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.

Projective Math Identities

January 22, 2016

I’m tired of students and colleagues telling me they’re not a “math person.” A little research into the subject reveals that in fact, everyone has more or less the same innate mathematical ability. Either we’re all “math people” or none of us are, but that won’t stop many people from arguing otherwise.

In truth, the whole “math person” conversation is more about identity than it is about aptitude. If I can get over this hump with a student, if I can convince them they are capable of doing math, then the rest of my job is a downhill battle. From this perspective, the work of a math professor is more about reinventing identities than it is about quadratic functions and irrational numbers.

James Paul Gee would agree with this idea. In his book, What Video Games Have to Teach Us about Learning and Literacy, Gee proposes that through role-play in video games, individuals can try on new identities in a safe, low-stakes environment.


In particular, Gee describes three types of identity that are at stake when someone plays a video game: a real-world identity, a virtual identity, and a projective identity. Here’s how each of these identities pan out in the case of the game Arcanum where Gee played as a female half-elf named “Bead Bead.”

  • virtual identity – This is the identity of your character in the virtual world as governed by the game’s rules. For example, half-eves are considered very intelligent in the world of Arcanum. At one point in the game, Gee needed to persuade a town meeting to fund the construction of a monument in order to please the mayor, and his character was able to do this precisely because half-eves have a high “intelligence” stat. This was a property of Bead Bead, the virtual identity.
  • real-world identity – This is the identify of the real person playing the game. In the case of Bead Bead, that would be James Paul Gee. He could be very unintelligent himself (he’s not) but this would have no direct bearing on how Bead Bead was perceived within the game.
  • projective identity – This is the most interesting of the three identities. In Gee’s own words, projective identity “plays on two sense of the word ‘project,’ meaning both ‘to project one’s values and desires onto the virtual characters’ (Bead Bead, in this case) and ‘seeing the virtual character as one’s own project in the making, a creature whom I imbue with a certain trajectory through time defined by my aspirations for what I want that character to be and become.'” As an example, Gee had Bead Bead sell a ring that an old man had given her to hold onto. There was no reason why a half-elf could not do this, but it just felt wrong to him. He felt he had betrayed the person that she was supposed to be.

As Gee observes,

A game like Arcanum allows me, the player, certain degrees of freedom (choices) in forming my virtual character and developing her throughout the game. In my projective identity I worry about what sort of “person” I want her to be, what type of history I want her to have had by the time I am done playing the game. I want this person and history to reflect my values though I have to think reflectively and critically about them, since I have never had to project a Half-Elf onto the world before. (pg. 56)

What about a student who has never had to “think reflectively and critically” about what it means to be a “math person” because they’ve never had to “project a math person onto the world before”? Wouldn’t that be a powerful and transformative experience for them if we provided a virtual environment through which they could try out this identity? I don’t think this virtual environment necessarily would have to take the form of a video game, but what would it entail? What ingredients are necessary for this projection to be possible?

Certainly, you need to be able to make choices within the game that modify or affect the identity of the virtual character. Athomas Goldberg makes the distinction in video games between an avatar and an agent. Both are representations of a character that you navigate through a virtual world, but in the case of the agent (as opposed to the avatar,) your choices have no effect on the character and it’s representation. For example, Pac-Man is an agent. You cannot customize Pac-Man; you cannot give him hair, change his color, make him smarter, faster, more attractive, more egotistical, kinder, braver, and so on. There’s no space for the player to think reflectively and critically about who they want Pac-Man to become.

Most role-playing games ask players to make explicit choices about their identity. They may give players a plethora of ways to modify the appearance of their avatar. They may ask them to select the gender and the class of their character or allot points to different stats categories like strength, dexterity, and intelligence. Sometimes though, games provide you with choices that will impact your character’s identity in less explicit ways. A good example of this is the (quite excellent) game 80 Days from Inkle Studios.

Based on the Jules Verne novel, Around the World in 80 Days, the game casts you as the French valet Passepartout who has reluctantly agreed to follow his master on a trip around the world in 80 days. Your master turns out to be quite useless, and you must begrudgingly go about navigating the world to fulfill all of his impractical whims. There are no stats, no way to modify the appearance of your character, no choice of class or gender, not much of anything really. You’re Passepartout and you’re stuck with it; that’s kind of the point of the game.

And yet…the very first choice the game asks you to make is to decide what items to bring with you on the trip. There’s only so much room in your luggage so you can’t take everything. Should you bring the European train timetable or an evening jacket? Would it be better to take the wool trousers or how about a top hat? I chose the top hat and the evening jacket because I decided I wanted to be a proper English French gentleman. See what’s happening here? I’ve made a choice about who Passepartout will be; I’ve projected an identity onto him!

80 Days is rife with opportunities like this one. You encounter countless people throughout your travels and how you respond to them will affect your master’s opinion of you. Were you a little to forward with the train captain? The master does not approve of that. A good English gentleman must be coy and composed. Did you show a little too much sympathy to the union strikers? No sir, the master is not fond of this either. The working class must learn to accept their role in life. Passepartout’s identity in the game is primarily negotiated through how he compares to his master, a sort of projective identity by way of social positioning.

At second glance, is the decision to play as a half-elf not also its own form of social positioning? The very choice of which virtual identity to explore, a half-elf versus say an orc, is a choice about positioning within the virtual world. Just as much as you’ve chosen to be a half-elf, you’ve also chosen not to be an orc. 80 Days may not give you an option about which virtual identity to take on (you’re always Passepartout,) but that doesn’t mean your decisions don’t affect who Passepartout becomes.

Getting back to the topic of mathematics, students need a space that feels safe, an environment in which they are not asked to be themselves but to take on a different role as a mathematician, a scientist, a cartographer, something fictional but relevant. More than that though, they need to asked to make choices that will affect what type of mathematician, scientist, cartographer, etc. they become. Maybe, they are given a word problem that embeds them in a story. Maybe they must decide whether to be a scientist who plays by the rules or one who goes rogue, and that decision must be tied in some real way to their ability to solve the world problem. Maybe, different decisions lead to different solutions or different problem-solving techniques.

Another place where this type of identity negotiation could go on is in a blog. Luehmann and Borasi appeal to Gee’s identity theory in their book Blogging as Change: Transforming Science and Math Education Through New Media Literacies.


They don’t go so far as to talk about virtual and projective identities, but they do make the case that blogging has an impact on a student’s identity. Blogs are basically narrative tools for exploring identity. Even though blogging does not ask you to make distinct choices like “should I bring the evening jacket or the European timetables?”, you could still use a blog to role play a character. The question then is how do you create a narrative that students will buy into and what will be their virtual identity in this narrative? Does it have to the narrative need a fantasy component or could it just be a mirror image of real life. If I play myself in a game, is that virtual self really the same as my real-world self? Am I negotiating a projective identity right now as I write this post?