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!

The User-Friendly Classroom

March 2, 2015

Mike Caulfield’s recent post about “users” has gotten me thinking about why I hate the phrase “user-friendly”:

People say they want a world of “producers not consumers” or “makers not takers”.  Peel back the assumptions under those statements and you’ll find some disturbing stuff.

And so it was when I returned to instructional design in 2009, fresh off the OCW experience, that I found these phrases, which used to seem so normal, now strange.

There was a time, after all, that we used to call lurkers “readers”. Users were “doers”.  These things had respect.

Mike’s point is that sharing and curating someone else’s work should be valued every bit as much as creating something from scratch. That’s a fair observation, but there may be other reasons why readers tend to “lurk.” As the term insinuates, reading can be a form of intellectual spectatorship, consuming others’ ideas without generating your own. Indeed, Marshall McLuhan considers print a form of “hot” media.

Hot media usually, but not always, provide complete involvement without considerable stimulus. For example, print occupies visual space, uses visual senses, but can immerse its reader. […] Cool media, on the other hand, are usually, but not always, those that provide little involvement with substantial stimulus. They require more active participation on the part of the user, including the perception of abstract patterning and simultaneous comprehension of all parts. —Wikipedia

Print tends to spell things out a little too much for people. It provides “complete involvement without considerable stimulus,” leaving less room for active participation on the part of the user. We should not judge “readers” for lurking, but we do need to consider if the medium itself is pushing users away from active participation. A lot of OER apologists will jump in here to point out that open educational resources aren’t limited to just print; they can include hypertext, images, video, and so on. That’s a fair point, but the reality is that most OER is just copy-left textbooks distributed through a digital medium, and even videos and hypertext are not “cool” media.

A lot of OER comes packaged as courseware modules that instructors are intended to remix. Pasting together modules does not make a textbook, and indeed, part of what makes this content so difficult to use is it’s inherently prescriptive nature. We are saying, “here’s a finished module. Go make a course out of it,” which is sort of like saying, “here’s an automobile. Go make a custom vehicle out of it.” OER needs to be “cooler,” more abstract, rougher around the edges. We can’t give “users” a finished product and ask them to be “makers.”

Union Docs has been running an amazing series on spectatorship called What You Get Is What You See. My favorite DJ, Jace Clayton, contributed a talk in which he told a sweet story about how he got into DJing. He was attending Jungle dance parties in dark Boston warehouses. The warehouses were so dark that you couldn’t see the DJ, and as it turns out, this was a good thing. In the absence of any visible performance, people danced carelessly together in the dark. It created a sort of shared space that wouldn’t have existed otherwise.

Right now, the value of DJ’s is a hotly contested topic much akin to the “makers” versus “takers” debate.

It seems that everyone is a DJ these days. Technology has made beat-matching as simple as pressing the “sync” button on a laptop, and without tour vans full of heavy equipment, DJ’s are cheaper and more versatile than a live band. The problem with DJ culture is that (as Derrick May has said) “it looks like people are checking their email.” It’s not fun to watch someone hide behind a laptop, and it calls into question how much the DJ is really doing beyond selecting the tunes and pressing play.

I’ve always valued musicians that blur the lines between DJ and live performer. To the extent that a DJ plays records, everything is literally rehearsed. In a live performance, there is no “sync” button. You are taking greater risks, and so there is a raw energy that’s missing with DJ’s. James Brown is definitely not “checking his email” on stage.

Two years ago, during LAMC, I saw Rafi eL perform live in a small bar in Williamsburg. He’d sketched out various pieces of his songs in a sequencer, and he triggered them live while singing over the mix. It was an invigorating performance, but midway through his set, a well-known DJ behind me commented that “he just ruined it by picking up the mic.” Whenever a musician sings, people stop what they’re doing to watch. They become instant spectators. As this DJ pointed out, “you should be dancing or hitting on someone, not watching the DJ.” The problem is you can’t get rid of spectators without getting rid of the performer.

Part of what made Jace Clayton’s experience in those Boston warehouses so great was that the DJ became invisible. A live performance demands your attention; it provides added stimulus, and in doing so it amounts to a “hotter” media. Spectatorship is the price we pay for that added stimulation.

Going back to the topic of OER, we talk at great length about making things “user-friendly,” and we take for granted that this is a good thing. The assumption is that the less someone needs to understand about how something works, the more likely they are to engage with it. Certainly, if something is easier to use, you will have more “users,” but how much depth of use does this invite? I’ve found that “user-friendly” is usually a coded way of saying “spectator-friendly.” To make something easier to use, we “provide complete involvement without considerable stimulus” so that “users” are encouraged to be “lurkers.” In McLuhan’s terms, “user-friendly” usually means “hot” media.

As a community college professor, I spend a lot of time designing class activities, and my number one concern is usually how to get students involved as “active participants.” Spectatorship is not a valued asset. If we want “readers” instead of “lurkers” and “doers” instead of “users” then we need to be invisible like the DJ’s in those Boston warehouses. We need to give students tools that are abstract and unfinished, and invite them to make something out of those tools. One thing we definitely do not need is a “user-friendly” classroom.

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