New Year, New Blog

Note: Julie Reulbach wrote this post on the ExploreMTBoS site! I’m copying it here for two reasons. One is that I want everyone to participate! Two, I have been in a crazy place this year, and I’ve thought “oh I should blog this” a bunch and never took the time to follow through. I am going to participate too! Join me!

Additional Note: Carl Oliver has also set up a blogging resolution challenge for 2017! And his post is inspirational.

Welcome to the Explore the MTBoS 2017 Blogging Initiative!

With the start of a new year, there is no better time to start a new blog!  For those of you who have blogs, it is also the perfect time to get inspired to write again!

Please join us to participate in this years blogging initiative!  To join, all you need to do is write just one post a week for the next four weeks.  To make it easier for you, we will post a new prompt every Sunday!  Once you have blogged, please fill out the form below.  Each week, your blogs will be posted on this site for all to enjoy!

This Week’s Theme:  My Favorites

This week, the blogging theme will be “My Favorites”, where you can post about one (or many) of your favorite things!  Called a “My Favorite,” it can be something that makes teaching a specific math topic work really well.  It does not have to be a lesson, but can be anything in teaching that you love!  It can also be something that you have blogged or tweeted about before.  Some ideas of favorites that have been shared are:

  • A lesson (or part of one) that went great
  • A game your students love to play
  • A fun and/or effective way to practice facts
  • A website or app you love to use in class
  • An organizational trick or tip that has been life changing
  • A product that you use in your classroom that you can’t live without!

Blog Newbies!

If you are brand new to blogging, you can read Starting A Blog from the 2015 initiative.  This post will give you specific instructions on how to start a blog.

Hot Tip!  Don’t stress about your blog name!

The hardest part about blogging is often coming up with a title.  Do not let this detail derail you!  A great suggestion is to make your blog address your name.  Then, you can title your blog later – or change the title anytime you want!  To see what this looks like, check out Sam Shah’s blog.  His web address is, but the site name is “Continuous Everywhere But Differentiable Nowhere“.  No one cares about your blog name, they just want to read interesting, inspiring, and helpful posts!

Hashtag it!  #MTBoS #MtbosBlogsplosion 

Don’t forget to tweet out your blog link and add hashtags so other teachers in the MTBoS community can easily find your post!  If you are not tweeting yet, you should be!  There is an amazing community of math educators there just waiting to inspire and support you!  Check out How To Start a Twitter Account to get started!  Also, if you are brand new to Twitter or just want to get more out of it, there are more Twitter tips on Julie Reulbach’s blogpost, Tweet, Connect, Repeat.

This year, we are joining up with the #mtbosblogsplosion.  Special thanks to  Carl Oliver@carloliwitter, for jump starting blogging for many people in our community!

Hashtags to add to your tweets:  #MTBoS #MtbosBlogsplosion

Also, if you have a wordpress blog, please re-blog this post to get the word out!

Deadline: Press submit by the end of the day Saturday, January 7, 2017.

Yes, this is a quick turn around this week – but we don’t want you to put it off or delay!  Once you are finished with your blog post, fill out this form and your blog post will be featured on this site [meaning the MTBoS site this is reblogged from] next week!


Twitter Math Camp 2017

We are starting to gear up for TMC17, which will be at Holy Innocents’ Episcopal School  in Atlanta, GA (map is here) from July 27-30, 2017. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc ( It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, however, this year we cannot guarantee that everyone who submits a proposal will be accepted. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Thursday, July 27 and 48 one hour sessions that will be either Thursday, July 27, Friday, July 28, or Saturday, July 29). That means we are looking for somewhere around 70 sessions for TMC17.

What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you!

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 16, 2017 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC17 – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Daniel Forrester, Megan Hayes-Golding, Cortni Muir, Jami Packer, Sam Shah, and Glenn Waddell

Teaching is hard work. Election aftermath.

Yesterday, I told one of my precalculus classes how it was an exciting day. I was setting them up because it was election day, and kids at my school are heavily interested in politics, so I thought they’d say “yes! Election!” And I would say: “Actually, it’s because one of my best friends from college is having a baby.”

Of course that setup didn’t work, because of course a kid asked “why is today exciting?” Thanks, kid. But I told the class about my friend’s baby.

Yesterday evening, as the election results came in, I got more and more anxious. And when it was clear that Trump won, I was destroyed. I am not going to use this blogpost to explain my love for Clinton, or why Trump makes my blood boil. Instead, I want to just share how my day has gone.

I teach at an independent school in Brooklyn, and the population of kids and parents we serve are (for the most part) liberal. The kids are politically active and aware and interested. Today, I came to school and kids were destroyed.

In my first class, I talked to kids a bit, and then asked them what they wanted to do. After hearing them, I came up with the following plan. The kids who woke up to the news and wanted to learn more and get informed could read articles online. (There were about 4 of those kids.) I just asked that before they started reading, they take 3 minutes to type out all the questions that they have — so help them start processing. (Like “How could this happen? What was wrong with the polling? Who was voting for Trump? What does this mean for issue X?”). For the others, we formed a circle with the desks and I let kids talk. At points, kids cried. I didn’t join in — I wanted this to be a space for them. They expressed real sadness, hopelessness, optimism, anger, frustration, embarrassment, terror, empathy. I really heard my kids, and when talking about this election, they were speaking their truth, about their hopes and dreams (and how those hopes and dreams were altering). It destroyed me inside to hear them. To see how much this election has affected them. I guess I hated the fact that my kids are feeling what I’m feeling. I don’t want that for them.

I went to my second class, that precalculus class that I told about my friend’s baby. The first thing a kid said to me was inquiring about my friend’s baby. That small gesture — that this student would remember that — lifted my spirits. In this class, more wanted to read the news, and a handful of us talked. This discussion tended to a bit more political punditry — about the what’s and the how’s and less about their emotional state. I suspect they got many of their feelings out in their previous classes.

In my third class, we watched Hillary’s concession speech.I teared up twice during the speech. One kid left to gather themselves for a few minutes after the speech. I didn’t know what to do after. Kids said they didn’t feel like discussing things anymore — they were discussed out — but they also didn’t see how they could focus on work. I made the executive decision to spend the last 20 minutes of class having my kids watch the pilot of the West Wing. I hoped that some optimism in politics might help.

I have one more class to go. It’s a 90-minute block. I’m drained, right now. I don’t have much more in me. I suspect kids are also drained, but I don’t know. I’ll suss out how things are, and try to get through it.

I’m exhausted. Yesterday I woke up at 5:30am to vote. Yesterday I didn’t get to bed until very late (maybe 1pm), and then woke up at 3am to watch Trump’s victory speech. I then read articles until I forced myself to sleep from 4-6am.

Teaching is hard work. Yes, there are lesson plans and grading and meetings and a zillion other things. But days like today, days like today keep me in check. And reminds me how hard the hard work can really be. Because the hard work is being an emotional support. To let kids cry. To let kids know you cry. And to get through the hard times together.

Update: My last class came in with bags under their eyes. I was also tired. I asked them what they wanted to do. A few wanted to continue talking, a couple wanted to do some math and do some talking about the election (a mix), and one just wanted to do math. I decided we would go over the nightly work first, and then talk about the election.

When going over the nightly work, kids were actually focusing better than expected. They asked questions. They were able to answer questions. It was going well. I then ended up going on a fascinating tangent about fractals (related to one of the questions we talked about). And when I realized kids had never heard of fractals before, I showed them a youtube fractal video. Then they wanted to know how it was made. So I gave a short 10-minute lecture on the complex plane, and how the Mandlebrot set is formed. Kids were entranced by the video. I gave a 5-minute break before we sat down to talk about the election. (During the break, kids were in the hall watching more of the fractal video on one of their phones!) When we returned, everyone was silent. No one spoke. I just let it hang there. Eventually one voice. Then another. It wasn’t a rowdy discussion. Not everyone was in it. But most kids had something to say. And then when the day was close to ending, and there was a natural lull, I used a comment about “voting systems” to show a video about alternative voting systems. And then I let kids go home.

I just made the first four slides for class tomorrow. They’re not fancy. I’m tired. But I think they encapsulate what I’ve taken away from today.



Now I must end. I now have to change all my lesson plans for the upcoming days, prepare for parent visiting day tomorrow, and write narrative comments. This feels impossible. But I needed to process today.


UPDATE: A student gave me a paper flower she made today, to thank me for facilitating a conversation about the election in our class on Wednesday. And that flower is going to stay on my desk all year to remind me of the other things we do as teachers that can be meaningful.


Waiters, Waiters, everywhere…

Today I was nerdsniped in the math office. My department head and two colleagues were working on this problem. I don’t know where it came from. But golly, did I enjoy it!

Imagine you have a row of waiters all facing forward. Each waiter has a beautiful silver platter that they are carrying. They have to choose: will they hold it directly in front of them, or on their left side, or on their right side? Here’s a diagram showing the three options (I imagine I’m looking down on the waiter.)


Okay, so there is one constraint. Remember the waiters are all standing in a row. So you can’t have the platters crash into each other. So here’s an example of an OK way the waiters could hold their platters, and then a NOT OK way the waiters could hold their platters.


So here’s the question… If you have n waiters standing in a row, how many different ways could they hold their platters?

I am not going to post the answer here, because I like to nerdsnipe! But if you want to check your answer, for 20 waiters, I calculate 267,914,296 different positions!

I bet you will have a lot of fun with this problem. One person in our office came up with many pages of work, and had a very complex approach which yielded some deep insights. She was super psyched about the intricate superstructure she was building. Another person got to review solving a particular type of thingie using matrices (I want to keep things vague so I’m going to use the word thingie to avoid giving anything away). I and another person had the same approach that led to a quick and elegant solution, but left me with rich conceptual questions to pursue. And as I started doing that, I realized that I had accidentally stumbled on the complex approach that the first person had taken.


My friend Edmund Harriss, who is a mathematician and artist, has a kickstarter. Usually I don’t blog about commercial things. And Edmund didn’t even ask me to do this.[1] I emailed him letting him know I’d be happy to write up a post on his kickstarter. Why?

It’s the same reason I think he’s an awesome person. He is invested in getting people to see the joy that is inherent in doing mathematical work, and the creativity associated with it. To the point that years ago, when he happened to be in NYC, he came and talked to my Algebra II classes about aperiodic tilings (and brought a ton of these tiles for my kids to work with).He co-authored/illustrated a fabulous adult coloring book, Patterns of the Universe, which inspired one of my students last year. (And led me and a colleague down a crazy fun rabbit hole this summer.)

And more recently, this year, when I started curating a math-art show at my school, he immediately agreed to be a featured artist and sent not only artwork, but some extra tiles that I requested just because I knew our lower school kids would enjoy playing with them. (A walkthrough video of the gallery is here.) Here are some of the tiles he sent… Yup… fractal tiles!


One of the pieces he sent for the show were these curvahedra. I had fun making and playing with them (and the sheets that the pieces popped out of).


So feel free to check out his kickstarter if you are interested…

As an added bonus, I emailed Edmund gads of questions about his work, his process, and his curvahedra. And below is our Q&A!

What are the feelings you hope kids/adults/students/teachers have while using Curvahedra

I hope they feel curious, but relaxed, looking to see what can be discovered and enjoying the pleasure of discovery. Many people report that it can also be quite a mindful experience, which I had not expected.

What are the different types of things that I can make with Curvahedra?

There are a whole collection of different balls and eggs that are the most obvious things to make. Donuts are the next option. I have made some exotic pieces which branch and rejoin. I discovered when making pieces for the kickstarter that you can make some nice cones. In theory though there are few limitations, anything that can be made as a mesh can be made from Curvahedra. As meshes make many of the creatures and objects in CGI the limits are hard to find!

What do you love about mathematics? How does that tie-in with your creation of Curvahedra?

I liked mathematics initially because I could do it, but that never gave passion. What fired my love of the subject was the way you could create little worlds and then explore them. Once the rules were fixed you had constrained the behaviour, yet that did not mean that you knew what could happen. I often feel that the most valuable thing I learnt from my PhD was flexibility of thinking. In mathematics you have to be flexible to think about what you can control and how that works with other rules. In this vein Curvahedra is a system that can do many things. The fun is in discovering the most beautiful and making mathematical rules to help find that.

What is the intellectual provenance of Curvahedra? In other words: how did you come up with the idea?

I wanted to make 3d objects on a machine that cuts essentially in 2d, a laser cutter. I knew that paper could bend, so that was my goto material. I tried a lot of options, not always successfully. Then with the right connector and the right idea suddenly I had Curvahedra. It was a revelation when I made the first ball.

When you were playing with Curvahedra, what unexpected discoveries did you make?

It has really helped my understanding of space and surfaces. In particular getting an idea of the curvature of a surface, and when that is rigid or not. That understanding helped when I was studying 2D crystals with a Physicist. I have also used Curvahedra to investigate minimal surfaces, these are surfaces that minimise strain and cannot be  improved by local change. This is exactly what the paper seeks out, you impose some geometry and the paper pieces find the most efficient form.

What do you do with your creations when you’re done making it?

They litter my house, many hang from string in my windows. Luckily my wife likes them too, she has been really supportive helping put together the kickstarter and will be playing a key role in running it. I could not have done it without her.

Where is the mathematics in Curvahedra?

The notions of curvature and minimal surfaces that I discussed above, as well as other concepts from multivariate calculus are really shown off well by the system. In fact many of these advanced topics can be easily introduced to elementary students with Curvahedra. There is also some excellent topology to discover. On the other hand that mathematics is maybe a little too advanced for a general audience. There is a lot of geometry to be done. The regular polyhedra turn up naturally and general ideas about space will come from playing with the system.

How can a math teacher use Curvahedra in their classrooms? What about an art teacher?

I would hope they would both use it in the same way. As a low direction activity exploring, where people initially make a basic ball and then are just left to it. This is how I usually run sessions with the system. As people discover things that can be done it is natural to try to find out if you can discover everything that could be done. That gives natural mathematics questions. I always feel that when people have a question and mathematics is the answer they are far more motivated to look at the mathematics.

What are your favorite mathematical ideas? Who are your favorite mathematicians?

I love all sorts of visual mathematics and geometry more generally. The discovery of non-Euclidean geometry was such a revelation and is incredibly beautiful to me. It is hard to list favourite ideas though, as it is all so cool. As an undergrad I swore off differential equations, for example. I did not look at any mathematics involving them for years. Then I finally started to see how they linked to geometry, made art with them and did research and now I love the ideas especially in Differential Geometry. The best answer is often that my favourite maths is what I have just been studying! For mathematicians, I have been fortunate enough to spend time with John Conway and both he and his work are phenomenal. The humour and insight in Archimedes’ writing is also a great favourite. For geometric insight Alicia Boole-Stott is incredible, she was able to work out all sorts of properties of four dimensional figures that I could not approach without a computer. Similarly Felix Klein’s visual insight is amazing. Finally you cannot be a geometer, interested in the more elementary side of the subject, without loving the work of Donald Coxeter.

Who are your favorite mathematical artists?

Max Bill is my favourite artist of any sort and I also love the work of Kenneth Martin. They both worked in concrete art which  attempts to create form out of nothing, rather than refining essentials from something real as in abstraction. That is a notion that works very well with mathematics. Moving in the direction of mathematics, George Hart has many inspiring pieces, his use of symmetry and shape to create intricate structures definitely played a role in the development of Curvahedra. Finally I am very lucky to be able to work regularly with Chaim Goodman-Strauss who brought me to Arkansas. His work which he describes as the illustration of mathematics was probably the single biggest inspiration for me. For a time I felt like all my best ideas were stolen from him.

Besides earning zillions upon zillions of dollars, why did you want to do this Kickstarter?

I want more people to get their hands on Curvahedra. Every place I have taken it is seems to inspire and delight and I wanted to spread that further. Sadly I doubt that Math Art will make anyone rich, but I do hope it can get into more hands and inspire more love and enjoyment of the subject.

What do you think — in an ideal world — would every math teacher have in their classrooms?

All sorts of geometry and math toys and games, and time to explore them. Things waiting to be explored by the students on their own terms rather than to satisfy a rigid curriculum. Though the problem with ideal worlds is that they also have ideal students! This might not be practical in all sorts of ways.


[1] To be very, very clear: I asked Edmund if I could promote his kickstarter. He did not approach me. And I am also getting nothing from him for posting this. In fact, I’m about to pledge money to his kickstarter!

Gaspable Moments

Sometime last year, I started thinking about why I love math so much. And when I kept on delving, deeper and deeper past all the adjectives (beauty, creativity, awe-inspiring, structure, …) to what really was beneath those adjectives, I came up with one fundamental, visceral answer. It is the spark of electricity you get when you think you start figuring something out, and the chase that happens, until things finally click into place, and you are so excited about your discovery that you want to share it with someone else. It’s that gaspable moment — that rush of endorphins — that feeling of sheer joy.

I’m guessing if you’re reading this blog, you’ve had that happen to you.

For me, this internal joy is everything. Why I love math, below all the flowery adjectives, is because of the emotional impact it has on me when things click.

It’s an internal thing. A private thing. But what if it didn’t have to be. [1]

I had this idea last year, and I’m terrified and thrilled about introducing it on Tuesday. This idea is designed to specifically make this joy visible and public. 

Here it is…


A bell. I’m going to ask kids to ring the bell on their group tables when they have some sort of insight or discovery or revelation that simply causes them joy. I want their internal joy to become external. I want my kids to recognize that this joy is something they should be conscious of and recognize it when it happens. I want a class culture where moments of joy are acknowledged and celebrated.

When a bell rings, what happens? Nothing. Kids continue working. Near the end of class, I will ask kids “anyone who rang a bell, can you share a bit about that moment and insight that you had? and can you describe that feeling with one adjective?” Or maybe I’ll call on someone who rang a bell? I do know that I don’t want the bell to interrupt the flow of class and thinking.  Will I have the bells out every day for that one class? I will probably have the bells out a lot after I first introduce them, to start building the classroom culture around them, but then I can see me probably make more conscious choices about when the bells will be out based on what we are doing in class.

Will this work? I don’t know. I’m going to roll it out in only one of my classes. Why? Today I had a killer class. The kids were persevering and having so many gaspable moments. It was ridiculous. I went into the math office after just to tell the other teacher of the class to get psyched for his next class (he hadn’t taught that lesson yet). But because this happened, it is a perfect time to have a conversation with my kids about the joy of mathematics. (I’ve had the bells since school started… I just hadn’t found the right time to introduce them.)

I can see a number of things happening.

  1. I can see kids being too “scared” to ring the bell. Because it is public. And they might feel like they’re insights or feelings aren’t “valid” or “good enough” (compared to their classmates). Not ringing the bell has no risks, so why do that?
  2. I can see kids being too “bell happy.”
  3. I can see this not going well in the first few days, and then me abandoning this idea.
  4. I can see this becoming a positive and normalized part of our classroom.

What do I suspect? Truth be told, I’m super excited about this, but I think #3 is the most likely outcome. It’s hard to be consistent with something that doesn’t get off to a solid start, because then keeping it up even though it isn’t working well feels fruitless, and finding ways to fix things and change course is way tough.

However I will say that I started using, with this class, the red/yellow/green solocup strategy for groups to self-assess where they are in terms of their own progress, and it’s been amazing. So that gives me hope for this idea working with these kids. Wish me luck!



[1] Okay, sometimes it isn’t private. I love when a group high fives when they figure something out. That happens when it is something hard-earned. Something they worked for. I also remember years ago a kid getting so worked up about understanding the sum of angles formula for sine that he literally fell in the floor. So sometimes the joy is visible. But I suspect that a lot of the joy that kids feel (when given the right kind of tasks, which put them at a place where they can have those hard-earned moments) often has a momentary and fleeting nature. I hope a ding! can give voice to those fleeting moments.




Technically Beautiful

Last week, Technically Beautiful opened. It is a math-art show that I helped put together (with another math teacher and a museum educator) at our school. We have a small teaching gallery at my school, and we wanted to do something special for this space which capitalized on our expertise. That is how Technically Beautiful came into being.

Technically Beautiful Card Art Draft 4 - 640px.jpgThe poster for the show. The name, by the way, came from a #MTBoS tweep!

In this post, I wanted to share with you the gallery virtually. (In a future post, whenever I get a chance to breathe, I’d love to talk about the programming we made around the show and possibly a bit about how the show actually came into being.)

Here’s a walkthrough video of the gallery:

Here’s our vision statement for the show:

The website for the show is here. The five artists featured in the show are: George Hart, Edmund Harriss, Veronika Irvine, the Oakes Twins, and Paul Salomon.

And lastly, here are photos of many of the pieces: