Consolidation

I'm part of a Facebook group for teachers who are implementing some components of Building Thinking Classrooms (BTC) based on the book by Peter Liljedahl. A question that comes up frequently in the group is how teachers are handling consolidation, which is the wrap up of big ideas that Peter recommends at the end of class, using student work from class as a means to summarize and help students make key connections. This is pretty similar to the final three practices in 5 Practices for Orchestrating Productive Mathematics Discussions (Selecting, Sequencing, and Connecting). We want students in all math classes to see beyond the specific problems they solved. This is especially important in a math classroom that doesn't follow the "I do, we do, you do" model since students are figuring out how to solve problems without being shown a general principle or model to follow and need to see how the work they did establishes something generalizable and useful. In both the BTC and 5 Practices approach, the teacher selects several pieces of student work, decides on what order to discuss them, and helps students make sense of this work and the big ideas of the lesson that they encompass. This is often the most challenging part of class since many students are much less interested in listening, summarizing, and comparing than in solving problems, which is inherently a more active process. 

I am by no means an expert on consolidation, but it is something that I worked a lot on this year, partly because I am working with a population of students right now where the vast majority have documented learning differences and who have historically struggled with motivation in Math. These challenges are most evident when students are asked to analyze classmates' work and participate in a class discussion about the day's problems. Here are some strategies that I have found helpful for this part of class:

• Keep it short

I try to never spend more than 10 minutes of a 50 minute class on consolidation (usually aiming for less than that). Middle school students and students with attentional challenges do not have the bandwidth for a class discussion that lasts longer than that. This forces me to be pretty strategic about what we discuss and does mean we can't talk about every interesting thing that comes up in class. Something that is especially interesting but that doesn't fit into the 10-minute time frame can be moved to a warm-up question for the next day. 

• Involve other students in discussing a group's work

I almost never have the students whose work is being discussed share their process since listening passively is not that engaging for the rest of the class. The students who did the work often know what they meant and why they did what they did and gloss over those parts. To involve everyone else, I usually first ask everyone in the class to quietly study the group's work we are looking at and give a small thumbs up when they finish reading it (props to @mpershan for this lovely teacher move). I then ask for volunteers to explain a few key parts (if there aren't many volunteers, I will first have students take turns explaining each step of the work with a partner), asking follow up questions, like "Why did they do ... here?," "What would they have done if the problem instead asked for...?" Other ways to involve more of the class is to ask for students who can restate what someone has said. This can give students who need more processing time an opportunity to participate and gives everyone the opportunity to hear big ideas a few times. This does mean that students have to show clear work so that it's understandable by others so that is definitely something I emphasize a lot. 

• Give students something to do during/after consolidation

Related to the point above, many students struggle with listening/talking for an extended period of time. My students engage more readily when given opportunities to discuss with a partner, write something down in their notes, or do a related problem using the method we just discussed. I often have them grab a mini-whiteboard so that everyone can write down their answer to a discussion question or solve a related problem similar to the one we just discussed. If the energy level is low, I have students find a partner to discuss with who is not standing next to them to build in a bit more movement. 

• Focus on connections and similarities/differences when looking at multiple pieces of student work

Consolidation often involves connecting different approaches or comparing related, but different problems. I have found it helpful to sequence the student work I want to discuss in such a way that we only focus on the step by step work of the first group. It can get repetitive if we go through each group's work in the same way. For the second or third piece of work, I ask students instead to identify what this group did the same or differently from the first group and hypothesize why they made those choices. Students can vote on their favorite method or say which method they would use in which situation and why.

• Set norms for class discussion

Something we have worked on all year is how to participate in academic discussions. This involves lots of modeling and practicing showing interest in a speaker with body language, eye contact, nodding along if things make sense, and looking quizzical when they don't. Every consolidation is an opportunity for students to practice these skills and give each other feedback. I have found it helpful to be very explicit about the behavior I am looking students to exhibit during class discussions and treat it as a skill that can be practiced and improved upon. 

Please share your best tips for helping students consolidate and participate in class discussions centered on student work!! This has been the most challenging aspect of running a problem-based math class for me and I'd love to learn more. 

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

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. 

Goals for second semester

As I've been wrapping up grading from semester 1 and planning semester 2 for my classes, I'm realizing that I did not set goals at the start of this year the way that I have in the past. Better late than never!

Changes for my personal teaching:

• Get back to individual feedback meetings. I blogged about them here, but the general idea is that I set aside 20 or so minutes to meet with each student approximately every two weeks in order to sit down together and look over their work and have a feedback conversation. I've found these incredibly helpful for students to actually attend to my feedback, understand what I mean and why I think it's important, and explain their thinking to me. This year has been very tricky since the schedule was changed and students lost a floating free period that I used to be able to use for these meetings. I am recommitting to instituting them again, using class time, if needed. It's been the best way for me to get through grading big projects in a timely manner since it's actually fun and rewarding to sit and discuss students' work with them rather than grading on my own after a long day (since, let's face it, grading gets put off and off).
• Be more on top of students who are struggling. I am committing to looking at work that is turned in every week to check up on students who are missing work or need additional support. If anyone has a good system for keeping track of interactions/observations/progress for all students and how they make sure that no one is falling through the cracks, I'd love to chat.
• More nuanced and thoughtful reflection questions - I think that the balance of reflection vs. doing math has been better this year, but I'd like to focus the questions I ask students in order to hone in on specific mathematical practices rather than just general "what's going well? what do you need to work on?" type questions. I also want to bring back, "what's one good thing that happened this week?" - it was a great way to regularly check in and connect with students.
• Collaboration quizzes to give more direct feedback to students on their groupwork and engagement and help them internalize expectations more effectively.
• More peer feedback. I've started doing this more this year, and love how much motivation it creates for students to express themselves more clearly and justify their thinking. I'm hoping to use peer feedback this semester to help students get better at analyzing strategy, getting positive feedback for extensions they create, and to deepen their understanding of different approaches. One of the lesson study groups worked on peer feedback last semester and I'm really excited to learn from them. I would also like to use a Slack channel for classes so that students can discuss and share ideas outside of class more easily.
• Better differentiation. I'd like to meet with students to set individual goals and do more follow up to help them stay on track with these. I think that there's already a fair amount of choice in problem sets and homework assignments, but I'd like to do a better job of teaching students how to use those choices better. One way will be to have them reflect at the end of class on the type of work they need to do to follow up on that day's learning (review of prior concepts, practice, connections, and/or reach problems). I know that they are learning project management skills in their other classes, but in Math, the product is the process, which is more abstract and harder for them to track and plan. 
• Continue and get better at classroom routines that foster reflection and a clear arc from start to finish. 
• I have often used Desmos Activity Builder to start and end class, but would like to do this more consistently and help students get better at constructing meaning from problem-based lessons by selecting useful reflections and comments to share. I still have work to do on making sure that meaning and connection emerges from students' own thinking and not ignoring times when they don't emerge or simply telling students what they should have learned. One way is to do more planning of student responses and how to connect these and have the main ideas of the lesson emerge from them, sharing methods and responses that did not emerge as part of that process. 
• This also connects to better note-taking. I have given feedback to students once or twice on their note-taking and organization and definitely need to do this again. I haven't really figured out a solution for sharing board work and "notes" from class since I've emphasized process and individual needs. I do share presentations, if they were used, but those generally do not contain worked solutions. If anyone has good ideas on this, I'm all ears. 
• I would also like to do this on a unit-level rather than just lesson-by-lesson by using student-generated essential questions, concept maps, and study-guides more this semester. There is still a fair amount of tension between student-generated conclusions/connections and teacher-generated ones that are more "efficient" and feel more comfortable and structured for students, especially if they're oriented towards maximizing content acquisition. I am working to help students get better at this and at understanding why I think that it's important, both of which are necessary to get more buy-in for the process and rewards that actualize when students do more of this work. One way is to be more transparent about the structures that I'm using and why - I observed a teacher recently giving an intro to a lesson by explaining the groupwork structure that he would be using and what he hoped it would achieve, and I think that enlisting students as teammates in this process is hugely beneficial. 
• Continue the following changes I implemented last year:
• Each assessment includes reassessment of previous content
• Visibly Random Groupings (new groups daily) and whiteboarding
• Homework that's spiraled and includes Retention, Review, Reflect, and Reach sections; students self-select problems to do (should sometimes group students by homework problems completed the next day though)
• Students submit all work digitally, all feedback is recorded digitally in one place (online gradebook)

Big Picture Curriculum:

• Decide on mathematical practices and habits that should be emphasized within a given year/semester/unit and link them to specific lessons and activities. I've been doing a much better job this year of giving students regular feedback on these, but haven't been very intentional about which habits will be emphasized when, noticing which ones students are making progress on and which ones need more work, and how (besides getting feedback) they might get better at them.
• Create more opportunities for interdisciplinary connections. I've put out some feelers to Science teachers and will do the same for Computer Science, English, and History to see where we can join forces and create projects that can support and enrich both disciplines.
• Formulate a more cohesive picture of our curriculum and mission so that our core sequence is less content-driven and so that we can explain to students and families why acceleration is not necessary or desirable. This will require a reducing/reworking of our acceleration pathways, enriching/differentiating core classes, and deciding how electives should support the overall program. 
• Start developing a portfolio assessment for one Math course. It might not be ready to go this semester, but if I can pilot a beta version in one class, I can work on tweaking/developing it more over the summer so that it's ready to go in more classes next year.
• Work on developing group assessments (and other differentiated assessments) for at least one unit in each class.

Professional Development:

• Continue lesson study this semester and figure out good systems for sharing the results that each group has found, both within the discipline team and with the school community more broadly. Possibly help other disciplines/divisions begin the lesson study process. Think about presenting about lesson study next year and the types of resources and supports teachers would need to get started with this.
• Figure out what I want to work on over the summer. Major contenders currently are:
• Attending PCMI
• Teaching at summer institutes for teachers
• Start compiling our existing curriculum into a more easily shared and edited form for students, families, and teachers
• Curriculum development for my school, focusing on alignment between courses, portfolio assessment, projects that connect to other disciplines and class trips, parent education, and developing new electives
• Summer math support for students who are doing independent work or working more directly on accelerating/remediating/enriching
• Coordinating with the middle school on curriculum, parent education, and development of mathematical practices

Starting Lesson Study and Update on Classes

This isn't going to be a very coherent post, but I need to get our current work into writing to better organize my thoughts. Here are the projects that are in progress right now:

• Lesson Study

• We've broken up our Upper School Math teachers into several groups of 3-4 teachers who teach across different grade levels. We considered doing a more traditional lesson study in which teachers plan a lesson centered around specific content, but decided to focus our efforts on developing our practice around a particular instructional routine that would be relevant across many grades and that would help us fine tune a specific pedagogical approach and learn from colleagues with whom we rarely get to work.
• Everyone read this article from KQED to orient themselves to the lesson study process in advance of our first meeting.
• Each group selected an instructional routine to plan out, teach, and refine this cycle. The routines selected were:
• Differentiation for students who learn at different paces
• Guided investigation
• Students giving feedback to each other

• In the next meeting, groups will plan a specific lesson around their instructional routine based on the first teacher who will be modeling it and decide on an observation time and what to look for when observing
• I'm super excited for this initiative to be gaining traction! We got some time to work on this while students were taking the PSAT or doing other activities, but I'm worried that if we don't get specific time off to work on this, people will become significantly less enthusiastic. 

• Parent Math Night
• Our team is working on developing an informational night to help parents better understand our program, available resources, and philosophy. It's just in the planning stages, but I think will be really helpful in getting on the same page with families. Right now, whatever information they receive when applying is the extent of it. 
• This needs to be thoughtful and informative for parents while also clearly conveying our position and getting buy-in and understanding of the program. If you have any resources or ideas to share, would love to have them.

• Math 1 is finishing our unit on Counting, Probability, and Sets, designing a game that has students analyzing probability and expected value to determine best strategy and fair outcomes. Next week, students will be playing each other's game and reflecting on what they've learned. This is a good opportunity to differentiate and identify gaps in understanding linear functions and algebraic manipulation skills as we prepare to move into a functions unit next.
• We're starting each unit with a "preview" assignment to look at prior knowledge and pre-requisite skills and concepts so that those students who have gaps can be identified and given extra support. Here is the preview assignment for linear functions.
• We're also starting the unit with another open investigation, this one more directly related to functions ("Cutting the Pie" task from IMP Year 1). I'm curious to see if students pursue a recursive or closed rule for this function. When we worked with Pascal's triangle patterns, students had a hard time moving from "each row is twice the previous row" to the rule f(x) = 2^(x-1).

• Math 2 is still in the depths of statistics, working through the Central Limit Theorem and connecting probability and the normal and binomial distributions. I'm realizing how much better I understand the material in my third year of teaching it and how much less formulaic and prescriptive my teaching is now that I have deeper content knowledge in this mathematical space. I have known that strong content knowledge is necessary, but it's amazing how much more depth is needed if you want students to explore and create and test their own theories, both in terms of creating those scenarios and in guiding the discussions that ensue. Maybe it's true then that teachers have to first progress through traditional pedagogy as they build up their depth of content knowledge before they can start to incorporate more problem-based or project-based learning. Would love some pushback on this though :)