# 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?

# 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.

# Reverse Engineering the Average Rate of Change

## March 16, 2015

Fawn Nguyen shared this clever task,

which I decided to copy for my Precalculus class.

The calculations build up to the formula for average rate of change. I’ve noticed that if you ask students to write down a formula for the distance from say x = 3 to x = 7, they will immediately state 4. This an example of Gray and Tall’s procepts at work. The formula 7 – 3 simultaneously represents both the process of subtracting 3 from 7 and the resulting number 4. The knee-jerk reaction of most students is to carry out that process immediately without thought. They are uncomfortable with 7 – 3 representing a number.

“Reversing the question” provides a deceptively simple solution to this problem. Students are forced to accept 7 – 3 as a number since the problem only asks what question the calculation is designed to answer. For example, calculation D combines calculations A, B, and C. I wasn’t sure how students would react to this, but they were remarkably comfortable combining their interpretations of A, B, and C. One student incredulously baulked, “but you’re just adding up the change so it’s just the total change.” He seemed to think his answer was too obvious, but I usually have to work much harder to coach more poorly articulated responses from students.

Since students want to simplify answers, I also found that they readily made the connection between calculation D and the numerator in calculation F. I didn’t have to spell this out for them! That’s an amazing amount of scaffolding to pack into such an open-ended problem, and I could imagine applying this trick to so many other situations as well. Basically, if you have a formula that you want to motivate, this is a trick that will almost always work.

# Mathimals

## March 6, 2015

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.

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!