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:



Binomial Expansion

This is going to be a super short blogpost. But I’m excited about a visualization I came up with today — as I was working on a lesson — for showing why Pascal’s Triangle works the way it does with binomial expansions.


I’m sure that someone has come up with this visualization before. It feels so obvious to me now. That that didn’t make me any less excited about coming up with it! I immediately showed it to two other teachers because I was so enthralled by it. #GEEKOUT

I am thinking how powerful a gif this would be. Start out with 1. Have two arrows emanate from that 1 (one arrow saying times x and one arrow saying times y) and then it generates the next row: 1x    1y. And again, two arrows emanate out of both 1x and the 1y (arrows saying times x and times y). And generating 1x^2    1xy     1xy     1y^2. Then then a “bloop” noise as the like terms combine so we see 1x^2     2xy     1y^2.

And this continues for 5 or so rows, as this sinks in.

Then at the very end, some light wind chime twinkling music comes up and all the variables disappear (while the coefficients stay the same).

Of course good color choices have to be made.

Who’s up for the challenge?

Okay, I’m guessing something similar to this already exists. So feel free to just pass that along to me. Now feel free to go back to your regularly scheduled program.

Visualizing Standard Deviation

A few days I got an email from someone (Jeremy Jones) who wanted me to look at their video on standard deviation. And then today, I was working with Mattie Baker at a coffeeshop. He was thinking about exactly the same thing — how to get standard deviation to make some sort of conceptual sense to his kids. He said they get that it’s a measure of spread, but he was wondering how to get them to see how it differs from the range of a data set (which also is a measure of spread).

Of course I was hitting a wall with my own work, so I started thinking about this. While watching Jeremy Jones’s video, I started thinking of what was happening graphically/visually with standard deviation.And I had an insight I never really had before.

So I made an applet to show others this insight! I link to the applet below, but first, the idea…

Let’s say we had the numbers 6, 7, 7, 7, 11. What is the standard deviation?

First I calculate the mean and plot/graph all five numbers. Then I create “squares” from the numbers to the mean:


The area of those squares is a visual representation of how far each point is from the mean.[1] So the total areas of all those five rainbow squares is a measure of how far the entire data set is from the mean.

Let’s add the area of all those squares together to create a massive square.


As I said, this total area is a measure of how far the entire data set is from the mean. How spread out the data is from the mean.

Now we are going to equalize this. We’re going to create five equal smaller squares which have an area that matches the big square.


We’re, in essence, “equalizing” the five rainbow colored squares so they are all equal. The side length of one of these small, blue, equal squares is the standard deviation of the data set. So instead of having five small rainbow colored squares with different measures from the mean, the five equal blue squares are like the average square distance from the mean. Instead of having five different numbers to represent how spread out the data is from the mean, this equalizing process lets us have a single average number. That’s the standard deviation.



I’m not totally clear on everything, but this visualization and typing this out has really help me grok standard deviation better than I had before.

I created a geogebra applet. You can either drag the red points up and down (for the five points in the data set), or manually enter the five numbers.

My recommendation is something like this:

  1. {4, 4, 4, 4, 4}. Make a prediction for what the standard deviation will be. Then set the five numbers and look at what you see. What is the standard deviation? Were you right?
  2. {8, 8, 8, 8, 8}. Make a prediction for what the standard deviation will be. Then set the five numbers and look at what you see. What is the standard deviation? Were you right?
  3. Set the five numbers to {2, 4, 4, 4, 6} and look at what you see. What is the standard deviation?
  4. Consider the number {5, 7, 7, 7, 9}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #3. Then set the five numbers to {5, 7, 7, 7, 9} and look what you see. What is the standard deviation? Were you right?
  5. Consider the numbers {3, 7, 7, 7, 11}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #4. Explain your thinking. Then set the five numbers to {3, 7, 7, 7, 11} and look at what you see. What is the standard deviation? Were you right?
  6. Consider the numbers {3, 6, 7, 8, 11}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #5. Explain your thinking. Then set the five numbers to {3, 6, 7, 8, 11} and look at what you see. What is the standard deviation? Were you right?
  7. What do you think the standard deviation of {4, 8, 8, 8, 12} be? Why? Check your answer with the applet.
  8. Can you come up with a different data set which matches the standard deviation in #6? Explain how you know it will work.
  9. Set the five numbers to {4, 4, 4, 4, 4}. Initially there are no squares visible. The standard deviation is 0. Now drag one of the numbers (red dots in the applet) up. Describe what the squares look like when they appear? Eventually drag that number to 15. What do you notice about the standard deviation? Use your understanding of what happened to describe how a single outlier in a data set can affect the standard deviation

Okay, I literally just whipped the applet up in 35 minutes, and only spent the last 15 minutes coming up with these scaffolded questions. I’m sure it could be better. But I enjoyed thinking through this! It has helped me get a geometric/visual sense of standard deviation.


Now time to eat dinner!!!

 Update: a few people have pointed out that the n in the denominator of the standard deviation formula should be n-1. However that would be for the standard deviation formula if you’re taking a sample of a population. This post is if you have an entire population and you’re figuring out the standard deviation for it. 

[1] One might ask why square the distance to the mean, instead of taking the straight up distance to the mean (so the absolute value of each number minus the mean). The answer gets a bit involved I think, but the short answer to my understanding is: the square function is “nice” and easy to work with, while an absolute value function is “not nice” because of the cusp.

Good Conversations and Nominations, Part II

This is a short continuation of the last blogpost.

In Advanced Precalculus, I start the year with kids working on a packet with a bunch of combinatorics/counting problems. There is no teaching. The kids discuss. You can hear me asking why a lot. Kids have procedures down, and they have intuition, but they can’t explain why they’re doing what they’re doing. For example, in the following questions…


…students pretty quickly write (4)(3)=12 and (4)(3)(5)=60 for the answers. But they just sort of know to multiply. And great conversations, and multiple visual representations pop up, when kids are asked “why multiply? why not add? why not do something else? convince me multiplication works.”

Now, similar to my standard Precalculus class (blogged in Nominations, Part I, inspired by Kathryn Belmonte), I had my kids critique each others’s writings. And I collected a writeup they did and gave them feedback.

But what I want to share today is a different way to use the “Nomination” structure. Last night I had kids work on the following question:


Today I had kids in a group exchange their notebooks clockwise. They read someone else’s explanations. They didn’t return the notebooks. Instead, I threw this slide up:


I was nervous. Would anyone want to give a shoutout to someone else’s work? Was this going to be a failed experiment? Instead, it was awesome. About a third of the class’s hands went in the air. These people wanted to share someone else’s work they found commendable. And so I threw four different writeups under the document projector, and had the nominator explain what they appreciated about the writeup. As we were talking through the problem, we saw similarities and differences in the solutions. And there were a-ha moments! I thought it was pretty awesome.

(Thought: I need to get candy for the classroom, and give some to the nominator and nominee!)

The best part — something Kathryn Belmonte noted when presenting this idea to math teachers — is that kids now see what makes a good writeup, and what their colleagues are doing. Their colleagues are setting the bar.


I also wanted to quickly share one of my favorite combinatorics problems, because of all the great discussion it promotes. Especially with someone I did this year. This is a problem kids get before learning about combinations and permutations.


Working in groups, almost all finish part (a). The different approaches kids take, and different ways they represent/codify/record information in part (a), is great fodder for discussion. Almost inevitably, kids work on part (b). They think they get the right answer. And then I shoot them down and have them continue to think.

This year was no different.

But I did do something slightly different this year, after each group attempted part (b). I gave them three wrong solutions to part (b).


The three wrong approaches were:

And it was awesome. Kids weren’t allowed to say “you’re wrong, let me show you know to do it.” The whole goal was to really take the different wrong approaches on their own terms. And though many students immediately saw the error in part (a), many struggled to find the errors in (b) and (c) and I loved watching them grapple and come through victorious.

And with that, time to zzz.

Nominations, Part I

At TMC this past summer, Kathryn Belmonte introduced an idea about sharing student work in the classroom. Something she termed “NOMINATIONS!” I loved the idea — and wanted to use it when kids do their explore-math project. But I saw it was so flexible, and pretty early on, the time was right to test it out. So I modified it slightly and this post is about that…

In all of my precalculus classes (I teach two standard sections and one advanced section), my kids are being asked to do tons of writing. A few who have had me before in geometry are used to this, but most are not. And honestly: getting down what mathematical writing is, and how to express ideas clearly, is hard.

So what do I do? I throw them into the deep end.

On day two of class, I ask them to write an answer to a problem for a seventh grader to understand. On the third day of class, they come in, and are given the name of the student who comes after them alphabetically (and the last person is given the name of the first person alphabetically). Then they read these instructions:


Everyone moves to the desk of the name they were given. Then I project on the board:


And I give students to read through a different student’s solution. They have to make sense of it — pretending to be new to the problem. And then they critique it. Eventually, probably after 3-5 minutes, I left them return to their seats. They read over the comments. I talk about why the feedback is important. And how specific feedback is useful (so “good explanation” is less useful than “your explanation of how the groups were made was easy for me to follow”). And then we continue on with class.

Here are examples of some post-its (front of a few, then back of a few):

To follow up: that night for nightly work, I gave students a writing problem — a simple probability problem. My hope was that this would help them pay attention to their explanations. I collected the problem and read through the writeups.

They weren’t so hot. Most of them didn’t talk about why and some didn’t have any diagrams or visuals to show what was happening with the problem. So I marked them up with my comments. (They got full credit for doing it.) The next day I handed them back and shared my thoughts. I also shared a copy of a solid writeup — one that I had created — along with four or five different possible visuals they could have used. (I realized –after talking with Mattie Baker about this — that I couldn’t really get my kids from point A to point B unless they saw what point B looked like, and what my expectations were.)

At this point, I wanted to figure out if they were taking anything away from all of this. So I created a page with three questions. A formative assessment for me to see what my kids understand and what they don’t about the content. But I also asked them to take all the feedback they’ve gotten about writing and explanations, and explain the heck out of these problems. Here’s an example of one of the problems (one I’m particularly proud of):


I collected them today. I haven’t looked through them carefully yet, but from a cursory glance, I saw some thoughtful and extensive writeups. And even from this cursory glance, I can see that these two activities — plus all the conversations we’re having about explaining our thinking in class — have already made an impact.

Yes, they’ve gotten some ideas of what a good writeup looks like. They know diagrams can be helpful. They know words to explain diagrams are important. They know the answer to why is what I’m constantly looking for when reading the explanations.

But more important to me is the implicit message I’m trying to send about my values in the classroom. I think a lot about implicit messaging to communicate my values, especially at the start of the year. And I am confident my kids know with certainty that I value all of us articulating our thinking as best as we can, both when speaking but also when doing written explanations.