Implementing Building Thinking Classrooms with Students with Learning Differences

Last week, I presented at the Building Thinking Classrooms Conference in Franklin, Indiana on implementing BTC with students with learning differences. I definitely tried to cram too many things into a 45-minute session so this is my attempt to unpack what will likely need to be split into (at least) two presentations going forward. Here's the original presentation, and now let's get into the unpacking.

Part 1: Why students with learning differences benefit from a BTC approach

At my previous school, I got a lot of pushback when using a problem-based approach with students with learning differences so I did my research when I changed schools and learned that I would be working primarily with this population. The benefits are very clear, when implemented thoughtfully and with supports: students with learning differences benefit immensely from teaching approaches that emphasize process and sense-making; meaningful contexts; connections to previous learning; opportunities to discuss and improve metacognition; frequent feedback; integration of concepts, procedures, and language; and a growth mindset (source: Teaching Mathematics Meaningfully). Oh hey, these are all built into BTC already! At the same time, they also benefit from opportunities to reduce math anxiety and learned helplessness, address misconceptions and unfinished learning from previous years, and receive more explicit directions and teacher-directed synthesis and instruction to make sense of the math they are working on and how to connect it to existing schema. And this is exactly where adjustments to BTC come in handy.

Part 2: Adjusting the first toolkit (what, where, and who should students work with)

The first toolkit is all about where students work, who they work with, and the types of problems they work on. The original practices have students working on vertical whiteboards, in random groups of three, and on tasks that require thinking and problem solving. The main adaptations that I have implemented provide more structure, fewer distractions, and supports for students with memory and visual processing challenges. For example, I found that assigning random pairs of students each day rather than trios along with having students take turns with clear roles, which I call driver and navigator (where the driver can only write what the navigator says), and using sentence starters were helpful for getting students to work together with greater focus and engagement.

Spending time at the start of the year teaching routines for getting supplies, finding your whiteboard, and working productively with others paid dividends for the rest of the year. 

It also proved extremely helpful to think very carefully about the level of challenge and explicitness in the tasks and problems provided. A lot of my students were coming in with high levels of math anxiety and a stated dislike for math. They needed to experience a lot of small successes early on to start to see themselves as having agency and be willing to try and persevere with challenge. I leveraged high-interest warm-ups that encouraged discussion, multiple viewpoints, and easy entry for everyone. Some fan favorites were visual patterns, estimation 180, fraction talks, slow reveal graphs, and which one doesn’t belong. When selecting non-curricular tasks to start the year, I used tasks recommended for a few grades below my students' actual grade level and started with tasks that had a clear, explicit goal and a very low floor. When using thin slicing to move students through content-learning, I started with review problems related to that day's learning (again, to lower the floor) and increased the difficulty very slightly between problems, often giving a few problems at the same level of difficulty before ramping up. I would sometimes also start with a worked example à la Michael Pershan as that day's warm-up to build student confidence and activate prior knowledge before asking them to solve a new, related problem. Structuring problems so that students experienced early success, as well as mixing in whiteboarding with other activities and gradually increasing the amount of time students spent in groups were all critical to building problem-solving endurance.

Part 3: Adjusting the second toolkit (teacher moves to start and maintain flow)

The second toolkit is all about teacher moves in giving the tasks, monitoring and supporting students while they work, and empowering student autonomy. Again, ramping up the explicitness and positive feedback went a long way in supporting struggling students. While Peter recommends giving tasks orally, with everyone in a huddle in the center of the room, I found it helpful to ask for volunteers and act out the task, if possible, checking for understanding along the way. 

My students also benefited from getting copies of the questions and key visuals in clear plastic sleeves so they could write on them with dry erase markers, taped up at the boards. To keep students in flow, I once again relied on routines and celebrations of small successes. Students were provided with a list of questions to ask yourself if you're feeling stuck and I frequently refered to these when checking on progress. 

We also spent time early on practicing several of the routines from Routines for Reasoning (book, website). Each routine combines ask yourself questions, sentence frames and starters for discussing with partners, and annotation to help students make sense of new problems in a structured, repeatable way. Combining these routines with the thinking classrooms framework has made problem-based learning significantly more accessible to my students with learning differences. 

A strategy that I used to implement, but which had fallen away during the pandemic and that I want to bring back, is giving each group three colored cups (green, yellow, and red) to help them monitor and reflect on their state of flow. I first learned about this strategy from Avery Pickford, but a quick Google search shows that a few others have blogged about it - here's a great description from the Math = Love blog. The idea is that every group starts with the green cup on top of their stack and shifts to yellow on top if they feel stuck, but haven't yet tried all of the routines and ask-yourself questions that could get them unstuck. They switch to red once they have exhausted all of their resources and need help from a teacher or classmate in another group to continue making progress. I really appreciate how this strategy makes visible where groups are at, as well as reminding students that there is a key step between "doing great" and "totally stuck," in which they have the tools and resources to move themselves back into flow.

To promote student autonomy, I relied heavily on collaboration rubrics and positive reinforcement. Instead of giving feedback on collaboration at the end of class, I would give feedback (or ask students to self-assess) 10 minutes in, which would give students a tighter feedback cycle and an opportunity to improve their collaboration that same day. 

I actively sent students to check out peers' whiteboards or to stand in the middle of the room and look around to get ideas if they were stuck. While circulating and looking at student work, I would also identify interesting work by students who were less confident or likely to share with others and ask them to help another group or tell them that I would like to use their work during consolidation. At the end of class, I would have students give shout-outs to classmates who contributed to their learning in a positive way that day (sometimes accompanied by a sticker reward from me... never underestimate the power of a sticker for students of just about any age).

Part 4: Adjusting the third toolkit (moving from collective to individual knowing)

I'm ignoring the hints/extensions part of this toolkit since that's more closely related to part 3. Consolidation has perhaps been the most challenging BTC practice for me to implement with struggling students. Even when I could reliably get students engaged and working hard with classmates on problems, the energy and enthusiasm would evaporate within minutes of trying to consolidate. I wrote a whole other post on consolidation, but the things that ended up making a big difference were basically more routines and positive feedback. We practiced how to stand (a nice semi-circle in front of the whiteboard we were looking at so everyone could see), how to comment and respond to questions about the work that was being shared (yep, back with the sentence stems: “I notice…”, “I wonder…”, “One difference between these methods is…”, “I like the first/second method more because…”), and how to find a new random partner for a stand-and-talk to discuss the work that was being shared. 

I also started using the 4 R's strategy from Routines for Reasoning (more here). In this strategy, a student shares an observation or summary and the teacher follows up by doing the following:

Repeat = ask someone to repeat what was shared (this helps to ensure that everyone heard)

Rephrase = ask someone to restate in their own words (this helps to ensure that everyone understood)

Reword = teacher states again, but inserting mathematical language (this helps to build academic vocabulary and increases precision) - this is also a key time to connect to previous concepts, if relevant

Record = annotate (add notes and vocabulary words, circle key components, and otherwise create a written record of what was shared)

Another strategy to try during this phase is to have students apply the method that was discussed to a new problem. Students can hold mini-whiteboards during consolidation and respond to checks for understanding individually or go back to their group whiteboards and try a new problem together using the specified strategy. I have also experimented with moving the consolidation phase to the beginning of the next class period rather than trying to rally the troops who are tired from 45 minutes of hard math work. Another option is to do a mini-consolidation half-way through before students tire out, leaving part 2 of the consolidation process for the next day. When all else fails, stickers for participating and engaging with this phase can be a clutch teacher move. Keeping consolidation short, snappy, teacher-directed, and fun and varying the strategies and questions used while still maintaining the key routines have made a big difference in its effectiveness in my classroom.

Let's talk about notes. This is another tricky thing to get right with students with learning differences who may be struggling with attention issues, dysgraphia, processing speed, and other challenges to traditional note-taking. At the same time, having a clear and easy to use reference may be especially helpful for students with these challenges. I have had some success with students building out a course pack, which organizes and summarizes key concepts from the entire year. Here is an example of a course pack for a class that's a mix of 8th grade Math and Algebra 1. You can see that each unit starts off with some review of key concepts students saw the previous year. 

I got this idea from Sara VanDerWerf, who blogged about providing reference sheets to students here. Because most lessons start with problems students have seen before as a way to review and pull in prior knowledge, it is really helpful for students to already have some notes on these topics. The rest of the course pack is organized into blank half-page sections labeled with content topics. When students take notes, I provide a half-sheet with sections for key concepts, examples, and vocabulary. Sometimes, I provide an example (like below). 

Other times (later in the year), students pick an example or two from the day's whiteboarding work to put into their notes. This half-sheet gets taped into the course pack and over time, that becomes a valuable reference tool for students. 

Notes is an area with which I am very much actively experimenting. I'm eager to try a new way of doing notes that was shared at the Building Thinking Classrooms Conference this past June, in which students work together in a group on whiteboards to first finish a partially completed example provided by the teacher, then solve an entire example from start to finish (where the teacher provides the initial question), then create their own example and solve it, and finally summarize key points from that day's lesson as "notes to my future forgetful self." They then have this model as well as the models created by other groups as a reference in order to write down their own individual notes on paper.

Part 5: Adjusting the fourth toolkit (grading aligned to values and to inform students)

There is a lot I can say about grading and the BTC model of aligning grades to goals and making progress  and areas of growth visible to students. However, most of it will not be very useful to others since every school has its own grading policies and expectations and this is an area of teaching where teachers have the least say. However, I will share two tweaks to the grading practices in BTC that have worked well for my students.

First, I know that Peter is very clear in the book about not grading homework or other "studenting behaviors" because we want students to be doing them for the right reasons and to develop their intrinsic motivation. However, I have found that it really helps my students if data regarding these behaviors is tracked because it makes the connection between "studenting behaviors" and progress in the class more obvious and allows for better goal setting and reflection. At my school, we are also required to give an "Approaches to Learning" grade that can affect the final semester grade so that is a good way for me to use a small amount of extrinsic motivation to help students who have not yet seen the value of "studenting." I organize the approaches to learning into a category separate from content understanding or mathematical practices and give students feedback on these using standards-based grading. There are four standards in this category: 

1. Turning work in on time, checking answers, and revising errors, if possible
2. Seeking out challenge, persevering, and asking for help
3. Coming to class on time and with supplies, staying engaged and participating in class activities
4. Collaborating with other students, giving constructive criticism, supporting a positive class culture

The second addition to this toolkit that has worked for my students is having each one keep a digital Math reflection journal. Approximately once every three weeks or so, students look over their digital gradebook to check on their progress, read over what they wrote the last time they reflected in their journal, and fill out a slide that has the following sentence stems:

1. The goal I have been working on is... 
2. I have or have not made progress on this and my evidence is... 
3. My next steps are... (continue working on previous goal or set a new goal and how you will try to reach it)
4. Pick one and delete the others:
1. Something that’s going well recently is… 
2. My teacher/parents can help me by…
3. I’d like Anna to know that…

They then email a link to their reflection journal to me and to their parents/guardians, which opens the door for communication about the student's progress between everyone. It does take 10 minutes of class time every few weeks, but has been invaluable in making sure that students are regularly reading and understanding my feedback and that they're making goals and connecting their progress to behavior that's under their control. 

It might be more evident now why this giant blog post dump did not work as a 45-minute presentation (I also tried to do a math task with participants to show some of the tweaks I was sharing, so yep, way too ambitious). If I do this presentation again, I will likely focus in on just one or two toolkits rather than trying to hit all four. And of course, I would love to revise and add on, if you have other ideas that have worked well for you or that you have read about and want to try. 

References:

• “Teaching Mathematics Meaningfully,” Allsopp, Lovin, and van Ingen
• “Routines for Reasoning,” Kelemanik, Lucenta, and Creighton
• “Dyscalculia Pocketbook,” Hornigold
• “Can I Tell You About… Dyscalculia,” Hornigold
• “11 Effective Strategies for Teaching Math to Students Who Have Given Up on Learning,” Smith

If you got this far, kudos to you! Please feel free to connect with me on here or via Twitter or Mastodon.

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.

Differentiation and the limitations of groupwork

It's important in any profession to stay humble, but teaching has a way of reminding you of this in particularly in-your-face ways, I believe. This semester has really brought home this issue for me in the challenges presented by my upper school Math 3 class. The issues have been around productive groupwork, an area in which I have felt particularly strong and well-trained, so it was perhaps an especially humbling experience to see all of my strategies and approaches come crumbling down and leave me turning to my Twitter network and colleagues to find new ways of helping students work together and feel confident in their progress. I wanted to share and summarize here some of the issues I've worked through that might perhaps be helpful to others.

First, some background.

I have been incorporating the essential elements of a Thinking Classroom in all of my Math classes for the past few years, but most notably in my high school class, where the focus on content and pressures to teach to the test are greater. This year, just like last, I had students read and reflect on Thinking Classrooms and we discussed why most of our time together is spent working on problems in random groups, sharing out ideas and conclusions, and using these to synthesize and summarize learning from the bottom up rather than top down via teacher-led instruction. Students initially seemed bought in and supportive of this type of classroom environment. We set up class norms and discussed the use of group roles, how to step up/step back in group environments, and how to be a skeptical peer and give respectful pushback on ideas.

Several weeks into the semester, however, I started noticing a troubling pattern - some students were disengaging from their collaborative work and seemed very hesitant in sharing their thinking within either small groups or the larger class. Then, I started to hear two different complaints from students - some were feeling that their work during class was unproductive because their groups moved too fast and they were feeling increasingly anxious and uncertain about their mathematical understanding and abilities. They were feeling unprepared to do problems independently on homework assignments or assessments and wanted more teacher guidance and structure, as well as more opportunities to go at their own pace and understand ideas more fully. Other students raised the opposite issue - they felt that the pace of the class was too slow, that they were doing too many problems that they already knew how to do or could figure out quickly and wanted more challenging and deeper problems, both during class and on homework assignments.

When group work goes wrong:

In reflecting on these issues and why they were coming up this year, I realized that we had actually made quite a large change to the Math program without making any changes to our curriculum or pedagogy. This was the first year that we had decided to mix all grades taking a particular Math course - Math had been the only discipline at the Upper School in which students were separated out by grade level. In the past, 9th and 10th graders taking Math 3 (students who had accelerated the normal sequence) were in different sections from 11th graders taking the class (students who were on grade level in terms of their progress through the sequence). This year really was different in terms of prior math experiences, expectations, and desire for challenge/acceleration for students in the same class and the normal groupwork structures were not sufficient to bring together students with such varying backgrounds and approaches.

My next step was to look for feedback from colleagues as well as the Twitter math teacher community. Some suggestions that I implemented that seemed to make a difference:

• Taking a break from random groups to help students regain their trust that the class would meet their needs; doing some work in pairs designed to foster productive collaboration; allowing students choice as to who to work with while also asking them to work with different students at times; being explicit when the goal of a task was to build collaborative skills
• Structuring activities so there was time at the start for individual exploration before asking students to share their thinking with others thus giving more processing time for students who worked more slowly; circulating and helping some students get started; building more optional challenge into tasks for students who worked very quickly or who had already had prior experience with a topic; creating tasks that could be approached with a greater variety of methods and building more writing into tasks so that different ways of thinking mathematically could be valued
• Meeting students where they were to regain trust and buy-in; this included at times splitting the class into two groups (students chose which group to join) - a more free-form exploratory group with more open and challenging problems and a more structured group where students would get some problems to activate prior knowledge and smaller, more concrete problems that would build over time to greater generalization and abstraction and more teacher guidance and reassurance that they were on the right track
• Noticing struggling students' successes and highlighting them publicly; selecting which students would share their thinking to make sure that different voices could be heard over time
• Make sure to leave time for synthesis and practice problems (at different levels) during class - this helped address student concerns that they were leaving class with lots of questions and feeling unsettled about the concepts they had explored that day
• Giving students more feedback during class about their understanding of a topic rather than relying more heavily on groupwork and self-assessment for students to know how they were doing and what might be helpful next steps
• Providing more problems at different levels and helping students navigate which problems might be more helpful for them to do during/after a particular lesson - here is an example of a tiered homework problem set.
• Providing more textbook resources - explicitly linking textbook sections to problem sets for students who wanted more references and examples

We are considering sorting students for next semester by grade level to decrease the heterogeneity of classes - while these strategies have alleviated the issues significantly, it does seem that productive collaboration and exploration is challenged when students in a class are so spread out. Despite these strategies, for example, it usually doesn't make sense for students who have worked quickly and deeply and have figured out challenging extensions to share their ideas with the whole class, most of whom have not even tried these problems. As a result, the class sometimes lacks cohesion and feeling like a true community. Additionally, the amount of work required to run a class with this much differentiation is really, really high. I'm essentially designing at least two different classes, creating both lesson plans and homework assignments that can reach the full spread of student interest and background, and giving individual and frequent feedback to students or small groups of students into which the class has fragmented. This is not really tenable for the whole year given my other preps and teaching responsibilities. However, breaking up students by grade level seems to run counter to our values of equitable access to challenging mathematics for all students and means that Math classes are essentially different from all other classes at the Upper School.

I would welcome any feedback or suggestions that others have around this issue - what strategies have worked for you in working with very heterogeneous groups? 

Connecting Math and CS with probability game simulations

One of my goals for this school year was to build out a few interesting and relevant projects into the 7th grade curriculum, which seemed a bit dry and skill-focused. One area that seemed to beg for an application project was the first unit on Data and Probability. Since one of my other goals was to incorporate more computer science into my classes, it was a no brainer. Developing a cross-over computer science project for this grade level proved to be a bit tricky because students are all over in terms of their experience with programming - we have students who have been coding for years as well as new students who have never coded anything before. I tried to develop a project that would differentiate appropriately and allow students to either explore the CS or the Math parts in greater depth, depending on their interest in and experience with programming.

Here is the project description. You'll notice that I created three distinct strands with different goals and let students select the one that was most appropriate and interesting for them. I was also lucky that the computer science teacher was able to come to my classes for some of the time that students worked on this project. Having many intermediate checkpoints for students to submit pieces of the project was very helpful here in ensuring that I could identify those who were behind or struggling and work with them during class.

Things that I would still like to build out:

• A more robust peer editing process -- I'd like students to be able to present their optimal winning strategy to peers and get critical feedback on how convincing their reasoning is that they would be able to incorporate into their final draft
• A revised rubric to make it more concise
• Move some pieces of this project out to computer science class - this definitely took up quite a bit of time, especially because I felt that most or all of the coding work should happen during class where students would have support
• A clearer division between group and individual aspects - this is always a challenge for me when designing group projects in terms of maximizing student learning and individual accountability. Students seemed to work well together during class, but this isn't an explicit part of the project currently. 
• Some sort of final presentation - for projects like this, I think that having the final product on display or presented to others creates a much more authentic need for clarity and functionality. I haven't figured out a good way to do that for this project. Should students do a gallery walk of projects within the class? Can this be presented or shared with students in other classes somehow? What about with parents?
• Other connections - is this something that can connect to students' work from previous years so that it feels less like a stand-alone project and more like a continuation of ongoing work and thinking? Are there other aspects of this project that can connect to other disciplines, like writing? Can we build on this in future years of either computer science or math curriculum?

Habits of Mind Unit - Math 1

We've had four whirlwind days of school so far - I'm really enjoying starting with a Habits of Mind unit in each of my classes as it means students are working on tasks and learning the routines of the class every time we meet and I am getting to know them and the flow of the new year.

In Math 1, we have been working on several different tasks, each of which is related to combinatorics, the first unit that we'll be officially starting next week. In each task, students start with an introductory question and then each group creates an extension to pursue next. The three tasks we've done so far are below. I'm still tweaking the fourth one and will post it when I'm done (hint: this is one of the things I need help deciding).

Task 1: How many paths from A to B if you can only travel down and to the right?
Extensions created by students: generalize for a grid of any size, allow travel up and to the left (without crossing over), allow traveling diagonally

Task 2: Consider a game in which you flip a coin four times. At the beginning of the game, your score is 0. Each time you get heads, you get a point. Each time you get tails, you lose a point. What are the different scores that are possible and how likely is each of these scores?
Extensions created by students: generalize for n flips, what about dice that have 4, 5, 6, etc sides?


Task 3: How many different monetary values can you make from these bills?
No extensions created yet, will have more time on this next week

Scroll down for the presentations from class for each of the investigations, which include slides about group/class norms.

Two big questions with which I'm wrestling in doing these tasks are:

1. How much, if any, content teaching should there be? Students are practically begging for more efficient methods than just listing out all of the options, but should this unit really be about helping students get better at exploring their own thinking or is it better to teach some content while they're hooked and eager rather than coming back to it when it actually comes up in the unit? For those who incorporate student-driven investigations along with teacher-led instruction, when do you do the latter? 
2. Relatedly, how much should I be pushing students to make the connections between these problems more explicit? I feel like I've been dropping some (subtle) hints and revisiting student work from previous problems in the hopes that some students will point out the underlying connections, but no such luck. Again, is it better to show these connections now, even if it means they will mostly be teacher-driven, or better to wait until later and let these problems simmer for a while longer?

My current thinking on these two questions is that I will require each student to work on generalizing one of the tasks and then have students present their generalizations and ask more explicitly about connections between them at that time. I have to now choose a fourth task that I hope will make the connection more obvious... suggestions? What are some tasks/problems you've liked for hooking students on combinations?

P.S. I am super happy with how group norms and vertical whiteboarding is going so far this year. Using the same routine with a new math task each day so far has created a really nice flow and students are interacting well and starting to independently leave their groups to find out what other groups are doing to bring those ideas back. It was definitely worth taking a few days out of the content rush to set things up.

Presentations from class

Math 1 Problem Solving Unit Day 1

Math 1 Problem Solving Unit Day 2

Math 1 Problem Solving Unit Day 3

Individual/Group/Class Norms Revised

In my previous post, I wrote about my updated group norms. But then, I got some great feedback in the form of comments, a few Twitter conversations, and a post by Sarah on her updated groupwork norms... damn you, MTBoS with your feedback, always making me want to change stuff to make it better.

I decided to break up my groupwork norms into three components:

1. Individual accountability
2. Group accountability
3. Class accountability

Here is the poster for each set of norms.

Individual Norms

Group Norms

Class Norms

Finally, I made a poster for the green/yellow/red cup strategy Avery uses in his middle school classes. I went back and forth a bunch of times to see if this was perhaps not going to work in high school, especially if students are usually working on whiteboards around the room rather than sitting at a desk, and if there was maybe a way to do this electronically, but eventually, decided to just do it the same way that Avery does it and then make changes, if needed.

The idea is that each group starts with a stack of three cups, with green on top, yellow in the middle, and red on the bottom.

If the group feels stuck or confused, they should move the green cup to the bottom of the stack. The yellow cup is a sign to the group to discuss their confusion together and try to get themselves unstuck using the various strategies we've discussed or by checking in with other groups. 

If they have discussed and tried to get themselves unstuck, but were not successful, then they come up with a single group question that every member of the group needs to be able to articulate, and can switch their cups so that the red one is on top, at which point a teacher will come over and ask a random member of the group what their question is.

Avery's selling points for me were:

• The yellow cup is an important step to prompt students to reflect on their confusion as a team and get better at the "unsticking" process that is such an important part of productive struggle.
• There is a clear visual for the teacher in scanning the room where groups are at and which might need attention soon (currently at yellow).
• You can hear cups switching so without even looking, have a sense of group need.
• The proportion of the time that various cups are on top gives you valuable information regarding the challenge level of the task you've given students that day. Ideally, cups are changing back and forth between green and yellow as groups become puzzled and then figure things out on their own.

Here's a poster summarizing this for students:

Cups

As always, feedback and suggestions for improvement are welcome!

Formalizing Routines

In my last post, I blogged about #TMC16 and how excited I was to take what I learned in @davidwees's workshop on instructional routines and apply it to what I do most in my class, which is guided investigations (aka problem sets that scaffold instruction) and open investigations, which are more focused on exploring connections and representations of student thinking. I've taken a first stab at writing out the steps and teacher moves involved in both types of investigations (links below), including writing prompts for students and class norms. The class norms were especially tricky to nail down because I've been thinking all summer about how to marry the norms that I learned in Complex Instruction, which are all about valuing different types of participation and making the group a cohesive and supportive unit, with what I'm seeing as emerging from the research on Visibly Random Groupings, which values flow and makes the entire class a unit of idea exchange and interdependence. Complex instruction often has assigned roles within the group and clear instructions on establishing a "group question" before a teacher can be called over for help. By contrast, in a VRG class structure, students are encouraged to share ideas with and ask for help from anyone in the class. Groups change daily and roles are eschewed in favor of flow of ideas and vertical whiteboards that encourage easy participation and engagement.

My attempt to merge these two cooperative structures (as well as my other goals for students) has resulted in the following group norms:

Group Norms

I am going to continue randomly assigning students to groups when working on problem sets or open investigations and avoid assigning roles. There will probably be one day every week or two when students are grouped homogeneously based on their self-assessment of their needs (more structure/support/direct instruction, same level (stay with guided inquiry), explore independently). I have to think about tweaks to the group norms that need to happen on those days.

I also wrote out the protocol for when a group can ask me for help. They need to first attempt the strategies posted in the classroom for getting unstuck (listed below), look around to see what other groups are doing and send a representative to another group to discuss and share ideas, and if they're still stuck, to formulate a single question to ask me... aka a group question. I should be able to ask anyone in the group what their question is and be assured that it was indeed a group decision to get help.

If your group is stuck:

I will try to remember to write another post discussing the various reflection prompts and closing questions that I've adapted, but here are the links to the two routines, which have all of the prompts I've thought of so far.

Guided Inquiry Routine

Open Investigation Routine

Feedback is super appreciated! These are still very much in the planning stages, but it's been immensely helpful to write out and formalize the routines that I normally use in my classes. My goal is to work on making these better this year, both in my classes and in those of my colleagues, through lesson study focused specifically on routines.

First two weeks of school

It's been a really fun first two weeks of the school year. Yes, exhausting as well, but super exhilarating and exciting too. This year, I started the year a bit differently, focusing more on how I wanted students to work together and think mathematically than on specific content. Because it was the first year I was doing this, I could do the exact same problems with all of my classes. Here's how it went:

Day 1:

• Students came in and saw a seating chart with randomly assigned groups of 3 or 4 and were directed to one of the vertical whiteboards. I wanted to establish this as the norm from the get go.
• Students filled out Google form describing a class in the past they've enjoyed, a class they have not enjoyed, questions that they have about this class, and questions that they have about me. I used the last two prompts as ways to discuss my expectations and structure for the class and to start building some personal relationships.
• Students worked individually for a few minutes and then discussed this problem, which I stole from IMP Year 2. Our first unit will be Statistics for all classes so I thought it would be good to do a fun, but challenging problems, that related to probabilities and ways of counting events.
• Homework was to fill out a Google form asking them about themselves and to keep working on the problem above.

Day 2:

• Students were grouped randomly anew and shared their work on the Tying the Knots problem. We spent the last half of class with group presentations sharing out their progress and practicing how to present and interact with presenters.
• Homework was to write a reflection on themselves as a learner and to start writing up the process and solution for the Tying the Knots problem (I used the Problem of the Week standard categories).

Day 3:

• New random groups, and I used one of @sophgermain's activities for helping students get to know each other. Nothing huge, kids just shared one thing they did over the weekend with their group.
• New problem! This one was incredibly fun. I originally thought that we would take it into proof by induction, but after input from @woutgeo and @hpicciotto decided to stick with a more intuitive visualization of the sequence.
• I basically let students work in their groups without too much guidance from me. Most realized it gave the Fibonacci sequence pretty quickly, but were not able to explain why. Many tried to develop a closed form rule, without much success (surprise, that's actually pretty hard to do). Most groups started trying the extensions, but didn't get super far. I stopped the class a few times and asked various students to explain their group's work. One of my classes this year doesn't have as many whiteboards as I'm used to having, but our desks can be written on so I'm going with that for now.
• Homework was a reflection on their process and feelings when working on these problems and presenting/watching presentations.

Day 4:

• New groups and I answered some more questions about the class and about me. I continued to have them share out a few personal tidbits in their groups as they are still very much getting to know each other (especially the freshmen). Today's questions were about favorite ice cream flavors and favorite movie.
• This was a slightly more structured day. I pushed students to be able to explain why the pattern that was produced matched the Fibonacci sequence. It was helpful to project pictures of their written work and explanations and use that to get more precise and tight in our language. I felt okay adding on to their explanations as needed since there were more extensions to explore (2 by 2 by n case and 3 by n case).
• Homework was to work on the two extensions and to start an integrated review problem set.

Students' work on explaining the derivation of the recursive formula

Some work from the first day on developing a closed form. I was not sure as to whether I should discourage students from going in this direction as finding the closed form rule is extremely challenging. 

More fun student work at the beginning of the exploration.

P.S. @daveinstpaul shared a great follow-up programming project in which students need to write a program to generate all of the possible ways to tile a 2 by n rectangle and then extend it to an m by n rectangle. I'm going to check in with one of the programming teachers tomorrow to see if this might make sense as a posible extension in her class.

P.P.S. A new teacher who I think is going to be amazing visited my class today and I got completely turned around in what I was saying and did not do a great job of moderating the discussion. It's been too long since anyone has observed me, and I just didn't feel comfortable with the kids yet to laugh it off so awkwardness ensued. Bah. We need to be visiting each other's classrooms much more frequently.

Quadratic Functions project

Thanks to @SweenWSweens and his M&M Catapult Project (explained here and here and updated here), we are ending the quadratic functions unit in my 10th grade class with a bang. Well, a whooooosh, but you know what I mean. Kids had a ton of fun with this activity and it gave great practice for writing and solving quadratic equations. The basic idea (but really, you should check out Sean's posts) is that groups launch an M&M and measure the horizontal distance traveled and approximate the vertical distance traveled by using the time, putting this together to create a quadratic function that models this relationship. Then, they apply this model when the launcher is placed a given height above the ground to figure out where to place a target.

First of all, Sean was super helpful, walking me through the lab and giving me great tips on how to adapt it for my students. Love that #MTBoS. My project description and follow-up questions are here.

Here are the changes that I made to Sean's awesome plans and why:

1. I let kids build their own launchers. I shared Sean's basic design (pictured at right), but let them tweak it or do their own thing altogether. It actually took kids only about 20 minutes using our engineering lab, which had all of the supplies already, except for clothespins, as opposed to the few hours I would have spent making all of them and then dealing with kid complaints that their launcher wasn't good. Next time, I will do this again, but will also share Sean's updated design, which I did not see in time (below).

Here is what my kids built (most just did the basic design, a few went nuts and did their own thing):







2. Little direction was provided about lab technique or how to find the equation relating the height vs. horizontal distance. We did discuss the equation relating vertical distance traveled and falling time, but next year, I will do a better job of integrating this concept into earlier problems so that students can generate this idea themselves. What I liked as a result of giving less structure:
• Students incorporated other topics, which I did not anticipate. A few groups did statistical analysis to look for outlier data, which was awesome since that was a concept learned way back in September. Others compared lab protocols from different science classes and their applicability to this project.
• There was much more variety in approaches, which allowed for richer discussions within and between groups and more connections made. Some groups used the vertex, some used intercepts, and others used quadratic regression on desmos to generate equations. There was likewise diversity in how to change the model to incorporate the new starting height for the final launch. 
• The intellectual rigor was higher - students had to figure out what to do and then for their write-up, remember and reflect on their approach.
3. I used some class time after the activity for groups to whiteboard their approaches and then share out with the class and get feedback on their thinking. I also used 15 minutes the day that the write-ups were due for students to peer edit each other's work. The goal was to have more cross-pollination of ideas and connections made, as well as a chance to justify their own and critique each other's reasoning. I'm hoping that this also helped to produce higher quality final products and deeper understanding. Next year, I hope to run a more structured peer-editing process with specific questions for students to address.
4. More individual accountability - students were asked to divvy up points to their group members and describe each person's contributions as well as complete individual follow-up questions. I need to think about this more to see if I think this overall contributed to students' learning and experience with this project and helped or hurt their collaboration.

And now, more pictures!!

Building the launchers:

Gathering data:

Final launch day:

A few student whiteboards:

Once again, huge thanks and shoutout to Sean for creating this!! it ended up being a great project for this unit. Students had a blast, but were also appropriately challenged. 

Feedback from students: