Remote Teaching - the community edition

 I just finished remotely teaching two one-week courses for rising 9th graders in which at least half the class was brand new to the school. Each group met for 3 hours every morning for a week and I had the luxury of creating my own curriculum that didn't have to cover any particular topics, but did need to be fun, engaging for students with a variety of math backgrounds, introduce new students to how we teach math at my school and how to learn math remotely, and most importantly, foster a sense of community.

Fortunately, Michael Pershan shared some words of wisdom about the need to build student-student connections over student-teacher connections in a remote space and this helped me rethink my original plan for each week.

Teachers have been worrying what online learning would look like when you don't have relationships with students. What I'm seeing in my camp teaching is the much bigger issue is kids not having relationships with each other.

— Michael Pershan (@mpershan) July 16, 2020

Based on his experiences with a virtual math camp this summer, I did a few things that I think helped students get to know each other and feel safer sharing and discussing than they would have otherwise. Here are some things that seemed to go well (at the end, I'll post some things that didn't go as well, not to worry).

• Low stakes whole class interactivity

I started the week with an activity in which students had to drag their name somewhere on a set of axes so we could learn about each other. 

I asked some chill follow-up questions about each set of axes as kids dragged their names, which they could answer in the chat or out-loud (almost everyone opted for the chat).

I then had kids drag their names somewhere onto an oval and go around the oval, saying their name out loud and answering another easy icebreaker question, the goal being - everyone knows how to unmute their mics, everyone gets to hear how each student wants their name pronounced. 

Every morning started with a low-stakes interactive component. We did a "Which One Doesn't Belong" with kids dragging their names into a quadrant and typing their reason for that choice into the chat box. We did a "Contemplate then Calculate," with kids typing their numerical expression into the chat box. We did an "Estimation 180" task, where kids typed their "too low/too high/best guess" into the chat box. Just something small and relatively low-stakes where every kid interacted with the whole class. As the week went on, kids were more likely to use their mics voluntarily to participate, especially if I sent them into breakout rooms for a few minutes to pair-share first. The chat box is maybe the best feature of teaching remotely though - getting kids comfortable using it and setting the norm that it's basically a backchannel for classroom discussions, where kids can type questions and ideas and respond to those of their classmates was a huge component of building community for my students. 

• Breakout rooms

Breakout rooms was where most of the community building happened though. I'm no remote teaching expert, but in my limited time doing this (spring + summer), kids are approximately 1500% more likely to talk out-loud in a breakout room than in a whole class setting. Each day, I created visibly random breakout rooms that stayed together for most of the day's activities. They started with an icebreaker here too. The first day, I used "personality coordinates" shared by Dan Meyer a while back, which translated really well to a remote space.

Each breakout room worked on one slide in the slideshow, putting their names next to the dots first, and then coming up with variables that could be placed on the axes to make this graph true for their group. Kids had a great time with this activity and came up with some clever and hilarious variables for their groups that we then shared out in the whole class. Following days had more traditional ice breakers, but I also had every student bring in a photo of something meaningful to them and add it to their group's slide and share about that photo, something I likely wouldn't have done in-person. That was another favorite.

• Games

Games are another low-stakes way for strangers to interact and build some familiarity and trust. I mostly used two-person games, borrowed from this list by Ben Orlin. Fortunately for me, Mike Flynn had already created online templates for two of the games (Black Hole and Ultimate Tic Tac Toe) and I made one for Magical Squares so we had a variety of games to play. I think games like Sprout, Nim, and Hex would translate well to remote space if you want more ideas. After students played a game against an opponent, I asked them to share out possible strategies and things they noticed in the whole group and got kids to participate who were quiet otherwise. 

• Norms

 Throughout these activities, but especially at the start, I was very explicit about the ways that I wanted students to engage with each other. The first time that students went into breakout rooms, I assigned a group leader and gave that person several tasks.

Each day, I had students reflect on how they supported each other, had students name peers who supported them along with what they did that was supportive, and asked students to share out strategies that were helpful in promoting effective collaboration. This was mostly done in a Google doc journal that students wrote in at the end of each day, but I also asked students to share out some of these things in breakout rooms at the start of the day. I frequently reminded them of our norms and why they were there. 

• Choice

My last hot-tip for building community remotely is about giving kids choice for how they interact with each other. While I really wanted kids to work together, I also gave them opportunities to work on their own, if they wanted to do so, or to pick specific peers to work with rather than be randomly assigned. Knowing that some of the time, they would have choice for how they worked and who they worked with seemed to increase buy-in and willingness to work with new peers during the times that I asked kids to collaborate with strangers. I also tried to give kids different ways to participate - even though the norms asked for participation, we talked about different ways that this could happen, whether by typing in the chat box, asking others questions, affirming or pushing back on ideas, writing out the group's work on a shared virtual whiteboard, or using nonverbal cues if their camera was on, like looking at the speaker, nodding along, giving a thumbs up. I didn't require kids to have their cameras on, but tried to make it a safe space to do so and where there would be a reason to be seen and heard by others.

This is not to say that everything was amazing and the students are now life-long friends. Creating a community virtually is going to be a challenge, even with all the tricks and best intentions because it's a weird, awkward space that's not conducive to vulnerability and intimacy. For example, the last hour of the last day was a choice project, and the majority of the new students chose to work on their own. I didn't force the issue, but I would have loved to see them choosing more collaboration. Something that I plan do differently in a few weeks when I meet my year-long classes is to have more opportunities for substantive sharing. I feel like we didn't really get past the easy icebreaker stage, and partly that's due to only being together for a week, and partly it's because I wasn't sure how much to push kids to share. I did have students share a Google doc with me in which they wrote a reflection at the end of every day, and those were much more substantive and raw. I'd like to have kids feel comfortable sharing those types of written reflections with each other and not just with me. With a class that I will have for longer, I'd also like to spend more time having students model and practice participation norms. 

I'd also love to hear your ideas of how you plan to foster community in your classes remotely this fall. Please reach out on Twitter or in comments to this blog!

Remote Teaching - prepping for next year

I'm in the same boat as a lot of others, without a concrete plan for the fall yet, but with schools in my area leaning more and more towards starting remotely. Even if not fully remote, we will be at least hybrid in order to reduce the number of students coming to school at any time so my current plan is to assume remote instruction and to have in-person students join via Zoom to work with at-home students, if we do end up hybrid for some of the time. If conditions improve beyond current expectations, it's a lot easier to roll back and move towards face-to-face instruction than the other way around. This past month of summer break has given me a bit more time time to play around with tech tools, listen to webinars, look at my curriculum, and build on the work I did in the spring in teaching synchronously while remote. This blog post is an attempt to organize some of the work and thinking I've done so far in preparing for next school year. It's pretty long so I have no expectation that others will read, but I need to write out my plans for my own sanity.

As I mentioned in my previous blog post, I'm working in a small school where all students have school-issued laptops and where classes will be run synchronously via Zoom, which influences the types of instruction I can do, but please don't hesitate to reach out here or on Twitter if you have any questions for how this might look at your school.

I'm jumping into tech tools first to get them out of the way, but the important stuff is below, my unpacking of the most difficult part of remote learning - students' need for relationships, understanding, and agency.

Tech Tools during Class
The most useful tech tools that I used during class in the spring and that I plan to keep using in the fall were Desmos Activity Builder, Classkick, and virtual collaborative whiteboards for breakout rooms. I used Desmos AB and Classkick for students working individually - both platforms allowed me to see students working in real time and to give them feedback via comments in Desmos and by writing directly on their papers in Classkick. Desmos was better for content that involved graphing and making and testing predictions, while Classkick was better for students writing out their steps, working especially well for the small minority of my students who had iPads or tablets and could write with a stylus instead of their trackpad (but it also worked pretty well for kids with laptops only).  I remade a number of my lessons as Desmos activities or simply imported pdfs of problems into Classkick. The drawbacks of both of these platforms were that they did not foster collaboration between students, even if I put them in breakout rooms and told them to talk to each other. This was very surprising as students had been used to collaborating effectively in my classes before we went remote so there's clearly something about a remote space that is much less conducive to working easily together. One strategy I plan to use in the fall (as shared by @mpershan a few days ago) is to assign one student in each breakout room the role of sharing their screen.

OK this went so much better this morning. Did a short icebreakers WITH each breakout rooms. (This is camp and I have counselors in each room. Would be tougher if it was me and 20 kids but I think possible.) Also made clearer expectations: someone is sharing screen in every room.

— Michael Pershan (@mpershan) July 16, 2020

Teaching students how to work together in breakout rooms is clearly a new skill and one we'll need to explicitly teach and practice in the fall rather than relying on their face-to-face collaboration skills to just extend into online interactions. I'm considering how to amend structures like group roles and participation quizzes to work in breakout rooms, since I can no longer observe multiple groups at once.  For example, new roles could be: 1. Someone who ensures that a screen is being shared and everyone knows what they're working on, 2. Someone who pauses the room every 5 minutes and checks for understanding and who can call in the teacher if there's a group question 3. Someone who ensures that everyone is writing out their work and there is documentation for the breakout room.

It might be good to shift teacher feedback on collaboration to a peer- or self-assessment model in which students set goals around collaboration, then reflect to themselves or to group members ("in what ways did you contribute to your group today?", "in what way could you be a better group member next time?", "tell your group one thing they did well today" , "give a specific shout-out to a peer who helped you learn today"). A very concrete thing might just be to ask students to track the number of times they asked or answered a question in their group. I think it might also be possible to do an amended form of a participation quiz where I pop into breakout rooms and record what I see in a shared document, although I won't be able to project it to them in real time.

I'm also going to be looking to inject more fun and interactivity into breakout rooms - icebreakers, sharing something non-academic, Anne's concentric circles activity, something small that gets kids talking and sharing their screens and builds their comfort level with digital participation. In whole class discussions, using the chat feature of Zoom (set to "chat to host only") was incredibly helpful in drawing in shy students in the spring and I will continue using it to invite more participation and to get insights into kid thinking in real time.

In the spring, I also used virtual whiteboards quite a bit when I wanted kids to work on novel problems together or to go over homework questions. I bounced around a few different ones - assigning a page in Jamboard or a slide in a shared Google slideshow per group were great for students adding photos of their handwritten work and incorporating typed comments, but not great for handwriting. Bitpaper was best for writing and graphing math, but unfortunately, due to increasing use, they removed their free version for new users a few months ago (if you made an account before this and had some boards, you can keep using these for free, which is what I'm doing). GoBoard is probably my second favorite for writing out math work and has handy integrations with Desmos and LaTeX. If you have some money to spend, either Bitpaper or Ziteboard work really well for writing out math work and integrating photos of work on paper with handwriting and typing on a computer. If not, Jamboard and GoBoard are decent options.

Online whiteboards are going to be a big part of my remote plan this year as well, and I need to also be explicit about norms there - the role of writer should rotate, everyone works on the same problem, students should look for multiple methods or connections between problems, there should be a check for understanding before moving on to the next problem, work must be clear enough that someone not in the group could understand your process. As these will be largely used in breakout rooms, these norms will need to be incorporated with the breakout room participation norms. So! Many! Norms! I will have to be very intentional about rolling these out sequentially and creating a small enough list that won't overwhelm students. But I know that time invested up front in fostering effective group work will pay huge dividends in how well students are able to learn from each other and work productively together for the rest of the year in a remote environment.

The big new tech thing that I plan to use in the fall is OneNote digital class notebooks. There was a pretty steep learning curve to figure out how they work from the teacher side, but I think they're now ready to go for the start of school and should greatly simplify the coordination of classwork and homework, as well as giving easy feedback to students in real time so that I may no longer need tools like Classkick or Google Classroom to organize assignments and feedback. It also means that I'll want to build in some time at the end of class for students to take photos of their handwritten and Desmos or whiteboard work to insert into their digital class notebook and reflect briefly on their understanding. One of my big takeaways from the spring was that everything takes 50% longer when teaching a class remotely, but it is also documented more thoroughly. There is potential here for deeper learning, but I will have to account for the amount of time that things take and be focused on the most essential topics in the curriculum.

It might also be helpful to state that I'm not planning on investing a lot of time and energy into making content videos. I have provided curated video resources for students in every class for several years now and based on student feedback and my own priorities, I'm going to continue outsourcing this. I don't think it's worth it for me to record a lot of videos teaching math content when there are already so many out there, many made by people with way better video recording technology and know-how. In the spring, I did often make short videos in response to student questions or common errors on their work, and I will make these as needed again this year, using the Notability app for iPad and iPad's native screen capture or by recording a Zoom call with just myself in it and screen sharing from my iPad so that my face is also in the video. But these videos are going to be in response to student work, not a replacement for synchronous class time.

Relationships/Communication/Support
I'm thinking a lot about teacher-student as well as student-student relationships for the coming year and while I list out individual ideas below, I know that a conversation with my department and school about values and priorities is going to be the most important. We need to plan out how to care for students remotely, how to know how they are doing and what we can do to support them as students, but also as kids and people who are lonely, bored, scared, and disconnected from their normal support networks.

I loved a suggestion from Audrey around students sending her photos of things that have meaning for them (sounds like pets were a crowd pleaser) and starting each class with a student talking about that photo. She then compiled all of the photos for an end-of-year slideshow. Several others have also proposed converting Sara VanDerWerf's Name Tents, which is how I usually start the school year, into a digital form where students respond to prompts either in writing or via short Flipgrid videos. Teachers and students could respond to these with their own videos. This also made me reflect on the power of audio or video feedback to foster teacher-student relationships as this was something mentioned by several teachers who regularly teach online. I'm excited that OneNote will allow me to easily record an audio response to student work. I had been planning to use Voicethread to do this before I committed to OneNote, but I know that students really appreciated video responses to their work in the spring and that they help to humanize what could otherwise feel like dry content-focused interactions.

Another idea that I liked that was shared this morning in response to Julie's post on teaching in a hybrid model (in which students are split into two groups and each week, they rotate which group is at school and which group is at home) was assigning each student in a group a buddy in the other group who could help them know what was going on during class or let the teacher know if there were issues when their buddy was learning from home. In the spring, I used Padlet for students to post and answer each others' homework questions and ran an after school homework help time over Zoom where students could drop in and work with peers and math teachers for a few hours each week. I will continue using Padlet and running after school Math Lab, but am also considering other outside-of-class structures that might encourage more interactions between students. Study groups? More group projects? This is an area where I could really use the wisdom of the collective - how are others planning to foster student-student relationships in their schools?

One of the things I took away from a "Designing for Online Learning" course I completed at the start of the summer from Global Online Academy is the importance of clearly organized course materials and easy to access supports for students. I used a Google Site in the spring with a daily agenda so that students could easily follow the sequence of a lesson and know what was going if their audio or video cut out or they lost their Zoom connection. Moving to OneNote will make it easier to share monthly, weekly, and daily plans with students so that they have a clear understanding of the content goals and work they are completing.

I am also going to cycle in one-on-one conferences with students to find out how things are going, build relationships, set goals, and go over feedback together. I conferenced with the majority of my students in the spring and although it was a lot of time and work to set up, I felt that they were incredibly worthwhile, even more so in a remote setting than in face-to-face school. My students benefited, but I also benefited tremendously in my ability to empathize and support specific students. In my experience, making these meetings required and ongoing (once a month is a good frequency, I've found) is key. My school will also be setting aside one day each week for tutorial slots so those will also be great opportunities for students to access support. I will also continue seeking feedback from students on how things are going and using that feedback to correct course. Short, anonymous student surveys once or twice each semester, rather than an end-of-year longer survey, have been more helpful for me in getting actionable feedback. Relationships and timely feedback were critical in the spring for motivating students to show up to class, engage with content, and reach out for support, and I know they will be even more so with a new crop of students who don't know me or each other very well yet.

Curriculum
My biggest content take-away from the spring was the importance of student agency in motivating students to stay engaged and work remotely, without the norms of being inside a school building. I built choice into assignments, I let students select their own breakout rooms every few class periods and let me know how they would like to work during class, and I designed the end-of-semester projects to have options and to include a variety of student interests. Student choice to support differentiation is something that's been important to me for a long time and I presented on it this summer, but in a remote environment, I need to be way more organized with helping students set goals, receive timely feedback, and revise. I've done a bunch of curriculum work this summer to hopefully be in a place where more of my time is spent giving feedback and conferencing with students and less on writing problem sets and planning lessons. And I'm hoping that OneNote is a platform that supports organization of assignments, quick feedback, and revisions.

With respect to deepening the curriculum, I've also revised several projects to include more choice and to bring an anti-racist lens to student mathematical thinking. For example, the first 8th grade project for the last several years has been to find a proof of the Pythagorean Theorem from the many options available and present it to the class.

5.3 Pythagorean Proof Project - Google Docs by Anna Blinstein on Scribd

The revised project will include more of a humanistic look at how different cultures have used and thought about this right triangle relationship and why it is that we have named it after a Greek mathematician instead of the many others who also explored it. Students will learn more context and history of the mathematician whose proof they are presenting and the work of non-majority culture mathematicians will be celebrated. A key understanding for this project this year will be to critically examine who gets the credit for a mathematical idea and how different cultures come to understand, apply, and prove mathematical ideas. Two later projects (one on modeling data and one on using concepts of standard deviation and z-scores to analyze outliers) will have a social justice lens this year - students will still have choice in their research questions, but will be working within the realm of social justice topics.

I will also be focusing more explicitly on retention this year since learning remotely may really impact how deeply students are learning content and there may be more gaps from last spring. Working on a curriculum team this summer, we revised the standards for 8th grade to more explicitly connect back to earlier content and we've rewritten homework problem sets to spiral in previous topics. Sara VanDerWerf has blogged about her use of green reference sheets to better support students with gaps in prior knowledge and I plan to use a form of this as well, since we already start each unit with a pre-assessment to help us and students know what topics in the upcoming unit will need the most support. I've been incorporating an "Important Concepts" section into notes packets, but am still thinking about how to best use this with students. Should students be creating their unit summary page? Should there be more explicit use of teacher-made reviews and references throughout each unit? Since many class periods are structured around problem-solving and student-led exploration, with some time spent synthesizing and applying at the end, rolling out a summary of what students will learn ahead of time seems detrimental to that process. At the same time, many middle school students are not great note-takers and having a clear summary that can help them see the big picture or review and solidify what they explored in class seems like a good idea. I'm going to play around with summary structures this year that build off student thinking and will hopefully, have more to report on this soon. I'd love others' input on how they've integrated review and content summaries with problem-based learning.

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. 

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.

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.