Gentrification at a Mile High

November 14, 2016

I’m giving another talk this Saturday, November 19 from 2:15-3:05pm at AMATYC‘s annual conference in Denver. I’m excited about attending a conference that’s focused exclusively on math education at the 2-year college level. This talk will be about the gentrification-themed board game that Jared Warner and I created to teach proportional reasoning skills. I will be bringing several copies of the game with me. In the meantime, here are the slides.

Space Range-r

September 18, 2016

Today, I’m releasing an early prototype of a new math game I’m making in GeoGebra. Introducing… Space Range…er. (It’s a pun. Get it?)


The idea for this game has been rolling around in the back of my head for a good 9 months. I thought I’d need Javascript to make it happen, but on Friday, I thought of an easier way. This version took me only one afternoon of work. GeoGebra makes it easy to quickly prototype new ideas!

Space Range-r prepares students to evaluate functions graphically and talk about their domain and range. It follows my idea of “messing around,” an idea I wrote about earlier on this blog and recently presented on  at the 4th Annual Southern Connecticut GeoGebra Conference. I’ve also written an article about this concept, which will appear in the conference proceedings; stay tuned for more info.

In the game, you control a space ship that moves back and forth along the x-axis. A button on the right lets you “fire your lasers,” which basically amounts to evaluating the function. Explosive stars appear along the y-axis. They heat up; turn red, and begin throbbing. If you don’t disarm them with your lasers, they’ll explode!


To win this game, one needs a keen understanding of how function evaluation works. That might seem like a simple thing, but even Calculus students have difficulty coordinating changes in the x-axis with changes in the y-axis. This leads to a lot of confusion when you introduce a concept like piecewise functions and ask a student a question like “what’s the range of this function?”

This is a very early version of the game, and also a very unplayable version. I need to do some work smoothing out the controls. One of the game’s key mechanics has not even been implemented yet. In the upper right corner, you will be able to switch between three different “weapons.” Each “weapon” switches to a different graph. Some graphs will be better than others for shooting at certain y-values. In most levels, some of the graphs will be piecewise functions or parabolas whose range doesn’t even include all the y-values shown in the image. Students need to make quick, smart decisions about which graph is best at which moment in time.

Here are a few other features I plan to implement:

  • Difficulty controls. A slider on the start screen will let you change the difficulty level. At higher difficulty levels, there will be more exploding stars and they’ll blow up quicker.
  • Accuracy controls. Right now, you have to be very precise with your shot, which can be frustrating. Students who want to play a more forgiving game will be able to use a slider to give themselves more leeway in their shots.
  • Music. Sound effects. Stars floating by in the background. I have to be careful about lag time over the internet, but some of this stuff is necessary to make the game feel alive.
  • A proper “you win!” screen!
  • Multiple levels with different combinations of graphs available to the player.
  • More ambitious ideas that may not happen:
    • More types of enemies. I want to add red herrings that look like stars but cause you to lose if you shoot them. Maybe some enemies will move around? Maybe others will shoot back at you? That could be a fun way to introduce the idea of inverse functions. I’m thinking about having later levels where you can transform the functions. I also was toying the idea of enemies that can shoot a hole in your graph, making it no longer use-able. I would love it if you could grab power-ups that float down.
    • Boss enemies. That would be really cool!
    • A story line?

The coolest thing about GeoGebra though is that it’s open source software. I encourage you to download it, mess around, and make your own levels! At the very least, stay tuned for more updates in the near future.

MAA Metro NYC Talk

April 27, 2016

Jared Warner and I will be giving another talk this Sunday about our gentrification-themed board game. If you’re going to be around at the MAA Metropolitan New York Section Meeting, come find us in room E105 at 3:35pm. As usual, the slides are available below.

Also, many people have asked us where they can get a copy of the game. We’re working on making it available as a free print-and-play through this website. Professionally printed copies will also be available from for a little under $50 a piece. We’re still working out the details though. We’re going to set the price on Game Crafter so that we have a $0 markup, meaning any money made off the game will go to Game Crafter for their services and not Jared or I. The intention is to release the game under a Creative Commons Attribution-NonCommerical license so that others can roll their own mods. For example, I’d love to see alternate boards show up for other U.S. cities, new cards, and other creative spins on the rules. I will post more details here on the blog when we’re ready to start distributing the game.

Categories: news • Tags: , , ,

MEC Math Club

March 2, 2016

My colleague Jared Warner and I will be presenting today at the Medgar Evers College Math Club. The talk is about a gentrification-themed board game that we designed to teach proportional reasoning in Guttman’s City Seminar course. This presentation will be similar to our talk from JMM in January, although thankfully will have more time to play the game and unpack some details. Here are the slides for the talk.

Categories: news • Tags: , , ,

JMM 2016

January 9, 2016

One of my New Year’s Resolutions is to update my blog more regularly.

Speaking of which, I’m at the Joint Maths Meetings in Seattle, and I’m giving a talk tomorrow today at 9:40am with my colleague Jared Warner. We’re in the MAA Session on Inquiry-Based Teaching and Learning, IV. We’ll be talking about a gentrification-theme board game we designed to teach proportional reasoning skills in our City Seminar class. Come say hi in Room 619, or take a look at the slides below if you’re interested…


Categories: news • Tags: , , ,

The Mario Way and the Link Way

August 31, 2015

I’m a big fan of Ludology, “a podcast about the why of gaming with Mike Fitzgerald and Geoff Engelstein.” In a recent episode (#108), Mike spoke with Scott Rogers, a video game designer well-known for his work on God of War, Maximo, and Pac-Man World, as well as his books on game design: Level Up! The Guide to Great Video Game Design and Swipe This! The Guide to Touchscreen Game Design. The whole episode is worth a listen, but I was particularly fascinated by the discussion starting at 21:51. Here’s a quick transcription.

Mike: Yeah, and maybe this is a good time to talk a little bit about–because I think it’s intriguing that–I know that Nintendo has a certain way that they like to introduce new hazards or new character abilities versus the way some other companies do it, and they try to do it in a very controlled fashion. Can you talk about those different systems for a minute?

Scott: Yeah, to my study, there’s actually two systems. There’s what I call the  “Mario way” and the “Link way.” The “Mario way” introduces a new ability and let’s you practice it and get really good at it. That’s part of the game-play, and then ultimately, you get to a challenge or a boss or something that allows you to use your mastery of the skill to defeat the enemies. Whereas in the Link method for introducing mechanics or more importantly abilities, they just kinda thrust it on you. One example comes to mind. In one of the–I think it’s in the Wind Waker. You get a bow and arrow, but the first time you get it is when you’re fighting a boss monster. So part of the ramping and the learning curve for the player in the boss fight is just how to use the bow and arrow and use the camera with the bow and arrow. When you eventually get good at using the bow and arrow, you realize that the boss is pretty easy, but part of that design was using the ramping of “how do I use this bow and arrow to fight this thing?” as part of the game design.

Mike: Right, so in the Mario system, usually when you get a new ability you can’t die when you’re first using it.

Scott: Oh yeah, they make it very safe.

Mike: And then you use it maybe for some very basic stuff, and then you have to use it in kinda like a more advanced way.

Scott: Yeah, they kind of show you an easy way to use it and then they start applying variations on that. The “Mario way” is very much like music to me. It’s a bit like, you know how the musical piece Peter and the Wolf keeps adding different instruments into the piece. You first have the duck and then you have the cat, and then you eventually get to the wolf, and the Mario system is really like music to me. It’s a very fascinating way to design, but I think it’s a very–it’s a system that people tend to understand immediately because as they’re playing it, they’re learning. I think that’s one of the great things about the Nintendo games is you’re always learning as you go. Even near the end of the game, you’re still learning new things.

Video game designers are like shadow educators. They have to teach hordes of new ideas and skills to players, and they have to do it in a way that never feels tedious or frustrating. Those are lofty goals that I’d love to see mimicked in the mathematics classroom. I’m not saying that classrooms need to be “gamified” (although, I’d be fine with that); I’m just saying that we could learn a lot by studying the tools and techniques that game designers have developed over the years.



I’m particularly fascinated with this distinction between the “Mario way” and the “Link way.” Just like regular level obstacles are tangibly different from boss characters, regular classroom work is also very different from exams and final projects. What exactly makes them so different? I could identify three main things.

  1. Safety: A boss character, just like a final exam, is seen as a riskier situation than regular level play. If a player dies while fighting the boss character, they must replay the entire level to get back to that same point. The boss fight is thus higher stakes than say the first obstacle you encounter in the level. Many modern games offer save points right before boss characters, thus making the sense of safety mostly a matter of perception. Even then, as Scott pointed out, many games like Mario create space within the levels for gamers to practice new skills in a way that they cannot die. Likewise, homework and regular classroom work are usually weighted less than exams, and students are allowed to consult books and work with other students on homework, whereas exams are supposed to be solely their own work.
  2. Purpose: A boss character, just like a final exam, often serves as a form of assessment for the skills you’ve acquired as you’ve played the level. In that case, regular level obstacles are intended to provide opportunities for practice before the big battle, much in the same way that homework and classroom discussions provide practice before an exam.
  3. Pivot-Point: The boss character marks a (temporary) peak in the story arc, preceding a dip in action as you pivot to a new level with new skills and new obstacles. Likewise, when you complete an exam, you move on to a new chapter with new skills and new challenges. Students have a moment to pause and catch their breath.

The “Mario way” follows all three of these conventions, but the “Link way” breaks convention 2, and to a lesser extent, convention 3. In the “Link way,” the boss fight is not exclusively about assessment. You may be asked to employ skills you’ve learned throughout the level, but you’re also introduced to a new skill, which you are not expected to have mastered. Although the boss character still marks a peak in the story arc, you will carry the bow and arrow into the next level and so there’s less of a clean pivot in skills and challenges as you transition from one scene to the next.


It seems like Scott is singing the praise of the “Mario way,” but there are certain affordances offered by the “Link way” that I find intriguing. As Scott observes, “when you eventually get good at using the bow and arrow, you realize that the boss is pretty easy.” The “Link way” can give you a sudden jolt of confidence that you might not have otherwise developed. The player expects the boss fight to be difficult. When they realize it’s actually quite easy, they feel a sudden sense of expertise and self-efficacy that the “Mario way” would not grant them.

The “Link way” also offers a less abrupt transition from one chapter into the next. You will continue to develop your skills with the bow and arrow in the next level so you’re not starting from scratch when you encounter the first challenge. There is a natural flow between the levels.

Finally, the “Link way” asks you to do something that I think is very important in the math classroom. It asks you to take old knowledge and apply it in new, unfamiliar settings in order to demonstrate mastery of your seasoned skills. Exam questions should not all be variations on homework problems; there should be at least a few questions that ask students to use their skills in unfamiliar ways. Students should not be able to get an A simply by mimicking their instructor; they must demonstrate that they understand the material well enough to solve new problems with it.

I would like to have been a fly on the wall in the Nintendo board room when they decided to introduce the bow and arrow in a boss fight. Why? What particular aspects of the bow and arrow made it appropriate to introduce in this way versus other skills that are developed in safer settings? Did the designers at Nintendo consider this? And are there certain types of players that favor the “Mario way” over the “Link way” or vica-versa? What lessons can we gleam from their differences?

I think these are questions worth answering not just for video games but also for the math classroom. Which topics or skills would benefit from being introduced in a riskier setting like on an exam? Are there certain types of students that would favor (or not favor) this type of question? What qualities do those students have? Are they qualities we’d like to see in other students, and if so, what kinds of interventions could bring those qualities out in everyone? It’s a scintillating and unexplored problem in the math ed literature.

Categories: Uncategorized • Tags: , , , , ,

Digital Math Narratives

March 30, 2015

I’m teaching a hybrid Precalculus class this semester based on ideas from Jim Groom’s DS106, Darren Kuropatwa’s blogging Precalculus students, Alan November’s Digital Learning Farm, and the CSCL literature. The students are maintaining their own WordPress blogs, which are aggregated to a main class website. They take on different roles managing the blogs, and they also respond to a weekly blog assignment. It’s a huge experiment. Some things are working really well. Others are not.

I’d say I have about a 60% buy-in from my students at this point. They are getting through it, but it’s a struggle and I often have trouble communicating my expectations to them. It only recently occurred to me that they have probably never written so much about math. We don’t usually ask our students to do that, especially not in such a public forum.

On the one hand, I feel relieved to know that (hybrid) time outside of class is being well-spent, that the tasks I’m giving them are challenging. I’m perpetually worried that we will not be able to cover the same amount of material with so little class time, and yet I’m also cognizant of the fact that this course is pushing my students in directions they didn’t know they could be pushed. Blogging requires the students to orient themselves to math in a completely new way. I’m not sure they appreciate just what that means.

The original idea behind the blogs was two-fold. First, I wanted to create an authentic experience, a place where people from outside the class could read and comment on my students’ work. Audience is one of the biggest missing pieces in a Blackboard forum, and what bigger audience could you ask for than the entire internet. Second, I wanted to create a participatory culture for online learning, a space where students could co-create meaning and help each other out in their struggles to make-sense of new mathematics.

Creating a participatory culture can be a fickle process. One unexpected place where I’ve found guidance is in the literature on video games. For example, Constance Steinkuehler used James Paul’s Gee Discourse Analysis to understand how gamers learn through the participatory culture in and around Massively Multiplayer Online Games. Video games are different from math though. Gamers have a different relationship to their craft than Precalculus students. Still the same, I think there is value in comparing the two cultures.

Take for example the ludology vs narratology debate in game studies. The “narratologists” argue that games should be thought of as new forms of narrative whereas the “ludologists” view games as systems of rules. I probably err on the side of the ludologists but in thinking about narratology, I couldn’t help but draw connections to DS106 and the fact that it is a digital storytelling class. Blogging is basically a narrative medium. I’m basically asking my students to write digital math narratives. Maybe games could bridge the gap for my students, bringing them into the fold of our fledgling participatory culture?

I haven’t quite figured this out yet. There is a lot of literature on storytelling in math class. Most of it focuses on storytelling for young kids and assumes the instructor will be the storyteller, not the student. There is also a lot of literature on video games in math class. Again, this literature seems to completely miss the narrative aspect of games, mostly looking at puzzles and systems of rules. Can we create games with engaging narratives out of which an understanding of mathematics will emerge?

I’ve already waxed poetic about the role of hyperlinks in developing overlapping goals. When I shared those thoughts with a game studies friend of mine, he mentioned Twine, an open-source platform for creating html-based choose-your-own-adventures. (The math behind CYOAs is pretty cool.)

I’m happy to report that Twine is amazing! It’s not just a tool for creating CYOAs; it’s also a great way to prototype games, and I could imagine removing the narrative component entirely and just using it for instructional design with branching questions. Twine is simple enough to use that I could actually ask students to make their own math narratives/games.

In short, stay tuned for some upcoming digital math narratives. We’re brewing something interesting in Guttman Precalculus, and even if not all the pieces have come together yet, I feel like we’re learning a lot while charting new territory.

Overlapping Goals

March 15, 2015

I just finished reading Kurt Squire’s, Video Games and Learning: Teaching and Participatory Culture in the Digital Age, and I’m still unpacking the overwhelming multitude of new ideas in this book.


My favorite part of the book was actually the discussion about video game design, and in particular, the design of overlapping goals.

A second design rule is to provide overlapping goals. When a Pirates! player sails into town for the first time, the governor instructs him or her to visit a neighboring city and receive a reward. So now the player has a long-term goal (earn fame and riches) and two short-term goals (attack a ship and visit a neighboring port). The short-term goals compete with one another, which gives the player an interesting choice: Do I attack that ship on the horizon, or do I sail to the next port? (pg. 7)

As I continued reading, I began to realize that the sidebars and footnotes in Squire’s own book serve as their own sort of overlapping goals. Should I finish reading this section or should I pause and read the sidebar on the next page? That got me thinking about my own half-baked criticisms of print as a participatory medium, and how maybe hyperlinks do invite participation in ways that regular print does not. After all, a hyperlink is the ultimate overlapping goal, whisking you away to another part of the internet to read a related article before you’ve finished this one.

Why are overlapping goals so important any ways? It seems that they create a situation where the user has to make a choice. Do I continue the course I’m on or chart a new one? Even if you choose to ignore the footnotes in a book, you are still making a choice not to engage with them, and that’s a choice you couldn’t have made in the absence of footnotes.

Many board game critics say that the best games require players to make interesting decisions. Hyperlinks, sidebars, and footnotes are just tiny snippets of what could be very interesting decisions. Looking back over Mike Caulfield’s original post about “users,” I began to realize that digging through hyperlinks is one way that “lurkers” participate in digital media. It’s not a form of participation that creates “makers,” but it’s still a form of participation.

So what about the decisions we ask our students to make in the classroom? My students usually work in groups on handouts during class. I try to design questions for these handouts with multiple points of entry and no right-or-wrong answers so that students will be forced to employ their classmate’s help. I’m thinking of differentiated instruction with menus, but that is giving students different choices about where to start a problem. We’ve completely overlooked choices about where a student ends a problem. What are their goals and do we want every student to have the same immediate goals?

I’m sort of imagining a choose your own adventure handout. Give the class a long-term goal (say developing their proportional reasoning,) and then let each group make decisions about competing short-term goals as they work through the handout. There could be sidebars and diverging projects that lead different groups off in different directions. That’s fine. It actually makes class discussions more interesting afterwards. Rather than rehashing work they’ve already completed, we’re telling stories about our own personal experiments. Your classmates might have taken a different path so there’s actually value in listening to them and hearing what they discovered along that alternative path.

Categories: Uncategorized • Tags: , , ,


March 6, 2015

I am crazy about this game.

Manimals is a good educational game for several reasons.

First off, the rules are stupid simple. If you’re going to use a game in your classroom, it better have simple rules. I learned this the hard way last Fall when I tried to use Sid Sackson’s, Can’t Stop, to explain relative frequencies in statistics. (I stole the idea from this guy.) The lesson was a raging success, but the rules explanation was not. Think about how hard it is to teach the rules of a board game to friends who’ve volunteered to play the game, and imagine that confusion multiplied across 25 restless students who do not want to be there.

Picture of Sid Sacson's classic game, Can't Stop.

The board in Can’t Stop even looks like a histogram…this just screams relative frequencies!

But forget simplicity, the best quality of Manimals is that it’s actually fun! A lot of educational games feel like homework…this one does not. The artwork is also gorgeous, and it packs up compactly. Manimals has a real-time, “race to be first” aspect to it, which will appeal to some of your more kinesthetically inclined and/or competitive students, but you don’t have to be fast to be successful.

This game is a versatile, educational tool. Instead of animals, you could use any concept that you wanted to teach. I’m imagining cards with different representations of functions on them (graphs, tables, equations, a description in words, etc.) and on the back would be icons representing properties of this function like “is linear,” “has an x-intercept,” “is always increasing,” “is one-to-one,” “domain contains [1, 3),” “is transcendental,” “has a vertical asymptote,” and so on. The game would otherwise be played exactly the same way!

It even gets more interesting because sometimes the representation may not provide enough information to determine if the function has the trait or not. This could be indicated on the back of the card with a broken icon, and the student could get 2 points (rather than 1) if they explain why there’s not enough information. This chance for extra points starts a conversation. Your classmates want to win. They’re going to challenge your explanation, and you better be ready to defend it.

Another reason I like this activity is because it reminds me a lot of Bruner’s Concept Attainment, which is a well-tested and widely used technique. We also know from APOS Theory that students struggle to think of functions as objects. By printing a different function on each card, you are basically presenting the functions as objects out of the box. A student can pick up and manipulate this card like a real-world physical object. One of the main weaknesses of APOS Theory is that it really only offers one intervention (computer programming with ISETL) for moving students to an object understanding of functions. This could be another intervention.

A lot has been written about the gamification of learning. I won’t rehash that literature here, but I will say that you can get a lot of mileage out of a simple game and your students will engage in ways you never imagined. That lesson on relative frequencies that didn’t go so well? I was out at a bar one night when my colleague called me from the office. My students were there, and they wanted her to lend them my copy of Can’t Stop. They chose to play a math game in their free time!