TMC in 2018: a personal reflection of where we are

Super rough draft thinking…

Last summer at the end of TMC17, there was a flex session on diversity. That brought up a host of questions that made us realize: as an organization/conference, we don’t even know who we are. That was both terrifying (do we not?!) and exciting (we get to tease out our core values!).

Here’s the thing. The conference runs pretty darn smoothly and appears organized to people attending, but that all comes as a result of a handful of people on the committee working on certain individual tasks (usually alone, sometimes with one other person). And Lisa, our amaaaaahzing Lisa, keeps her eyes on everything and does the bulk of the infrastructure/logistical work that makes the conference actually happen.

But — and this may come as a surprise to y’all — I’ve been on the organizing committee for a few years, and never once in that time did we all get together in person or on a google hangout to have a discussion. Mostly the committee members worked on their individual tasks and we would occasionally send an email out or ask a question… most people knew what they were working on, but didn’t know what everyone else was doing. And honestly, there probably wasn’t a need. The conference was successful in that it, for many attendees, provided something that they didn’t get in other places professionally, and people were on the whole happy.

So when that flex session on diversity came up, it opened my eyes to something. We were doing the logistics of recreating the conference from year to year. But it was now year six, and although still organized by math educators for math educators, we (as organizers) were still considering ourselves to be a ragtag conference that we cobble together. Maybe I shouldn’t speak for everyone. I should probably say…

… at least that’s how I looked at TMC until then. It was perhaps a vestige of the thinking when it all first happened in 2012, and we were cobbling everything together from scratch (here’s the original website that I put together but don’t even remember putting together!: TMC12). And perhaps it’s because it was just a merry band of us making spreadsheets and organizing ourselves in dorms and trying to read contracts and figure out what they meant and fumbling around trying to make things work over the years… it just felt like we weren’t professional conference organizers. We were just trying to make things happen as best as we could. And so that was enough. I thought we’re volunteering our time for this… so who could ask for more? 

But the flex session made me realize we are not that anymore. We can’t be that anymore. Because the conference is real and sustained and impacts people. That’s it: we were responsible because we were putting things together and what we did impacted people. And it was clear that for all our work to be welcoming, we weren’t always being successful. Jenise Sexton wrote in 2017:

So when I walked in the Dining Hall of Holy Innocent Episcopal School along with around 200 other participants, I wasn’t surprised to see like 4 or 5 other brown skinned people. But that evening I received a DM from another participant which read, “Where are all the black people?”

In that moment, reading that message, I realized it wasn’t my role to be the sole representative. It isn’t supposed to be normal for me to be the only black face in the room.

And in the flex session, people had lots of questions about how things happened to make the conference what it was. How did we think about registration and who can attend? What outreach do we do? Who chooses the keynote speakers and how do they get chosen — and does diversity (in any form) come into play? How do sessions get selected/accepted and organized? Why do we really want diversity, and what kinds of diversity are we even talking about? What is diversity anyway?

People cared about this. And conversations (good conversations!) happened at TMC around this. Marian Dingle wrote wonderfully about it here (“Yet, I don’t feel that I was able to fully let my hair down at TMC. For one, I was a first-timer, and the few people I “knew”, I only knew through Twitter. For another, I was one of a few black women, which although not an unfamiliar situation, still was not comfortable.”) and I’m glad that she was selected to be on the TMC Board this year. [1]

And this is precisely where I started to realize we were responsible for answering these questions. We had to think more than simply about logistics and getting the conference to run smoothly. We had to think about what the conference really was, and what a vision for it was. We can’t hide behind “don’t expect too much from us… we’re just putting this thing together on our spare time!”

And from this, Tina (with me as a trusty sidekick) took the reins on coming up with a mission statement for TMC — asking for input from all constituents (anyone who attended TMC, the committee, the board) and then asking for more feedback from the community during a Global Math Department session before it took its final form.


That was a huge step forward for us, I think. Because it helped us define our core values as best as we could, and then refine them. It is perfect? No. But does it give us a place to start working from? Yes.

Now let’s fast forward to TMC18. Another flex session on diversity. And the first question that comes up:

Why do we value diversity as TMC? The mission says “we value diversity” but it doesn’t go deeper. We need to go deeper. 

And in this hour and a quarter, many things got raised. Some of my key notes:

  • We are at a place where TMC is a grassroots organization whose grass has grown too high. We are just starting to grapple with how to deal with that mindfully, inclusively, awesomely.
  • Do people see diversity as important for TMC because: (a) diversity helps TMC (it makes TMC stronger), or (b) we want to create a TMC that’s valuable for all math educators?
  • We all blog and tweet to connect with people, and sometimes it is comforting to connect with someone like you in a way other than just math — such as in terms of race or sexual orientation. “If white people/cis people can find it, I should be able to find it too.” [One thought that was thrown out for those who want to connect in safe spaces at TMC was to form “homerooms” or “affinity spaces.”]
  • “It’s not just about inviting people, it’s about what they’re being invited into.”
  • As math educators, we know that numbers can be powerful. But the number of people of color attending TMC each year might not be the metric to use to measure us with “diversity.” We want to be careful not to try to “get” people of color (or diverse participants) just for numbers, to make us feel like we’re filling a quota and doing the right thing. We don’t want diversity to be trendy — to make ourselves feel better, or so we’re trying to make the “picture” look better. Maybe other metrics (more qualitative) are better. [What we do at TMC? Action steps we’ve taken? Things like pronouns on tags. Sessions. Keynotes. Big and small. We tried some these things — what can we measure? What does success look like?] Numbers don’t speak; they lie in a vacuum when given without context.
  • We need to be more strategic than being generally welcoming (which we do really well). We need to be specifically welcoming.
  • If TMC is going to double down on addressing diversity and encouraging participants to engage, it needs to build that into the schedule (perhaps by having a diversity/equity strand in the program, and perhaps other ways to encourage conversations around these issues).
  • TMC and the #MTBoS are intertwined because the #MTBoS is the primary pipeline for people coming to TMC. So one possible thing that could happen is that people could start having more sustained and organized discussions about diversity and equity in the #MTBoS. One idea was about having a regular chat (e.g. #MTBoSequity) and having it center around shorter things like articles/blogposts instead of books. Questions were raised about twitter being super public and challenging to have those sorts of conversations — and the idea of having semi-private discussion boards or not-saved-video-conferences similar to Global Math Department were raised. (Tina just blogged about the idea here.)
  • Having people write their pronouns on their nametags was a way TMC forced people to confront diversity in a small way. Someone asked if there were other small things like that that we could do.
  • We have a lot of people at TMC who are supportive of the ideas of diversity. 

In the flex session, people asked a lot of questions about how we plan the logistics, just like last year. How are the sessions chosen? How are the keynotes chosen? How does registration work? At heart, the reason these questions were asked is because people were interested in seeing if there was intentionality in how they were being done. [2]

And that to me is where we need to go next as we think about organizing TMC. Intentionality. I wrote earlier that:

I’ve been on the organizing committee for a few years, and never once in that time did we all get together in person or on a google hangout to have a discussion.

TMC is at a place, at least to me, where we have successfully figured out many of the logistical aspects of putting the conference together. We were able to do that as a committee without being a cohesive whole — we could work in our little silos. However we’re at a critical juncture as an organization. We need to figure out how we can ask big questions, have sustained and challenging discussions that push us, have a process for moving from discussions to making decisions so we can move forward, and come up with a clear and shared vision that we’re moving towards. 

After writing that, I only have one word that comes to mind: overwhelmed.

After TMC18, the committee and the board did all sit down together. We had four hours. And though we could have used four more, I was proud that we identified that we needed a way to communicate and have discussions, and we are in the beginning stages of learning how to do this all virtually and asynchronously. In the few weeks since TMC18, we’re actually starting to do this. It’s a lot we have to do, and we don’t have a roadmap on how to do it. It would be way easier to just go back to how things were done. But that’s not in line with what we want to be. Earlier, I wrote:

The conference was successful in that it, for many attendees, provided something that they didn’t get in other places professionally, and people were on the whole happy.

But I wrote that to reflect my thoughts then. Now I see that this is problematic. Is that a good metric of success for us? Who gets counted in the “many” and what does “on the whole” really mean? We have to be intentional to make sure that everyone feels at home and that we are working towards our mission of reaching “all learners.” Which means we need to think about teachers who are feeling left out. When we’re thinking about the conference as a whole, who are we designing it for? Is it for people like ourselves (those on the organizing committee/board)? [3] Or do we need to start de-centering ourselves before asking this question?

As I said: overwhelming.

I don’t know how this all will happen. I know it won’t be easy, and it won’t be quick. I’m imagining years. [4] And honestly, a big part of me wants to shy away from all this change. Those who know me know that change is really hard for me. I’m a little  (big) hobbit who likes to live in my little hobbit hole. But I also know that if I were one of my students who came to me for advice on all this, I would say to them “doing what’s easy is not always doing what’s right.” And I’ve never been scared of hard work.

I should conclude by saying these are just things I’ve been thinking a lot about. But they are my thoughts. They aren’t fully formed (rarely are my thoughts fully formed), and certainly not cohesive. But more importantly, they aren’t official thoughts of the committee or the board. I don’t really even know if other TMC organizers will agree with my thinking about this! But I promised @BeckyNftP I would blog about the flex session around diversity, but the only way to keep that promise was to embed that in the much larger thing that I’ve been mulling over in my head.

[1] Please don’t ask me what the Board does and what the Committee does. Remember when I said ragtag? These are things we’re still figuring out.

[2] And things were being done.  Those selecting the keynotes or organizing the conference program did think about bigger picture things. But they took that up on their own, not because they were working towards a shared vision we all had.

[3] As TMC started, it had to be. That’s why it was started! For a small group of #MTBoS people to meet in real life. But now we have to ask the question if our answer is still the same?

[4] We’ll probably need at least a year just to figure out how to communicate with each other semi-effectively.


Desmos Pre-Conference 2018 Recap

This is a quick blogpost that I’m using to recap just some of the information from the Desmos Preconference before TMC18. I was dealing with some other stuff when I returned from TMC, and then I had to take a short few-day jaunt to see my parents/aunt/uncle. Now I’m finally home and starting to do things like write college recommendations and think about my new class for next year (Algebra II). But I’m afraid if I don’t take the time to reflect on some of what I took away from the conference, I will not end up using it. But at the same time, I feel like it’s so much stuff that to do it comprehensively, it will take too long and that’s keeping me from starting. So here’s my pledge: I’m just going to do what I can, and not worry about being incomplete, and then I’m going to #pushsend.

Tonight, I’m going to #pushsend on the desmos preconference day.


I went to one session, led by Heather Kohn, David Sabol, and Mary Bourassa. These three desmos fellows shared how they use Desmos in the classrooms. Here are a few gems:

  • Heather often creates handouts to accompany activities. For example, for Will it hit the hoop? she has a spreadsheet for kids to fill in (e.g. “Predict, Screens 5-11” “Analyze, Screens 12-19” and “Verify, Screens 20-26”). 
  • I shy away from doing cardsorts (or even short activities) on desmos because I tend to have some groups finish way earlier than others. But this would happen even for paper cardsorts! So here are some tips. First, just so all groups start at the same time, you can pause the activity on the first screen (which you can have be an introductory screen). When everyone is ready and logged in, you can then unpause the activity which allows everyone to start at the same time. More importantly, you should create a slide after the cardsort/activity which links to another activity or has some extra practice for those kids to work on. And for extra fun, you can have this slide be a “marbleslides challenge.” But one tip is to use the teacher dashboard to pace the activity to the slide before the challenge, so that you can make sure kids aren’t rushing. (You can check in with the first group done and ask them a few questions to make sure they’re getting things.)
  • You can do a Which One Does Belong on Desmos (example: go to https://student.desmos.com and enter 5CK W7N). Have kids vote on which one doesn’t belong. You can then display how they voted! If no one picks one, after they finish and you discuss, you can have them go back and everyone has to pick the one that wasn’t picked… and then explain why that last one might also “not belong.”
  • David was worried about how kids will access desmos activity knowledge later. There’s a lot of digital work and verbal work in class, but then things aren’t archived. So here’s a great example of how David deals with this. He used Andrew Stadel’s “Math Mistakes with Exponent Rules.” On day 1, he used the first day PDF to have kids work the problems in class. Then on day 2, he screen grabbed the second day PDF and made a desmos cardsort (sorting them into true/false) and used the dashboard to showcase wrong answers and have class discussions. Also, after the cardsort, he had a screen that said: “Make a FALSE statement that a classmate may think is actually TRUE.” Then that night he created — using what kids wrote for their false statements — a paper copy with all these FALSE statements (sometimes there’s a true statement that a person wrote!) where kids had to identify the errors!
  •  A great question in a desmos activity is to show a lot of work/visualizations/etc. and write: “What would you tell this student to reinforce what they know and correct their errors?” If the student work has some nice thinking and some subtle not-so-good thinking, this often will lead to solid class discussions.
  • Mary uses Desmos occasionally for assessments. There were only a few questions, but they involved deeper thinking (e.g. given a graph of part of a parabola, can you come up with the equation for the parabola?). The presenter asked her kids to do all their written work on paper handed out for the test. Yes, students could revise their work/answers based on what they saw on Desmos, but that had to be reflected in words/notes/changes on the written paper. So a student guessing-and-checking on desmos with no supporting work will not garner credit. (For students who finish early, put a screen with marbleslides challenge.) One big note: make sure that at the end of the test, every kid goes to a blank last screen, and then PAUSE the activity. That way kids can’t come back and rework problems or show other students particular questions on the assessment.
  • Rachel K. (attending the session) said that she often had kids project their laptops up to the airplay and lead the class through something they found/built/figured-out on the Desmos calculator, or will have one kid lead a desmos activity on the big screen.
  • I often worry about how to lead effective discussions on activities that kids are doing. For pre-existing Desmos built activities, there are “teacher tips” that help teachers figure out what to focus on and how to facilitate conversations. But more importantly, whether Desmos built or random-person built, every activity has a teacher PDF guide (Click on “Teacher guide” in the top right hand of the screen for the activity.) You can print this out and use this to help you come up with a specific list of things you want to talk about, and stop at those places (e.g. questions, places to pause, etc.)
  • After the session, I talked with Heather about this feeling I had when doing long activities with Desmos. Although I was constantly checking the dashboard, and walking around listening for conversations, I often felt useless and bored and like I was doing something wrong because I wasn’t … doing much. She let me know that she also feels this, but that’s part of it. Letting kids engage. But I realized that some of my best classes (without desmos) have me circulating and listening but not doing too much beyond that. I was “being less helpful.” So I think I just have to make sure that when I’m not doing much, it’s because kids are doing good things mathematically and conversationally, and that’s because I’ve orchestrated things to be that way.

As an interlude to this wall of text, here’s my favorite nerdy math picture from the day.

20180718_141146.jpgYes, indeed, you see a 3-4-5 right triangle, and a visualization of the oft-taught “Pool Problem.” In Starburst. My kind of math manipulative!

For the remaining two sessions, I worked on playing with Computation Layer and refamiliarizing myself with it (I spent 3 days earlier this summer spending huge swaths of time on this… a huge shoutout to Jay Chow who helped immensely with this). Having CL experts in the room and granting myself three hours to play with CL was amaaahzing. I first reacquainted myself with some of the basics (a lot of which I had forgotten, but it came back fairly quickly) and then I decided to start trying to “desmosify” this calculus optimization activity.). I didn’t get too far in, and so far this is no better than the paper version of the activity, but I am proud of what I was able to do with my CL chops! (You can see what I made here.)

The keynote session was given by Robert Berry (the new NCTM president) and he gave an overview of the recent NCTM book Catalyzing Change (which I have bought but haven’t yet read!), talked about some big picture NCTM things (advocacy, membership, financial health), and then told us what has been happening on the ground level. He ended his session talking about technology and what excites him about that. He said that “Technology that supports and advance mathematical sense-making, reasoning, problem solving, and communication excites me” and that “Competence is about being participatory in mathematics – with each other, with the teacher, and with the mathematics.” He then said technology can be used for good or evil based on how technology affects the following things in the classroom: 

  • Positionality [how students engage with each other, their teacher, the curriculum, the technology, etc.]
  • Identity [how students see themselves]
  • Agency [how students present themselves to the world? how do we create structures for that to happen?]
  • Authority [“shared intellectual authority”]

His latest NCTM President’s Message is precisely on this. Also, Robert is a totally awesome guy.


That’s me on the left, him in the middle, and friend and TMC keynote speaker Glenn Waddell on the right.

Lastly, Eli (founder of Desmos and super nice guy) showcased a new desmos feature for teachers: SNAPSHOTS. You can read about it here, but what I love is that it allows teachers to facilitate discussions more thoughtfully in line with the 5 practices. (I’d love any help finding or coming up with problems at the high school level that work well with the 5 practices… Most examples that I’ve seen are at the middle school level so it’s been hard to wrap my mind around how to find/create problems for a precalculus or calculus class that might make this approach work super well.)

My favorite slide of his was:


Eli keeps things simple, which allows me to read slides like this and think: “wait, in what ways does my teaching do that?”

And with that, it’s time to #pushsend.


The State of the 2018 #MTBoS (to me)

Every so often, I like to take reflect on where I think we are as a community of math educators who collaborate online. Usually after attending TMC — a math conference run for and by math educators. But please forgive me in advance. This is the ramblings of someone writing late into the night and someone who isn’t revising. Musings, at best.

Taking Stock Previously

May 2009: Why Twitter?
November 2009: How BlogBuddies Became Friends
July 2012: 40 Choose 2 First Dates, or Initial Impressions of TMC12
March 2013: Some New Things On the Interwebs & HOLY COW What is Happening!
June 2013: My Thoughts on the #MTBoS
August 2013: TMC13: The State of Things For Me
July 2014: Teacher Growth, the MTBoS, and TMC14
July 2015: My Thoughts about the Evolution of the #MTBoS: 2015 Edition

But I haven’t taken stock for a while. Partly it was because I didn’t see much change in 2016 from what I wrote in 2015. And then I made a decision to consciously avoid jumping in the numerous conversations that happened during and after TMC17, although I had lots of thoughts. [1] So now it’s time to take stock again.

Taking Stock and Making Predictions: Teacher Leader Edition

The biggest thing by far that I’m noticing is that people in the community are becoming (and seeing themselves) as teacher leaders.

When I started teaching a bit over a decade ago, I didn’t see many pathways for classroom teachers to have huge impacts on math education. (That wasn’t something I was looking for, to be clear, it just wasn’t something that I saw in my world.) In the past five years, as social media has fueled things, our classrooms aren’t silos anymore and our practice have become more public, and as this community has grown bigger, I’ve seen the rise of various pathways that teachers can become teacher leaders.

There are many ways people can become leaders from within the MTBoS community (for example, Annie Perkins/Megan Schmidt/Dan Anderson/Justin Aion inspire playful mathematics through art for so many math teachers. And people are planning and running mini-TMCs and tweet ups. New things like Desmos’ Computational Layer come around and people like Jay Chow become the expert teacher of us teachers who want to learn it. There are always people running and presenting at the Global Math Department.) That has actually been something I’ve found true since the early days of the MTBoS. It’s just that there are more and more of them now as the community has grown. I’ve found that the MTBoS provides numerous ways for people to gain their voice as a teacher leader. Tweeting is one way, short and sweet. Blogging is another. And TMC also gives some opportunities for honing a face-to-face voice (from the short and sweet my favorites, to the 30 or 60 minute sessions, to the six hour morning sessions). In a small community which attempts to be supportive and people try to raise each other up, it is easier to identify and amplify your voice, see that others find what you have to offer valuable, and gain confidence in yourself.

But we now have community members that have become recognized as teacher leaders outside of our MTBoS community. For a long time, there was Dan Meyer who fit this definition. But now there are a group of people who are in that category — MTBoS people who have kinda broken out of the MTBoS in terms of their impact on math education. Fawn Nguyen is always traveling to give talks! Julie Reulbach publishes a post and almost 10K people get it in their inboxes! John Stevens and Matt Vaudry published a book around their approach to the classroom that resonates with people! Heck, so many books are out now or coming out now by MTBoS people *cough* Christopher Danielson! Chris Shore! Edmund Harriss! Denis Sheeran! Wait, so many white men! In any case, I’ve compiled a list of all MTBoS books that I know about here.) [2] In fact, there are now a small group of MTBoS people who seem to be making the conference circuit giving keynotes. There are so many people who are now presenting or even keynoting regularly at NCTM and other conferences. And there are a lot of MTBoS folk who are on NCTM committees. (I don’t know if I’m in the minority or majority, but I see NCTM as acting as a pretty great partner to this online math teacher community.) And probably most exciting to me, I’m reading more and more tweets of people in the community taking on math coaching roles for districts, or becoming department chairs! Teachers who love math teaching leading math teachers! The MTBoS cultivates internal leaders who then can become leaders more broadly. A lot of that has happened in the past three years.

I have done a lot of thinking about the rise of the teacher leader, and what even a teacher leader is. These thoughts started in 2015, and became more solidified as I started talking with Julie Reulbach in 2018 about her TMC keynote address which is specifically about teacher leadership. (YOU SHOULD WATCH IT!) I came to the conclusion that I personally would define a teacher leader broadly… as someone whose work has a positive impact on kids outside of their own classroom.

So that includes sharing materials/lessons with other teachers, supporting other teachers when they’re down, writing a book that changes someone’s practice, running (non-sucky) PD which pushes participants forward in their thinking about teaching, creating spaces and structures for teachers to reflect about their practice (ahem, shameless plug for The Virtual Conference of Mathematical Flavors that I’m hosting right now!), tweeting out answers to questions people ask, writing blogpost reflections about your classroom that inspire others, whatever. As Julie said in her talk, teacher leadership is not a ladder you climb. (I think it used to be.) You don’t have to leave the classroom to have that broader impact. And that a tweet that hits someone at the right time might be way more impactful and useful than a book. [3] This isn’t something “new” but it is something that I think people in the MTBoS are starting to be aware of.

With the smaller things we’re doing (cheerleading each other, having book chats online, pushing people to think about diversity and equity in their classrooms, etc.), we used to think of it as just helping each other out. But there’s a reframing that is happening. We’re doing the same things we’ve always been doing, but people are starting to recognize that this work is the work of a teacher leader. That, “OMG, I am actually a teacher leader… wait, no way… I am not… but… maybe I kinda am a little bit?” It might slowly be entering some MTBoS folk’s understanding of who they are. And that’s pretty awesome. There may be nothing different in what people are doing, but I sense the beginnings of a shift in how people are interpreting what they are doing. (And acknowledging and really accepting that what they are doing has real value to people.)

It’s like — I don’t know — I see the pathways for teachers to become leaders as numerous, crisscrossing, and some pathways are only just beginning to show themselves. Maybe vines on a trellis is the image that I’m trying to go for? All I know is that there are now a zillion different ways for someone to have a positive impact on kids outside of their own classroom… and the existence of the internet and the online math teacher community is making that even easier. I predict that there will be some cool ways people figure out how to have an impact outside of their classrooms that we can’t even image. Who knows? Find ways to get involved with new (math) teacher training programs — or create resources to help new math teachers? Argh. I literally can’t come up with good ideas… but that’s precisely the point. Who could have predicted 5 years ago that there would be a math space in the Minnesota State Fair?! I suspect in 5 years there will be a number of different awesome pathways that people will carve out for themselves to become a teacher leader in a way that aligns with their passion/interests that I totally wouldn’t have anticipated today. Which means we’re at a cusp. We’re still trying to figure out what precisely teacher leadership can look like. We have a number of examples that have emerged. But we’re likely going to see a bunch more.

We have people who are in math ed and also part and parcel of our MTBoS community. Specifically, I’m thinking of Lani Horn and Tracy Zager. Their books Motivated and Becoming the Math Teacher You Wish You’d Had tout the work of people in the MTBoS do and outline pedagogy that most everyone in the community can rally behind. But also their work also engages deeply with the idea of identity and status. And importantly, although in the past I saw certain beams of light here and there emerge about the idea of identity and status in the classroom (in the vein of complex instruction), this kind of thinking is becoming much more normal in the conversations that are being had in the MTBoS. More and more, I think people (myself totally included) that the basis of an awesome classroom has to be rooted in consideration of these things. Thinking about “how to teach completing the square,” for example, isn’t just about scaffolding and conceptual development of the idea. Being able to do that well is also predicated on building an atmosphere in the classroom that the scaffolding and conceptual development can happen within. I think more and more people (myself included) are starting to recognize this. Like Lani and Tracy do, I predict more people in math education are going to start engaging with our community, as they will come to see us as a powerful resource, and we will engage with them if we see them as an ally in our classroom work.

In general, the MTBoS conversations aren’t entirely consumed with content or sharing resources anymore. It hasn’t been that for a while. Broader discussions about identity, status, diversity, equity, social justice are now more commonplace. Not everyone is engaging in these conversations, but they’re happening and I bet this is going to turn into another pathway that the MTBoS will be creating teacher leaders that are recognized outside of the community. And I’ve seen that these conversations of identity, status, diversity, equity, social justice have been turned back onto us. Not just things to consider for our classroom community, but also for us to consider as an online math teacher community. Right now there is a small but growing recognition that we can’t brush these conversations under the rug — for the sake of our students, and for the sake of our own community. 

I don’t quite know what to make of this… but there are a ton of #MTBoS classroom teachers that have left the classroom to work full time at Desmos or Illustrative Mathematics. I went to dinner with a group of IM people in Cleveland recently and there were only two people at the giant table who I wasn’t friends with from the #MTBoS. Similarly, I know a lot of the people working at Desmos (and I heart them all!). Although I don’t know what to make of this, what I can say is that these people are working at places I believe in, and that will have a large-scale positive impact on math education. And I probably can say that if it wasn’t for the MTBoS, many of them might not have this opportunity that they decided to take.


This post feels incomplete. I feel like I have so much more to think about and write. When rereading this post before I #pushsend, the ideas seem inchoate, and somewhat incoherent, to me. I can’t underline anything and say “THIS. THIS IS WHAT I’M TRYING TO GET AT.” Usually when I sit down to write something like this, all the blathering helps me refine my thinking and gets me to clarify what I really think. And at some point, I usually hit upon that thing that makes me feel excited and like I have found something TRUE and A REAL INSIGHT. But I am not leaving this post feeling any more lucid. No big insights emerged. So I suppose I leave this post as late night musings, as I started it as.


[1] If you don’t know what this is all about, but me being all cryptic has piqued your interest, you can get a flavor of it by reading these two posts: here and here.  This moment was important for me in one huge way. It highlighted how scaling up the online math teaching community without a central authority could lead to problems with intentionality/messaging/impact. How does a decentralized community with community-builders who are volunteers grow bigger with a coherent set of values and vision and positivity that are not only agreed upon but also enacted? Where intentionality in things that happen have a place? I don’t have answers.

[2] In 2015, I noticed there were “brands” (for lack of a better word). I still see that. The number of people with a narrow/specific idea/vision and a coherent voice that really resonates with people is growing. These people are gaining a lot of cache outside of the MTBoS. Often times, these “brands” are related to projects that were just gaining traction in 2015 (like Visual Patterns or Open Middle or Which One Doesn’t Belong or Clothesline Math or …). Conferences, books, at the MTBoS spread the word of these “brands.” And now from their rising popularity in 2015, they’ve become something bigger in math education. And with that, they carved out one pathway of “teacher leadership” that didn’t really exist before. But a number of people seem to be following it in some form. And that’s cool to see.

[3] But what I like about the MTBoS is that, taking my definition of teacher leader, it’s all a community of teachers who are trying to support each other. And so in the ways we all prop each other up, push each others’ thinking, share resources, etc., we all are having a positive impact on each others’ classrooms and each others’ kids. And so we all in small ways can consider ourselves teacher leaders. I love that it’s not a zero-sum game.

I wrote this and it doesn’t quite fit above, but I didn’t want to delete it. So I’m sticking it at the bottom of this post here. When I started teaching, I wanted to become a “master teacher.” That was my goal. Feel like I know I’m an expert in what I’m doing. I have since realized I will never consider myself a master teacher. That’s probably due to my own psychological makeup. But in the #MTBoS we have this idea that “everyone has valuable things to share.” And I truly believe that. But I also wonder if — now that I’m seeing so many ways people teach —  I don’t believe in the idea of a master teacher anymore? As opposed to my first few years of teaching,  I now don’t really think about it at all. Because over the years in the MTBoS I’ve see so many different ways people reach their kids, I recently realized that I don’t have a single ideal/standard to live up to. There are so many ways to be an awesome teacher. (Hence, my virtual conference of mathematical flavors was born. I wanted to showcase this diversity!) I just have to figure out who I am as a teacher as best as I can and just keep on trucking.


Senior Letter 2017-2018

Every year that I’ve been teaching, And at the end of each year, when I start growing wistful (but also a little bit glad to get them out of school because their second semester-ness starts to take over), I write them a letter which I give to them on the last day of classes. The letter usually always says the same thing, hits on similar themes, but I write it from the heart.

Sometimes I remember to post it on my blog, sometimes I don’t. This year (obvs) I remembered!

Even though they got to be a bit punchy at the end of the year, I’m kinda missing my seniors right now.


Part III: The Final Turn

I was in the middle of writing how I made the transition from Anti-Derivatives to Integrals, and then I got swept up with school things. So a month and a half later, at the start of my summer vacay (!), I’m going to write up the very last post in this installment, where I make The Final Turn.

As a recap: I taught anti-derivatives as “the backwards question” of derivatives, but didn’t give them any meaning other than a mathematical puzzle. And my first post (Part I) starts when I tell kids we’re going to put anti-derivatives aside for a while and we’re going to think about a road trip. In this, students recognize that velocity graphs can tell you about displacement, and we are forced to contend with left handed and right handed riemann sums. The conclusion is that “the displacement of an object can be calculated by estimating the area under a velocity graph.” Importantly, for kids, this has nothing to do with anti-derivatives. In my second post (Part II), students explore this idea in more detail. They are given a bunch of step functions for velocity and they come to recognize the difference between distance and displacement, and also why we need to specify a starting position to know an ending position. They also see the need for negative signed area to represent moving backwards. Again, this has nothing to do with anti-derivatives.

This is where our story begins. The final part of this tale, where we connect signed areas (which students are comfortable with) to anti-derivatives in such a way that the connection makes sense and is actually kinda obvious. That was my goal, anyway.

At the end of the same packet students were working on in Part II, I gave the following velocity and position graphs. Like my students, you can probably see that the velocity graphs are approximations of “nice” curves (e.g. quadratic, square root, linear, sine, exponential, cubic).


After being asked to identify some specific things on each graph (find what happens to the corresponding position piece when the velocity was greatest… find what happens to the corresponding position piece when the velocity was zero… etc.), I asked students to write six different things they noticed/observed. Kids came up with great observations! Here are some of them taken from their nightly work (note: “palley” is our class terminology for peak or valley):

  • Zero on the velocity graph yields a slope of 0 on the position graph (the object didn’t move for that time period)
  • Most positive value on the velocity graph yields the largest positive slope on the position graph
  • Most positive value on the velocity graph corresponded with the largest change in position (from low to high) on the position graph
  • The position graph usually has one more palley than the velocity graph (when we’re dealing with polynomial velocity graphs)
  • When the velocity graph is negative, the position graph is decreasing
  • When the velocity graph is positive, the position graph is increasing
  • For the 4th graph, the velocity graph looks like a sine graph and the position graph looks like a cosine graph
  • Many of the relationships between graphs look similar to derivatives we’ve been studying
  • At the point of highest velocity, the position graph usually has a palley
  • The area under the velocity graph gives the height of the position graph
  • Velocity graph is the derivative of the position graph

Ooooh! I just found the smartboard page where we codified these in one class:


To debrief this, I had kids call out their observations and we talked about them/refined them using the graphs I was projecting at the front of the room. The most important thing I remember doing was making sure we started with “simple” observations and only then had the class move to more “complex” observations. That way kids who only noticed basic things would have a chance to voice their thinking.

From this we finally got to the conclusion we were looking for: the velocity graph is the derivative of the position graph. I had to nudge kids to make the counter-observation they get for free: the position graph is the anti-derivative of the velocity graph.

We saw all the things we had previously learned when studying the shape of a graph come out (e.g. when the derivative is negative, the original function is decreasing; when the derivative was zero, the original function often had a “palley”). Everything fit together!

We then carefully looked at small pieces of one of the graphs and discussed what we saw going from left to right, and also going from right to left.


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From left to right: we saw that the signed area under the curve on the left corresponded to the change in position on the right.

From right to left: we saw that the slopiness of the curve on the right corresponded with the value/height of the function on the left.

So we codified this. But importantly, we finally got to say “antiderivative” but with meaning. HERE’S THE KEY TURN: We knew that the derivative of position was velocity. That means by definition, the anti-derivative of velocity was position (ok, technically displacement, but let’s not cut hairs right now). But we already knew from our work previously that to get from velocity to position, we had to find signed areas. THUS, accumulated signed areas and anti-derivatives were one and the same!!!

In one class, we outlined this as such.


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In my other class, I had us codify things more formally.


Both ways of codifying this knowledge seemed to make sense for kids. And to be perfectly honest, the fact that antiderivatives were accumulated signed areas was always a bit magical to me. I know I truly understand something conceptually when something magical/mysterious changes and becomes a “well, duh, of course it has to be that way.” I don’t think in all my years of doing and teaching calculus that I had ever totally gotten rid of that feeling of magic/mystery until this year, when I really changed how I approached this with my kids.

Now I know you might be saying: but Sam, you never PROVED this relationship, that an antiderivative of a function and the signed area under that function were the same. Agreed. I wanted to have kids see and make sense of this relationship first. But I did do a basic proof of this (that the antiderivative of a function is the same as the accumulated signed area of that function) soon after. If you’re dying to know what came after, I had kids use a Riemann Sum applet (I slightly modified this one to not have the integral symbol on it or the “show actual area” button on it) to fill this sheet out [docx version: 2018-05-03 Riemann Sums]:

And at the very end of this packet is our “proof”:



Now this page/proof should have appeared super familiar to my kids. We actually had made this exact same argument earlier when seeing the derivative of the volume of the sphere is the surface area of the sphere, and when seeing the derivative of the area of a circle is the circumference of the circle (see post here). Although this proof did require us to work together as a class with a bit more handholding than I would have liked, it went pretty well.

And with that, I can put this three part beast to bed. If nothing else, this series just reminds me how much I really think about how students build and construct knowledge. It is something that I think about so naturally when designing new materials, but to explain all that thinking and how it unfolds in the classroom takes way more words than I anticpated! No more three part blog post series for a while!


Polar. Graph. Contest.

Here’s what I hung up last week:


Here’s a closeup of some of them…



These are polar graphs that students designed using Desmos. Then I printed them out on photopaper and hung them up.

This was something I wanted to do after introducing polar graphing. Why? Because one day during the polar unit, I started playing around with desmos and accidentally created:


… from something so simple …


(Now to be fair, desmos isn’t great with creating great complicated polar graphs… and it’s better to write them parametrically to get a bit more accuracy… so this is a bit of a lie of a graph in that it isn’t totally accurate… but it’s oh so pretty.) [1]

So after our unit on polar graphing, I took 10 minutes at the start of class to introduce this idea of a Polar. Graph. Contest. First I threw this image up:


I then pulled up desmos and asked my kids to shout out some polar function. I graphed it. Then I put in a slider or two. So for example, if they said r=\cos(\theta), I might have added the slider r=\cos(a\theta)+b. And then I started changing the sliders. Then I might have altered the function a bit more, like r=\cos(a\theta)+b\theta and we saw what happened. Then I gave everyone 7 minutes to just come up with something pretty.

It was magical.

Kids just started playing. They dug into old functions they had learned about. They got excited by what they were seeing. They gasped and turned their screens to show their friends. Some who were getting boring graphs saw the cool graphs their classmates were getting and were inspired to mix things up since they knew they could make neat things. #mathjoy in the house.

My heart was singing.

Then I showed kids a google doc which had all the info for the contest — and the link to the google form to submit their entries. There were initially two contests. Students needed to create the coolest polar graph with one equation. And students needed to create the coolest polar art using multiple equations. However, some students were animating the sliders and coming up with fun animations (like this or this… watch both for a while). So I added an optional third animated polar graph category.

I haven’t yet told my kids who the winners are. I want to just let them appreciate the work of their classmates for now.

After creating the bulletin board, I’ve seen kids look at the artwork. Kids from my class, but also kids from other grades. And what I’ve found fascinating is that so far, very few kids pick the same polar art pieces as their favorites. I expected everyone to love the same ones I do. But it just isn’t the case. I think when I announce the winners, I’ll have the class go to the board, have everyone point out a few that they like, and then I’ll make my grand pronouncement.

Student Feedback:

I asked my kids, when submitting their artwork, “This is something new I came up with this year. I want to know if you enjoyed doing it or not. No judgments if you didn’t. Y’all tend to be honest when I ask for feedback, and I appreciate it! I genuinely want to know. I also am a bit curious if you had any mathematical thought as you were playing on Desmos? You don’t have to say what thoughts you had (if any) — just if you had any.”

Every student responded positively. Some responses included:

  • I think this was so awesome! I love art and this felt like art to me. It is so fun when art and math intersect, I loved it!!!!!
  • Messing around with the graphs was actually more entertaining than I thought it would be. I spent a lot more time on this than I thought I would, and I feel like I’ll probably spend more time on this trying to find a really cool design I like (and possibly gaining a better understanding why the graphs look the way they do…).
  • I had so much more fun doing this than I thought I would, honestly. Once I finished my multi-equation graph, I looked at the clock on my computer and realized I had been working on it for nearly 20 minutes; it had seemed like maybe 5.
  • I really enjoyed doing this assignment. I felt that I learned a lot about polar through it. I didn’t think too much about math while making my graphs, however I thought about math a lot in order to observe and think about patterns I found in my graphs.

Now I want to be frank: there isn’t much “learning” that happens when kids are doing this assignment. This isn’t a way to teach polar. But it is a way to get kids to appreciate the power of polar when they are done working with polar, and what sorts of different kinds of graphs compared to the boring ol’ rectangular coordinate system. I just wanted kids to play, like I played, and get excited, like I got excited. It’s a slightly different way to appreciate the power of math, and I am good for that. Especially since it only took 10 minutes of classtime!


As an aside, I love that when I tweeted this out, a tweep said he was going to be doing this in his class after his kids learn about circles. Um, hell yeah!



[1] So there are two ways to graph polar in desmos. First is the straight up polar way, and the second is the parametric way. It turns out that the polar way is solid for most things, but it loses refinement at times. Let me show an example. If we graphed r=\cos(57\theta), we should get a flower with 57 petals.

And happily, if we graphed it in both polar and parametric, we get the same looking graph:


However if we zoom in a bunch, we can see that the red graph (the polar equation) is interesting and stunning, but just isn’t correct. While the zoomed in blue graph (the parametric equation) is more boring, but is technically correct.


It turns out desmos samples more points using parametrics than polar.

As a result, a few of the polar artworks my kids made aren’t “true.” Their pieces are a desmos quirk, like the red graph is above. But what a lovely desmos quirk.

Part II: Transition from Anti-Derivatives to Integrals

In my last post, I outlined the road trip scenario I used to prime kids to think about areas under curves. It had nothing to do with anti-derivatives. And that’s important to keep in mind. This post is going to try to outline how we made the intellectual leap from areas under curves to anti-derivatives. [Update: I wrote this during 40 minutes of free time I had in school today where I didn’t want to do other things I was supposed to be doing. So I didn’t get to writing the part of the lesson where we make The Leap. This post ends where we’re literally all primed to make the leap. But indeed! I will make the leap with you! But in Part III. Which will be written. Soon. I hope.]

The road trip introduced this idea that kids can approximate how far someone traveled using a left-right-midpoint Riemann Sum approximation (we did not give it that name…). It arose naturally from the roadtrip scenario.


We also made the conclusion that if we had more data, we could get a better approximation for how far someone traveled. To remind you, we started with this data:


and then we got more data:


That’s going to be our transition. We are now going to give infinite velocity data! [1]


Wonderfully, kids had no problem with doing this. The reason I highlighted question 1c is because I was very intentional about including that question. When students graphed 1e, they often would draw:


I didn’t correct anyone while they were working. And it was nice to hear a few groups have the requisite conversation about why we needed to connect the points. Afterwards, when we debriefed as a whole, this was something I highlighted. We knew the position at every moment in time, including at t=2.31 (as asked in 1c).

Kids continued on with Questions 2, 3, and 4. They flew through these, actually.


These were golden. Let me say that again: these were golden. [2] It was amazing to watch kids:

  • Parse the connection between velocity graphs and position graphs
  • Understand the idea of negative velocity
  • Think about the fact that we have to specify an initial position in order to create a position graph
  • Draw a connection between motion on a number line and a graph of position v. time
  • Understand what distance and displacement are, and see the difference between the two

Seriously, just watching kids work through these problems was… well, I’ll just say this feels like something I’m super proud of creating. It didn’t take much time to do but gave us so much fodder.

We didn’t need a lot of time to debrief these four questions. I had students highlight a few things, and I made sure we brought up the fact that we were drawing line segments for the position graphs, and not something curvy. Because constant velocity means position is changing at a constant rate, it’s linear. So for example, the position graph for 3c would look like what I have below. It isn’t a parabola.


But there was one huge thing we had to go over. With the roadtrip, we drew a connection between area and how far someone went. Most kids, as they were doing these problems, didn’t think about area. I wanted kids to think about area. So in our debrief, I explicitly asked them what our huge insight from the roadtrip was (area as distance travelled!), and if we could apply it to one of the problems.

So first I went back to question 2c. And I asked students how they calculated their answer:

pic6.pngKids said she went backwards a total of 24 units. So they did 100-24. And then I explicitly had them draw the connection to what we did with the roadtrip. This is when we talked about it being area, but “signed” to represent the backwardsness.


To be clear, some students had already been thinking in this way (about area/signed area) when working on these problems. But most hadn’t been, so we had to bring that idea to the forefront.

Then I had a student talk through Question 3 with areas in mind:


And finally, I asked groups to discuss how we could understand distance and displacement from the velocity graph.



There is one more part to this packet I had my kids work on that I will outline in my next blogpost in the series! But here’s an editable .docx of the file I made [2018-05-02 Velocity Graphs]. And here’s the document to view here:

Stay tuned for Part III.


[1] Both @calcdave and I stumbled upon the same approach for this!

[2] The only note is that a few students didn’t realize the time interval was 1/2 hour for Question 4. And it involves a fair amount of calculation.