Math Hombre

If at some point in this post you don't say "that's a bit of a stretch," I will not have done my job here.

I got an iPod Touch for work this summer. Despite being aggressively pro-tech, my personal tech level is loooow. No cell phone, no iPod, no iPad, no video game system ... ridiculous, really. But I'm working with Alejandro Montoya, a computer science grad student, this summer (following my colleague Char Beckmann) as he develops a cool iPhone quadratics game. (Due in the app store for free any day now.)  One of the problems was a ridiculously low number of devices, so, time to invest. My colleague Paul Yu and I have an NSF proposal in to equip a classroom with iPod Touches (among other tech) and I'm a believer in really using things before asking students to do so. Paul and Dave Coffey use their iPhones well to support their class.

While developing the game, Jon Engelsma, director of my university's mobile development lab, wanted the game to use more of the iPhone specific capability. Each phone/pod is equipped with an accelerometer - which is why it can detect orientation and movement like tilting.  In response, Alejo added a new aspect to the game where you're making the parabola in the air to show orientation. But along the way, Jon mentioned Doodle Jump as an example of a game that used it well. "Oh, only 99¢," I say.

Ruh roh.







At the time of writing this, I'm at a high score of about 20,000. I swear I don't use real time to play, just moments where I'm stuck somewhere.

There's math in that game?

My son asked that in surprise. As Alejo mentioned to some of the high school students playing ParabolaX, there's a LOT of math in programming and game design. Frank Noschese has written so much good stuff on the physics of Angry Birds. Doodle Jump has that. I've thought about how far up and across you can get on a jump, use constant speed estimation of moving platforms, etc.  But other than doing some modeling (which would be interesting I think) of the jumps, there's not a lot of explicit math for players. There is, though, an understanding that the game is on a cylinder (go off the left, return on the right, etc.). That's made me wonder if anyone has developed an iPhone game that really uses the accelerometer to explore topologically interesting surfaces. A Möbius maze that you navigate with the accelerometer? A Klein bottle version of Othello? Who knows where that could lead.

There's math in that game
The Common Core State Standards Mathematical Practices:
1. Make sense of problems and persevere in solving them.
At its heart, like many games, Doodle Jump is a big problem (get as high as possible) composed of smaller problems (how do I get past the pink monster reliably). My perseverence, like most students, is very high for games.

2. Reason abstractly and quantitatively.
The constraints are simple: you can't stop jumping, and if you land on nothing or bump into an obstacle, game over. There's not much quantitative reasoning, unless you're fixated on a score. Then there's some nice linear programming on the fly to figure out how to improve your score and what's required. I have wondered why, being extremely right handed, I seem to be better with my left on this game.  Specific game questions too, like why don't I see rocket packs anymore?

3. Construct viable arguments and critique the reasoning of others.
Hasn't happened for me yet, because I haven't discussed the game with anyone other than my son, who's even more of a novice than I am.  But there's definitely opportunity, as we've discussed: Does a back lean enable higher jumps? That's my son's conjecture but I haven't experienced it yet.

4. Model with mathematics.
Also not yet. Though I am interested in trying to model the jumps, and I'm curious about how to even start.  There's good reason to measure to improve your game play, but the modeling is the kind of idle mathematical reasoning that mathematicians love. I'd also love to know how the accelerometer data feeds into the shape of the jump. Constant horizontal velocity if tilted or does it depend on the angle of tilt?

5. Use appropriate tools strategically.
Not yet. But the modeling will definitely be helped by mathematical tools.

6. Attend to precision.
Hmmmm. Kinesthetic precision is definitely required. It matters how you move and how much. The engagement of this again has me wondering how to make math more kinesthetic more often.

7. Look for and make use of structure.
Very rich context for this. Even at my lower levels, there are many patterns to notice, and noticing them is crucial to survival in the game. 

8. Look for and express regularity in repeated reasoning.
I think it's a new thought to me that this is a crucial part of video games, and I want to use this connection in math class. The skills required early on become automatic and no longer a problem. This is just natural as you get better, you gain automaticity with tasks that used to require thought. Even though you may make occasional mistakes.  Does that describe math or video games?

What am I learning about teaching?
Half of learning about teaching is learning about learning.  This game has been good for me to think about because I'm not very good. There was a brilliant teacher educator at Siena Heights, Sr. Eileen Rice, required her advisees to take a class in a subject with which they struggled.  Great idea. Just like with a math problem, we can't problem solve if it's too easy for us.  It's struggle that affords an opportunity for growth.

In the game I've had to figure out specific challenges, wonder about how things worked in general, identify areas where I need improvement, and make some realizations about my limitations. I can not shoot effectively. My videogame dexterity and reaction time is probably below average.  But I've had nice moments of achievement. Figuring out how to get past a few beasts, then that I can jump on them, specifically working on using the cylindrical aspect, etc. Things I couldn't do before that I can do now.

The other half of learning about teaching is tackling the question of how do I support learners? That's the heart of instruction to me, and what motivates gathering data (assessment), giving feedback (evaluation) and problem and resource selection (planning).  This game has made me wonder about all these.

Do I look up directions? Cheats? The game is popular and there are lots of tip sites out there.  Even Doodle Jump cheats. This requires me to think about my purposes. Do I need the highest possible score? Do I want to figure it out for myself? What's the purpose of playing?  Once I was a teacher that told students this information without asking. I was good at it, and got good evaluations, and most of my students did really well on tests. Then I was a teacher that would never tell this information, even if students basically begged for it.  Students did some amazing work, found out things I hadn't known, and most were successful.  But some students were frustrated, including some of the successful students.  Now, I try to assess student purposes and provide relevant information.  But it's harder than having a simple extreme policy.

Wow, I made it through that whole paragraph without mentioning how Khan Academy can be like those tips and cheats YouTube videos. (Shoot.)

Talk. It's also better with other people.  I've been hampered by doing this game alone. Looking at the practices made me think about the richness I'm missing.  I've seen this in Alejo's pilots of ParabolaX also - radical differences in what students get out of the game based on how much they discuss it.  I want a healthy balance for my students between 'let me do it for myself' and 'how are you thinking about it?'

How do I measure success? I like the score as a measure of how far I've gotten, or as a measure of whether I've gotten better. But as I consider that, there's really better things to notice. The game territory changes as you climb, so that's a good measure of how far.  (When I get to the bounce-once platforms currently, that's a good game! In the jungle setting.)  Can I make difficult moves? Have I gotten better at getting better?  It feels like I have better learned how to identify problems in the game and am more efficient at addressing them.

Then I got wondering what the scores even really mean.  Is my high score the best measure of how good a player I am? Should it be my average or median score? Weighted somehow between the two?  If the goal is the maximum score, it encourages high risk behavior that's bad on the average but when it works garners big points.  Score-based thinking also pushes me towards tips and cheats, which are not in my best interest as a learner or enjoyer of the game.  The best use of the scores is a nice mathematical problem, like a simpler version of trying to figure out what are the relevant baseball statistics.  In general, though, it encourages me to go farther in the direction of SBG and portfolios.

So...
If you made it this far, thanks for sticking with the rambling. I'd love to know what you think about this, or other thoughts you have about teaching and learning from games or other odd contexts.

P.S.  Yes the title is a little poke at the now ubiquitous Jump Math, which has some super-proponents. (There are samples at jumpmath1.org)

P.P.S. We need some games this engaging with more math content.  Waker and MangaHigh are a start, but the accelerometer using games will be a big step up.

Image credits: there were no good CC images for this post, so if I used one of yours and you wish it not, just let me know. All images click through to original source.  The ragecomic was too true not to include.

There was a nice (if long-windedly titled) article in the Decemeber '03 Teaching Children Mathematics called "Metaphors as a Vehicle for Exploring Preservice Teachers' Perceptions of Mathematics," by Brenda Wolodoko, Katherine Wilson and Richard Johnson.  In the article the preservice teachers made images to display themselves as teachers or learners of mathematics.  The majority of their images revealed anxieties about the content as learners, but hope for themselves as teachers.  The researchers liked the way that the images created an opportunity for dialogue and created a potential for change.  One interesting sidenote is that students used the idea of puzzles both positively and negatively, modeling both frustration and engagement.

My preservice elementary teachers recently made images of what does it mean to teach mathematics in small groups, and our secondary student teachers envisioned their future classrooms.  There were many neat ideas to share.

 The mullet came later, but I think they thought the "Business in the front, party in the back," slogan did relate to teaching.














 This and the next chart were imagined classrooms, which you'll see a lot of with the secondary teachers as well.  They value group work, technology, manipulatives and whole class time.  I often wish college classrooms had room for a carpet section for students to sit down.


 These next two charts are more like concept maps.  This group focused on the most important aspects to them.

While this group saw their concept map evolving into a hierarchy.  There's some pretty interesting connections here to look into.














Somehow missed my favorite poster here.  Clever use of Facebook, and really made me think about that page as a representation of who someone is.  I think there are lots of idealized people I'd be interested in seeing Facebook pages for.

The secondary teacher assistants made quick individual sketches at the end of a seminar.  So don't expect the artistic commitment we got from the elementary teachers.  One thing that came across in classroom images is the presence of the kind of technology to which they've been exposed.  It's becoming clear to me that we need to do a better job of teaching technological pedagogical content knowledge, primarily by explicit modeling.

Let's start off with a few of the text descriptions.  This teacher is worried about the content they will be forced to cover.







This teacher is thinking about classroom management as the start of learning.
I'm not sure if this teacher was describing life as it is or as they envision.  Somebody definitely considering the different models with which they've been presented.













Next come several visions of cooperative learning.  There seems to be a clear value on student discussion, and varying images of what the teacher's role is in relation.













And this sketch merges a vision of the classroom with a concept map of what is important to them.
































I'd be very interested in knowing what you think about the images here and what you notice, if you'd care to leave a comment or drop an email.  Thanks!