Friday, July 27, 2018

Narrative Equity

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.)


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 ended up submitting this to Sam Shah's Festival of Flavors, a blog conference of people thinking about the flavor of math in their classroom.  Just the keynotes he has lined up are spectacular, and I'd expect there to be many more worthwhile reads. So head on over. "Kicking the flavor, getting busy
You're going out, I think you're dizzy."

I can quote several of Flavor's raps by heart, lest you think I take the name in vain.  


  1. John, I want to take your classes.

  2. Great post John. Naturally, I was drawn in by the MTG card on the front; I used to love the game but kind of lost touch with it. For me, this quote from Mark really sparked my interest and got me thinking about motivation: "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." The connections to teaching mathematics are great. I ended up circling back to that quote and thinking about how students can only alter their behavior to create a personal story if they have options and influence over their learning experience. If everyone is doing the same thing, in the same way, there is no opportunity to make it their own. Boring. With options, I can be "the guy who used Desmos," or "the girl who solved the hardest problem." Now all we have to do is help students realize that they want to make mathematics a part of their story.

    1. I love your response, and the connection with choice has to be key.