What’s MArTH? It’s the art in math, of course!
The logo for our all-school mathematical art event
Mathematical art has infected my school! It’s connecting students with mathematics in powerful ways, and it’s high time I write about it. I’m going to tell the whole story, and there’s a LOT to say, so I’ll try to break it up into sections. Maybe I’ll condense later on, but I want to get it out there. Enjoy the pictures!
Long story short – What started with a half year mathematical art seminar for high schoolers became a thriving community of artists and boiled over into a math art festival for over 650 students, called MArTH Madness, the largest, most successful math event in school history.
Justin and Anna have each written about it already, so I’m late to the game, but here goes.
* * *
A culture of mathematical creativity [Our Seminar]
By Hudson
This semester Justin, Anna, Max, and I are co-teaching a Mathematical Art Seminar, and “MArTH” is our pet name for the subject. Mondays, from 4:20-6:00, we get together with 10 or 15 students and explore the enormous world of mathematical art. We look at cool examples we find in books or on the internet. We discuss the mathematical concepts at play, but most of all, we spend time making and sharing mathematical art of our own. (We’re planning a gallery show this month.)
By Julien
We’ve played with photography, paper folding, music, video, computer programming, tiling patterns, polyhedral sculptures, and on and on. It’s been amazing so far how seemingly endless the world of mathematical art is. Better yet, the students just keep on digging it. Their enthusiasm flows over into the facebook group we set up to share our work and amazing links. Students post. Teachers post. Students comment. Teachers comment. Everyone’s there to engage in our little MArTH community in whatever way they want. It’s been incredibly positive and very lively. (social media note: ours is a secret group. No one sees our posts but us, and membership is by invite only.)
* * *
An Opuning [All-school event?]
Geofix Rhombicosidodecahedron
At Saint Ann’s, we have a Reading Marathon and a Poetry Marathon and Scene Marathon, but no such math event. We’d been talking about a Polyhedra Marathon for a year or so, but just talk. Given my belief that schools teach culture first and foremost, all-school events like these feel pretty important. Culture is built through common experience, after all, but apart from the SAT, we have very few shared math experiences. This is especially true at Saint Ann’s, where the math classes are highly individualized.
Sensing the opportunity for a terrific pun, I suggested we do some kind of “MArTH Madness” event at school. It was mid-March when I had the idea, so time was running out on this little pun, but Justin said, “we should at least do something this year.” Within the hour we made a preliminary pitch to the math department.
Pattern block spiral
The idea was this: We all teach in the same 4 periods of the day, so let’s have a sort of in-school field trip. Let’s bring our classes together and do all kinds of different mathematical art activities! Sometimes great ideas are that simple. They just need people to not say, “no.” In this case, about half the teachers said they would bring a class, so we had our vote of confidence, and I got to work.
* * *
2 weeks of pre-madness [PLanning and Prep]
3D Construction in the cafeteria
{Where}: Space was a major concern, especially since the initial numbers showed over 100 students per period, and teachers kept asking if they could come. (Even some science teachers brought their classes.) Worst case scenario, we could repurpose our classrooms, but we were hoping for a huge space to fill with people. The gyms were entirely reserved for gym classes, but the cafeteria staff was refreshingly cooperative and let us have the space as needed. Phew! With only a few exceptions, all 6 rooms in the undercroft (our basement) were used for math during our periods, so we made a few room switches with study halls and other teachers, so we could hold all of the other activities down there. My own classroom (in the undercroft) has a removable wall, so we were able to combine the rooms and get another big space for about 70 students.
Quintuple Helix – side view
{When}: We ended up choosing April 3rd for the event date, all but killing the pun, but it was the date we needed. Math periods were the first two of the day and the first two after lunch, so that was our time frame. That gave us 8:00-10:10 and 1:00-2:35. We had to move quickly in the lunchroom to clean up in between breakfast, lunch, and snack, but we made it work.
{Who}: Students from every grade 4-12 mixed together in the same places, working on projects of their own. Justin hung out in the cafeteria, Max taught in the large room in the undercroft, and students led every other session! Lots of seminar students helped out (9th-12th graders) and we even had a pair of 6th graders help lead a session on mathematical doodling. Anna and I floated around, and Sam Shah even came over from Packer. Other teachers were all around, following their students or just working on some art of their own.
Quintuple Helix – axis view
{What}: If we were going to be making art, we needed materials. We used money from the department’s supply budget to order scissors, straws, balloons, colored pencils, and markers from Quill, which came very quickly. We were able to get card stock, paper, and a few other things from the school supply closet. For computer-aided design we were able to use our netbooks and chromebooks as well as the department’s Makerbot. For the 3D construction space we used Geofix and some other stuff we already had, but mostly Zome. Zometool will mail out loaner workshop kits for a nominal fee, so we were able to get roughly 8,000 pieces of Zometool (2 kits of 4 boxes each) for about $150 after shipping. We also got to keep the Zome for the rest of the month and display the amazing works. There was a small mix up with the order, but the kind people at Zometool were willing to ship on short notice with rush delivery, and we got everything in time! All together material costs came in under $300.
{How}: The day before, I gave the teachers a flyer to share with students explaining the activities. I split up the classes so that half of the teachers would begin up in the cafeteria doing 3D construction while the other half chose between the activities downstairs. Half way through the period we would switch. Each period, new kids.
The “menu” we handed out to students the day before.
MArTH Shirt
{Bonus}: We wanted to get MArTH Madness T-shirts for the presenters, but the best price we could find online was $18/shirt, and they wouldn’t arrive until well after the event. Luckily a talented colleague prints shirts at home! He agreed to print our shirts for $14 each and in time, with only 4 days notice!!! Shirts always make an event look legit. The design comes from some art I was making for the seminar exploring the structure of star patterns. The back reads:
“Beauty is the first test. There is no permanent place in this world for ugly mathematics.” -G.H. Hardy
* * *
MARTH MADness! [The Big day]
The school was full of energy as kids were swept up in the mathematical creative process. They came from study halls and asked if they could miss classes or come during lunch just to make more math art. I walked around facilitating and squeezing in a little art of my own where I could. I’ll describe what went on in each room.
Sam Shah, proud of his tiling
{Escher Tessellations} – This was led by Hudson, an awesome 9th grader, based on some stuff he noticed about how to create tiling patterns a la M.C. Escher. We printed out squares and hexagons on card stock, which participants could cut out. These shapes each tile the plane, of course, and Hudson explained how to convert them to new shapes that would create customized tiling patterns.
With a square for instance, if you cut a shape out of one side and glue it onto the opposite side of the square, the piece will fit nicely with a copy of itself. Repeat as desired until you have some beautiful shape that you want to cover the plane with. Then trace it over and over to cover some paper and color to your heart’s content.
A basic hexagonal tiling
You can do the same thing with a hexagon, and there are several other variations. On the square, for example, if you flip the cutout over before taping it to the other side, you get a tiling with glide flip symmetry. Lots of students wanted to try things of their own and explore what was possible. What if I flip them on the hexagon? What if I do it like this? Will it work?
Max teaches Sonobe
Best part, when they were finished, we stapled the patterns to the wall outside, just underneath an AWESOME sign Hudson had made that morning. My fifth graders stare at them every day before class, and some have continued working on new designs. We stapled a new tiling to the wall on Friday during free-choice time!
{Unit Origami} – Max (seminar co-teacher) led groups of about 40 or 50 students in some paper folding. He showed them all (and they showed each other) how to fold a Sonobe unit, a very basic design, as well as how to combine them to make a cube.
Icosahedral sonobe ball
With only about 20 minutes to work, it was hard to get into anything more than this, but Max had some other models on display like an octahedron and icosahedron, and the kids got a sense that a lot would be possible from this basic little structure. Amazingly, the excitement from this session carried over into the following weeks, and I helped several students put together larger structures, which we left on display in the undercroft.
A page of doodles
{Mathematical Doodling} – This session was led by 4 experienced math doodlers (junior, sophomore, and 2 sixth graders), who shared their work and helped students come up with their own clever doodles. Usually one of them would be explaining a specific kind of doodle to a few students, while another two helped with a large chalk doodle at the board, and the fourth worked on a fresh design at a desk. We also had chromebooks available so that kids could watch some of Vi Hart’s “doodling in math class” videos for inspiration and jumping off points.
Chalk board Apollonian gasket
Doodles included Apollonian gaskets, fractals, polygons, stars, coloring patterns, and whatever else occurred to the artists. Once again, we were able to staple some of the work to the wall for display. There is something so simply satisfying about filling your school’s walls with student work.
Symmetry Artist piece 1
{Computer-Aided Design} – We had two rooms devoted to two two different activities using computers, both led by students. In one room we had chromebooks set up so that kids could use Symmetry Artist, a wonderfully beautiful applet. It let’s you select various rotational and flip symmetries, then it copies whatever you draw so that the result has the desired symmetry. The kids used the “line” option to make stars and polygons incredibly easily. As they drew, the image changed symmetrically, possibly suggesting mathematical dance. When I stopped by, I showed off a few of my ambigrams, and a few played with the idea. If you’ve never played with Symmetry Artist, you should give it a try. The symmetry itself is so beautiful you almost can’t go wrong.
Symmetry Artist piece 2
In the other room we had our set of 10 netbooks set up so that kids could do some algorithmic programming in Scratch, but the Makerbot stole the show. Our department got a Makerbot 3D printer earlier this year, and we’ve been playing with it since. We brought it down to the undercroft so kids could watch it print and print their own pieces, which they designed on 3DTin. We used that site mostly because we hadn’t yet installed OpenSCAD, and we didn’t yet realize how much better Tinkercad is. (Which is A LOT better.) Unfortunately, the ReplicatorG was glitchy after the move, so we didn’t get anything printed that day. Nevertheless, Noah, one of the seminar students, spent Monday afternoon in the computer center, printing out the MArTH Madness designs for the people that came to his activity.
I’ll just add here that the kids were unbelievable teachers for each other. They were patient, kind, and totally passionate. They were an enormous help, and the event would not have been such a success without them. Students led 4/6 sessions, after all!
Artists at play
{3D Construction} – Justin was in charge of this space, but most of the work was completely unguided. After wiping off the lunch tables, we just set out boxes of Zome, Geofix, Pattern blocks, Unifix Cubes, and Polydron. Free build! Kids jumped in and built towers and solids and all sorts of spikey balls. A simple start often led to an idea and then a plan, at which point the student has to wrestle with the mathematical possibilities until they arrive at a satisfying end result. Then the process spirals on, as one good turn yields another. Against one of the walls we set up a display (just stuff we had already made) to give inspiration, and more pieces got added throughout the course of the day. I loved watching kids snap pictures with their phones. There’s nothing like the feeling of satisfaction and success you feel after building something you like!
MetaZome Cube
Justin spent his time working with a coming and going group of students to build a metaZome structure. They used the zome pieces to build larger versions of the zome pieces. (You know I love that kind of thing.) He wrote about it here. I love that they didn’t quite know what it would ultimately be. The project had space to grow and build steam in whatever direction it might, and in the end they put together a beautiful metaZome Cube with diagonals and a center node. After the Madness had subsided, we carried the structure to the undercroft, where it stayed on display for the rest of the month. We had to return the pieces last Friday, and student after student walked by saying, “Awwwww. Don’t take it apart!”
“Don’t worry,” we said, “next year we’re going to make something even better!”
* * *
MArTH on! [Reflections]
I am incredibly proud of the work we do with math at Saint Ann’s, and this newborn mathematical art program exemplifies a lot of my educational beliefs: (I cannot speak for the department.)
1) That math is something people do, and that we should be getting kids doing mathematics as much as we can. Well that happens every day in our seminar, and students are still riding the MArTH Madness doing-wave.
2) That math can be done by everyone – regardless of age, grade, race, gender, and to some extent, regardless of mathematical preparation. Whether you’re a seminar student, a 4th grade girl, a 10th grade boy, or a 50 year-old teacher, you can make beautiful mathematical art, starting immediately. And man, does it feel good!
3) That math classrooms should be learning communities – groups of people that work together to learn and create. This is exactly what our seminar is, and the all-school event was a summit that brought us together in unprecedented ways, around beautiful mathematical work.
4) That the determining factor in the success or failure of many endeavors is personal significance, and that students can thrive in an environment of free creative choice. Mathematical art is entirely about finding some part of the inexhaustible mathematical world that you can explore and call your own. Original mathematical art provides personal attachment and identity with a subject that desperately needs it.
5) That there’s more to an education than the transmission of data and technique, and that learning should be highly individualized. When we made Escher Tilings, we shared in the experience, but everyone was creating something unique, for themselves. What a student takes away from a room full of 3D constructors is highly nonstandard, beyond their own unique creation. Maybe they learn about Platonic solids, polygons, symmetry, or graphs or something. Or maybe they just learn that they are capable of bringing new things into being. They themselves can create mathematics.
The ideals of MArTH Madness can absolutely carry over into our students’ lives and into our math classrooms. After experiences like these, a student’s relationship with math can change drastically. When presented with new concepts, they may see a chance to play and create where previously they saw nothing but work. Far too often, math is math class to our students, and it stops at the door. It rarely extends beyond our own class, or grade, or school, but MArTH Madness showed us all that math can break these boundaries. We can work together at any age. We can all have original math ideas. We can all create beautiful things. We can make it on our own, we can notice it in the world around us, and we can all find ourselves in mathematics.
MArTH Madness teaches us that we are all mathematical.
* * *
Now, the seminar carries on with full-steam, in its basement room, even on beautiful afternoons. Next year we’re offering it full-year, as well as a middle school elective, which has heavy enrollment. Students and teachers are already looking forward to next year’s “Madness,” which may last over several days. We bring MArTH in and out of our classes, but best of all, it’s in the hands of our students, right where math belongs!
For more pictures visit the MArTH Madness Facebook page.
Paul Salomon
Lost In Recursion 