Two Problems that Got Me To Think

Here are two problems that have gotten me to think a lot.

The first one came from my Precalculus co-teacher James. We had been finishing up our unit on combinatorics and also creating new groups, and he devised a great question. So here’s the two-part problem I posed to my kids:

First Problem: We have a class of 14 students, with two groups of 3 and two groups of 4. If I were to have a computer program randomly create new groups: (a) what is the total number of different configurations/outcomes we could have? (b) what is the probability that your entire group was the exact same if you were in a 4-person group? 

I thought I solved it successfully and was feeling really confident. Then James told me I was wrong. Then I tried but didn’t understand his logic. So I made a simpler case, and then I thought I understood it. My brain hurt so much. I kept switching back and forth between a couple different answers. It was marvelous! Finally, I felt like I understood things and felt confident. I shared it with my class, and lo and behold, a couple students got what I got, and a couple students didn’t. But the students who didn’t convinced me with their logic. And then I shared their thinking with James, who didn’t have the same answer, and he too was convinced. And I thoroughly enjoyed being wrong and telling the kids that this problem messed with my head, and they helped me see the light!

The second problem came from a student who emailed me about wanting to become a better problem solver. And they shared this old entrance exam for this summer camp they were thinking of possibly applying for, and wanted some guidance. The problem that I got nerdsniped by and ended up spending hours working on over Thanksgiving break was as follows:

Second Problem:

This is from the 2019 entrance questions for a summer program. I think I was able to successfully solve (a) and (b). And then I think I solve (c) for n=3 and n=4 (and got an answer for n=5, but haven’t proved it is optimal). And I have no way to even start thinking about (d). But what I thought was lovely is how many different places my brain when went trying to think through this problem. And the neat geometric structure that arises out of the setup. (Even though I wasn’t able to fully exploit this structure in my thinking.)

I hope you enjoy thinking about these!

An excerpt from an essay

I received an email from a former student (R.L.) who I taught a few years ago. She’s a senior now in college and is taking an education class. She wrote a paper that she wanted to share with me, because half of it centered on her time in our Advanced Precalculus class during her junior year. I know I’m a warm-demander teacher (or at least that’s what I strive for). I try to make my classes a little bit harder than kids think they can do (but exactly at the level I know they can do). Reading her essay made me feel a lot of things, but I love how in so many ways she captured some of the things I strive for and do in class. The fact that she noticed them and remembered them years later means a lot.

She said I could share that part of her essay here on my blog when I asked. I like to archive good things in teaching, and this is something I’d like to archive. So here it is.


I’ve been thinking about ways that coaching, questioning, and telling played out in my education at Packer. Packer was the ideal setting for these methods of learning, as we had small, seminar-style classes and teachers with the capacity to work with students one-on-one and develop individual relationships. Upon reflection, my eleventh grade math class, Advanced Pre-Calculus with Mr. Shah, exemplified coaching, student-telling, and questioning, and also included exhibition-style projects.

This was the hardest class I took in high school. The content was very challenging, and Mr. Shah’s approach to math drove me crazy. Mr. Shah sees math as a creative field, one that demands critical thinking and a deep conceptual understanding of topics that many see as surface-level and robotic. Mr. Shah’s packets used broad questions as the benchmarks of understanding, pushing students to explain concepts using their own words. His problems had a playful tone, and we spent most of our class time working through material in small groups while he played music in the background and buzzed around answering questions and challenging students to think more deeply. Why are conics important? What is the meaning behind this geometric sequence? Why did you choose to solve this combinatorics problem this way? At the time, these questions made the class a nightmare for me – the content was already challenging enough (it was in this class that I failed the only test I’ve ever failed), and the way he forced us to think about it made it even harder. But, after reading Sizer and thinking about coaching, telling, and questioning, I see what Mr. Shah was doing.

In addition to trying to make math more fun and meaningful, he was pushing us to develop mathematical skills that built upon each other. Sizer writes that the “subject matter chosen should lead somewhere, in the eyes and mind of the student” (Sizer, 111). The curriculum progressed when we made “mathematical discoveries,” and those discoveries led us to mastery of complicated skills and a deeper understanding of concepts. Mr. Shah never talked at us. Discoveries came through telling, but, it was table-mate to table-mate telling. As the teacher, Mr. Shah’s role was to encourage our discoveries and offer support as we worked through our own questions and explained concepts to one another.

Through his approach, we were also practicing broader skills like critical thinking, creativity, perseverance, and thoughtful reflection. He centered the class around group work and we spent a significant amount of time reflecting on our individual contributions to the group and the strengths and weaknesses of our team. I honed these skills in my other classes at Packer, and I am sure they are part of my academic success at Tufts. Mr. Shah’s class was enormously challenging for me, but in writing this paper I have found an appreciation for his approach, and I know I owe him a thank-you.

Lastly, Mr. Shah also incorporated exhibition-style projects into our curriculum. He called them “Math Explorations,” and we had to do four of them throughout the year. They were not as big as these example exhibitions and they did not center around a presentation [like someone mentioned earlier in the paper], but they provided an opportunity for individual exploration in a subject area of our choosing. Being the English-lover I am, for one of my Math Explorations I wrote a series of math poems – a sonnet, some haikus, an ode, and a free-form poem. I was proud of these poems. It was exciting to take ownership in a class in which I often felt overwhelmed, and pursue something that made the content relevant to my interests. These projects were empowering, and they helped me feel connected to the material and confident in the class. If this is the power of exhibition-style learning, then I’m in full support, because the takeaways made a difference in my learning. If only more schools had teachers like Mr. Shah and the resources and the capacity to make classes like his more widely available.


Hispanic Heritage Month – Math

I am going to type a super quick post, because I need a break from thinking about school and classes, and I decided that this will be it. I realize that some of you might be celebrating Hispanic Heritage Month in your schools, and wonder how you can bring that into your math classrooms. I’m sure there are lots of ways. But I wanted to share — or remind some of you, if you forgot — of this amazing website called

It has profiles and posters of Latinx and Hispanic Mathematicians! And so many interview podcasts, where kids can hear what mathematicians do, but also hear stories that either might resonate with them because they’ve experienced similar things as the interviewee, or inform them of something they might not have experienced or even knew was an experience.

At our school, another teacher and I created a “Math Space” outside of our math office. (Read more about that here.) Since we’re now in COVID-times, we don’t have the table and stools there for students. But we still have the bulletin board. So I printed out two of the posters from the Lathisms site, and created my own poster of a passage from Evelyn Lamb about why we should care about these stories.

It’s a little barren, but this is what it looks like:

I haven’t decided what I’m going to do in my classes yet… But I am thinking of having them leave class 7 minutes early to come to the hallway, and have different students read different paragraphs aloud for the poster I made and also for one of the mathematicians. (And of course share the Lathisms site with them, and encourage them to listen to a podcast or read more.)

Wait, maybe the reading aloud in the hallway thing isn’t a good idea because with the masks it’s just hard to hear things… I might need to re-think! But now, I have to go back to planning for classes and looking at nightly work. Adios!

Mathematical Habits of Mind

I haven’t been blogging for a long time. As you can imagine, the pandemic took a toll on teachers, and at least for me and my teacher friends, we were working insane amounts of time, and it was so hard. Emotionally, physically, intellectually. At the time, I just didn’t have it in me to blog about the experience.

But now we’re about to start a new school year. And I’m vaccinated. And my students are vaccinated. And we’re wearing masks. And my classes are going to be with all my kids together in a single room [1], which is such an awesome thing compared to last year.

One of the classes I’m teaching this year is Advanced Precalculus. Another teacher, my friend James, is also teaching the same course. And he’s new to my school this year, and so when talking about the course, he shared with me how he formally incorporated Mathematical Habits of Mind in his teaching in previous years. And interestingly, last year, I toyed with the idea of formally getting kids to be metacognitive about problem solving strategies — but decided to focus on something else instead. So when James shared this idea with me, I got excited.

Right now I have an inchoate idea of how this is going to unfold. Hopefully I’ll blog about it! But for now, I wanted to share with you posters I made using James’ Mathematical Habits of Mind. Most importantly, here is a link to James’ original blogpost with his habits of mind and rubric.

Photo of the posters hung up in one of my rooms:

I know, I know, the lighting is terrible. The key words are:

Experimenter, Guesser, Conjecturer, Visualizer, Describer, Pattern Hunter, Tinkerer, Inventor

If you want these posters, the PDF file is here.

And here are all of them shared as a single sheet, and not as a poster.

Of course, if you’re a math teacher, you know there are a lot of lists of mathematical habits of mind. We agreed to use the ones James had already been using. But there are many alternative or additional things we could have included.

At the very least, I know that as we get kids to think about what strategies they’re using to solve problems, we’ll also see where there are lacuna in our curricula in terms of using those strategies. Or maybe we’ll discover it doesn’t have as much problem solving as I imagined in it. All entirely possible, since we — the kids and James and I — will all be looking through what we’re doing through our metacognitive Mathematical Habits of Mind lens.

[1] The reason I note this is because at the end of last year, I was teaching students live simultaneously in three places: they were in two different classrooms and there were a few at home on zoom. Yes, seriously. When I mention that to teachers and non-teachers alike, they asked how that was even possible. It was… a lot.

Concerns about MoMATH

I was so excited with the Museum of Mathematics opened in NYC. I attended a zillion lectures they had with math teacher friends before they had the space ready and a zillion lectures after they had their space ready. I even convinced some of my students to attend with me and another teacher, and we got to bond over math.

I encouraged some of my students to enter the contest for the Strogatz Prize for Math Communication at the end of last year. Even recently, I went to a wonderful virtual event they held called Bending the Arcwhich featured a panel of Black mathematicians and scientists and allowed participants to talk with them more intimately in smaller groups in breakout rooms. The name for the session came from Martin Luther King Jr.’s quotation which I sincerely hope is true: “The arc of the moral universe is long, but it bends toward justice.”

(The reason I mention that is because previously the museum had planned an event in honor of Martin Luther King Jr. day which somehow connected his “Letter from a Birminham Jail” to a session devoted to the Prisoner’s Dilemma problem. For so many reasons, this was a poor decision that I was surprised no one caught when setting it up.)

Also, over the years, I had heard grumblings about the museum from people who worked there or volunteered there as interns. Recently someone shared with me a letter that was sent to the Museum of Math’s Board of Directors that was concerning.

Letter to MoMath Board

My understanding was it was written a while ago, but only recently shared with the Board. I’ve been told that it is now officially okay to make this letter public. It’s short, but packs a punch. Here is the top-line conclusion, written in a signed letter by two former “Chiefs of Mathematics” at the Museum along with others who work/ed there. 

“With respect to our educational mission, race and class discrimination are embedded in the Museum’s practices.”

To me, knowing that there are multiple people — including people who were high up in the organization — who felt the need to write an open letter to describe some of their concerns speaks volumes to me. They didn’t have to. It is easier not to. It puts them at some risk, publicly speaking out. 

To me, one of the most problematic charges in the letter is that students from Title 1 schools who visit MoMATH often get lessons that end up being 20-25 minutes instead of the normal 45 minute sessions. The letter states “We cannot remain silent while the Museum chooses to offer sub par services forthe least fortunate students who are vastly more likely to be people of color.” I hope that with this letter, those at the Museum take a close look at their practices to ensure there is equity for all the students visiting the Museum. To me, more than anything else, this is of paramount importance before the museum opens its doors again. 

Also damning is this paragraph:

Unfortunately, the Museum actively discourages any form of negative feedback, and the staff has virtually no autonomy. This repressive culture has been described on social media and in the many letters you have received from former employees about mismanagement, abuse of hours, and general lack of respect for staff. In fact, the Museum’s formal Employee Policies Document warns staff members against contacting the Board. Staff members who have advocated for improvements in the Museum’s operations have seen obstacles set up in their paths and have been pressured out or fired. The rapid turnover of MoMath staff, which has an average tenure of less than a year and a half, is evidence of this. Joe Quinn, former Chief of Mathematics for MoMath, was fired shortly after expressing his opposition to the discriminatory provision of education services to Title 1 schools. We believe this to be unlawful retaliation against him

It is important for an organization which promotes diversity and inclusion to make sure that concerns can be heard safely, that feedback can be given. To these letter writers, it sounds like that hasn’t been the case and there is a culture at the museum which sounds, frankly, oppressive.

I went to GlassDoor, to see reviews of the museum from people who work there.


Most museums I looked at had star ratings in the 3+. (I recognize that those who leave reviews on GlassDoor are likely to be those who have a lot to say in either direction.) It was disheartening to read through the reviews

I only hope that this letter prompts some sort of investigation into the working conditions at the museum. How long do people work there? Why do they leave? Are employees being taken advantage of in terms of their hours worked? At the very least, this letter suggests that the Board can and should look into this. 


How to have kids set up an individual zoom meeting online for office hours using google calendar

This is going to be a really quick blogpost. On twitter, I was talking with someone about how to set up individual check in meetings with kids. I shared briefly how I had kids sign up to see me during office hours, and I thought I’d quickly write a blogpost about it to make it clearer. A lot of high school teachers at my school did this, and found it helpful. (I learned it from them.)

First, you need to install the Zoom Scheduler extension for chrome.

Then, all you do to set up appointment slots for kids is to go to the day you want to on your google calendar and create a new event. Now to make appointment slots for kids to use, it’s just this simple!


That’s literally it. You’ve done it! Once you’re done, your google calendar will look like this:


So how do students sign up for an appointment? You just send them a link to your appointment calendar, and they click on the appointment slot they want. To be clear, your appointment calendar is different from your regular calendar. The appointment calendar only lists the things that you have set up appointment slots for, not your entire calendar. (So you don’t have to worry that kids will see the happy hour that you included on your google calendar!)

To get the link to your appointment calendar to share with kids, just click the appointments you just created on your google calendar and you’ll see this:



Clicking “go to appointment page for this calendar” will lead you to a page with just appointments. The URL on that page is the link that you give to your kiddos. So they use that one link for the whole year! That’s it!!!

But if you’re like me, you want a bit more info… so…

This is what the appointment page looks like for your kiddos:


They click on the time slot they want, and it will reserve it in their name. This pops up when you click on it [note: it shows my name and not a student name because I signed up for an appointment with me! It will really show your student’s name.]


Now on the appointment page, that slot has been taken away so no other student can claim it:


When that happens, I as a teacher get an email that alerts me to the fact that a student has signed up. That email gives me the zoom link to the meeting — but I can get that zoom link when I go to my regular google calendar and click on the time slot that was taken. This is what my calendar looks like after someone signs up for a meeting:


Also, the student who signed up gets an email in their inbox with the information for the meeting (including the zoom link)… and the appointment automatically shows up on their own personal google calendar. AWESOMENESS!

Okay, that’s all!

In case this is helpful! It worked for a lot of us at my school.

PS. A couple pieces of advice…

I did this during my required “office hours” which were usually 3-4pm. I created 15 minute windows for them to meet with me. But I told kids they had to sign up before 2pm on the day of. That way I wasn’t just waiting around for them in case they signed up at 3:44pm.

Even though these were individual meetings, I figured sometimes kids might want to come in a pair or trio (if they were working on something together and got stuck). I told them one of them had to sign up but they could share the zoom link with others, so we could all meet at once! No one actually did that in the spring, but I suspect that that’s because I only mentioned it once or twice… I might make more of an effort to have kids to do that in the fall.

UPDATE: Recurring Office Hour Meetings

If you want to create Office Hours that kids can sign up for every Monday, you don’t have to do this every week. You can do this when setting up your office hours…

After entering the basic information, click “MORE OPTIONS”


You’ll “Make it a zoom meeting,” and then you’ll click on the “DOES NOT REPEAT” option to bring up the options to create a repeating meeting. I always choose Custom — because even if you want office hours weekly every Monday, if you click “Weekly on Monday,” they’re going to appear on your calendar every Monday until the end of time. :)


So then you just decide how you want the office hours to recur! (Yup, you could even say Monday, Wednesday, and Friday if you want! Or every other week!)


This screenshot above shows office hours being created every Monday but will stop showing up on the calendar on December 14th (so, for example, at the end of a semester or some reasonable date).

But WAIT Sam! What if there is a particular Monday you can’t have office hours?

Just click on the office hours on your google calendar and press the delete button (trash can). It’ll ask you if you want to delete ALL of the office hours, or just this one day of office hours. Just pick “This event”:



How I Schooled During the Spring of 2020

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:

5Initially, 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.



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:

Here are all my Desmos Activities for Algebra II:

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:8I’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.)11
    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.]



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.14I 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:15
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.