Teaching Mathematics Classically

Students typically walk into my class believing that authority is the source of mathematical truth – the authority in the back of the book. It's perplexing to them not to have that authority available. The refrain I use is that the goal of the discussion is to first convince yourself that you’re correct, then convince your peers, and then convince me. So our discussion is about reasoning and persuasion, also stunningly counter–cultural in math.

The central image I use when talking to my students about how to form their mathematical community is the body of Christ. As St. Paul writes in his letter to the Corinthians:

[T]here are varieties of gifts, but the same Spirit; and there are varieties of service, but the same Lord; and there are varieties of activities, but it is the same God who empowers them all in everyone. To each is given the manifestation of the Spirit for the common good.

Each student in the classroom has gifts that will allow them to contribute to the common understanding and common good. As a teacher you have a positive role and a negative role to play in bringing those gifts to bear. First, the positive role.

Setting the physical tone of the classroom is really important. I actively cede control of the whole board to the students, which means putting the markers down, and moving away from the board. I usually sit with them in a student’s seat, and tell them explicitly that the board and the room are now theirs to use well. They then freely move about the room, as do I. My goal here is to convince them that I’m no longer leading the class: they are. If they forget that I’m in the room, so much the better.

Students typically walk into my class believing that the only way to succeed in discussion is by saying something new, true, and clever. My job is to convince them that there are many, many more ways to successfully contribute to a discussion in mathematics. On the first day of class, I give my students a handout outlining some of the ways they can contribute. I’ve included it in your handout. That starts a conversation about how to participate in discussion. I then spend much time teaching the kids to discuss.

Almost all of the comments I make during one of our discussions are not about math, but about discussing. So, if a student throws up a hand and declares that they are lost, I’ll encourage that. If a student says something interesting that the class ignores, I’ll wait a few minutes and then point out that they should pay more attention to everyone’s comments. If a student is running way ahead of the class, I’ll restrict them to only asking questions to help their peers understand their ideas. If a student seems disengaged, I’ll sit next to them and ask what their take on the discussion is. Sometimes they have been pondering something really interesting, and I can encourage them to share it!

Now the negative role. Once you’ve fostered that sort of working environment you must preserve it. It is fragile, and many things can destroy it. Here are some of them:

• If students believe, even for a minute, you will do the thinking for them, they will not think. If you insist that they think, they will call your bluff by sitting in silence. I’ve never had a group of students make it past about twenty minutes without someone breaking and saying something interesting. Especially early, you must be very patient with students. Even once they’ve realized that you do want them doing the thinking, they will test you on this every so often. Don’t give in. The group that made it to 20 minutes, I reminded them that their paper was due in a week, and it’s much easier if they talk about it together before they write it. That encouraged them.
• Seriously though - you have to not help them. It’s really hard. They are going to be expert weaselers.
• If students believe that you only care about the ideas of the best mathematician in the room, then they will wait for that mathematician to do all the work. You’ll probably have a student who just gets it. It’s important that that student not dominate your attention. That means actively engaging with quiet and struggling students, calling out good ideas that flow from other students, and sometimes even imposing restrictions on your best mathematician. It’s amazing how well they learn to articulate their ideas when they’re not allowed to write anything on the board and only allowed to ask questions.
• If students believe their peers will think less of them for venturing ideas, they will not venture ideas. I mercilessly quash any talk that cuts down a student for expressing wrong ideas. Getting kids to separate their ego from their mathematical ideas takes constant and thoughtful encouragement and can be destroyed in a single ripple of laughter. It’s hard to hit the balance where students feel comfortable arguing against genuinely bad ideas without imputing moral blame to the student who has articulated the bad idea. Finding the right balance takes lots of open dialogue and trust. The students must come to know and trust one another as mathematicians, so the classroom must be rooted in Christian love.