Math Hombre

It should come as no surprise to anyone who spends more than 5 minutes on this site that I see a lot of connections among math, games and art.

My favorite game of all time is Magic: the Gathering. I love it in concept and in play. Amazing strategy, accessible at several levels and varieties and terrific flavor and art for bonus. But I'm not trying to convince you to try the cardboard crack - unless you're interested? - I'm just letting you know what I'm about to try to riff on. Mark Rosewater is the long time lead designer on MtG, and is very generous at sharing his design thinking, on Tumblr, in longer form blogposts and in podcast.  He is a serious student of game design, and focused on engaging play, so there are often connections to teaching and learning. A recent article is on narrative equity. One of the ways games engage players is the opportunity to make a story. It's a rich payoff, and can be significant to identity. I'd encourage you to read his post, but the examples after the intro story are in terms of Magic, so may not be accessible. Mark wraps up the introductory stories about his daughter and himself with this:

What do these two stories have in common? In both, Rachel and I prioritized having an experience. Our personal story carried enough value that it influenced how we behaved. It was an interesting concept, that people will give weight to choices based upon the ability to later tell a story about it. I call this idea "narrative equity."

The next step for me was applying this idea to game design. What does narrative equity mean to a game? Well, games are built to create experiences. I talk all the time about trying to tap into emotional resonance and capture a sense of fun. Narrative equity should be one of the tools available to a game designer to do this.

After thinking it through, I came up with seven things a game designer can do to help maximize narrative equity in their game.

What follows here is his list of game design connections to this idea, and why I felt like he was talking about teaching mathematics.

#1 – Create components with enough flexibility that players can use them in unintended ways

Math, to me, is ultimately about doing. We often make it about acquiring facts and techniques, and can lose track of why we are asking learners to do that. When learners are exploring these ideas, these powerful, culture changing ideas, which we are teaching, there are going to be ways to combine them to get new places. When we front load mathematical ideas, so that in the next section we can use them to solve this kind of problem, we're working against this.

The big shift for me on this was going from that linear learning curriculum model to a landscape approach like those in the Fosnot & Dolk work. (Image from this workshop.) They create a distinction among models, strategies and ideas, and realize there is a progression, but there are so many paths that learners can take from place to place. Formally or mentally, this is how I see curriculum now.

#2 – Create open-ended components that can be mixed and matched in unforeseen ways

To some extent, for me in math, this is about tools and representations. I am a deep believer that learners being able to represent (in the old NCTM process standard sense; create, move among and choose representations) magnifying their problem solving capacity. Given the ability to create graphs, diagrams, written/verbal descriptions, contexts, tables, equations or expressions... that creates excitement. I cannot tell you how often I learn something new or see a new idea and need to make it in GeoGebra or Desmos. And am delighted by the result. Or to write down a function to model a behavior. Or see a pattern in a table that was hidden from other perspectives...

Naturally, this dovetails with tool use. We live in the future, people, with free tech that gives capacities to everyone once reserved for super-geniuses. To some extent, I think why I stood out as a young math student was that I could do that in my head. Now everyone can! Why hide it from learners? Several of my Calc 2 students this summer had Calc 1 with NO TECH.  Augh! On the flipside, I felt like learning Desmos, GeoGebra and Wolfram|Alpha was a goal in my course, and was frequently happy to see them used in ways that we had not done. A good sign the learner is making it a tool of their own. We also had programmers making things, and a student from another U sharing his Mathematica programming, which they are required to use.

#3 – Design in unbounded challenges that allow the ability to create memorable moments

THIS. I want to get much better at this. The twist is that math does this naturally, so we've had to contort it to hide that aspect. I ask students to do this, but don't know how to support them. Especially when I see them, they have learned that the teacher always has an end in mind. Show us how! Show us an example! There are, of course, times for this. But when I ask you to see what you make with this, I really want to leave the door open.

I hit a pretty good middle ground with the quarter the cross assignment in Calc 2 this summer. We used David Butler's examples to launch it and model, but then opened the calculus door by connecting to how we had great area calculating power. Many exciting results. Not all of them, but I don't think we can require creativity. Just make space for it, and celebrate it. For the assignment, we had a little experimentation in class, a bit more in the takehome and then a lot for the people who chose that for a writing assignment.

#4 – Create near-impossible challenges that can become a badge of honor

Mark sees #3 and #4 as related. And this is something I do not do much of in my teaching. I do give SBAR grades for good progress on hard problems, instead of credit for right answers. I propose extensions for writing, and have optional assignments that can be very challenging. Is that enough?

I think near-impossible is affecting me as a mathy type. The idea of a challenge, that a learner would remember solving or trying is probably the goal. How do we support them to give these a try, though? Much like #3, I think sharing student work on such things is probably a key part.

At Twitter Math Camp, Sasha Fradkin had a session on impossible problems. She didn't mean this kind of impossible, but I think by coincidence, it might fit the category. Something like: using three straight cuts, divide a circle up into 3, 4, 5, 6, 7, and 8 pieces. (Not all of those are possible.) One of her takeaways was to consider what do we want the learners to mean when they say 'this is impossible.'

#5 – Create alternate ways to win

In a game, of course, you're trying to win. If there is only one way to win, the game becomes boring and narrow quickly. If a multitude of strategies is available, the game is richer as people pursue different resources and strategies.

In class, this feels to me like assessment. The goal is demonstrated understanding. If the only way to do that is timed tests, I think that narrows the game. Now it's not competitive, maybe, and the people who are better at that don't necessarily bar others from success... unless it creeps into your test writing. Or you curve. Or you measure the middle and less successful students by those who are good test takers.

For me in college there was a strange thing. My first two years I was adjusting from high school's low expectation tests to honors courses where they wanted some version of deep understanding. I got some Bs. The high school tests just wanted recall, which due to no credit of my own was easy. I couldn't not know a lot of those things. But then, beginning of my junior year, tests just made sense. I wasn't any better of a student, but I think I went almost two years without missing a question. It was weird.  When I started teaching, this got me to include a lot about test taking strategies in my classes and review days.

Eventually, though, I realized that this meant that tests weren't doing what I wanted them to do. So now my learners know the standards they're being assessed on and there are multiple ways to demonstrate understanding. And they can reassess.

#6 – Allow players opportunities to interact with other people where the outcome is based on the interaction

I think this is a regular feature of classes that feature cooperative learning.  It does require communication that is not teacher <-> student. If your classroom communication is you talking or asking questions and people answering you or asking you questions, it is one dimensional in a three dimensional world.

#7 – Give players the ability to customize, allowing them opportunities for creativity

This is sooo hard. But, ultimately, necessary. Dave Coffey likes to say that if the only choice students have is to do something or not to do it, of course some will choose not to do it. Even if the choice is as simple as choose the even or odd problems to do can increase engagement. Is it possible to let students choose a topic? Form of an assessment? Application? Which question to investigate in a 3-Act?

I love Elizabeth Statmore's emphasis on returning authority to the learners. This is part of that. Give choices and ask them why they chose as they did. Math class does not have to be everyone doing the same thing at the same time. Choices imply there is self-assessment to do. To me, this is the holy grail of assessment: learners start to think for themselves about what do they understand and what do they not get yet. And what should they do about it.

Sometimes I describe Magic as chess where you get to build your own pieces and bring your half of the board. (Plus a layer of variability from being a card game.)

Endgame

Mark's last words:

Narrative equity isn't a lens you have to view every game component through, but it is something you should view some of them through. When putting your game together, be aware that you have a lot of control over what the end experience will be. By making certain choices, you can maximize those choices that lead to your players forming stories, which in turn will change how your players emotionally bind with your game.

I am left with questions. What stories will my learners tell about the course they had with me? Will they be the hero or at least the protagonist in those stories? Will it change their view of the mathematics genre? Will every learner get an opportunity to weave a tale?

PS: Flavor Flav

I must have three unfinished blogposts to get through, but this is what I keep coming back to this week.

Natasha Lewis Harrington is a doctoral psychology student who writes about my favorite game (Magic: the Gathering) in her spare time. Sometimes she crosses the stream to great effect. Like this week, when she wrote about why this game is so good at encouraging creativity among players. It's applying the work of (let me copy and paste here) Mihaly Csikzentmihalyi (specifically Creativity: Flow and the Psychology of Discovery and Invention [Google book preview]) to the question of how can we learn to engage more. I think it's well readable by non-Magic players, so please do peruse.

Here's the quick take:
(The little bit of art is from Flickr, Paolo Colacino who does what he calls generative art. Quite neat.)

Csikzentmihalyi has a TED talk about leaving boredom:


Why is this gripping me so? Because of the divide between math as taught and math as it could be.

Math, as it is often taught, violates all three of these principles. (1) We tell you the problems to do, (3) we insist on solo mastery and uniformity of method.

Wait, that's only two.

I'm wondering if I have, in my need to change (1) and (3), more than occasionally neglected (2). Is that the procedural knowledge which I de-emphasize?  I usually do that in an attempt to get the pendulum swinging in the other direction, but in doing so am I denying needed support?

Maybe not. Maybe Learning the System in mathematics is not the procedural stuff. Maybe it's the processes, hidden behind the procedural emphasis. (The processes now appearing with their new band, the Standards for Mathematical Practice.)

Of course, there's hope. Teachers like Fawn Nguyen, Michael Pershan and Andrew Stadel are knocking this engagement issue out of the park on all three principles.

But, as Dave Coffey has cautioned, and convinced me, we need to teach our students to take control of their own engagement. So when they leave Jim Pai's classroom, they can be engaged the next year, too.

That's empowerment, and that's what I want for my students.

Mark Rosewater, head designer for Magic: the Gathering, has up the his intro to game design article, so it must be time for my 2nd part of my commentary thinking about educational games.

The first five principles were:

1. Goal(s). Easiest part for educational games.
2. Rules.  
3. Interaction. 
4. Catch-Up. Most subtle, maybe because so many games lack this.
5. Inertia. Hard for teachers.

The second five:

6. Surprise. The game should have some unpredictability for players.

To me, this connects strongly to Interaction and Catch-Up. One way to get surprise is hidden information - which often can contribute to interaction amongst players.  Information can be hidden from both or the players can hide it from each other. The new game Flip Out has a good element of this with two sided cards of which each player sees different sides. A benefit for math and literacy is that this makes inference a part of the game.

The other easy way to add surprise is random events - which can contribute to making catch up possible.  The only thing that makes Monopoly playable is the dice rolling.  In Euchre, no matter how good you are, you need cards to play. The math benefit is the addition of probability, even if informal, to game play. It's no surprise that the two most common game pieces are dice and cards.

7. Strategy.

Interesting to me that this is so low on the list, which makes me wonder what he was ordering them to achieve.

by mhuang @ Flickr

This is the biggest add-on for educational games over other activities. The problem solving inherent in any game with strategy is such fantastic grist for mathematics.  Mathematicians often see math as a game because of this strong connection. How do we achieve a result with allowable moves? Using games with K-12 students,asking for their strategies always makes for an amazing summary and unearths most of the math content of the games. It also helps build Inertia as then students are more interested in playing again, trying our others' strategies or designing ways to beat them.

There's a natural tension between Surprise and Strategy.  If things are too random, strategy loses all impact. If things aren't random at all, it is chess or go.  Both great games, obviously, but also both games that struggle with Catch-Up and Inertia for many players. Plug for Magic: the balance of these two elements is a large part of what makes the game so bloody amazing. Also applies to Bridge, to a lesser extent. (Yes, I'm claiming Magic > Bridge.)

8. Fun.

I struggle with this. Because I find interaction, surprise and strategy so engaging, I love games in general. I'll play anything. But what makes a game fun to kids is often a surprise to me. It's not uncommon for me to take a game to kids, and let them add the context. I wrote about this a bit with my Division into Decimals game.  Games like Decimal Point Pickle and Power Up had this in spades. Probably this is the difference between a game being good, and the game being a smash hit.

To some extent I think the last two principles are really subcategories of this one. Did they get pulled out to make ten or - more likely - is there something I'm missing that makes them truly distinct?

9. Flavor.


10. Hook.

Flavor is about the context and setting for your game, which heavily influences the fun aspect for players, in my experience. At least on entry, and Mark connects this to the barrier or entry cost to your game. The other principles determine long term fun. In our house, this gets us to play a game fr the first time, but won't sustain interest. One neat point he makes about flavor, though, is how it can influence design. My youth Bible study is making a return of the Lord card game based on the 10 bridesmaids parable. (Yes, really.) But the context for the game is inspiring a four horseman of the apocalypse feature that will definitely add interest to the game. Probably shouldn't have shared this story.

The idea of constructive flavor reminds me of my colleague Jacqui Melinn talking about integrated units.  A marine biology integrated unit is not when you put your math practice problems on a whale-themed sheet, it's when your questions about whales require math to think about and solve. Good flavor isn't an add on, but supports the game mechanics. For a math game, this gets at the structure of the game supporting the mathematical objective. Cheap flavor is the hallmark of flashcard/drill math games. "Look you're doing lots of multiplication, but it's on a baseball diamond!"

Hook is what gets people to try your game. This is less important in educational games to me as we have a built-in market (students), but I'm also not trying to sell my games to a publisher. (So maybe my hook is that my games are free?)  However it does remind me of Dan Meyer talking about a hook for a lesson, and could well be linked to engagement. I just don't know how to tease it out from flavor and fun. Maybe hook is a measure of whether the game has things that make you wonder?

My Nine
Looking over the list, I think I'd order them more like the following to get at my process.

1. Goal(s). Design starts with objectives.
2. Structure. (Not in his list!) What is the essential nature of your set of learning objectives and how can that show up in the game?
3. Strategy. These three have to go into the primary design phase as well, or the game will just not have them.
4. Interaction.
5. Surprise.
6. Catch-Up. As you start to playtest, these two are important to attend to for good design.
   
7. Inertia.
8. 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.
9. Context: Fun-Flavor-Hook. To me this can't really be evaluated fully till you're out with the intended audience. You need a first take on this before that, but should be open to major changes in this area.

Boy, I enjoyed these two articles. Thanks, Mark, for  writing them, and giving us fledgling designers something to think about. Note that Mark has a Tumblr where he answers many questions and gives the behind the scenes story at Wizards of the Coast. I asked, for example, about the order of his list, and he wrote: "I left it up to my subconscious. That was the order that felt organically correct. I’ll be honest, that I can’t exactly explain why."

Images: All Magic: the Gathering stuff is very heavily (c)'d by Hasbro.

Hosting the Play Date
I am hosting the Math Teacher's at Play this month.  Please submit your own blog if you have something to share.  But also consider submitting something you've found valuable at someone else's blog.  It's the same form for either, and it just takes a second.  The link can always be found at the bottom of the right side column here.

Complex Instruction
I've shared before how NRICH is probably my favorite source for rich problems.  They also share bits of scholarly writing, host discussions and contribute in other ways to the math ed community.  This month, they've posted a wealth of resources on problem selection for group environments, leaning heavily on Jo Boaler's work with several of her articles or chapters; Dr. Boaler calls it complex instruction.  Take a look!


Concentration
Last and most oddly, I wanted to share some quotes from a Japanese gamer.  I play this game called Magic the Gathering, which is really the best strategy game ever invented.  (With the possible exception of global thermonuclear war, but you know the only way to win that...)  Players construct their own (or copy someone else's) deck of cards, and then face off against one or more other players.  It has a pro tour, and a small number of players who actually make their living playing.  The best analogy I've heard for it is a chess game with modifiable board, rules and pieces.  Each year the game expands and changes, yet remains simple enough for interested people to pick it up.  It has probability situations that will break your brain.  Frankly, it's amazing that I've gone this long without geeking out about it in this blog.

The author, Tomaharu Saito, is one of the best players ever.  He is known for his concentration, and wrote an article on how to learn to concentrate.  I love when people share their understanding on how to learn, and really feel like that the best reason to teach math (for most students) is that it is such a good context to learn how to learn.  The full article is here; selected quotes below.

I decided to write an article that would always be helpful to players, one whose ideas would not fade with time or format changes.

One of the best reasons to write!  That's one of the reasons I like so much for my students to write.  But then we need to give them ways and opportunities to share.  I'm still working on that.

In order to show 100% of your ability, concentration is crucial.  Frankly, as far as concentration is concerned, each person is different. And I feel like there are some people, although it may be only a handful, who can concentrate extremely well without training.
However, this does not mean that most players are naturally sufficiently focused. I think that time spent training concentration can affect one’s ability to refocus when their concentration is broken. This will boost your deck’s power, and is itself a way of taking countermeasures against weaker decks.

One of the things I am currently struggling with as a teacher is how to develop and support my students in becoming long thinkers, better able to apply themselves to significant problems, and to persevere through being stuck.  This feels relevant.

When practicing concentration, I find it particularly effective to pretend I am playing in a real event and focus accordingly. If my only goal were to boost my concentration, it would be good to always play as though I were in a tournament, but I also like to watch deck development, tune my build, learn match-up odds, and work on other goals which can capture my attention and cause a distraction. Because this can also lower my efficiency, I have found it is not a good idea to always split my focus. It is for this reason that I make time to challenge myself to work on concentration.

Remembering overall purpose, creating authentic conditions, assessing barriers to improvement.  Choosing to challenge yourself.  My teaching question is how to create purpose like that within the classroom.  You wouldn't do this for an exercise.  It needs to be relevant to your life.

Also after reading a book or watching movies or television, make sure you are able to explain the subject to a third party. This is always effective in improving awareness. When it comes to explaining the subject to someone else after reading or watching, you need to have watched it carefully and tried to concentrate in order to pin down the main point.

This is transfer!  He's talking about applying concentration in another circumstance.  The point about awareness in relation to concentration is new to me, but makes sense.

There are various methods for training your concentration, but there are none that allow you to master the skill in a short period of time. There is a feeling that steadily bit by bit the skill increases, so persist in your attempts. 

Also, one characteristic of concentration ability is that it is greatly affected by your every day life. In particular, people who stay up late should be attentive to this. If an individual does not spend enough time in sunlight various problems can ensue, and concentration ability is no exception.

Clearly true.  He's talking about life long learning, and how lifestyle impinges on performance.

Indeed, circumstances where concentration is broken and misplays are made are frequent, and naturally they make winning more difficult. I recommend self-confidence, but you can also learn my own method for recovering concentration.

Have you heard of the Saito Slap?  If there is bothersome noise around me, I will slap my cheek hard to recover my concentration.

He's not kidding.  He's not recommending everyone slap themselves, but instead talks about the usefulness of a routine that you can rely on, that by training triggers a response.  I think about that in my prayer life, but haven't thought about it for teaching.

What are the times when it is easy to break concentration? When you recognize the times when it is easy for problems to arise, it becomes easier to cope with them.

• The moment you think you’ve won
• When you’ve lost a previous game due to a play error
• When you’ve lost a previous round due to a play error
• When you’ve lost a previous game due to a poor draw, mana troubles, etc.
• When you’ve lost a previous round due to a poor draw, mana troubles, etc.
• After some kind of trouble occurs
• When something causes you to become irritated
• When you think about things other than the match

Perfect for an anchor chart!  I love the idea of thinking about ways you get stuck or when you give up on problems or...  What would this list look like in math class?

Magic is a wonderful game. Right now, I am betting my livelihood on it, but even misplays and losses do not cost me my life. I don’t let other people get to me.

If you relax a little, you can move forward.

That deserves an amen for brother Tomaharu.

Now to get more people to reflect metacognitively on their passions!