed is to be the one with the right answer, first if possible.
• We have learned to keep our mouths shut if we’re not sure.
• We have learned to pretend to understand, copying what we can from those who seem to know (teacher or other students), as if the right inscription will someday make sense on its own or be enough to get credit and move on.
• We have learned to limit our inquiry to the right answers being asked for on the test.
• We have learned that education is about remembering or deriving those right answers on our own, or at least appearing to.
• We have forgotten how to be curious.
• We have forgotten that learning is a social act, not an individual one.
• We have forgotten socially active ways of knowing.
• We have forgotten that we are in this together.

It is these dark social patterns, more than anything intrinsic to the subject itself, that puts us at such a disadvantage as we come together. These are what we have to work against if we hope to make this time meaningful and actually learn something.

Concretely, for our time in class, this means not only being ready to speak but being ready to listen. More than telling someone they are wrong, we need to find ways to contribute to a conversation that make it impossible for anyone to remain wrong. We need to broaden our utility with the subject manner beyond answering test questions. We will know if we are getting there once there are questions that have different perspectives, or which we may only be able to partially consider. We will see that everyone feels comfortable moving our discussions forward, and pointing us to the next confusion. We will notice that good ideas come from everywhere and when we are able to move on as a group to all see how these ideas work together.

Overcoming these dark patterns is about more than getting better at math too. Truth be told, math, at least the math you do in this and other courses, isn’t that important for most people to do well. If it were, more people would remember algebra after the tests are over. They don’t. I don’t mean to devalue it entirely. Once you get to know me, one thing that will be obvious is how much meaning and joy I get from this aspect of humanity. I’d say instead that it’s more like an art form, something that one can study as one of the many ways of emboldening the spirit of humanity, and not an arbitrary one. But probably about as relevant to day-to-day life, even in technical professions, as ballet.

Anyway, my point is that what we do here in the name of math is really practice for all of our shared endeavors. The patterns I’m setting up for us to learn are those I imagine to sit at the center of a more functional humanity generally. Rather than being bound by point totals and pedantic rules, we have a responsibility to each other that is derived from the fact that we breathe the same air and inhabit the same sphere. We are together on this planet, as neighbors, and need to grow to treat each other as fellow beings, not cogs in a machine.

Team Teaching

A special wrinkle for this semester is that there are two of us teaching the course. I (Chris) am the usual professor, and Bibiana is my student co-teacher. Last semester, she studied all our material and more, and together we planned out this semester. She’ll also be here each day, and will help provide support for and feedback on your work. Some days we will be following her lead entirely. Like the other kinds of teamwork, this does complicate things a bit, but is worth that because we have a chance to do something richer than normal. She and I see the material, its purpose, and learning differently, and so the resulting perspective you feel will be that much more nuanced.

As we’ll learn, learning math really isn’t about seeing someone else take the path from question to answer as painlessly as possible. Instead it’s about people learning together to see in the same way. So even though there is bound to be some momentary confusion around who is at the head of the class at particular moments, or which approach to take in a given case, with the patience to actually work these things out, we will—all of us, myself included—arrive at a more complete understanding by the end.

One nice thing you can count on is a good role model for telling me when I’ve got something wrong. Again, talent in this subject is not the same as a lack of mistakes, but that can be hard to feel when you’re used to being at the back of the classroom.

Teams for presentations

Successful teamwork here is not the same thing as dividing the content up equally and hoping each person does their part. Real teamwork takes effort and time. And it pays dividends. You can always tell in the end when a team has unity and when they worked little together. For these tasks, working together on the material allows you to

• Avoid embarrassing errors of understanding the material
• Have better luck encountering good approaches to understanding and heading off likely misunderstandings
• Practicing something unfamiliar (teaching math) at least once
• More chances to relate your material to the class as a whole and the context of the original work.
• There’s not a lot more boring than watching three people give unrelated presentations that should go together.

The other reason to push for real teamwork despite its apparent inefficiency is for the sake of practicing something you will need to do later and which is hard. Working well together is work on its own and takes different skills than doing math. I want you to care about and develop these skills.

The difficulty of teamwork is not a mirage. One big problem likely to come up here is meeting times. Nothing is worse than finding out that a group never met before the presentation, but it’s understandable that students have different schedules. So instead of choosing chapters based entirely on subject matter preference, I suggest using schedule as the first determiner. Find a group you can actually meet with. The topic matters much less in the end than actually being able to meet.