Messing Around in GeoGebra

August 16, 2016

This blog has basically become a last-minute repository for all my talks. I’m not happy about that; I’m going to start blogging for real again, but in the meantime, here’s another talk for the repository.

I’m speaking tomorrow (today?) at the Fourth Annual Southern Connecticut GeoGebra Conference. This is an exciting talk for me for a lot of reasons. It’s inspired by a blog post I wrote back in February (back when I still wrote blog posts,) and it will be part of the conference proceedings. Most importantly though, I gave a talk at last year’s conference, which set in motion several exciting collaborations with some really wonderful people. Wow, I cannot believe it’s only been a year! It feels like a decades. In any case, here are my slides. I hope the coming decade year can be every bit as meangingful as the last one.

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MathFest Talk

August 5, 2016

I’m giving a talk at MathFest today about the hybrid Precalculus class I taught last Spring. If you’re around, come say hi. My talk is at 1:45pm in Union D. As usual, the slides are available below.

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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.

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Innovative Practices in Developmental Mathematics

April 14, 2016

I’m giving a talk tomorrow at LaGuardia Community College’s new Innovative Practices in Developmental Mathematics conference. This is another talk about the gentrification-themed board game that Jared Warner and I created to teach proportional reasoning in Guttman’s City Seminar course. I’m excited to be speaking at LaGuardia, which has a reputation at CUNY for doing cool, new, innovative things…kinda like another Community College I know.

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Good Ideas in Teaching Precalculus And…

March 18, 2016

I’m giving a talk early tomorrow (Friday) morning at the “Good Ideas in Teaching Precalculus“conference at Rutgers. I’ll be talking about a hybrid Precalculus course that I created at Guttman, which uses blogs to engage students in mathematical inquiry. The course was inspired by Jim Groom’s DS106 course, and it’s open, meaning you can read our blogs and join in on the discussion if you’d like. I am very excited to finally get a chance to share this work with other people. It’s definitely an experiment, and I have as many questions as I have answers, but I guess that’s what makes it a lot of fun to teach. Here are the slides.

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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.

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Messing Around with Math

February 3, 2016

Here’s a puzzle that my office mate shared over the cubicle wall last week.


Now, Guttman is unconventional in the sense that I share cubicle space with English professors, a librarian, secretaries to the Deans, an academic advisor, a sociologist, and so on. Two of these “non-math” people immediately became obsessed with this problem. They spent every free moment of the next two days drawing stars and lines until they’d found a workable solution.

This is basically a math problem. Let’s a consider a second, very different math problem and the very different reactions it might evoke.

The profit for a certain brand of MP3 player can be described by the function P(x) = 40x – 3000 – 0.01x^2 dollars, where x is the number of MP3 players sold. To maximize profit, how many MP3 players must be produced and sold?

I took this problem from what I consider to be a top-notch College Algebra textbook, but it’s pretty difficult to imagine my office mates rushing to find the optimal numbers of MP3 players. What makes these two problems and the reactions they evoke so different? How do we get our students engaged in the second type of problem?

It might be worth stepping back for a moment to consider the problem-solving strategies our students employ. I’ve been reading this book.


Through three years of collaborative, ethnographic work, Mizuko Ito and fourteen other researchers have identified three major “genres of participation”: “hanging out,” “messing around,” and “geeking out.” These genres describe how young people use digital technology. It’s important to note that these are genres of participation, not genres of people. Instead of casting individuals into broad categories like “geek” or “luddite,” they acknowledge that one person can participate as a geek in one setting and a luddite in another. Our identities are neither singular nor fixed.

The genre of “hanging out” refers to friendship-driven usage of digital technology, activities like sharing photos from a party on Facebook or exchanging texts with a friend. On the other hand, “messing around” represents the first steps into more interest-driven uses of technology. This category includes what Ito et al. call “fortuitous searching,” “moving from link to link, looking around for what many teenagers describe as ‘random’ information” (pg. 54). Note how this type of search is open-ended rather than goal-driven. In general, “messing around” describes casual, open-ended experimentation with new media that’s guided by interests in a certain topic, but does not necessarily have a specific end-goal. Finally, “geeking out” refers to intense, interest-driven engagement with new media such as maintaining a blog about a particular interest or editing a video. It tends to be less open-ended than “messing around” and requires a more deliberate investment on the part pf the participant.

When my office-mates engrossed themselves in the stars and triangles problem above, they were “messing around” with math. Although it ended up consuming a lot of their time, the puzzle didn’t require any sort of “geeky” investment. There is something about this puzzle that lends itself to “fortuitous searching,” experimenting with different arrangements of lines without knowing exactly what you’re after. Compare this with the MP3 player problem, which has a very clear goal (the optimal number of MP3 players to produce) and cannot be solved by simply playing with lines on a page. You have to “geek out” on this problem or there’s no hope.

Goal-oriented tasks are difficult for my students. They don’t always have the metacognitive skills to visualize the end-goal. Even if a student has a casual interest in mathematics, it can be difficult to keep them focused enough to persist and solve a difficult problem. They may start the MP3 player problem, realize that it requires them to coordinate a lot of different ideas, and then they lose interest.

I wonder if “messing around” could be the gateway drug to “geeking out” about math. How would you do this? Let’s say I have a list of major learning objectives for my math class, and one of them is “identifying the vertex of a parabola.” I need students to master this specific concept so I cannot have them just “messing around” with any idea that piques their interest. Is there a “messing around” activity that I can create, which will help students achieve this learning objective?

It seems like a good first step would be identifying precisely what differentiates these two types of tasks. Then, I can take something like the MP3 player problem and craft it’s features to be more like the stars and triangles problem. When I was talking about this with my office-mate, he mentioned casual games like Threes or Theseus and the Minotaur.


What makes these games so addictive? How is that people can swallow hours and hours of time in them and yet it feels like no big investment to pick them up and play them? Here’s what I’ve come up with so far; I’d love to hear your ideas as well.

  1. The “messing around” tasks require very little domain-specific knowledge. You don’t need to know what a quadratic function is or a vertex to understand the stars and triangles problem. You don’t need to have any fluency with algebraic symbols and manipulations; the meaning of the problem is immediately clear and so are the tools with which you’re working.
  2. Each time you fail a “messing around” task, you always feel like you were just about to get it. You draw some lines on the star, count up the triangles, and realize you’re two short of ten, but you’re almost there! Next time, you’ll get it. This motivates you to keep on trying; the answer is just around the corner! Every time I play Threes, I’m convinced that this will be the time I get the 1024 block. On the other hand, when you get stuck on the MP3 problem, there’s the sense that you haven’t made any progress and you don’t know what you’re doing. You feel lost. You never feel lost on a “messing around” task.
  3. A “messing around” task allows for “fortuitous searches.” You can stumble around and experiment without knowing exactly where you’re going or what you’re after. I never have a very deliberate strategy in mind when I play Threes, but that quadratic function problem, you need to stay very focused on the goal. I think this is closely related to (or equivalent to) the previous property because by definition, fortuitous searches always give you the feeling that you’re just about to find what you’re looking for.
  4. They’re portable. This is really important. There’s a reason why mobile phone games tend to be more casual and puzzle-like. If participation requires you to be in a specific place at a specific time then it requires another level of determination to engage with the task. The beauty of the stars and triangles problem is that you can work on it anywhere, doodling on a napkin or a piece of paper while you’re chatting with a friend. Speaking of which…
  5. A “messing around” task does not require your full attention. You can play Threes while listening to music, while riding on the subway, while watching TV, while talking to a friend; it can ride on autopilot in the background while you do other things. There is no autopilot for the quadratic functions problem above. This could be an important property as Ito et al. note that “certain forms of participation also act to bridge the divide between friendship-driven and interest-driven modes. In chapter 5, we will describe how friendship-driven modes of ‘hanging out’ with friends while gaming can transition to more interest-driven genres of what we call recreational gaming” (pg. 17). Maybe, “hanging out” with friends while solving math puzzles can transition to more serious mathematical problem-solving.

There must be other properties that I have not thought of yet, or maybe some of these properties are not as essential as they seem? I worry that property 2 means students will keep trying the same false strategy (because they’re just about to get it) and thus they’ll persistent with bad strategies rather than reflecting on and finding new ones.

Nonetheless, my idea is that you might take a goal-oriented problem like the MP3 player example and modify it according to the five properties listed above, creating something more appropriate for “messing around.” Once a student’s interest is piqued, you start slowly shedding these five properties. Maybe it becomes less portable as you introduce more complicated elements. Maybe, you introduce domain-specific terminology and tools like variables and functions, or maybe you give them more and more specific goals to work towards.What do you think?

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?

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…


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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.

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