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

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

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

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

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

Now on the left side we have this:

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

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

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

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

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

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

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

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

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

And that… is about it!

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

*Vision: To create an unstructured public space where kids can relax and fiddle/tinker around with fun math things that might not be related to things in the formal curriculum. The hope is that this allows for the experiencing of math as something casual and playful. We want this space to encourage students to want to talk mathematically with each other.
Through this space, which will be curated and changed periodically, we want to widen the umbrella of what gets counted as “math” and “doing math,” and who gets to be counted as a mathematician. *

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

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

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

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

With that, I’m out!

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If you’d like to read my post and my competitor’s post and vote, I’d appreciate it:

The Big Internet Math-Off: The final – Sameer Shah vs Sophie Carr

It will only take a short time (no need to login or anything to vote, the only time it will really take is the reading).

My mathematical tidbit today attempts to have you look at these two squares, a 17×17 colorful square and a 127×127 greyscale square.

Both are… slightly uninteresting.

My goal, through the post, is to show you that both of these squares are *insanely *interesting. I call them the most beautiful 17×17 and 127×127 squares ever. And my conclusion: once you learn about the mathematics embedded in these squares, you’ll never look at them the same way again. You can’t.

It’s like having a huge a-ha moment when learning something. It completely transforms the way you look at something, so you can’t see it in its original form again.

I hope you enjoy!

If you want to see my five entries into the Big Internet Math Off 2019:

Entry 1: a counfounding conundrum: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-2-jorge-nuno-silva-vs-sameer-shah/

Entry 2: a card trick: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-2-vincent-pantaloni-vs-sameer-shah/

Entry 3: a magical property of circles: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-1-marianne-and-rachel-vs-sameer-shah/

Entry 4: an unexpected break in a mathematical pattern: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-semi-final-1-lucy-rycroft-smith-vs-sameer-shah/

Entry 5 (the one outlined in this post): two beautiful squares: https://aperiodical.com/2019/07/the-big-internet-math-off-the-final-sameer-shah-vs-sophie-carr/

]]>I taught Advanced Geometry at my school for two years (2014-2016), and I wrote the curriculum with a good friend and dear colleague. We both hadn’t taught geometry before and decided we’d do a super deep dive and come up with a sequencing that made sense to us, and that prioritized conjecturing and noticing. In fact, we were so excited by this process that we shared our thinking both about how we built up the curriculum but also how we collaborated at a conference. Below are our slides, but you can also click here and go to the slideshow and read some of our presenter notes for each slide for more detail.

We were super intentional about everything. We carefully thought through how we wanted to motivate everything, and we didn’t want to give anything away throughout the course. In other words, we wanted kids to do all the heavy lifting and to be the mathematicians that we knew they could be.

Below is a word document with all our skills/topics (you can download the .docx file here: All Topic Lists Combined). The order might seem a little strange (we end, for example, the year with triangle congruence), but it worked for us! Everything was done on purpose (in this case, congruence is just a special case of similarity… so that came beforehand, along with trig which is all about exploiting similarity!). We eschewed two-column proofs for different forms (paragraph proofs, flowchart proofs, and anything else that showed logical reasoning).

Oh wait! For some reason our work on Area and Volume didn’t have a topic list. And I just looked and my core packet for Area and Volume derivations (where kids just figure things out on their own) has handdrawn images in it, but I didn’t scan a PDF of those. Well, at some point in the future if I remember, I’ll try to write a post to share that. (We did it after kids learned trigonometry, so they had a lot of flexibility. For example, I think kids came up with like 6 different methods to find the area of a trapezoid when they were asked to create a formula and justify it!)

I hope this is helpful for anyone trying to think through geometry. As I said before, the best thing might be to just read the blogposts, but this is a bit of an overview.

]]>One of my students this year told me I could put her senior speech on my blog. So here it is, for posterity.

**How I Have Changed the Way I Have Thought about Myself and the World**

When I was a lot younger, I was more hyperactive than I am now and I was always doing something to take up my time. I mainly always liked to keep myself out of my house because then I wouldn’t have to be in the middle of what was going on inside.

There was a lot of fighting that happened between my parents. They’d get physical with each other or throw things at each other. They never seemed to think of my brother and I being there.

I realize now that I was fearful of what was going on, that it put a lot of stress on me. When I was younger my thoughts were not so developed. When I would be with the rest of my family, my brother would be there to protect me. He could not prevent me from seeing or hearing certain things, but he would take my mind off of it the best he could. I don’t know what I figured the behavior between my parents meant or was- but if they were loving to me, I forgot about it and moved on. This was the way life went for me for many years. It was a contorted way of thinking.

At this time I also felt that the person I was defined by was my home life and my immediate family. I felt this way yet I didn’t think moments when my parents were violent towards each other or embarrassing my brother and I were moments that meant anything. I was always taught to let it pass and start new. That way of thinking ran me into trouble. I took what was happening in my home outside of it and I would have unkind thoughts/ engage in unkind behavior towards people I loved and cared for. Afterwards, whether I identified if it was mean or not, I would move on. I didn’t think it said anything about who I was.

Then in seventh grade I was in Ms. Oster’s health class and she showed us this speech by an author named George Saunders. It’s titled Advice to Graduates, and it is centered around the questions why aren’t we kinder? and how do we become more kind? It touched on the idea that becoming kinder comes with age but there are ways you can speed up the process. I didn’t feel like It wasn’t as if I hadn’t heard these words before, or as though this was a new concept to me but they processed differently this time and I began to question who I was.

I realized that I had always thought I was a really nice person but if I wasn’t genuinely acting kind, meaning my actions weren’t speaking for themselves, how could I be a nice person? This caused me to think back to things I had done and ask myself a series of questions like: Did I do the right thing? What did I feel that made me act the way I did? What caused that feeling to arise? Why could that feeling have been brought up?

Seeing that my actions didn’t reflect who I knew I was, I continued in life being very analytical of instinctual tendencies, thoughts, and behaviors. This way of questioning took over and continues to this day. Questioning myself allowed for me to realize the roots of my actions so I can allow myself to change. If I can identify the root, then maybe I can change the way I do things.

And so, through this process of questioning, I came across the concept of time and how I spend my time. I thought to myself what I spend my time doing makes me who I am. I thought, what had I been spending my time doing? School came to mind. I go to school most days out of the year and I am in school most hours out of each of those days. Again, it wasn’t that I didn’t know these facts before this moment but I had never paid attention to the concept. When I began to pay attention I realized that school was a place that nurtured me in all positive ways and I was taking it for granted. I could look to the building as a safe place, the teachers as people who could listen and give me advice and, my peers as those who stand equal to me in our journey to understand and live with ourselves. I am very grateful that I am able to attend school, have teachers who care for their students, students who engage deeply in class conversation, and a place that I can actually call my home.

At this point in my life I feel I have come a long way. Since 7th grade I have realized that a lot of what I was seeing when I was younger was my mom struggling with her mental health. In 8th grade child services took me from living at my mom’s house. They had been in and out of my life until this past summer when I permanently started living with my dad. It has been difficult accepting that my mom was not able to guide me through big developmental parts of my life but I love my mother and I am so thankful she is the woman I call my mom. Now here’s a George Saunders quote that is important to me.

“If we’re going to become kinder, that process has to include taking ourselves seriously — as doers, as accomplishers, as dreamers. We *have* to do that, to be our best selves.”

Now also, Over these years of high school I have come to understand that change is inevitable and a big part of our job is to adapt, learn, and accept it. Through that process I am continuously discovering my best self.

]]>I know the irony of being scared of public speaking and being a teacher, but I know a lot of other teachers also feel this. But my biggest fear was just not being good enough. I wrote to Mary:

*The truth is I can’t help feeling like this might be too big a leap for me. I don’t know if I could do what Fawn or Julie to me when I hear them talk, or do what Chris Shore or John Stevens do when they present. And I don’t want to commit unless I knew I wouldn’t be wasting anyone’s time. *

She replied:

*Firstly, and most importantly, you would not be wasting anyone’s time.*

*I am a big proponent of elevating classroom teachers and giving them a voice and I hope that you will find yours for this event. *

As someone who feels like an evangelist of the online math teacher community, I’m always saying to people hesitant to dip their toes into the water that their voices and perspectives are important and valuable. And when I say this, I mean it with every fiber of my being. So why was I doubting the value of my own voice? I agreed to do the talk as long as I could do it with a collaborator and friend. It was a 75 minute talk (and an associated 75 minute workshop) and planning that individually seemed so not fun. But I thought working collaboratively would be so much more intellectually fun! So I dragged my friend Mattie Baker (@stoodle) into the presentation. We brainstormed for ages, but in the end, finally decided that the idea of “The Teacher Voice” was exactly what we needed to talk about.

It’s now over, so I’ll start with the ending. The lecture hall of 200 people for our talk was almost full. And the 75 minute talk went fabulously. I had to save some tweets for posterity.

And OMG, we got a standing ovation. That was unexpected. And people were crying. That too was unexpected. I am not someone who feels proud about things easily. I usually focus on all that went wrong or ways I could have been better. But when we took a bow at the end and people stood up, *my heart was bursting*. All the work Mattie and I put into the talk for the previous 10 months, the weekends we sacrificed to write and practice and edit felt purposeful because at least for some teachers in the audience, our message was at least temporarily valuable. Weirdly my fear of public speaking disappeared the day of the talk after we had a solid rehearsal the previous day, and my fear of wasting peoples’ time disappeared after the talk ended and people came up to say such nice things to us.

The talk was broken into two parts. First, Mattie and I shared something we each did in our classrooms that was inspired by other teachers, and then adopted by other teachers. We wanted the audience to have something concrete to walk away with in case the rest of our talk didn’t resonate with them. We were breaking down the silos of our classrooms. Second, we each talked about the emotional life of a teacher. We wanted to break down the silos of our emotional worlds. There were so many messages we included in this part of the talk. Mattie shared his first year in teaching, which he previously shared on the Story Collider podcast. But here is one takeaway from my section of the talk:

Teaching is hard. We are going to *feel* bad. We’re going to *be* bad. And that’s okay. It’s okay to not love what you’re doing all the time. I’ve never met a teacher who is putting themselves out there in the important but hard ways who does. But we can be brought closer as we become vulnerable and share these things and realize we aren’t all alone in this.

And a second related takeaway:

Often times, we’re so critical about ourselves, we think of all that isn’t going right, all that we aren’t doing… that we lose sight of* all that we are*…. It’s so easy to be critical of yourself, to set the bar high, to see all the ways you’re not succeeding.

You see yourself in one way. But the reality of the situation is: ** We aren’t really all that good at seeing ourselves.** That’s my big realization, and it only took twelve years. When we’re down and think we suck, yeah, we probably definitely maybe can be doing better. But hell if we aren’t already doing good, and we need to acknowledge that and spread it. We need to believe our friends when they tell us that our ideas our good, that something we did was good… we need to believe our kids when they say something spontaneous and positive about something happening in the classroom… and… we need to be sharing the good and the positive that we see in others. We need to help others see how important they are to you. We need to give cupcakes, send the random email, prop each other up, and help others see how they make your life better.

The talk focused on the hard times in teaching, and what we do when we hit them. At one point I asked the audience to share their coping strategies at the low points. I promised I would share them online, so here are what the audience typed. It’s amazing how similar the responses are…

(All the references to “coffee” in quotations comes from part of Mattie’s talk. You can interpret that to be getting a drink at happy hour.)

I shared my coping strategies afterward, and so many of them were covered by what people in the audience typed! Except for those people who talked about exercise and running and the gym and other evil things like that. Some of my favorites!

In the talk, I also shared the* Explore Math* project that I do with students. The website that I created for the project is here: https://explore-math.weebly.com/

I posted about it early on when I first started it, but haven’t done any additional posts on how I’ve changed it or how it’s evolved or what I’ve noticed when doing it with different grades (sorry, I should). The posts are here, here, and here. The most important thing I can suggest is that you need to adapt it to work for your kids and your school. For example, this year I tried this in 10th grade and it wasn’t as successful overall because I think the kids needed more structure and hand-holding. So I’m going to take that into account for next year.

Two teachers shared their experiences with the project, which I couldn’t fit into the talk. So I’m posting them here in case it entices you to do the project or some variation in your classroom.

At the conference, Mattie and I also gave a 75 minute workshop on the online math teacher community designed for people who were interested in joining in but didn’t know how (our slides for that are here).

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- The core part of what I did to get the number to pop up was to use @lukeselfwalker’s Desmos activity. I like it for so many reasons, but I’ll list a few here. It starts by “building up” a more and more complicated polynomial of the form , but in a super concrete way so kids can see the polynomial for different
*n*-values. It shows why the x-intercept travels more and more left as you increase*n*, so when you finally (in the class discussion) talk about what happens when*n*goes to infinity, you can have kids understand this is how to “build” a horizontal asymptote. It gets kid saying trying to articulate sentences like “this number is increasing, but slower and slower” (when talking about the value of the polynomial when . And they see how this polynomial gets to look more and more like an exponential function as you increase the value of*n*. If you want to introduce*e*, this is one fantastic way to do it. - A few days later, I had everyone put their stuff down and take only a calculator with them. They paired up. (If someone didn’t have a pair, it would be fine… they just sit out the first round.) On the count of three, both people say a number between 0 and 5. (I reinforce the number doesn’t have to be an integer, so it can be 4.5 or something.)Then using their calculators, they calculate their score: they take their number and raise it to their competitor’s number. The winner has the higher number. (If it’s a tie, they go again until there is a winner.)
Then the loser is done. They “tag” along with the winner and cheer them on as they find another winner to play. This goes on. By the end, you have the class divided into two groups each cheering on one person. (I learned this game this year as an ice breaker for a large group… it’s awesome. This is the best youtube video I could find showing it.)

Finally there is a class winner.

So I then went up against them.

And when we both said our numbers, I said:

*e*.The class groans, realizing it was all a trick and I was going to win. We did the calculations. I obviously won.

We sit down and I show them on my laptop how this works:

The red graph is my score, for any student number chosen ().

The blue graph is the student score, for any student number chosen ().Clearly I will always win, except for if my opponent picks

*e*.I tell kids they can win money off of their parents by playing this game for quarters, losing a few times, and then doing a triple or nothing contest where they then play 2.718. WINNER WINNER CHICKEN DINNER!

- After this, I show kids these additionally cool things (from the blogpost), saying I just learned them and don’t know why they work (yet), but that’s what makes them so intriguing to me! And more importantly, they all seem to have nothing to do with one another, but
*e*pops up in all of them!I re-emphasize

*e*is a number like and I showed them this to explain that it pops up in all these places in math that seem to have nothing to do with that polynomial we saw. And that even though we don’t have time to explore*e*in depth, that I wanted them to get a glimpse of why it was important enough to have a mathematical constant for it, and why their calculators have built in*e*and*ln.*

That is all. I honestly really just wrote this just because I was excited by the “game” I made out of one of the properties of *e *and wanted to archive it so I would remember it. (And in case someone out there in the blogoversesphere might want to try it.)

**UPDATE: **Coconspirator in math teaching at my school, Tom James (blogs here) created the checkerboard experiment using some code. You can access the code/alter the code here. The darker the square, the more times the number for the square has been called by the random number generator. And with some updates, you can make more squares! In the future, we can give this to kids and have them figure out an approximation for *e*.

[1] And introducing it with compound interest means you have to assume 100% interest compounded continuously. Where are you going to get 100% interest?!?!

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Desmos writes interesting job descriptions when they have openings. When someone pointed that out to them, they mentioned that this article on reducing unconscious bias helped informed how they write their job descriptions. It’s pretty great and I highly recommend it if you’re hiring. I have thought a lot about “fit” in the past few years when doing hiring, but it’s tricky to think about it well. I have come to recognize that someone entering our department needs to be open and willing to collaborate and compromise, but also have sympathetic pedagogical beliefs with what our department values (and can’t compromise on those). One way I have tried to avoid it is thinking about these things:

But also I have found it harder to balance these thoughts, which I admittedly have a lot:

Not quite those things, but similar thoughts that get at my own personal views on the what persona/personality traits make an effective teacher. Which I tend to think mirror my own traits. But that’s only because I have these traits because I think they make an effective teacher. But I have worked with enough amazing teachers to know that amazing teachers come in all personas! Just like amazing students don’t all have to have the same personas. But this type of bias is something I am trying to be super cognizant about when on hiring committees.

***

I saved this just because I like the question and wanted to work on it. And I can see all kinds of extensions. A formula for *n* circles? What about spheres? I’m guessing (without working on this problem yet) that this is a classic “low entry point, high ceiling” type problem.

***

I just really liked this quotation, and I need to think about the ways that students can see themselves in the mathematics they do. It is part of a larger thing I want to do which is “humanize math” — but I’m not very good at making it a core part of what I do in the classroom. Small bits here and there humanize and expand what kids think about math, but I’m not there yet. I want to one year leave the classroom and know that kids have looked in the mirror and saw something. (It kind of reminds me in a super literal way of how Elissa Miller put a mirror in her classroom, and I think on the bottom she wrote “mathematician.”)

***

Okay, I love this so much. If you’ve never seen it before, it a great trick. You have someone pick any number between 1 and 63 secretly. They just point to the cards that number is on. In about three seconds, I can tell you your number.

I actually made a set of these cards where the numbers are more jumbled up, so kids don’t see a pattern to it. I do put the powers of 2 in one of the four corners though to make things easier for me. Oh wait, have I said too much?

If you don’t know this trick, or how or why it works, I’m sure you can google it. But I’m going to recommend the awesome book “Math Girls Talk About Integers” (there are a lot of great “Math Girls” books out there, so make sure you get the Integer one.

Not only is the book awesome (and great for kids to read), but it breaks down this trick so well. *Shivers with joy*

***

I was excited with Karen Uhlenbeck won this year’s Abel Prize, the first woman to win it ever! I had my kids read this article in the NYTimes about it, and write down three notes about the article. We started the next class with a “popcorn sharing” of what people wrote down. (I also said that although I liked the article, it was a bit dense and thought it could have been written more lucidly.) One thing that came up in both classes I did this in was what a “minimal surface” was — so I told kids it is a surface with minimal area.

I then showed my kids this short youtube video:

And explained that bubbles, though not “central” to all higher level mathematics, do come up. And then I gave them a question. I’m too lazy to type it out, but watch the first 1 minute and 45 seconds of this video (https://www.youtube.com/watch?v=dAyDi1aa40E) and you’ll see it. Then we talked about some basic solutions. And THEN I revealed the best answer was the answer shown in the video we all watched together.

Of course @toddf9 (Todd Feitelson) used this as inspiration to create his own bubble thingies:

but he also explained how he made them…

and then he EVEN created an awesome desmos activity on this very problem, which I want to archive here for use later: https://teacher.desmos.com/activitybuilder/custom/5cb50bed4dcd045435210d29

(Oh! And Mike Lawler (@mikeandallie) made a mobius strip bubble!)

***

Dylan Kane wrote a nice blogpost about calling on students (and the “popsicle sticks of destiny” — though he doesn’t call them that). My favorite line is this simple question that isn’t about right or wrong:

- After students attempt a problem in groups, or reflect on an idea and share with partners, I call on students asking, “How did your group approach the problem?” or “What is something useful that you or your partner shared?”

It’s so obvious, but even after so many years of teaching, I forget to ask things like this. Or my curriculum isn’t group problem solving based enough for things like this to make sense asking. Or whatever.

***

There’s nothing special about this one… I’ve read it a few places before and it always makes me laugh.

Questions are good. I might have a kid read this at the start of the year and then have a short conversation about why we’re reading it.

It will get at the problematic idea of “obvious,” and when and how learning happens and more importantly when and how learning doesn’t happen.

***

In case you didn’t know, Desmos has a list of all their mathematicians they use when they anonymize in Activity Builder.

https://docs.google.com/document/d/1OY-8dk6vYW1Cags8E6_v3I8YZ-RYROzgsCauW5CZt9w/edit

***

I can imagine putting this picture on a geometry test as a bonus question and asking them why it makes math teachers all angsty… Plus it made me chuckle!

***

I’m so not here yet. Anyone who knows me as a teacher will probably know I’ll probably never get here. I’m such a stickler for making the use of every second of classtime.

***

Crystal Lancour (@lancour28) tweeted out a slide from a session led by Robert Berry (NCTM president) which had this very powerful slide:

Four rights of the learner in the mathematics classroom

- The right to be confused and to share their confusions with each other and the teacher
- The right to claim a mistake
- The right to speak, listen, and be heard
- The right to write, do, and represent only what makes sense to you

***

Love the idea of using marbles/paint to draw parabolas (click here to go to the original tweet and watch the video — it’s not a static picture).

***

Bree Pickford-Murray (@btwnthenumbers) gave a talk at NCTM about a team-taught math and humanities course called “Math and Democracy.” Not only did she share her slides (like *right after* the talk) but also she links to her entire curriculum in a google folder. SUPERSTAR!!!

I’ve gone to a few talks about math and gerrymandering (both at MoMATH and NYU) and listened to a number of supreme court oral arguments on these cases. It’s fascinating!

***

I just finished teaching “shape of a graph” in calculus. But I wish I had developed some activities like this, to make it interactive:

***

I’ve literally been preparing to give a talk next month for… months now. And this one stupid tweet summarized the talk. Thanks.

***

I have so many more things I can post, but I’m now tired. So this will be the end.

]]>It’s basically a number line, that’s all. But it’s a nice public giant number line which can get kids talking. Today I came back from spring break and before break, students learned about logarithms. However I wanted to have them recall what precisely logarithms were… so I created a quick Clothesline Math activity.

I hung a string in the classroom. I highlighted it in yellow because you can’t really see it in the photo…

I then showed them this slide – explaining the string is a number line…

I then showed them this slide, which explains what they have to do if they get two of the same number. (I brought cute little clothespins, but mini binder clips or paperclips would have worked just as well):

And then I gave them the rules of play:

I handed out the cards and let kids go. It was nice to see they didn’t get tripped up as a class on too many of them, but I got to listen to debates over a few trickier ones, which we collectively resolved at the end.

Here are the cards I handed out: .DOC FORM: 2019-04-01 Clothesline Math – Logarithms

Here is a picture of some of the cards. The two on the left are average level of difficulty. The two in the middle caused my kids to pause… it took them time to think things through (they haven’t learned any log properties yet). The one on the right doesn’t belong on the number one (it is undefined) and the kid who got that card immediately knew that. Huzzah!

Here’s a picture of the numberline at the end.

And… that’s it!

I was excited to try it out as a quick review activity. And it worked perfectly for that!

(Other things of note: Mary Bourassa made a clothesline math for log properties and shares that here. The author of Give Me A Sine blog does something similar here, but has *kids* create the cards. I couldn’t find anything with basic log expressions — so I made ’em and am sharing them in this post. Chris Hunter has a nice tarsia puzzle that sticks with basic log expressions here, but I wanted to try out clothesline math so I didn’t use that!) But if anyone has others out there involving logs, I’d love to see them in the comments!)

Last week I met up with my co-teacher in Algebra 2. We’re working on our unit of exponential functions and logarithms, and we were talking about spending a short amount of time introducing “e” to our kids. Personally, this question has haunted me because when I taught Algebra 2 at the start of my career, I couldn’t ever find a motivation for it — except for interest being compounded continuously. That never quite sat well with me because you have to assume that you have an interest rate of 100%.

Like as gets larger and larger.

But where are you getting an interest rate of 100%?!? It isn’t a terrible way to introduce “e” (getting kids to understand the structure of that equation above, there’s a lot of deep thinking that goes on in there). There is also the idea that there is a limiting value for that expression above — instead of the value just going up infinitely — that can be exploited and discussed.

But I never thought “e” and “ln” really belonged in Algebra 2, precisely because I couldn’t motivate them in a way that was intellectually satisfactory. So I tweeted out:

Little did I know I was going to get so many responses! I wanted to archive them here, which is why I’m writing this post, but then share where I’m landing on this whole “e” thing right now.

**Ways To Introduce “e”**

- @retaneri linked to this question from “Play With Your Math” I
*love*that site, but hadn’t worked on this problem before.

Interestingly, @bowmanimal and @averypickford came up with the same problem to share with me! Apparently the answer is to break up the number into a bunch of*e*s … I haven’t figured out why this works yet. But for 25, if I understand this correctly, I think it means which has a product of . (Or to be super precise: .)

And @benjamindickman shared some articles about this: - @mikeandallie shared with me this approach by throwing darts. Which includes this gem:

and this instantiation:

- @jensilvermath suggested just looking at for larger and larger , without reference to a limit. Have kids make predictions about what is going to happen and why, and then let them explore it. At first, I was like “hmmm, would this work?” but I love the idea of kids stumbling upon and wrestling with (1+almost zero number)^(super huge number) might be tricky. Does it have to be a huge result? What data could they collect? What would they do with it?
And @LukeSelfwalker shared this simple but stunning Desmos activity which gets kids to see how polynomials can start approximating exponential functions — a beautiful visual connection to all of this.

- Of course @dandersod showed me a connection between Pascal’s Triangle and e, which I didn’t know about (or if I did, I totally forgot). He sent me this link:

Whaaaa?! But okay! WHOA?! - @roughlynormal suggests:

I have to think about this… Basically the differences relate to the derivative… But I did a quick 5 minute look with a google spreadsheet and I couldn’t make this work. I think for it to work, you need to divide the first differences by the change in x, and also divide the second differences by the change in x, and look for them to be equal. In other words, the derivative… - @jdyer gave this gem which I’ve never heard of before:

“You have a full glass with 1 liter of water. You take gulps from the glass; each gulp is a random real number of liters from 0 to 1. On average, how many gulps do you expect it to take before the glass is empty?”And his discrete version is you start with 100 and kids take away a random number (generated from 0 to 100) per step.

- @bowmanimal also talked about this “game”:

Whoa! TOTALLY new to me. Since we can’t put up e fingers, we could play the game where we each pick a number (doesn’t have to be an integer) from 0 to 5. I had to see this for myself!

The red line is always greater than the blue line (I think… I want to prove it algebraically!). [Note: related, this set of tweets on which @BenjaminASmith alerted me to]. And @mathillustrated shared this amazing presentation (read it!) on scaffolding and formalizing this game with kids. - @CmonMattTHINK shared this fact which I LUUURVE but forgot about:

The probability of a random permutation of n objects being a derangement (no object remaining in its original position) approaches 1/e as n->oo.And @DavidKButlerUoA shared a wonderful presentation he made on where the derangement formula comes from. - @bobloch also shared with me a fact I don’t remember ever learning!
*e*shows up in the harmonic series.Apparently, if we look at which*n*values bring this series to an additional integer, we get:And if you go further and further, and take ratios of these

*n*values, you get a better and better approximation for*e.*I calculated 227/83 and got about 2.735. - @mathgeek76 reminded me of the chapter in Steve Strogatz’s excellent book
*The Joy of X*which talks about how you find the right partner, mathematically.

The answer involves . (Spoiler here.)

And then a wonderful conversation about where “e” belongs in a high school curriculum popped up on my feed, launched off by this tweet:

I should say this was Steve’s launching tweet, but he was open to thoughts on both sides of the discussion that ensued. My favorite part of the conversation was when someone (I can’t remember who) brought up and when that should be introduced and why.

**My conclusions**

First and foremost, I knew there was a lot to — it bridges continuous and discontinuous phenomena. It is important in calculus and (to some degree) combinatorics. There’s a lot I don’t fully understand yet about what people shared with me in their tweets — and I have to work through them to see their connections. I do see lots of connections but I haven’t worked through the math of any of these things to draw them in fully.

Second, in my musings on this, I think I’ve come to recognize that why I have always found so fascinating is that is keeps on popping up in unexpected places. In my mathematical career, my jaw has hit the floor a number of times when I see it suddenly emerge when I never thought there would be a connection. I mean, think of the first time you saw . ‘Nuff said.

So if I want my kids to see what I see when I encounter , that’s what I want them to encounter. A surprise. That it can pop up in totally unexpected places, and you might not initially know why, but it eventually can become clear.

I don’t think I have time to pull this together for this year, but here’s what I’ve decided I want to do in some future year… I want to have a period of time where kids are told “this is a problem solving day” (or set of days). And I give the class different problems that will result in them approximating … but they don’t know it. Like Pascal’s triangle… they can just do the calculations and see pop out. Or the chess board experiment. Or taking random numbers away from 100. Or the harmonic series. Or the product challenge. Or the vs game. Or the compound interest problem. And have them work on them. They all seem unrelated. Yes, this is contrived. Yes, I’m telling them what to do in many of them.

But BOOM. Soon this number that keeps popping up in all these unrelated problems.

For each of these, they are going to get numbers close to . And for me, I can say

*This, *my friends, this is what is so beautiful about this number for me. It is a universal constant. It is like . It pops up in so many unexpected places. There is an underlying structure to why this is all happening, why it pops up everywhere. That is going to start to be revealed in calculus, but that’s only the bare beginnings… It goes much deeper. But I wanted you to get the experience of wonderment and have something that’s you know is true that *begs *the question *why… WHY? WHAAAA? WHYYYYYYY! *Because this desire to know, to figure out why something is true when you know it must be and it feels too unbelievable to be true, that’s a feeling mathematicians get that drive them forward in their work. And making those connections, and we know they must exist, it’s awesome when it happens. So yeah… I wanted to introduce you to this important number which we’ll just take as that… an important number that we’ll get to play around with like we do … but know it’s more than just a number. For you, now, it’s a question that’s begging an answer.

**A realistic ending to an idealist exhortation from me to my students**

Okay, I don’t know if I could really pull this off. But I’d love to.

]]>I asked them to do the following. Think about . It’s going to be a long number when it is all written out. I wanted them to come up with a guesstimate about how many digits there are in the expansion. To scaffold, I asked them for three things:

a) What’s a guess (for the number of digits) that is too low? How do you know? (Can you come up with a larger low estimate?)

b) What’s a guess (for the number of digits) that is too high? How do you know? (Can you come up with a smaller higher estimate?)

c) Based on your work and your intuition, if you had to make a guess, how many digits are in the expansion of ?

Honestly, it was one of the best things I’ve done recently. Kids were showing grit and so much flexibility in their thinking! I had to correct a few misconceptions and nudge a little here and there, but it was all on them how they wanted to go about this. It was beautiful. (At one point, a kid said they wanted to give up, but I came back around a few minutes later and they were rapidly making progress and *hadn’t* given up.)

At first, kids didn’t know where to start. I told them they were going to* get time* to work on this, so they could take on strategies that might take a while. (Normally, we start class with something short and quick. I wanted to indicate this wasn’t that.) Initially, I gave 7 minutes, but since so many kids were on a roll, I expanded it to 14 or 20 minutes. I honestly don’t remember how long.

What I adored is that this problem was definitely in their wheelhouse. Most groups were gung ho, and just started writing stuff down — and eventually (sometimes with a *little *encouragement/prompting from me), they came up with SUPER awesome solutions. Seriously, things I had never thought of.

The main two approaches I saw were:

- Kids noticing that . Which is close to . So . So that puts us at around 19 digits.
- Kids noticing this pattern:

So after going up about every 3 exponents, we add an additional digit to the number. (I say about 3 because all groups who did this method saw that a few times, you’d get 4 exponents in a row which keep the same number of digits instead of 3. But it was usually 3.)Assuming the number of digits increases after going up every 3 exponents, that means that exponent 12 has 4 digits, exponent 15 has 5 digits, exponent 18 has 6 digits, exponent 21 has 7 digits… etc. So exponent 60 has 20 digits.So that puts us around 20 digits (or maybe a little lower because of those occasional 4 exponents in a row).

That’s about all I wanted to share. I was a little out of my comfort zone because I didn’t know if they would all just throw their hands up and give up. But they didn’t, and instead did some phenomenal thinking.

I just realized… you might want to see how this relates to logarithms. It turns out that the number of digits is equal to doing the following: take the log of the number, and then take the floor function of that result, and then you add one. I won’t spoil it by explaining why, though. See if you can figure it out!

]]>That being said, this past week I had one of the very few times that one class was about half a day in front of the other and I didn’t feel the need to forge forward as acutely as I usually do. And in that short exciting window, I saw a student’s shirt which I thought was mathematically beautiful…

… which of course I told to the student. “Math?” He didn’t see it, nor did others in the class. It was an 90 minute class, and when having kids work on some problems together, my mind started thinking… I want to *show kids how I see this shirt. The glasses I use to see the world are different from theirs.* We take a break in the middle of class for kids to get water or quickly grab a snack, and so 5 minutes before the break I stop everyone. I have them look up. And I say something like: “I know many of you don’t see math in [Stu]’s shirt, but I think if you start looking at the world with math in mind, something like that shirt will pop out as beautiful mathematics. So grant me 5 minutes where I’m on stage, you’re the audience, and I *live* give you my thought process for creating a version of that shirt on desmos. It may not work, but I think it will be neat to try.”

So they are sitting watching me sit at a laptop. I start by graphing . They ooh. Maybe they don’t. But in my mind’s recreation of the event, I hear them ooh. I explain that this isn’t a random thing I created. I ask who is taking statistics. I mention the “normal distribution.” A few nod knowingly, and for those who don’t, I say “this isn’t genius, this is me seeing [Stu]’s shirt and recognizing one of the most famous equation shapes in the world.” Then I graph the reflection over the x-axis. They understood that. But then I said I need more lines.

So I say: “I am now going to try to make the same curve with a higher peak. I think I should do a slight vertical stretch.”

But then I note that it isn’t just that each curve gets slightly stretched, but also the width of the bump gets slightly widened too. I go to the board and explain how I’m going to do a horizontal stretch too, and write up how I’m going to alter the x-variable in the equation to do that.

I flipped that over the x-axis and then manually entered a bunch more equations that did the same thing — slightly higher peaks, slightly wider bumps. Kids asked me to add in the two circles in the middle, so I did. It looked meh because I only had 6 or 8 curves. I sent them on break and promised them I’d get it to look a bit better when they returned. And that I would do this with just *one equation*.

During break, I whipped this up using lists.

When they returned, I explained how the list worked in Desmos — so one equation actually plotted a bunch of equations.

I didn’t know what to make of doing this. I wanted them to see how I saw things, how I thought about things. That math is in lots of places if you just look for it. That playing with math can be fun and what they already know mathematically are quite powerful tools. If for just a second one of the kids was like “Oh, yeah, wait, math *is* pretty neat,” I’d be happy. It might have happened because the next day I was talking with a science teacher who was telling me that my kids who she also taught were talking about it in her class.

Also, you know, I always find that when I deviate from my plans for something I’m excited about, I always feel *so good about doing what I’m doing with my life*. I have to keep this in mind and try to go off the beaten path more…

PS. Of course when I saw the shirt, I didn’t initially “see” the normal distribution. I saw fluid flow around a cylinder:

But I forgot everything mathematical I know about that. :) So normal distribution it is!

]]>

I posted this tweet yesterday:

I’ve been on the organizing committee for Twitter Math Camp for the last three TMCs and also for this upcoming one in Berkeley, CA. It was initially a surprise — and also upon reflection not as much of a surprise — to me that board member Marian D. stepped down from the board of TMC a few days ago. When I called Tina C., also a board member, to talk about possible next steps, she told me she was ready to step down also.

I will let Marian and Tina talk about their reasons themselves if they wish. But without much information out there, I figure people in the TMC community are confused and filling in the empty space with their own conclusions. I mean three people leaving organizing TMC in the span of a few days? That is a message encoded in an action, right?

Here’s what I want to share about that, from my perspective. **I love TMC. I love the people who put on TMC. **If I had to point to one thing in my teaching career that has grown me the most as an educator, it would be MTBoS generally and TMC specifically. I don’t have adequate words for my love and appreciation for it.

And mostly, I worry that because of these departures, people are going to vilify TMC.** I don’t want TMC to be vilified. **And in my opinion, it doesn’t deserve to be vilified.

It was born out of desire and necessity, to have math teachers who knew each other on twitter to meet in person. It provided a space for teacher voice to have value, and a place of emotional support for many who didn’t have any locally. It created one pathway for classroom teachers to become teacher leaders. I can’t say it enough. **I love TMC, I love TMC, I love TMC, I love TMC, I love TMC, I love TMC. **And for many, many, many people who have attended and have had amazing experiences, they too have a special place in their hearts for TMC.

So know that although I am not going to be a committee member, I am not anti-TMC.

So why, then, would I leave? Before that, I need to say it was a terrifying decision to make. I had committed to helping out, and shirking on something I committed to is a huge sin in my book. The good news is that the work that I commit to doing every year is quite simple logistically and not a huge time suck (organize a new-attendee/supporter mentorship, plan a new-attendee dinner, design some buttons, organize a homespun photobooth). And we’re five months before the conference and I told the committee that I’d work with someone on helping them learn to do these things with all my documentation. I wanted to leave as responsibly as I could.

But yes, why did I leave? As much as I love TMC, I didn’t enjoy the process of planning TMC. I’ve been in so many awesome collaborations where we have the opportunity to plan something awesome (whether it be a lesson or a workshop or whatever). And TMC is one of those. I mean, imagine you got a bunch of dedicated, passionate, awesome, thoughtful math teacher friends together (the committee/board is full of people I call friends!) and you get to put on this awesome event. I mean can you even imagine how *fun and exciting* that would be? Brainstorming, debating, revising, throwing things out, having a genius moment, outlining, revising? Simultaneously intellectually draining and intellectually invigorating. All the good! But the reality is that the process of planning TMC was broken, and over the years, it has remained so. It shouldn’t be so, and I tried to help change things, but without going into specifics, it never became unbroken. TMC, for me, was amazing. But *planning* TMC is a different story.

I was on the phone talking with Tina about Marian’s sudden departure, and Tina told me she was thinking of stepping down too. For the past few years, but more so in the last year, I’ve been working alongside Tina on a number of projects. We worked on creating a TMC mission statement with core values (engaging with the community in multiple ways to do this), we worked on improving communication and defining roles in the TMC leadership (beforehand, everything was haphazard and no one knew what everyone else was doing), we worked on figuring out where TMC needed to improve (diversity) and putting together a plan to address that, we worked on spearheading the first TMC fundraiser, and we were working on implementing ideas from the diversity proposal that the TMC committee and board had approved. My earlier work — things like calling a few restaurants and sending out emails pairing up new attendees and supporters — started to feel less important to me. I was excited about working alongside Tina with professionalizing TMC and addressing our deficits. That was the work that felt difficult and juicy but so much more important. Working with Tina on all this *was* that exhilarating collaboration that I had talked about above. So when Tina told me she was leaving, I sat down and wrote and wrote and wrote nonstop 40 minutes just trying to figure where my head was at with all of this.

What I realized I was losing my mentor-leader who brought meaning to my work. And she was my rock when trying to navigate our dysfunctional planning process. I tried to imagine doing our work without her *and* thinking about the frustration level of planning with the committee. I *couldn’t* imagine it. And that’s when I realized I had to leave with her. I thought: better all at once, a little less than half a year before the conference, so the committee/board can regroup.

I was excited about the directions we were taking this year in terms of diversity and equity. Marian reached out to educators of color to present and attend. In the program, a strand of sessions throughout the conference was focused on equity. Affinity spaces for educators of color and LGBTQ+ folk were built into the schedule. Marian kept the idea of “safety” for educators of color on our minds and we were explicitly thinking about people with mobility problems, deafness, etc. In other words, we as a conference knew we were generally inviting, but we were working on being specifically inviting.

The people organizing TMC are friends. They might not create a synergistic working group, but they are doing selfless work which has helped so many. There are still people organizing TMC who are hoping to try to make these things listed above a reality. Any *one* of these things — well-crafted affinity spaces, a strand of strong equity sessions, thinking deeply on small and big ways to put on a conference where marginalized groups are valued and welcome and feel safe — would have pushed TMC forward leaps and bounds. It is a fair statement to say the committee is not strong at thinking about equity and diversity as a whole yet (I’m honestly very far below where I know I need to be at this point, but I’m learning from every conversation I have and pushback I am given). There is a steep learning curve and it’s not always easy to see through these lenses. Mistakes are par for the course. (We’re math teachers, we know this!)** But there are those who are invested in this work and they are going to continue pushing TMC forward. They deserve all our gratitude and thanks. And support. If they ask for help, I hope many of you who want to see TMC do better on these fronts charge in and give them an assist. Help make TMC the place you want it to be.** This is what TMC has always been to me: a space for passionate people to do good things together.

]]>

But today I wanted to write a short but sweet post. Every so often, I ask for feedback from my classes. I’ll create a google form and ask how things are going, if kids’ pronouns have changed, how long their nightly work takes, and other thing I’m curious about. Sometimes I have kids reflect about their own work or their groupwork.

But last year, I started occasionally including this question:

I love that it gives kids a chance to think about who has helped them out. I don’t make it a required question. Only about 1/2 or 1/3 of my kids filled that question out this time. But I really loved the short bits I did get to see. I learned who might have been studying together for tests, or who worked super patiently with another person who might have been struggling, or whatever. And kids got to have a moment where they got to be grateful for someone else.

What was nice is that I actually asked for this survey a week before parent-teacher conferences, so I was able to share with parents who came some shoutouts about their kids (if they got any). Parents *really* appreciated hearing that their kid received praise from another kid (and why).

And today, I sent short emails to any kid who got a shoutout…

Hi Stu,

In our last check-in survey, I asked students to give a shoutout to someone who was an important part of their learning experience. I wanted to privately share these with students…

Stu2 wrote:

Stu! We work great together because we have different strengths and weaknesses, so when we do a problem together I’m able to understand the whole problem, not just the aspects I’m especially strong at.

Stu3 wrote:

Stu! He is always great at explaining things to me!

Hope this brightens your day!

Always,

Mr. Shah

And… that’s it! A little sweet thing that I came up with that I really like. Short, simple, but for the right kid at the right time, it can be meaningful. (A few kids emailed me back saying that reading the email did make them smile or brought some light in a dark day…)

]]>PS. Once, I had a bulletin board in my room that I had reserved for “shoutouts” or “notices of gratitude.” Where kids could post index cards with shoutouts for other students. I wanted it to be public and for it to “grow.” I would occasionally build in time for students to reflect and add to the bulletin board. That was years ago. It didn’t really take off, which means I didn’t roll it out and implement it well. But what I’m doing with this google form is really nice because it isn’t intensive or involve much planning!

PPS. If you’re at my school and want advice on how to do something like this, feel free to ask. I’m happy to brainstorm with you. I just don’t want everyone doing this which then will take away the “specialness” of when I do it!

However at the end of each year, around November and December, I think about what I truly value and I find an organization or two that fit a good number of things that I care about. This year, I have two organizations that I am donating to. I’m writing this short post in case any of y’all out there are thinking about places to make end-of-year donations to, and these organizations speak to you in some way.

(1) **BEAM: Bridge to Enter Advanced Mathematics.
**

This is an organization that I started volunteering for last year. It was created by a friend of mine. The idea is that there are a lot of underserved kids in New York City who like and can do mathematical thinking, but aren’t getting what they need to excel. BEAM is a summer mathcamp for underserved *middle school kids. *The organization goes out and finds these kids, and then creates a summer program that’s *free* for them.

The reason the organization focuses on middle school kids is that’s where the most good can happen from an intervention. And these kids who go through the program get support *for many years afterwards*. It isn’t a “one and done” experience. Often times, kid return to the camp a second year. But more importantly, the organization helps kids navigate the high school admissions process (which is super complicated in NYC), helps them succeed with open office hours where kids can drop by to ask for help with any subject (this is how I help out… I staff the office hours!), and helps to with the college application and financial aid process. It’s a long-term program that has a *huge* impact on kids. (If you want to learn more about it, this New York Times profile is amazing.)

So yeah. A thoughtful, wonderful, amazing program. I’ve worked with these kids and they rock. Right now donating $1 will bring in $3.14 dollars through a matching campaign. If this organization sounds like it fits your values, and you have some bucks to spare at the end of the year, consider donating!

(2) **Twitter Math Camp (TMC/TMathC)
**

This organization hits close to home for me. This is a math conference that started up in 2012. It was literally the most grassroots thing ever. A bunch of us math teachers who had been communicating online decided we wanted to all meet and do math together… and that morphed into an embryonic conference. A school donated space, people offered to present on things they were doing in their classrooms, and a small conference was born. It has been running every year since, and has been entirely run through volunteer efforts. Schools or colleges donate space. We have people volunteer to present. Organizers (disclosure: I’m an organizer) work out the logistics, while they’re also working/teaching/coaching/etc-ing full time. It’s pretty awesome. And because of the goodwill of all these people, synergy happens. It is an incredible space for math teachers to grow their own practice and forge life-long relationships with other math teachers.

So as an organization, financially, it’s pretty efficient. We only charge a $20 registration fee because we want to keep costs low. But there are costs associated with the conference (e.g. buses from the hotel to the school/university where the conference is held, custodial charges from the school/university, renting a conference room in the hotel for registration and game night, providing supplies for speakers, etc.). Amazingly, with the few thousand we get from the registration fee, we’ve been able to pay for everything.

But for the first time ever, TMC is hosting its first fundraiser (read more about it here) and is asking for donations. And like with BEAM, TMC has a matching donors program, so every $1 donated will actually be worth $2! Now here’s the thing. When deciding to donate my own money to TMC, I had to take a beat. I thought: “Wait, I’m giving my money to a conference. Which I go to and pay hotel fees and transportation fees for. And I’m asking my *friends* to give money to a conference. A conference they likely are not even going attend. Does this make sense?”

But here’s what I realized… The conference encapsulates things I value. It gives classroom teachers a voice. It creates a space for teachers to share their thoughts, and maybe realize for the first time that their thoughts are useful to others. By having a powerful conference experience (which often lasts far beyond the conference itself through the relationships forged and communication that happens on social media), TMC is having a positive impact on math education for *so many students*. My donation is going to go to help increase access to educators of colors and seed a scholarship fund to help provide access to the conference for teachers whose schools might not be able to afford it.

I want to help the conference become financially stable and not have to rely on only the the small $20 registration fees it collects. I want to help increase access to the conference for teachers of color and teachers who can’t pay out of pocket and whose schools won’t help them out financially. It isn’t about *me* and the *conference*. *I’m giving because I want to make sure that this conference can exist for others, and provide for them the same sort of experiences that helped shaped me as a teacher. [1]*

So there you go. I wanted to share these in case you were wanting to donate to some causes, but hadn’t quite hit on the ones that made your heart pitter-patter yet. (Both are non-profits so donations are tax-deductable.) These are the ones that do it for me. And I’d encourage you to give a few bucks if you happen to have them at the end of the year. (You’re probably a teacher if you’re reading this, so you probably don’t have a few bucks to spare, so in that case forget everything you just read!) And if you have any organizations you think I’d be interested in, please throw them in the comments!

[1] That’s precisely why I give to the mathcamp I attended when I was in high school, and which made such a large impact on me. Or occasionally why I’ll give to my college (even though I don’t usually give a lot because they are pretty darn flush and already have a lot of wealthy donors… I like to give to smaller organizations where I know a small amount of money will help a lot.).

]]>This year I decided to do something different. A colleague of mine did this for a class we both co-taught years ago, and I really thought it would be a great way to start this year.

**Part I: The Initial Card Sort that Sorted My Kids Into Their First Groups**

I said hello for literally only one or two minutes, and then I shared the activity we were going to do for 15-20 minutes. We were going to do a puzzle-y card sort to figure out who was grouped with whom. But in order for the class to be successful, they all needed to work together. I projected a sample card. I said anyone is allowed to use a calculator. But some of the cards might require some laptop assistance. So they had a little laptop symbol on it.

So in this case, for example, I knew almost none of my kids would know what binary numbers are, but using google they could find a converter online that would say this was actually “170.”

Each card had a kid’s name written on back. So each kid got “their” card. And their goal was to find others who were in their group because their cards formed a logical group. Here’s a sample group to show you what the cards looked like and how they link:

See if you can tell what the link is among these four cards…

I’ll give you a second.

I will reveal the answer in the next line, so don’t keep reading until you are sure you want to know.

Okay, the link is the number “ten.” So 10! is the number of seconds in six weeks. When the kids type those equations into desmos, they will see the number 10 show up. Neon is the 10th atomic element. And “X” is 10 in roman numerals.

You can see why kids are going to need each other and the class is going to have to work together. Because until someone recognizes that “ten” is a category, these all seem very unconnected. But as soon as you know someone’s card represents “ten,” then things like the neon symbol or the “x” make sense.

I’m kinda proud of these, so I’ll show you another:

The theme? “Pi.” The first one is circumference over diameter, the second is a recipe for pie crust, the third is an approximation for pi, and the fourth is a world record holder for reciting the digits of pi.

(If you want to download my cards, here you go: Group Card Sort! And the explanations are Group Card Sort Explanations.)

I only had allotted 15-20 minutes for this. I had no idea if this would go quickly or take forever. In all four classes I did this in, I was able to get them to finish in 20 minutes but only through some careful prodding/help. If I were a bit more hands off, I could see this easily taking 40 minutes and it being time well spent. But alas, I didn’t. Here’s how I intervened:

- After 7 minutes, I stopped everyone. I asked who knew what they had. A few people did.
- Throughout the time, I gave a “few” hints where I could, but mainly I was acting as facilitator to help others help each other. So for the pie crust recipe one, I had the person go around asking if anyone was a baker (or I would shout out to the room if anyone liked to bake, and had them come to us).
- When someone wasn’t doing anything, I had them go help others. They might have been confused about their card, but they could help others (and get help from others).
- Sometimes when a kid “got it” but still had some uncertainty, I would put them out of their frustration and tell them they got it. If I didn’t have time pressure, I wouldn’t have done this, but it didn’t ruin the activity or anything.
- After 15 minutes, with my proddings and connecting, kids were doing pretty well. So I stopped everyone and had people who
*knew*what their card represented be quiet. There were always three or four people who were stuck. So I had them share their card or write their puzzle on the board and see if*anyone*could figure it out in the remaining few minutes. (We wrote the different “solved” categories on the board, so sometimes they could figure out their card by seeing what it might be.) They gathered, talked, and some classes barely finished in time and others didn’t. I didn’t focus on that. For the ones that didn’t get them all in 20 minutes, I quickly went through the explanation of the remaining few cards.

It was really fun for me to watch, and I saw kids really getting into the puzzle-aspect of things. The first time a kid figures out their card and finds someone else with the same thing, it’s just a wonderful feeling. It honestly feels impossible to kids at the beginning. They literally start looking for anyone with the exact same card as them, or if they have a picture they’re looking for other people with pictures. But as soon as they realize it’s more challenging and more interesting, I get to see how they react and what they do. Do they sort of back down? Do they go help others? Do they hope someone comes to them? My big goal was having kids realize *they can’t do this alone* and *most cards won’t tell you what they are so you need to hear about others and help others*.

Oh! One big thing. I realized in the first class that kids were just kinda sitting with their cards. So I made a rule that until the card sort was over and everyone in the class figured out their cards, *no one was allowed to sit down — not even when using their laptops*. This actually got kids up and moving. It was a small thing, but I know it was super helpful to making this a success.

I wish we had time for kids to say hi to their first group and do a little group norm setting, but alas with only 10 minutes left, I had to transition.

**Part II: New Years**

So I totally saw Howie Hua’s first day post and was in love. It was positively inspired. Often times, people post awesome things they do in their classrooms that are awesome but just *not me*. When I read this, I felt: “OMG THIS IS ME!” He celebrated *new years* with his classes. Here’s one of his students’ videos/tweets:

And it really got me thinking. The first day IS my new years. My life doesn’t go in January-December cycles. It goes in September-August cycles! And it was the perfect time for kids to make a new years resolution. They had 90 seconds of thinking to come up with *something*.

Then after 90 seconds, I threw up this screen, obliquely referencing the Maurader’s Map from Harry Potter (but opposite-ish) and I had them recite this pledge:

Then I gave each kid a baggie that I prepared. In it was a super fancy piece of origami paper, a mardi-gras necklace that someone had a zillion of and was throwing them away, and a noisemaker I bought from amazon. It mabye took me 45 minutes to put these all together. But totally worth it. For some reason, I believe that being given your own personal goody bag is way more exciting than having someone pass out necklaces, noise makers, and origami paper individually.

I then handed out party hats too (but those had to be returned to me). I actually always keep a stack of party hats in my office, and when it’s a kids birthday, I give them a hat, candy, and we sing a short birthday song. As I said, this idea of Howie’s fit me!!! Anyway, kids had to write their name and their resolution on the origami paper which I collected. (Later that day, I put them together in a ziploc bag and hung them visibly in the room so this doesn’t become a thing we did but wouldn’t return to. I was thinking I’d give them back to kids after the end of the first semester so they can see how they’re doing on their resolution. But I might have another brilliant idea. Who knows!) As soon as the bags were out, the noise makers were making noise. And that was a lovely cacophony of BWWWAAAPP and BAAAAAAAA noises. (That was also why I had kids pledge to do no evil with what they were given… *grin*)

In any case, I was standing at the front of the room when they worked on writing their resolutions. When they were done, they had to bring up the resolution to me and wait at the front of the room with me (with the necklace, hat, and noisemaker). After 2-3 minutes everyone was up. And then… we took a class picture, all decked out, blowing on the noisemakers and just being amazing. And oh yeah, we also took a class boomerang (which is an app that lets you take a 2 second video and plays it over and over).

The boomerang was my favorite part because kids were jumping up and ducking down and doing fun things. And I kind of am obsessed with boomerangs. So there’s that.

I think I’m going to get these photos printed and framed, and hang them up in the classroom. I don’t know what to do about kids who were missing (there were a few) or who transfer in after some schedule change, but maybe I’ll list them missing on a caption instead of some awkward photoshop job?

Our first day together. (I did post the boomerang video and our class photograph on the google classroom site in case any kid wanted it.) [2]

And then it was the end of our first 30 minutes together. I was really happy with how it went. I like the feeling that I left each class starting the year with good vibes. Thanks go to my chemistry teacher colleague and friend for the card sort idea which I made into something my style (with my kind of clues!), and to Howie Hua for helping me make a memorable moment to start the year.

***

[1] We do a lot of the logistics things in the following week. They read the course expectations at home and fill out a “get to know you” google form which also asks them questions that require the expectations to finish. And then each day or day, I talk about one or two things I want to be explicit about (like how to write me an email, or that’s it’s okay to go to the bathroom and they don’t need to ask, but they do have to discretely let me know they’re leaving if I don’t see them, or that they need to bring a waterbottle to class because they can’t leave to get a drink).

[2] I just realized this photo could be fun to have up on the screen on parent night, when parents/caretakers come in two weeks to hear me talk about our class.

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