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.

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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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?

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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Unimaginable Mathematics

May 5, 2015

It’s Springtime in NYC, and Guttman statistics students are learning about Sampling Distributions and the Central Limit Theorem.


A sampling distribution.

This has been a tough semester. It’s the first the time I’ve taught Statistics B at Guttman. I’ve been reminded repeatedly of Mathews and Clark’s paper where they used APOS Theory to assess students’ understanding of the Central Limit Theorem. The short version of their story is that none of the students they interviewed had any clue what was going on. The authors had hoped to identify the objects and processes that students must construct to understand the Central Limit Theorem. Instead, they found that “none of these eight students was able to discuss the Central Limit Theorem in a meaningful way.” This included A+ students and at least one class where the instructor said they put “major emphasis” on the Central Limit Theorem.

Part of the problem here is that we tend to avoid the difficult problems in education. In most classrooms, statistics is taught procedurally. Procedural understanding is easier to measure, and it’s what our students expect. Can you calculate the standard error? Did you use the correct formula? Did you plug in the correct numbers? Those are easy questions to answer for both the student and the professor. Here’s a more difficult question: do you know what those things mean?

The other problem is that Central Limit Theorem is really hard. At it’s core, it’s a theorem about the sampling distribution, which is in itself a very difficult topic.

My class meets 4 times a week, and we spent two weeks discussing the sampling distribution and Central Limit Theorem. In our first discussion, we took a “population” of 5 students in the class, asked them what borough they lived in, constructed all possible samples of size 3, and calculated the proportion from Brooklyn (the estimator) in each of these samples. The students used a carefully scaffolded handout to construct the sampling distribution from this information, and then reflected on how the standard deviation and mean might be used to measure error and bias.

This took about two days. The hardest part for them was just delineating all of the samples of size 3. They had no algorithm to enumerate the samples in an orderly fashion so many students repeated samples or stopped when they ran out of ideas without any clear way to confirm they’d exhausted all the options.

Next, we imagined a population of six M&M’s, 3 blue, 1 green, and 2 red. This time, I gave a list of all samples of size 4, and asked them to calculate the proportions that were blue and green in each sample. Again, through a carefully scaffolded process, they constructed sampling distributions for the proportion of green and blue M&M’s and reflected on the error and bias in these distributions.

When you roll a 6-sided die, it’s easy to think of this as a random process with a list of different outcomes. Students don’t think of selecting a sample as a random process with a list of different outcomes. Part of this is because they usually only get to see one sample whereas a die can be rolled several times.

After our first two discussions, I was thinking that students needed some assistance in seeing the sample selection process as something random. I gave them a population of 2 “cat people” and 5 “dog people.” Students were asked to calculate the parameter, and to think about possible outcomes. For example, is it possible for 25% of the sample to be cat people? What about 100% of the sample? Can you describe a sample with an estimator of 50%?

I wrote each of the 35 possible samples on a piece of paper, and came around the class with the samples in a cup. Each student randomly drew one of the samples from the cup, calculated it’s estimator, and wrote this information on a post-it note. They were then asked to attach their post-it note to the white board so that we created one big sampling distribution together as a class. We repeated this process with different colored post-it notes and a different, more biased sampling method, and then we discussed how the shape and standard error of the two distributions were different.

You’d think students were getting it pretty good by now, right? They weren’t, but I had to go on so we launched into a discussion about the Central Limit Theorem. I was tired of creating sampling distributions by hand; it was tedious and sucked the life out of class so we used this Central Limit Theorem app. The setup was to imagine creating random samples of NBA basketball players. According to the NBA Tattoos Tumblr blog, 55% of the 442 players in the 2013-2014 season had tattoos. That gave us a parameter of p = 0.55, and a population of 442 players.


I’m a Trail Blazers fan.

I gave students a handout that helped orient themselves to the information on the website, setting the slider for p = 0.55, and making sense of the different graphs. Once they were comfortable with the app, I had them adjust the slider, slowly increasing the sample size, and taking notes about the standard deviation and the shape of the sampling distribution as they went. The goal was for them to see for themselves that the distribution becomes normal and the standard deviation becomes equal to the standard error.

The app shows you estimators for a random 8 of the samples. This turned out to be the best part of the lesson. As students looked at their friend’s computers, they wanted to know why they’d “gotten it wrong,” why their friends had different answers. Of course, the point is that the selection of samples is random so they didn’t do it wrong. Just like the roll of a die comes up different every time, no two students are going to get the exact same random selection of samples. They were at least starting to understand that fact.

Finally, we had a discussion more focused on when you can and cannot use Central Limit Theorem. I laid out the 3 conditions of the theorem and gave students different scenarios in which they could check the conditions.

As you can see, the sampling distribution was a “major emphasis” in my class, but the depressing fact is that students still don’t understand it. After some dismal test performances, I decided to return on Friday to the topic one last time. I prepared a carefully laid out worksheet showing all of the samples piling up on a graph so that students could literally see the sampling distribution. One table of especially diligent students got through the handout and made sense of it, but I was shocked with just how difficult it was for them.

Even my best students had trouble graphing the probabilities in the sampling distribution. Instead of counting the 12 samples in which 67% of the respondents said yes, they counted the 24 respondents who said yes. In other words, they were thinking of respondents rather than samples as the objects in the sampling distribution. They got hung up on some very simple calculations, and it seemed like two parallel narratives developed. On the one hand, there was a narrative about calculating numbers, a sort of obstacle course of formulas and complicated hoops to jump through in order to get the “right” answer. Parallel to this was a narrative about statistical meaning. It was like these two narratives were on separate paths that never crossed. They got further down the calculation path, but never stopped to ask how the calculations they had just done might inform or give meaning to the statistics.

In hindsight, I should have broken chapter 7 into two tests, the first on the sampling distribution and the second on the Central Limit Theorem and confidence intervals. It’s crazy that all three of those concepts are in one chapter. I also could have made my lessons more focused, introducing a single idea each class and pounding home that one point one day at a time. I could have spent two classes having students construct sampling distributions without discussing error, bias, sampling methods, etc. Later, we could have used sampling distributions to talk about error and bias.

There are always things you could have done differently, but there is also something especially difficult about the sampling distribution. From the perspective of APOS theory, the sampling distribution treats samples as objects. Until this point, the objects in a probability distribution are always respondents, but now they are sets of respondents. According to Dubinsky and crew, this means that students have to develop an object understanding of a sample.

I wonder if this is really the issue. My students are comfortable acting on samples, removing or adding respondents to create new samples. Plus, a sample is really just a subset of a fixed size. Is that such a difficult concept to reify? It seems like the bigger issue is that students need to be able to conceive of the set of all possible samples of a fixed size. That is a very big set.

APOS theory is based on Piaget’s concept of reflective abstraction. It’s founder, Ed Dubinsky, has long been an advocate of using the programming language ISETL as a way to teach mathematics. Seymour Papert is another pupil of Piaget who has also advocated for the use of programming in math education. Papert’s version of Piaget is a little strange; at times he sounds more like Vygotskyite, especially in the significance he attributes to cultural artifacts. For example, in Mindstorms Papert describes a “typical experiment in combinatorial thinking” in which children are asked to form “all possible combinations of beads of assorted colors.” He notes that “it is really quite remarkable that most children are unable to do this systematically and accurately until they are in the fifth or sixth grades,” and then provides the following analysis.

The task of making families of beads can be looked at as constructing and executing a program, a very common sort of program, in which two loops are nested: Fix a first color and run through all possible second colors: then repeat until all possible first colors have been run through. For someone who is thoroughly used to computers and programming there is nothing “formal” or abstract about this task. For a child in a computer culture it would be as concrete as matching up knives and forks at the dinner table. Even the common “bug” of including some families twice (for example red-blue and blue-red) would be well-known. Our culture is rich in pairs, couples, and one-to-one correspondences of all sorts, and it is rich in language for talking about such things. This richness provides both incentive and a supply of models and tools for children to build ways to think about such issues as whether three large pieces of candy are more or less than four much smaller pieces. For such problems our children acquire an excellent intuitive sense of quantity. But our culture is relatively poor in models of systematic procedures. Until recently there was not even a name in popular language for programming let alone for the ideas needed to do so successfully. There is no word for “nested loops” and no word for the double-counting bug. (pg. 22)

In short, students have never encountered anything like a set of all combinations before. It is difficult for them to imagine this set.

In his short story, The Library of Babel, Borges describes a library “composed of an indefinite, perhaps infinite number of hexagonal galleries.” The books in this library are composed of “twenty-five orthographic symbols.” William Goldbloom Bloch has a wonderful book of essays exposing the “unimaginable mathematics” behind Borges’ story. In the first of these essays, he works through a rough estimate of the number of books in the library: 25^131200, a number that he notes is several magnitudes larger than the number of particles in the universe.

Somehow, it feels all too easy, even anticlimactic, as though instead we should have had to write pages and pages of dense, technical high-level mathematics, overcoming one complex puzzle after another, before arriving at the answer. But most of the beauty–the elegance–of mathematics is this: applying potent ideas and clean notation to a problem much as the precise taps of a diamond-cutter cleave and husk the dispensable parts of the crystal, ultimately revealing the fire within. (pg. 17)

As Bloch notes, “the number of books in the library, although easily notated, is unimaginable.”

It seems that the problem with the sampling distribution is just the same. Although easily notated, it is unimaginable. At best, I can show my students a small sampling distribution for a population of 7 and sample sizes of 3.


A “small” sampling distribution.

This example completely contradicts the point of the sampling distribution. If there were only 7 people in the population then you would just talk to all 7 of them. There would be no need to select a sample and use statistical inference. The precise point of the sampling distribution is that we cannot survey every person in the population so we need to how know reliable a small sample of these people will be. Just as children struggle to systematically arrange beads of different colors in Papert’s example, my students struggle to conceive of the sampling distribution. It’s unimaginably large and they have no cultural artifacts or prior knowledge to build upon. What we really need is a programming language that is visual and accessible in which students could develop a concrete understanding of the sampling distribution through systematic thinking.

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.

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.

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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!