Sunday, March 29, 2020

Remote Teaching

As I end the first full week of remote teaching, I wanted to quickly jot down some of the things that I've tried this week and what is and isn't working for me. First, a bit of context:

- I'm in San Francisco and teach Math to 7th and 8th graders at a K-12 independent school
- We are 1-to-1 with laptops in the middle and high school and have reached out to students with internet connection issues to help them access remote classes
- We don't have textbooks or give grades, which means that we need to think differently about structures and motivation for kids to engage in remote learning
- Two weeks ago, school rolled out a Remote Learning Plan that involves synchronous video classes over Zoom for most core classes, along with assignments and drop-in hours, but no synchronous class, for P.E., music, art, and electives. We are on an A/B block schedule and classes have been shortened from 75 to 60 minutes so kids get a 20-minute break in between, but otherwise, have 3-4 remote classes four days a week, and then a day to work on homework, projects, work from their asynchronous classes, and meet with teachers one-on-one.

Some things that I've found really helpful so far:

  • A class webpage... I use this to structure what we're going to do during each remote class. It was super, duper easy to make a webpage using Google Sites. Mine is not fancy, but has a tab for the day's agenda, another one with assignments students complete outside of class (which are also posted to Google Classroom), and a final tab that organizes class notes and online resources for each topic, since we don't have a physical textbook.

Here's last Wednesday's agenda for 8th grade:

Kids can use this to get to the different parts of that day's lesson, which is especially helpful if they lose their connection and have to reconnect to the Zoom class or have bad audio.
  • Structures that break up the class into different types of learning environments seem to be going well. We usually start with a warm-up problem that is linked in Google Slides that we either discuss as a class or in breakout rooms. Then, I have each breakout room work on problems together in a slide on Jamboard, a virtual whiteboard. 
Here's a whiteboard from one breakout room in a 7th grade class from last week:

Some kids are writing using their mouse and trackpad, some kids are using text, others are doing work in a notebook and uploading photos of their work. Having the virtual whiteboard means that I know which groups are struggling and can be more efficient in which breakout rooms I visit while they work together, as well as identifying responses I want to highlight when we debrief as a whole class.

The last part of class is usually a Desmos activity I can use to see how individual kids are doing with the topic and decide on whether I want to see any for one-on-one time. I'm mostly taking Illustrative Mathematics lessons and converting them into Desmos Activities or creating quick exit tickets that I can have students complete in the last five minutes of class.
  • Using Google Classroom to get homework out and collect written work is going really well. Students are submitting photos or scans of their handwritten work on homework assignments and the Classroom app for iPad lets me annotate submissions so that I can give them written feedback on their work. This is another important avenue for me to see how individual kids are doing and to identify kids I want to follow up with one-on-one. That might mean asking a student to see me during office hours or recording and sending them a short video addressing something I am seeing in their homework. I just figured out that my iPad has its own screen recording function so I don't need to download and use a different app, I can just jot and talk through a quick note in Notability, upload it to Google Drive, and send the student a link.

Things I still want to work on as California will remain "sheltered in place" for at least for the next month:
  • One-on-one meetings with every student; I have been mostly using office hours as an optional drop-in time for students, although I have reached out to those that I can see are struggling, but I want to schedule a one-on-one meeting with each and every student in the next few weeks to check in and find out how things are going for them and to look over their work together. I'm finding that middle school students are just not great at taking in written feedback over the computer and knowing what to do with that information or what their next step should be. 
  • More variety in the activities I am doing with students during class; right now, they seem to like the mix of class discussion, breakout rooms, virtual whiteboarding, and Desmos activities, but I imagine that these will start to get old soon. These are pretty similar to activities we did in class when regular school was in session, but we also did collaborative projects, labs where students gathered and modeled data, student presentations, and explorations with manipulatives, which are all missing from remote Math class. I'd love ideas of what others are doing that might create more balance and variety in working with students remotely. 
  • Ways to check in with every student during class; I really miss the ability to walk around the room, scan what students are doing, and ask quick questions to probe their thinking or identify cool ideas that I could then ask them to share with the class. It's really hard to informally see the work of individual students and and talk to them without drawing a lot of attention to it in a video meeting while still keeping an eye on the rest of the class. 
  • I want to build more community and connections between students because I know this is something they're really missing in this remote learning space and is one of the main reasons that kids are excited to go to school and learn. Initially, I focused on structures that would help with content and organization because that's how I deal with stressful situations, but now that classes are running pretty smoothly and we have a system, I want to develop more ways for us to be human and connected together. 

Thursday, January 16, 2020

Proof in a non-Geometry classroom

A beef that I have long had with the standard math curriculum is that for many students who don't take college math courses, proof is a weird one-off that you do in a Geometry class for one year, writing algorithmically in two-columns full of acronyms. You know it's coming, but certainly aren't expected to do anything about it until you get to Geometry, and then, just as enigmatically as it appeared, it vanishes from the curriculum again. Truly, a mystery.

I began to rethink my own views on proof and how its definition might be broadened to make it a more regular component of ALL math classrooms when I read Avery's blog post on redefining proof several years ago. More recently, I've been thinking a lot about how to help students become more rigorous and formal in their proof writing, while still treating their informal reasoning and ideas as valid and interesting in their own right, not just as a stepping stone to "more correct" proof.

Teaching an integrated math course to 8th graders this year gave me some unique opportunities to play around with proof and formalization. The year starts with a unit centered on the Pythagorean Theorem and culminates with a project in which students choose a proof of the theorem (we used a collection of proofs at Cut the Knot) to study and then present their proof of choice to peers. There's a great variety of proofs there in terms of complexity and usage of algebra/geometry/trig/similar triangle concepts and focusing on analysis of existing proofs emphasizes what it means to understand a proof and be convinced by its reasoning. We also had a great class discussion about the difference between an example or demonstration and a proof, looking at several videos "proving" the Pythagorean Theorem that really only showed it to be true for a specific instance.

Image result for prove pythagorean theorem
Look, it works!

That was pretty great, but there was nothing in the curriculum that built on this for the rest of the year so I decided that halfway through the next unit, we would do an exploration of polygon areas that would lead students to Pick's Theorem where they could start to write their own proofs, but I really struggled with structuring it in a way that honored students' existing reasoning, but also pushed them to formalize more and consider proof techniques, such as casework. If you haven't seen Pick's Theorem, it's a neat little formula that connects the area of a polygon whose vertices are points on a lattice to the number of lattice dots on the perimeter of the polygon and the number of lattice dots in interior of the polygon. 

The structure I eventually created seemed okay, but definitely produced mixed results. All students were able to come up with a conjecture for the area of a lattice polygon, but even with my hints (which I thought were maybe too helpful), virtually no students were able to make progress on proving the theorem on their own.

Questions I now have:
  • Was it too big of a jump to go from analyzing proofs to having students write their own proof, even with lots of hints?
  • Is this problem perhaps not the right one for first proof writing?
  • What other problems or structures could I have used to transition more effectively towards proof-writing that still build on students' original reasoning and perspectives?
  • Would students have benefited from writing a proof (or the start of a proof) together as a class first?
  • Where should I go next to develop students' formal proof-writing skills?

I do have some thoughts on these, but would love to get more input and ideas from the community.