Math Circle: MacMahon Squares

When I got an invitation from Judy Wheeler to come lead a math circle activity I jumped at it. I've never been, but have wanted to for so long. Sue Van Hattum's influence, no doubt, with the great math circle stories on her blog and Playing With Math.

Judy asked for my topic during #tiles week in the #MathPhoto15 challenge. That had provoked an in depth discussion on tiling vs tessellation, and whether aperiodic tilings were tessellations, and then somebody mentioned Wang Tiles. Wang Tiles? Oh are those cool. Digging around about those led to finding a very fun series of blogposts from Steve Natusiak on MacMahon tiles. (Here’s the first. One cool sequence. The whole schmegegge.) These tiles were introduced early in the 20th century by Percy MacMahon and are just a lovely construction. MacMahon's idea was squares with colored edges, that you could tile if the matching edges colors matched.

I read up on math circles protocols, and headed for Kalamazoo. (After some printing troubles, which had the Xerox actually spewing curled up sheets of paper into the air like in a sitcom.) I prepared a Google doc, mostly for follow up resources, and some blank tiles (pdf). Judy said they would have scissors and markers.

In the break before my session, LuAnn Murray posed a sticky not problem. How many of the numbers between 1 and 100 can you make with 4 9's  and any operation found on a calculator. (So exponentiation and square roots - despite the implied 2 - are in.)

I showed a couple tiles (P,P,R,G & P,P,G,R), and asked what they noticed. First question: do all three colors need to be present? So I also showed all Green.  The observed the properties and I asked what I was going to ask them. (So meta. But a room full of teachers, so...) Correct: how many tiles? I asked for estimates ahead of time, which ranged from 12 to 1296. They jumped to working right away and then the clarifying questions began. Biggest: if you rotate them and they match are they different? I asked what did they think? Unanimously, they thought those should be the same. What about flipping? Different. "So they're not colored on both sides!" Is no color an option? (No.) Is red, red, green, purple different than red, green, purple, red? (Yes.)

People worked on lists, making diagrams, a few trees and a couple purely combinatorial approaches. A few were actually coloring them out. I let them know that each table would need a set for the next part, which encouraged some more actual coloring.  A couple times I polled the tables for how many they thought, and answers started to converge. When there was agreement but not yet unanimous, I brought them together to share. One teacher jumped up right away: these were all the ones with four red sections, three red sections and two red sections. She didn't do one red, because that would show up in the other tiles. The green, watching out for repeats then purple. Went down by two each time. One person brought up her list with less, and we worked together to figure out what was missing. "It's hard to figure out what's left out!" So how do we do it?

Then I asked: now what? We've got these tiles figured out. What should we do next?

I was really curious to see what kind of problems were posed. Here's what they suggested...

 

Michael Tanoff takes off when I start...
comes back and he's got the book!
Very cool.

Good extensions! Very representative of usual math teacher extensions. But I wanted play with the tiles we had, so I put on a restriction of using these tiles and the rules we were given. Immediately they posed the rectangle problem - which is what I wanted to get to, and which was MacMahon's original puzzle. I gave them his extra condition, that all the colors match on the outside edge. I want to think more about the kind of extensions we do in math class, because it seems to me we extend to big general ideas versus the kind of closely related problems where mathematicians are more likely to start. As a profession we do more of the 'let's make this harder' extensions than 'here's a parallel problem.' I think.

Now they were playing! 

They had several different approaches to this, as well, but it was much more collaborative in general. Maybe because most tables only had one set of tiles made from the first half. They posed conjectures pretty quickly. They gathered data about how many triangles of each color. They got close and tried small swaps, but also realized that some configurations were a dead end and required starting over.

Good problem.

Nobody was quite done when our time was up. I assured them there was a solution, then stole a couple minutes from working for a reflection and explained why reflection is so important to me. They'd been focused on the SMP, so I asked them to think about SMP1 - especially the perseverance. I asked them to share at table and then just a couple shared with the whole group. They pointed out how I asked more questions than told answers, but encouraged, too. They mentioned how working together helped with perseverance and the problem solving. They appreciated the different methods that people had.

All in all, I was pleased. I think this problem got at the spirit of the math circle, and had plenty of problem solving opportunity. The teachers were great and showed a lot of strong mathematical thinking and practice. And they continued to work on the puzzle while I was leaving.

P.S. Totally an aside & a plug: one of the other benefits of the tiling discussions was that I finally got around to making a sketch for all 17 wallpaper groups in GeoGebra. 

I'm pretty happy with how it turned out, but am very open to suggestions.

Making Whoopee at TMC15

Whoopee means games, of course. (For the song, Ella or Ray are the best options. Though I suppose I should go with Dr. John - no relation, despite the resemblance.)

Spoiler: I got rambly here. If you're going to not read this, here's two quick takeaways: the four new games and James' start of a game/curriculum alignment.

I was delighted when James Cleveland asked if I'd do a Twitter Math Camp morning session (meaning three 2 hour sessions) on math games with him. He'd led a one hour session at TMC13, and Sebastian Speer led another at TMC14, but this would be the first one with time to really make new games.

The format seemed pretty natural and intuitive: look at some games together, introduce a few principles, and get making. Just in case, we each had some ideas for games in case people didn't have any urges, and James had a couple of neat statistics games ideas burning a hole in his pocket in particular.  I had a mechanic; my family recently discovered Sushi Go, which has a simple and elegant drafting mechanic. There was also this Tug of War that I had been discussing with Nora Oswald  based on a Daniel Solis idea.

The plan:

• Day 1 - play good games and discuss.
• Day 2 - start design.
• Day 3 - playtest.

Writes itself, eh? Materials from the first day are mostly here on the TMC wiki, and the 2nd & 3rd days in a Google doc. (Including rules for the developed games.) 

Day 1 Games
I brought Linear War  , but we didn't actually play! I considered it for day 1 because I like how the students make the cards for the game (learning part one) and then play (learning part two) working on vocabulary, concept, quick recognition and computation. We did play: (in order of complexity)

• Product Game: Illuminations, handout (original & integers), post (decimal). My nominee for best math game ever. Comes at the content from multiple ways, amazing replay value due to the deep strategy, quick to learn, structure supports students in learning the content, adaptable... The only thing missing is context, but this would distract from it as a strategy game. Teachers thought of several different uses for this immediately.
• Quod Game/Metasquares (app not currently in US app store) All you need is a grid, and the strategy is deceptively deep. Subtle approach to content, though, as there is a great mathematical structure, but it's more about noticing it than learning it.
• Factor Draft, James' great game. Interesting in that you can parallel play or interact a lot. Really requires the mathematics that it concerns. Needs its own pieces manufactured, but they can be used for multiple purposes. Great example of development in balancing the pieces for interesting play. High cognitive load game, lots of challenge.
• Domain Ranger, post 1 and post 2. Norah's serious game. It's an intense strategy game, for which you need the math ideas of domain and range, and the ability to compare different graphs. Participants had awesome suggestions about this. Recognizing the difficulties in learning such a complex game, they thought about doing a 1-dimensional board set up learning game. And also the great idea of doing a preset first game, Settlers of Catan style. I'll try to work one up the next chance I have to use the game.

It became clear pretty quickly that this was going to be a good couple of days. We picked these games because they all are content focused. I do not have any problem with review games that fit any content. (In fact, here's my list.) Instead of that, we were looking to design games for specific content. Where the game play was the learning activity. Day 1 was promising because the group as a whole was really able to focus on what aspect of the content the games addressed, and where in the lesson/unit on that  content it would be appropriate.

Harvey Mudd had CHALK boards. Deja view.

Day 2 Design
We started off this day with a look at Decimal Pickle, maybe my best game, with a focus on desing thinking, the mathematical goal, and how the mechanic works in the game. One of the most interesting parts about preparing with James was thinking about classic game mechanics are use of them in math games.

Classic Game	Math Game version: K-7	8-16
Apples to Apples, Dixit		Graphles to Graphles
Blackjack	Zero!
Connect 4/Tic tac toe		Exponent & factor block game
Guess Who		Polygraph
Racko	Decimal Pickle
Rummy/Concentration	Unit Rummy	Trig Rummy
Taboo		Algebra Taboo
Uno		like terms UNO
War		Log War, Trig War,
Wits & Wagers		Ideas for Use

 
There's a lot of room for addition in there. I'd also like to hear your thoughts about what's missing. Even just writing this I got thinking of Farkle and Yahtzee. (King of Tokyo is an example of a tabletop game that uses that great Yahtzee mechanic. I have an upper el math game that's a direct rip off adaptation of Yahtzee, too.)

For design principles, I have this goofy list of 9 I use as a framework (adapted from Mark Rosewater of Magic fame). We emphasized just a few-

1. Goal(s). Design starts with objectives. (Whole point of Day 1.)

4. Interaction.

5. Surprise.

6. Catch-Up. As you start to playtest, these two are important to attend to for good design.

 
What's really promising from the prep for this day, though, is James' start of a spreadsheet for curriculum aligned games. Here's  a Google spreadsheet version - open for editing. If you know of things to fill in, PLEASE DO. If you have a hole you especially want addressed, let us know on Twitter.

People got designing pretty quickly. We divided into 4 groups, working on statistics (James was in this group), Fraction operations, Arithmetic Sequences, and Unit Conversion. I floated amongst the groups. This was a bit of a breakthrough for me. I design mostly in isolation. But (like for most things) collaboration was energizing, powerful and fast. Between this part of day 2 and some wrap up on day 3, we finished four good games. 2-3 hours of work. My contributions floating were questions, connections with other game experiences, and the occasional idea.

Day 3 Playing
Also today, we took time to do some rule writing:

1. Rules. For me this comes late; kind of a synthesis step as you think about how to communicate the game. It will often result in design revision, though.

 James knew of a good blank template for rules writing.

People needed a little time to finish. We had a good Skype chat with Nora, who shared her experiences playtesting, took people's questions, then discussed some of their interesting feedback on Domain Ranger. Dave Chamberlain (participant) shared his published game of Team Up! which is a 4-12 common core review board game, and some of his process. Also what it took to get it in commercial finished form. 

James gives a good write up of the statistics game, Fighting for the Center. Use playing cards, players build a data set that meets some goals (measures of central tendency) hidden from the other players. It's great at making players think about how changing a data point affects those measures. Lots of interaction, since you both are playing on the same data set, catch up is not an issue, and students will find more means and medians than they ever would in a homework set.

The fraction game is about addition and subtraction, modeled a bit on the Connect 4/Product Game framework. The board is really interesting, by asking for ranges, which really leads students to using representation (on the fraction cards we had). The teacher may want a way to get students to add precisely. I think there's some more playtesting to do here, too, as the placement of the various squares was more about coming up with them. 








Honeycomb Madness, unit conversion game. This is a classic board game, and the closest to being a general review game. You start in the outer ring, and are trying to get to the inner ring by answering questions. The ring level serves as a kind of rubric, though, and might support some kind of awareness in students as to different levels of understanding of the material. There's a nice bit of randomness that's reminiscent of Trivial Pursuit. I liked that it is not the kind of game I might design; I think it might be quite popular with students, too.

Arithmetic Sequence Game. This one is right up my alley, though. Deck of playing cards. Deal three: starting value, common difference, step number. That determines a target. Each player is dealt 5 cards, and tries to get as close as possible. Then the idea that complete changes it: you bet on your play, 1-6 points. Closest gets the points everyone bet. 2nd closest gets their bet back.  Wow. Plays great. I'll be trying this out, next algebra class for sure. I made a GeoGebra sketch to help with the calculation and to practice. 

Thanks to James and all the participants. I feel like I learned a lot about collaborating in game design, and broadened my tastes a bit, too. This more than ever makes me want to get students designing games, so if you're in the area and wouldn't mind a mathematician in the room...

 

Where Do I Start?

Oh, that's longer than a tweet my friend.

Jim Doherty, @mrdardy to you, recently got a lot of great responses to this (basically) question on Twitter (link to conversation). So I want to archive them someplace. My preservice HS teachers have a pretty good collection, but maybe I should think how I use them. The lists below are in order of how frequently I use them. I'm a weird one, though, so it's not meant to be a ranking on quality.

So I want to teach a new lesson about something. I go to...

Search

• John Stevens' MTBoS Search Engine bit.ly/MTBOSS. Helps me find that thing I read that I should have bookmarked and who was it... 
• Robert Kaplinsky's Problem Based Lesson search engine. More focus for when you need it.

The Big Ones

• NRICH - Cambridge's problem site that continues to grow and expand. Searchable by topic, age level, and challenge level.
• Illustrative Mathematics, free ever-more-complete curriculum 
• Shells Center/Mathematics Assessment Project, good as lessons, problems or assessments. Often with sample student work.
• Georgia Standards, even if they're sadly going away from them. A complete, free Common Core curriculum, generally of more consistent quality than Engage NY.
• Mathalicious, paysite but worth it.
• Math Forum, in process of changing to being an NCTM program/ally/thing. I have high expectations.

Community Collections

• Fawn Nguyen's Visual Patterns and Math Talks. I use Visual patterns a lot. Problems, investigations, resource for students... Math Talks I should use more. Great bits of dialogue and student thinking, on simple but rich questions.
• Michael Pershan's Math Mistakes. Valuable as a teacher for thinking about misconceptions, good reasource for mistakes for students to look at to try to fix or look for reasoning.
• Dan Meyer's Google spreadsheet of 3 Acts lessons 
• Mary Bourassa's Which One Doesn't Belong. Built on the idea in Christopher Danielson's great shapes book, but expanded to many content areas. These are hard to make and great to have collected. 
• Open Middle problems by Nanette, Robert and Bryan.
• Desmos Activity Bank (new but I'm confident in greatness-to-come)
• MTBoS Activity Bank (spreadsheet; we need something like this. Maybe this is it?)
• #MTBoS collection of community efforts (some of the above and more...). Estimation 180 and Would You Rather? are both good sources of warm up/cool down questions.

Amazing Personal Work

• Don Steward's Median. Lots of lessons, often with an interesting visual component, and very clever. Awesome variations on a them within each set of problems or activities.
• James Tanton's Curriculum Resources. YouTube videos that are problem & thinking centered. I like that he separates the problem, and gives learners a chance to do it.
• Brian Mark's Yummy Math, real real-life problems.

Outstanding Curation

• Geof Krall's Problem Based Curriculum Maps, MS and HS, traditional and integrated. I use these regularly and recommend them frequently. So important, and done so well.
• Sam Shah's Virtual Filing Cabinet, so many great lessons from the #MTBOS. Sam's taste is impeccable.
• Tina Cardone's #matheme page (bunches of math teachers writing on selected, typically practical, themes.)
• Jo Morgan @mathsjem's Resourceaholic. Weekly blogposts highlighting up to 5 new resources for teachers.

More Teachers' Virtual Filing Cabinet (Link Collections)

• Nathan Kraft's VFC
• Jim Doherty's VFC 
• Timon Piccini's VFC
• Bowman Dickson's VFC
• Kristin Fouss's Diigo links 
• Wendy Menard's VFC 
• Chris Hunter's VFC 
• Lisa Bejarano's VFC of Warm Up Activities

Miscellaneous

• Inside Mathematics' Problems of the Month. Generally interesting, sorted by grade level. More at the Inside Mathematics site than these, too.
• Brilliant.org. User submitted problems in multiple choice format. Inconsistent, but some are... well, you know.

New to Me (both from Jim's tweet, but solid recommenders.)

• Math Bits via Wendy Menard
• APlusClick via Megan Schmidt, problems sorted by grade and content

Games

• I'll plug my math games page here, especially the links to others' games and review games.
• James Cleveland's spreadsheet, trying to find math games for major content topics throughout high school. 

And that's not all, folks.  But at least it's a start, right? Please feel free to mention your starting points in the comments.

P.S. Glen Waddell wrote a nice Teachers Share with Teachers response to a poor NYT article. It has a lot of great resources with descriptions.

An Allegory of Passion

Saw this painting, An Allegory of Passion, at the Getty Museum on my way out of LA from TMC15. (By the way, the Getty's AMAZING. I'd go back to LA just to see it again.) Thought immediately that it was a good summation of camp.

My spouse gets endless amusement from
sending me to camp.

Last year I was looking forward to Twitter Math Camp, but nervous. I'm a stereotypically introverted math geek, and people... <shudder>. Of course, these are my people, and it was amazing. Things gleaned from that meeting include Counting Circles and Talking Points, which had huge impact on my practice last year. The subtler effect was making these people whose ideas and opinions matter to me more real, and the whole community as a consequence. I'm kind of a teacher fanboy, anyway, but these are special folks.

And if I thought expectations were high last year, sheesh. I spent a lot of the year telling people that this was the best professional gathering I had ever been to. How could it come close?

By being all these amazing people from this community. Such good folk, sharing their hearts for teaching and students.

Twitter Math Camp has a morning session that runs for 2 hours all three days. Goal is to collaborate on larger projects. Gathered as a whole group, there are My Favorites, 5 minute quick shares of something that makes a difference to your teaching. Each of the first 3 days has a keynote, and then two 1 hour sessions to follow up on work from the morning or on separate topics. Again, much more participatory than at your run of the mill conference. The Twitter Math Camp Wiki has a large portion of materials and resources from all the sessions, and this year a lot of video.

James Cleveland and I ran a morning session on making math games. It was great fun, and a learning experience for us, too. Separate post on that. (I hope.) The only drag about running a session was not being able to attend a session. I'd hear Elizabeth Statmore talk about anything, but the session on Exploratory Talk would be great. Talking points (I first saw at TMC14) was a big addition to my PD and teaching last year. Or to be in a room with the Desmos gurus...

I just want to share some of what has stuck with me.

Chris Shore did a My Favorites on Neuron Stickers. Here's a tip: if you ever have a chance to interact with this guy, do so. Amazing teacher and person. In this quick pitch, he described work with a group of students who had all failed 8th grade math, in an algebra class. He focused all year on growth mindset, and helped students develop their own culture of recognizing big ideas. Not just in math, either. Spoiler of the story: 100% pass rate. See also his Rally for Roatan.

Lani Horn gave the day 1 keynote. She's the very model of a modern math ed researcher. She focuses on important issues, goes about it in a holistic way that respects teachers, and has keen insights into the data she collects and conversations she holds. In particular here, she presented her best idea as to what separates good teachers and great teachers.

• how do teachers describe teaching problems? Great teachers think of them in actionable terms.
• how do teachers represent instruction? Great teachers are heavy on context and student voice.
• how do teachers interpret their situations? Great teachers have beliefs built on connections amongst student learning, mathematics understanding, and ideas about teaching. She described this as ecological thinking. Research shows this is the most difficult thing to affect in teacher candidates.

The afternoon sessions were basically time with my favorite teachers. Megan Schmidt  led a session on powerlessness in teaching. It was a deeply moving session. I'm a big believer in 12 step programs, AA gave me 30 years with my father I would not have had otherwise, and Megan has been brave enough to bring that kind of grace and support to the MTBoS. Read her post about it. Sadie Estrella led another confidential session afterward, where she conducted a half hour of excellent and energizing teacher stories.  It was indeed a half hour of cool.

Day 1 closed with Art Benjamin's Mathemagic. I'm not super-keen on his TED talks, but wanted to give him a try in person. I still don't know what to make of him, and wonder whether he's confirming more biases, ruining more good problems, or just putting a fun face on math. He didn't seem to have a sense of the audience, though. I did make a GeoGebra version of his guessing game that doesn't give away the trick, though.

Some of the Day 2 favorites that stuck with me include: John Stevens and his MTBoS search engine. Great bit.ly address: http://bit.ly/MTBOSS. I use it all the time. Glen Waddell talked about the deep and relationship building practice of high fiving his students on the way into the class. Not because of a school policy, but because he commited to them to do it. When do you get a high five? When you do something awesome. Heather Kohn did a bit on her use of 3-D printing in a Desmos project. I've got to look into that. (That link goes into an effort for 3-D teachers to share resources.)

Day 2's keynote was Christopher Danielson. (Video, text) The most productive man in math ed. Books, new math memes, Desmos activities, Oreo controversies... what hasn't he done? His message was simple:

Find what you love.

Do more of that.

Which he then proceeded to model for us by showing how much of his work grows out of his love for ambiguity. This gets to the heart of one of my beliefs about teaching: it is personal. It's a deeply human activity. The best teachers are ones who have found a way to make their teaching true to themselves. Giving up authority - in the you're an author of your teaching sense - is a soul killer.

I think what I love about math is play and puzzle. In grad school I used to tell people they paid me (not much, of course) to play. There's a lot of play in my classes, but I need to work on the sharing of it.

In the afternoon, I went Lani's session on eliciting student thinking. She's writing a prequel (based on real classrooms, of course) to the great 5 Practices book. This will be good - especially for my preservice teachers, I think. Bottom line: foster intellectual risk taking. I followed that up with Robert Kaplinsky's session on formative assessment by questioning. Two pointers: avoid yes/no questions and hit the hows and whys. David Coffey (via Twitter) thought her list of "Belonging, competence, meaningful, autonomy" sounded a lot like Brian Cambourne's list of requirements for engagement.

This afternoon also saw the big news of Math Forum moving to the NCTM and Nguyen and Vaudrey as Barbie and Ken.

For some reason no one was taking pictures of Ken.

That night the California guys and Mathalicious put on a Bar-B-Que. I got to listen to Lisa Henry relate the tale of the Twitter Math Camp. This is a special community, but I don't think these would happen without her and her husband. Mil gracias, compañeros.

The third day keynote was Barbie herself. In an amazing performance of comedy merged with move you to tears love of students and community, Fawn presented teaching as a web of relationships. Parents, administrators, colleagues and students. Focus on what is important, make no excuses, and give your best. Because these parents have given you their best. If you have had any exchanges with Fawn, you respect her. If you only watch one of these, watch Fawn. You have to make it through some MTBoS stuff, but that is really making it clear how we are this community of colleagues. "Of course it's hard. It's not worth it if it's not hard"

In the afternoon, I got to meet Bruce Cohen who has some beautiful GeoGebra. And watch Dan Anderson recreate it in processing on the spot - man's got skills. Then we sat down in a mini GeoGebra user's group. Despite MTBoS being Desmos first, GeoGebra is still there.

Sunday is the wrap up, a few more favorites. I did a favorites on Tumblr, at Hedge's request. Made it in a Tumblr post. Are you on Tumblr? Let me know. There's the start of an Activity Bank - hope that develops. Curation is really an open problem to me still. Princess Choi gave a hilarious pitch for her student created videos. Lisa summed up, well and emotionally. And announced TMC16 in Minneapolis.

So see you next year in Minnesota!

PS. Many better reflection posts than this one at the archive. Elissa Miller was prolific in reflection. And Sam Shah's turned into a state of the MTBoS. Also has the TMC15 song...