At PCMI we’ve been talking a lot in our “reflection on practice” sessions about assessments. Starting broadly, we wandered in the realm of epistemology and started talking about what we mean by knowledge, and how we know if we know something. Which then led to conversations about how we have evidence of knowing (and why it might matter that we teachers think about this and reach a consensus on this).

From this more abstract beginning (where we did try to bring it down to the classroom level), we then started to get more concrete. A very generative question was “what does it mean to be mathematically competent/proficient?” Everyone interpreted the question in a different way, and initially I was stuck – do we mean in a particular skill? or overall, when can we call a person’s body of work “mathematically competent/proficient”? procedurally or conceptually? isn’t competent < proficient? So many layers to unpack in the question itself. However I still started brainstorming and the very first thing I came up with was “can see the idea in a larger web of ideas/connections.” I often think that’s a hallmark of a strong mathematical thinker – where things aren’t this hodgepodge of ideas and procedures but they are tied together in a larger web. And what’s lovely is that one of the most powerful parts for me of Francis Su’s Mathematics for Human Flourishing book so far was in the Meaning chapter:

mathematical ideas, too, are metaphors. Think about the number 7. To say anything interesting about 7, you have to place it in conversation with other things. To say that 7 is a prime is to talk about its relationship with its factors: those numbers that divide evenly into 7. To say that 7 is 111 in binary notation is to have it dialogue with the number 2. To say that 7 is the number of days in a week is to make it converse with the calendar. Thus, the number 7 is both an abstract idea and several concrete metaphors: a prime, a binary number, and days in a week. Similarly the Pythagorean theorem is a statement relating the three sides of a right triangle but it is also, metaphorically, every proof you learn that illuminates why it is true and every application you see that shows you why it is useful. So the theorem grows in meaning for you each time you see a new proof or see it used in a new way. Every mathematical idea carries with it metaphors that shape its meaning. No idea can survive in isolation — it will die. (37) [italics mine]

In addition to this being part of being mathematically competent (which, overall, I think I ascribed to a body of work that a student did instead of with a single skill), I also immediately thought of various mathematical habits of mind.

Something that struck me, after my brainstorming, was that much of what I wrote was about a world of ideas, but it was disembodied from the physical world… and so I ended my brainstorming: “One thing I’m thinking about when I think about “mathematical competency/proficiency” is that those terms seem to be pretty clinical… I’m wondering if joy and appreciation and an emotional connection to math would exist in this or not.” Looking back at this reflection of mine, I think of Rochelle Gutierrez’s eight dimensions of rehumanizing math, and my wondering seems to fit squarely in the “Body/Emotions” dimension.

As I’m writing this blog post, I’m enjoying seeing how lots of different ideas in this conference seem to all be strung together loosely and I’m only now seeing them braid together.

One great exercise we had was just writing down all the different ways we collect evidence of student understanding. All the ideas we came up as a group are here, and a word cloud of what we came up with is:

We talked about bias in various ways we collect and interpret evidence, the ease/difficulty of various ways we collect evidence, and how confident we were in our interpretation. What I liked about this is that we went beyond “paper tests” and “exit slips” and allowed the true range of things that we as real teachers do to get a sense of what a student or class knows… I mean, I do actually use facial expressions to help me get a sense of student understanding, but I also recognize it’s not always the best indicator. So this broad list we generated didn’t seem like a trite exercise because it valued all the ways we as teachers do truly get a sense of things on a day-to-day basis. In other words, it broadened our sense of “assessment” to go beyond “the things that we grade, that often are done on paper.” Assessing is just getting a sense of what kids know – whether formative, summative, or something else (e.g. self-assessing their own confidence on something).

We read a couple articles on assessments as well as having kids self-assess or peer-assess. I liked NCTM’s section on assessments from their Principles to Action — reframing assessments to be something that provides feedback to students but also informs instruction moving forward.

Shifting the primary focus and function of assessment from accountability to effective instructional practice is an essential component of ensuring mathematical success for all students (p. 98)

For me, the big reminder was that we teachers traditionally think about assessment as a noun. That paper thing we give. A static snapshot. But we should think about it as a verb, assessing, and that is part of a learning cycle, a journey — both for student and teacher. However, of course, there’s the ideal and there’s the reality.

There are ways I’ve thought more as a verb, like when I’m did standard-based grading when I taught calculus. But usually, when I give summative tests (which is what I normally do, and I don’t really do projects), I don’t think of them as a part of a cycle or journey. They are the static snapshot.

But I also know in reality, I do lots of assessing in small ungraded ways (self-assessing, start of class problems and walk around, listen intently to students talking, looking at nightly work, etc.), and use that to inform my next teaching move or plan my next class. For one example, I have a general idea of what the nightly work will be each day, but based on what I see in class, I often will alter it based on what makes sense… or generate some new problems to address a misconception or gap I’m seeing… or to have students think about a particular insight that came up in class.

I’m now getting tired, so I want to end with three things before I lose steam.

First, we watched a video of Max Ray-Riek on why 2 > 4. It is an ignite talk, and I’d seen it before, and I realized after watching it that during then pandemic, I started listening for instead of listening to because I felt so stressed for time. I think this is a video every math teacher should watch!

Second, we brainstormed ways that we get real-time evidence of student understanding in our normal everyday classes. And then we each chose a few to share out. I like this document of collective knowledge that we generated, and I want to come back and read through it again to be inspired.

Third, we talked about ways we had students self-assess themselves and their own understanding as a way to become more independent learners, and think metacognatively. I really enjoyed brainstorming this individually and listening to everyone else’s ideas. I wish we had more time for this activity — because I love talking about concrete things we do in the classroom so I can get more ideas and rethink things I already do!

And with that, I’m done! Tomorrow begins our third and final week at PCMI!

I kept on wanting to write up a short post outlining how I dealt with online teaching in the Spring and reflect. But the year ended with a bang, and I wasn’t in a headspace to do this. I’m going to do that now, but without too much reflection, since now there’s too much distance — the details are lost.

What I can say is that I did similar things in both Algebra II and in Advanced Precalculus. I would say based on the regular feedback I was collecting that all students really appreciated my organization, consistency, and clarity. That being said, even though I had pretty much the same structure in both classes, things seemed to go better in Advanced Precalculus. I have some conjectures as to why, but I don’t really know the cause.

Update: A Disclaimer and Caveat

I wanted to write this up for me, to archive my process/thinking. It helps me. But I also want to make clear that this isn’t a how-to guide for anyone else. This is just how I figured out things to work in the situation I was in. In teaching, context is key. There is never a one-size-fits-all approach. I have so many friends who had to teach “but not any new content” or weren’t allowed to expect that kids would be able to join class “live,” or had to do everything asynchronous. Of course much of this wouldn’t work in many situations. And more importantly, I have so many friends who have kids or other obligations that took up much of their time. If I had, for example, a kid, I would need to come up with a totally different plan. To be clear, this was not sustainable for much longer. It worked for me for three months, hopefully for my kids, and I say overall it went “fine.” But I couldn’t do this over the course of a year. I didn’t talk about my mental state in the original post because I did a lot of that processing with friends in the moment. But let’s just say this whole sudden online teaching took its toll. There were so many evenings I wanted to break down and cry. I was frustrated, mad, angry, overwhelmed, drowning, and felt like no one could really understand. Like this was a Sisyphean task. If I shared with you some of the texts I wrote to friends, I’m guessing even though we were in different situations, you would totally point and say “yeah… me… that was me.”

The Planning

I spent a good part of my Spring Break trying to envision what class online could be. I used my friend Alice as a sounding board and I realized I had to figure out what my core values are that I wasn’t willing to compromise on — and build from there in the space we had available. I hit on these three things:

Initially, we were given very short classes (30 minutes) and then later they were extended to 35 minutes.

The Setup

I opened a document to write a revised set of course expectations. And as I thought about each section, I started to be forced to imagine what our class was going to look like, how students were going to be assessed, how I was going to make things manageable for me, how I was going to provide support for my students. I didn’t quite know what to expect before we entered this phase. It could have been only for a few weeks, or (as it turned out) it could last to the end of the year.

Here’s what I came up with:

Online Learning! – Adv. Precalculus – Google Docs

The main highlight of this is that I switched our courses to Standards Based Grading. Our school went to Pass/Fail for the second semester and I wanted a way to assess that would support my kiddos. This also gave me a way to determine Pass vs. Fail. I’m really familiar with making SBG work because for years I taught standard Calculus and I learned how to change the flavor of SBG so it worked for me in my particular school.

Most importantly, although we switched to virtual school, my goal was to keep our classes as consistent as possible in terms of how students would learn. I didn’t want to immediately make students work individually since they were used to collaborating in teams. I didn’t want to give them videos showing them how to solve some sort of mathematical problem since they were used to figuring that out themselves.

Everything wasn’t perfect, but I can say that overall the feedback was pretty positive. Here are a few comments from a reflection/feedback form I gave to kids a few weeks into our online learning:

Honestly, it’s working so well for me. This class feels the most structured and like I’m engaged and getting something out of each class. Thank you for all the effort you put into making the Demos activities!!! I also really like the structure of watching videos outside of class, and then coming back to any questions and building off of what we watched.

I personally find that math class is working really well for me. We’ll see how the upcoming assessment goes, but I feel like I’m understanding the material we are being taught almost as well as I did in live school.

I think the structure of our virtual math classes is pretty successful. I really enjoy working in breakout rooms together with my classmates. It allows me to “spend time” with people and work on math together, which is awesome. I also really like when you spend a few minutes explaining concepts by sharing your screen and using a virtual whiteboard. It feels pretty close to the normal organization of our math classes.

The Constraints

Our schedule allowed us 30 minute classes for the first few weeks we did online learning. Then, when we refined the schedule, we were given 35 minute classes. They were short.

Almost all of my kids had working internet and a school-issued laptop. Access wasn’t a huge problem.

I decided I had an obligation to hit all the major ideas I would have covered, but I felt comfortable paring things down to smaller and more essential bits, and eliminating the things that felt more minor.

We have a weird 7 day rotating schedule where we had 5 times we could meet. We had a choice for how many of those classes we wanted to be “live.” (Some teachers, like history or English teachers, would have kids read or work on papers for some of the days and then meet live only a couple times each cycle. Or they would set longer one-on-one meetings up with their kids to talk through ideas.) For the structure I set up, I usually had my kids meet “live” 4 out of every 5 classes — and I had something for them to do that didn’t require collaboration for the fifth day. It’s important to note that these classes were a mere 30 minutes (later extended to 35 minutes), so every minute was precious.

We were using zoom as our communication/video platform.

I didn’t want to give a lot of nightly work, since kids were going to be on their computers a lot. Since we were meeting live a lot, my goal was in the range of 5-30 minutes, depending on the day and their level of understanding.

Organization

I used the “Classwork” tab on Google Classroom to be our central hub. At the top of the page was:

I had a revised version of our course expectations, an ongoing skill list for what we were learning, and a link to my google calendar where kids could reserve a time to meet with me individually. (The other links aren’t as important.)

Then below that I created a different “topic” for each week of learning:

We were asked to create an assignment for each live class meeting we had — so that it would appear on student’s google calendars (since we had the option of holding a live online class or not). Then each day I would add the nightly work. Notice I would have the nightly work due by 7am the next day we had a live class. I’ll explain why that was so below.

The Planning

Here’s how it worked. I centered the learning using Desmos Activities. I didn’t want kids to have to learn a new platform (they had used Desmos Activities a number of times before). And Desmos had instituted a way to give students feedback.

So the crux of every live class was students working on Desmos Activities that I had adapted or created from scratch. They worked together in breakout rooms, where one kid would share their screen and they would work through the activity together. Some of the slides were “practice” — so not much talk would happen — but some of the slides included exploration and investigation and conjecturing and explaining conceptually what’s happening.

Here are all my Desmos Activities for Advanced Precalculus used during remote learning: https://teacher.desmos.com/collection/5e80e25ec9089c33af3d954f

Here are all my Desmos Activities for Algebra II: https://teacher.desmos.com/collection/5e80e247431047086cf42c54

I kept two evolving separate google documents with my lesson plans for each day. They looked something like this — with easy access to links that I could copy and paste quickly into the zoom chat box when I needed them to go to an activity.

I’m a teacher that likes to go at the pace of my students — so my different sections weren’t always perfectly aligned. I would design the next class based on where kids got.

Here’s what a “normal” class might look like from a student perspective (remembering we only had 30 or 35 minutes):

1. Kids join the zoom. Near the end of the year, they started hearing me playing music as they were admitted into the class. It gave me something to bop along to and put me in a good mood! :)
2. Kids hear me say “hi!” I send kids (in the chat box) the link to the Desmos Activity they had been working on and ask them to go there and spend a couple minutes silently looking at the feedback I left them.  I do this for just a couple of minutes — most of the feedback is short, and I tell them to look more seriously at it after class. We don’t have much time together.
3. Kids hear me outline what I took away from the work they did during the previous class and what they for nightly work after the class. If there were issues that more than a couple kids in the class had, I made sure to address it in the whole class. I would do this by screensharing a particular slide of a desmos activity and talk through it, or sharing my iPad and talking through an idea. During this time, I might occasionally preview an idea or remind students of something they had seen previously that might come in handy. This would take 2-7 minutes. (But with 30 minute classes, I wanted to have kids work together during the majority of the time.)
4. Before kids go to the breakout room, they hear me say: “Okay, you’re going to log into this Desmos activity. Write this down in your notebooks — today you’re going to call me over so I can talk with your group at Slides X and Slide Y. Remember if I’m busy to keep working and I’ll come by when I’m done with the group I’m with.”
5. Kids work together in their breakout rooms. Sometimes they’ll see me pop in when I’m following along on their work on Desmos and see something I want to point out, correct, or compliment. (I didn’t have much time to compliment, honestly, though I tried to do that so me popping in always didn’t seem like it would be a critique.) When they get to particular screens where they were asked to call me over, I’ll join and give them feedback, ask a few questions I’ve prepared to assess they know what they’re talking about, and then have them contiinue on (or ask them to discuss more after I nudge them forward, and then call me over if they didn’t seem they got an idea).
6. Three minutes before the end of our time together, I’ll either send kids in breakout rooms a message saying they can leave at the end of the class straight from the breakout room, or I’ll call them back to the main room to say something and then dismiss them.
7. The nightly work will be posted on google classroom pretty soon after class. The assignment will look like this:I’ll ask them to review my feedback from the previous night’s work some more. Sometimes I share with them a resource if they struggled with that work (usually a video I created going over some of the problems.) I post what they’re supposed to do. Sometimes I’d include DeltaMath practice for more routine problems, which I love because it gives students feedback on how they’re doing.

On my end, this is what my side of things look like to make this all happen

1. School starts at 9am, so I wake up at 7am and shower and get ready by 7:30am. I sit down at my table and look at my calendar. I tell me Amazon Echo to send me reminders 5 minutes before every class/meeting I have.
2. Then I look through the Desmos activities for the classes I was seeing that day. I always made the nightly work due by 7am of the day we’re having class, so I could look it over and understand where kids were at, and give them immediate feedback on their work/thinking. I go over every student’s slides (choosing key slides to give feedback on). As I do this, I make a note of which topics are worth bringing up in class — if anything. There were a good number of days when kids seemed to get most of the material!
3. As I do this work, I also fill in a nightly work spreadsheet to keep track of whose doing the work. I also had a column where I started keeping information that might be useful about things I noticed in their work, but truth be told, keeping that additional column wasn’t sustainable so I ended up using it for notes about when kids didn’t do their work — if I emailed them, what they said, etc. (As a side note, if a kid didn’t do their work, I let them complete it later.) Here’s a sample of what my spreadsheet looked like.
   
4. If I see a bunch of students are struggling with an idea, I quickly prep a short iPad presentation to talk about a concept or work a problem — a mini-lecture I’m going to deliver. I add that into my lesson plan for the day. From start to finish, looking through the nightly work for the kids and doing any last minute mini-lecture prepping usually takes me a little over an hour.
5. I open the classroom 5 minutes before class starts. As I admit kids into the zoom room (two or three at a time), I mark them present on my attendance spreadsheet. Sometimes when kids come too quickly right at the start time of class, I’ll just admit everyone and fill this in when kids are in their breakout rooms. (This is a fake spreadsheet to illustrate.)
   When everyone has arrived, I say hi and then tell kids to check the feedback I left for them on their nightly work (on a Desmos activity). I put the link in the chat box.
6. After they look at the feedback, I gather us together. I go over the things I noticed from their work in the morning, and give any mini-lectures I feel is necessary for that section. I send them off to work on the Desmos activities — telling them to call me over at one or two pre-chosen slides. Usually, I set them up in random groups of 3-4 students, though occasionally I’d do pairs for certain activities.
7. At the very start when they’re working in breakout rooms, I’ll take a piece of paper and write down the answers for each slide I think they might get to. That way when they enter their work into desmos, I can quickly check it. This usually only takes me a few minutes and kids are still settling into working together.
8. Then I start keeping tabs of what’s going on by using the teacher dashboard. I can see which group is on which slides. I write down on a piece of paper the name of one student per group, and I tend to follow along using that student’s work as a representative for the group. As they’re working, I’m noting down which slides they’ve completed correctly and if there is anything I need to talk with them about when their group calls me over. I’ll occasionally pop into breakout rooms when I see a group is stuck and needs some help.
9. After groups get to a particular slide and call me over, I’ll look over their work (if I wasn’t able to keep up as they were working) and ask them questions I had pre-scripted to check their understanding. I ask if they have any questions for me, and then they go on. These pop-ins are short — as short as a minute, but if we start discussing, we can get to three or four minutes.
10. I’m following along and checking in for pretty much the entirety of the time kids are working on the Desmos activity.
11. At the end of class, if I call kids back to the main room, I remind them of anything that might be upcoming or encouraging them to see me in office hours if they were feeling lost, and then I dismiss them. I always remind them I’ll stay after if anyone needs to talk about anything, and a few times students did hang back and ask some questions to shore something up.
12. After class ends, the first opportunity I have, I go to google classroom and think about what I saw, where students got to in the activity, and then decide what the appropriate nightly work should be. I would often have kids work up to a particular slide in the Desmos activity. I might choose a DeltaMath assignment. I might make and include a video of me working through a slide or two that kids had difficulty on the night before (if any) so they would have something to look at if they struggled.
13. After that, when I have another small expanse of free time, I’ll look through where we are and whip up a new lesson plan for our next class based on where we got, and add it to my ongoing lesson planning document.

And the cycle starts over again the next day.

I’d use evenings and weekends to think through and create the Desmos Activities, and create videos of me working through specific slides that I thought kids would need help with. (That way after kids attempted them, I could lead them to the video and have them watch it for assistance.) (The videos were more for Algebra 2. I didn’t find I needed many of them for Advanced Precalculus since kids seemed to get the ideas fairly quickly.)

[Note: at the start of the time we had online, I would include “check in” screens to see how kids were doing/feeling. I would also reach out to individual kids if I thought something was wrong, or check in with their advisor or dean. Trying to understand and attend to the emotional well-being of my kids was really tough. But that’s a post for another time.]

Assessments

The assessments I gave were fairly traditional. I kept an ongoing skill list, like the one below:

Adv. Precalculus Skill List (Ongoing) – Google Docs

Then on assessment day, I would upload a test for kids to work on. I was pretty standard in terms of what the test would look like — though I was super duper extra explicit about everything in terms of how I wanted students to format their answers. (For example, I wanted the work for each skill to be written on its own page. So for a five skill assessment, they’d submit five pages.) Students were given a fair amount of time to take it on the honor system in one sitting. I didn’t have the energy to think of all the ways kids might cheat — it felt like such a low priority in terms of what I wanted to give my mental energy to. I figured it was better to just trust my kiddos, because they hadn’t given me any reason not to trust them during my time with them in-person.

Kids used the CamScanner app on their phones (they used this throughout the year to submit their nightly work, so the process was familiar to them) to submit their test on Google Classroom.

I would mark it up and give feedback using the iPad and pencil that my school got for me (bless them!), and then email it back to kids after I had marked them all up and recorded their scores.

Since we were doing Standards Based Grading, if kids didn’t show a solid understanding of the material, they had the opportunity to sign up to reassess that skill. I had a system set up that was easy to manage, but it did mean that for every test I created, I had to create two versions (one for the original go-around, one for the reassessment).

Feedback Loops

I was very intentional to make sure that I had a way for kids to understand what they knew and what they didn’t. Here are the ways that played out:

When given assignments on DeltaMath, if students got something wrong, they immediately know and they also are given a complete solution to the problem to learn from. The way DeltaMath is set up is that you keep working problems until you show competency — which could be doing a few problems or it could mean doing a bunch.

When given assignments on a Desmos Activity, I would go through each morning it was due and give feedback. I’ll leave no feedback on slides that kids were getting right, but on a slide where kids did a bunch of work, if they got it all right, I’d make a note of that. I’d also point out if there were mistakes. I also would have videos made (more for Algebra 2) with me working through particular key slides, so if I saw a student was struggling with something, my feedback in Desmos would include “Look at the video I created and will post on the nightly work today! I think that will help!” I would also encourage kids to meet with me in office hours to talk through things that I saw they were struggling with.

Based on looking at the whole class’s work, I would address common misconceptions or point out different interesting approaches at the start of every class.

If students messed up on a skill on an assessment and didn’t show a solid understanding, they could look at my feedback, go back to our Desmos activities, set up a time to meet with me, or talk with friends… and then ask to rework it to show a stronger level of understanding.

Close to the end of our online learning, two weeks before we ended, I asked kids explicitly about the feedback I was providing them. Here are some of their responses:

I love this structure! I love having structure in general. It’s so helpful when you go over common errors at the beginning of class, and I’m able to take notes on it. I also like the little desmos feedbacks if it was just a personal issue.

I feel as though this feedback look is extremely helpful. I particularly like the specific comments you leave on our Desmos activities – I find them super targeted and helpful. Additionally, I really like it when you share your iPad/give general feedback pertaining to the whole class in class (and sometimes start w/practice problems if you think that we need them).

It has been working really well! The comments on desmos at the beginning of class have really helped direct my questions that I ask in breakout room, and my meeting with you after school really helped me understand the material on the first test better.

I chose some of the ones that were more detailed, but almost all students said they found the feedback system helpful. It was awesome to read.

My Own Organization

I had everything for online learning in a single Google Drive folder that I linked to from my bookmarks bar.

In here, I had a folder for everything related to assessments, folders for reflection forms and feedback forms, my attendance/nightly work/check-in-with-kids spreadsheets, a google doc keeping all my individual meetings with kids and what we talked about, my ongoing lesson planning documents for both classes, and my course expectations for online learning.

I kept all attendence, nightly work, and notes on individual check-ins for students all in the same Google spreadsheet. Each got different tabs. So I would open a spreadsheet and see this at the bottom.I didn’t want to have information spread out over a thousand documents. My goal was to be as consolidated as I possibly could be.

Five minutes before I taught each class, when Amazon’s Alexa reminded me, I opened the following windows to get prepared and ready to go:
I’d have my google calendar up, because I often needed to refer to it to find the time the class ended. I had my ongoing lesson plan document open so I could execute the plan I came up with. I had the attendance spreadsheet ready so I could take attendance, and I had any Desmos Activities tabs open (for what kids did the previous day and/or any new activities we’d be doing).

I’ve never been a person who scheduled my life using google calendar, but during this time, I came to fully rely on it. Here’s a screenshot of what a random week looked like on my calendar:

Final Thoughts

Wow, that took longer than I anticipated to type out. I honestly figured this would just be a 30 minute blogpost where I throw up a few screenshots. Sadly I think most of this structure won’t be useable next year if we’re in hybrid mode. And I wouldn’t say it was perfect or even great. It was… fine.

The biggest thing that I felt was after a few weeks, it started to feel monotonous to me — and so I assume it was the same for kids. We’d do the same thing in most classes. I needed to find ways to break things up — different activities or ways to learn or engage with the material. But I was so fried from juggling everything and creating everything and worrying about covering key content that I didn’t have the opportunity to mix things up in the ways my kids deserved.

I should also mention that this was a lot of work that isn’t outlined here. Planning and creating the desmos activities took massive amounts of time. I had to collaborate with my teaching partner. Reach out to kids and adults when I was worried about kids. Create the skill lists and plan out the content we’d get through for the year. Write assessments and mark up assessments. Write reassessments, set them up, and mark up reassessments. Work with kids during office hours. Not to mention plan our daily advisory and attend meetings (including some of my own doing… like a book club I helped kids organize). There were many days where I’d be on my laptop every moment from 7:30am to 7 or 8pm with only a short break for lunch and dinner. Being on my laptop so long gave me headaches sometimes. Weekends were super important for me to organize myself and get as much preparation as I could for the following week. It was a lot. I found ways to make it streamlined and sustainable, but doing this work — even just “fine” work — took a lot out of me.

‘For the last year or two, I’ve seen so many people tweet about they have tables in their math classrooms where they put math or math-adjacent things for kids to fiddle around with before/after school or during their free time. Here is a recent tweet thread:

So of course, color me insanely jealous. (I think I first heard of this idea from Sara Van Der Werf on this blogpost.) The thing is… I really want kids to see math as something that exists outside of the math classroom. And anytime I see an opportunity to do that, I go for it. So things like math club and math team, yes, I’ve led those in the past. Independent studies/work with kids, yes. But I like the idea of opening up the umbrella of what counts as math. So a few years ago I helped organize a math-art gallery (with real mathematical artists!) at my school — with an exhibition called Technically Beautiful. Or organizing math-related book clubs with kids (from Flatland, to Hidden Figures, to How Not To Be Wrong, to whatever.) Or assigning my “explore math” project to some of my classes.

The appeal of the math play space was so strong that last year I decided I would make one for this year. The tricky part is that in my school, we don’t have our own classrooms. Last year, I taught in four different classrooms. But luckily outside of the math office, we used to have a long bench where only a few kids sat on when waiting for class or a meeting. So my plan: remove the bench and make a math play table/space right there.

My colleague and friend Danielle was interested in the idea, so we basically just did it. We asked maintenance to remove the bench. We set up three card tables. And we had the space ready for the first day of school. Ready to see what it looks like?!?

I’ll go through what exists in our space now.

When discussing the space, we agreed that it had to look cozy and inviting. So with our limited artistic skills, we put together this beautiful sign. We tried hard to come up with a better name, but we kept on converging on this simple one… so we went with it. We literally crumpled paper of different colors and tacked them up to write the word space. I’m actually in love with the way it looks. It was what we had around, and we got creative!

Now on the left side we have this:

This little cart was being thrown away by a third grade teacher, so we stole it! We put showerboard on it so it can act as a whiteboard, and if you look closely, we have some whiteboard markers below for students to us. On the board itself is a number game lifted totally wholesale from David Butler (his post about it is here). The idea is that with four small numbers (e.g. 1, 10, 10, 7) and two large numbers (e.g. 60, 120), students should attempt to make the target number 121.

After showing this to one of my precalculus classes, a student was obsessed with trying to get the target number using all six numbers, and came up quickly with a way to do it. He was super proud, and rightfully so!

Next we have two card tables covered with some fun cloth I found at home.

These are books that I brought in for kids to thumb through (though they just have to ask and they can take it home to read!). I have a zillion books that could go here… My criteria was nothing that could turn off a student easily. So a book of math poems, a childrens book about Sophie Germain, a math book based in funny comic strips, women in mathematics book, and a couple “math novels.” I even had a math department colleague/friend write a “recommendation note” that we stuck in The Housekeeper and the Professor, like this was a book store! (I asked our school librarians if they had the little book stands, and they were happy to give me some!)

Ikea had some $1 picture frames, so we used them to post some puzzles and jokes!

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We also put out some puzzles from Play With Your Math which we thought had a low barrier of entry but that kids might enjoy!

We also have a little estimation station (currently of jars with rice in them):

And of course we saw that Sarah Carter had provided us with a lot of math jokes that we could steal and use in our math space… So we have that up also! Because how could we not?!?!

Lastly, we have a “tinker table” where we have some tiling turtles, other tiles, and a weird set of puzzle pieces which need to get put back into a square shape.

And that… is about it!

At the start of putting things together, we realized we needed a bit of a formal vision for us to stick to… so we drafted this super quickly, but it was something we both felt was approximately right:

Vision: To create an unstructured public space where kids can relax and fiddle/tinker around with fun math things that might not be related to things in the formal curriculum. The hope is that this allows for the experiencing of math as something casual and playful. We want this space to encourage students to want to talk mathematically with each other.

Through this space, which will be curated and changed periodically, we want to widen the umbrella of what gets counted as “math” and “doing math,” and who gets to be counted as a mathematician. 

We encouraged teachers at the start of the year to share information about the math space with their classes, even writing them a blurb they could read in their classroom but also encouraging them to leave their class five minutes early to bring kids over to just look around. What we wanted teachers to emphasize? “Most importantly, we don’t want you to be scared to sit down there. We spent time making this space for you. We want to say that again — this space is for you! Pick up books and see what they’re about. Make designs with the tiles. Flip the joke page over to see what the groan-worthy punchline is. Try the number game puzzle out, or pick up the paper folding puzzle that we have there for you. Make an estimate for the estimation challenge. We want you to feel comfortable here — not treat it like a museum.”

Lastly, you might have noticed that in the vision we mentioned that the math space is designed to changed periodically. That’s the goal. Of course the jokes will change each week as will the numbers for the number challenge. But everything else — books, estimation, picture frames, tiles — will be swapped out. We have a giant list we’ve brainstormed of things that we could put in this space, and we’ll make decisions as we see if and how kids are using it. Some ideas include:

• Instructions for the game of SET, and space for kids to play the game!
• Wooden “put these together to form this neat shape” puzzles
• Legos
• A variety of math poems that students can take and put in their pockets during Poetry Week at our school
• A spirograph or two
• Math and Climate Change coloring books with lots of colored pencils (where we hang up the pages on the bulletin board after things get filled in)
• Towers of Hannoi
• Origami paper and instructions
• 3D printed mathematical objects, including cool math based optical illusions (like these!)
• A museum of WEB Du Bois stunning and eye-opening infographics involving race in America
• Geoboards
• Information on women mathematicians and mathematicians of color and mathematicians that are LGBTQ+ and…
• Fun little math problems (the size of a business card) that kids can pick up and bring with them

I actually have so many more ideas on my list, but it’s all written so informally no one would ever fully make sense of things. But these are just some. But if you have ANY other ideas that you think would make sense here, I’d love to get a nice long public list for math play spaces — so throw any ideas down in the comments.

With that, I’m out!