school visits, part 3: Exeter

I loved my visit to Exeter, most of all because it just felt so good to be in an environment that was so focused on learning-- the faculty were intent on learning (and on facilitating the learning of their students), the students were intent on learning (at least, so it appeared during the classes we observed), and the physical space of the school was designed to support and promote learning. It was so wonderful to see the Harkness method in action, after having heard and read so much about it, but I was most impressed by the variation in which we saw it implemented. Here's a sample of what we saw:

• In one classroom, the teacher drove the direction and flow of the class period by telling students which problems to put on the board and offering extensions and generalizations wherever possible. Students critiqued one another's responses, and asked questions like "would that still work if it were negative?" or "what if you used this form instead?"
• In another classroom, the teacher did much more of the work; students explained their work but often faltered and looked to the teacher for affirmation or to catch errors, and the teacher asked almost all of the questions. Students mostly spoke up when they were confused or needed help.
• In a third classroom, the teacher spoke only three times in the entire class period: once, to introduce us as the visitors, and twice to pause the conversation (she literally just said "wait--") until a student picked up on what she wanted them to talk more about and restarted the conversation. In this class, students not only continually pushed the mathematics forward, but also pushed each other to explain better or differently, to consider an alternative, or to make sure the whole class was on board.

The first classroom, in particular, made me rethink how quickly we jump to "teacher-centered" as shorthand for bad and "student-centered" as shorthand for good, and led me to quickly draw up the following table of what I saw students and teachers doing in the classrooms that felt most joyful and engaged (the ones I didn't want to leave):

What STUDENTS do

What TEACHERS do

Advocate for themselves and their own understanding Clarifying someone else's thinking : "Could you explain…" Checking their own thinking: "Could it be this instead?" "These two are the same, right? So shouldn't you…" Contribute to others' learning Direct the conversation so that it can be productive for others: "For the record, so far we've said…" "To come back to the original problem…" "as we discussed earlier…" Look out for their peers: "Wait, Nika has a question" Catch errors (own and others) Explain their thinking (e.g. "here's how I knew it was different from that other problem") Demonstrate confidence and agency & a belief that things should make sense Willingness to put problems on the board and say "I don't think this is right but I wanted to share what I was thinking so you can help me understand" or "I think my method makes sense but I don't think I got the right answer" Want to know why they're learning what they're learning-- not at the level of superficial concrete applications, but at a more sophisticated mathematical level: "Isn't this method more complicated? What's the advantage of this method instead of something else?" Do not wait to be invited to speak Students who are not speaking whole group are actively writing, figuring, testing on their paper and/or with a partner Pull out calculator to double check or to explore Listen to and follow multiple conversations at once Express joy and delight: "ohhh!"  "whoa… look at that!" "omg."

Generate counterexamples that allow students to answer their own questions (e.g. student asks "if you know the hypotenuse, can you find the other two sides of a right triangle?" and rather than answering, teacher draws two right triangles with the same hypotenuse and different side lengths, student says "ohh never mind") Pose extension problems or questions:  "Could you do that with four dots instead of three? With five? Pose hypotheticals: "Given that, do you think you could find b? What if I just gave you the three sides; what do you think we could do then? This is instead of explicitly posing an extension problem; through a series of hypotheticals around the Pythagorean Theorem, students see how to find b, how to use sides to determine whether something is a right triangle, whether right triangles are similar when sides are increased proportionally, and whether the result is still a right triangle. This would have taken at least a week to teach in a traditional objective-driven sequence Ensure that students have identified all possible routes and methods: "Why else is that true?" Generalize from problems to algebraic representation Bring up old ideas (factor trees) that are relevant to what's at hand, both to support students and to ensure mathematical connections are being made Reinforce mathematical habits: drawing a picture, checking your answer, etc. Ask whether something makes sense

I'm not pretending this is a complete list, but it was a really helpful exercise to do this inductively, while sitting in the back of classrooms, rather than deductively, particularly since it gave me concrete examples that I didn't have to make up. Old hat to you? Any surprises? Now, to figure out what (if anything) I should do with this table... any ideas?