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tl;dr – This is about a new digital lesson I made with Christopher Danielson and our friends at Desmos. It's called Waterline and its best feature is that it shares data from student to student rather than just from student to teacher. I'll show you what I mean while simultaneously badgering publishers of digital textbooks. (As I do.)

Think about the stretches of time when your smartphone or tablet is in airplane mode.

Without any connection to the Internet, you can read articles you've saved but you can't visit any links inside those articles. You can't text your friends. You can't share photos of cats wearing mittens or tweet your funny thoughts to anybody.

In airplane mode, your phone is worth less. You paid for the wireless antenna in your tablet. Perhaps you're paying for an extra data plan. Airplane mode shuts both of them down and dials the return on those investments down to zero.

Airplane mode sucks.

Most digital textbooks are in airplane mode:

  • Textbooks authored in Apple's iBooks Author don't send data from the student's iPad anywhere else. Not to her teacher and not to other students.
  • HMH Fuse includes some basic student response functionality, sending data from the student to the teacher, but not between students.
  • In the Los Angeles Unified iPad rollout, administrators were surprised to find that "300 students at three high schools almost immediately removed security filters so they could freely browse the Internet." Well of course they did. Airplane mode sucks.

The prize I'm chasing is curriculum where students share with other students, where I see your thoughts and you see mine and we both become smarter and life becomes more interesting because of that interaction. That's how the rest of the Internet works because the Internet is out of airplane mode.

Here's one example. In Waterline we ask students first to draw the height of the water in a glass against time. We echo their graph back to them in the same way we did in Function Carnival.


But then we ask the students to create their own glass.


Once they successfully draw the graph of their own glass, they get to put it in the class cupboard.


Now they see their glass in a cupboard right alongside glasses invented by their friends. They can click on those new glasses and graph them. The teacher sees all of this from her dashboard. Everyone can see which glasses are harder to graph and which are easier, setting up a useful conversation later about why.

We piloted this lesson in a local school and asked them what their favorite part of the lesson was. This creating and sharing feature was the consensus winner.

A selection:

  • Making my own because it was my own.
  • Trying to create your own glass because you can make it into any size you want.
  • Designing my own glass because I was able to experiment and see how different shapes of the glass affects how fast the glass filled up.
  • My favorite part of the activity was making my own glass and making my other peers and try and estimate my glass.
  • My favorite part of the activity was solving other people's glasses because some were weird shapes and I wanted to challenge myself.

Jere Confrey claimed in her NCSM session that "students are our most underutilized resource in schools." I'd like to know exactly what she meant by that very tweetable quotation, but I think I see it in the student who said, "I also liked trying out other's glasses because we could see other's glasses and see how other people solved the problem."

I know we aren't suffering from too many interactions like that in our digital curricula. They're hard to create and they're hard to find. I also know we won't get more of them until teachers and administrators like you ask publishers more often to take their textbooks out of airplane mode.

David Cox sent his students through Function Carnival where they tried to graph the motion of different carnival rides. (Try it!)

Every student's initial graph was wrong. No one got it exactly right the first time. But Function Carnival doesn't display a percent score or hint tokens or some kind of Bayesian probability they'll get the next graph right. It just shows students what their graph means for that ride. Then it lets them revise.

David Cox screen-recorded the teacher view of all his students' graphs. This is the result. I love it.

BTW. I'm hardly unbiased here, having played a supporting role in the development of Function Carnival.

Today Desmos is releasing Function Carnival, an online math happytime we spent several months developing in collaboration with Christopher Danielson. Christopher and I drafted an announcement over at Desmos which summarizes some research on function misconceptions and details our efforts at addressing them. I hope you'll read it but I don't want to recap it here.

Instead, I'd like to be explicit about three claims we're making about online math education with Function Carnival.

1. We can ask students to do lots more than fill in blanks and select from multiple choices.

Currently, students select from a very limited buffet line of experiences when they try to learn math online. They watch videos. They answer questions about what they watched in the videos. If the answer is a real number, they're asked to fill in a blank. If the answer is less structured than a real number, we often turn to multiple choice items. If the answer is something even less structured, something like an argument or a conjecture … well … students don't really do those kinds of things when they learn math online, do they?

With Function Carnival, we ask students to graph something they see, to draw a graph by clicking with their mouse or tapping with their finger.

We also ask students to make arguments about incorrect graphs.

I'd like to know another online math curriculum that assigns students the tasks of drawing graphs and arguing about them. I'm sure it exists. I'm sure it isn't common.

2. We can give students more useful feedback than "right/wrong" with structured hints.

Currently, students submit an answer and they're told if it's right or wrong. If it's wrong, they're given an algorithmically generated hint (the computer recognizes you probably got your answer by multiplying by a fraction instead of by its reciprocal and suggests you check that) or they're shown one step at a time of a worked example ("Here's the first step for solving a proportion. Do you want another?").

This is fine to a certain extent. The answers to many mathematical questions are either right or wrong and worked examples can be helpful. But a lot of math questions have many correct answers with many ways to find those answers and many better ways to help students with wrong answers than by showing them steps from a worked example.

For example, with Function Carnival, when students draw an incorrect graph, we don't tell them they're right or wrong, though that'd be pretty simple. Instead, we echo their graph back at them. We bring in a second cannon man that floats along with their graph and they watch the difference between their cannon man and the target cannon man. Echoing. (Or "recursive feedback" to use Okita and Schwartz's term.)

When I taught with Function Carnival in two San Jose classrooms, the result was students who would iterate and refine their graphs and often experience useful realizations along the way that made future graphs easier to draw.

3. We can give teachers better feedback than columns filled with percentages and colors.

Our goal here isn't to distill student learning into percentages and colors but to empower teachers with good data that help them remediate student misconceptions during class and orchestrate productive mathematical discussions at the end of class. So we take in all these student graphs and instead of calculating a best-fit score and allowing teachers to sort it, we built filters for common misconceptions. We can quickly show a teacher which students evoke those misconceptions about function graphs and then suggest conversation starters.

A bonus claim to play us out:

4. This stuff is really hard to do well.

Maybe capturing 50% the quality of our best brick-and-mortar classrooms at 25% the cost and offering it to 10,000% more people will win the day. Before we reach that point, though, let's put together some existence proofs of online math activities that capture more quality, if also at greater cost. Let's run hard and bury a shoulder in the mushy boundary of what we call online math education, then back up a few feet and explore the territory we just revealed. Function Carnival is our contribution today.

Let's speculate that before this year's cohort of first-year teachers retires from math education more than 50% of American classrooms will feature 1:1 technology. That's a conservative prediction – both in the timeline and the percentage – and it's more than enough to make me wonder what makes for good curricula in a 1:1 classroom. What are useful questions to ask?

Here's the question I ask myself whenever I see new curricula crop up for digital networked devices like computer, laptops, tablets, and phones.

Is it any different?

That isn't a rhetorical or abstract question. I mean it in two separate and specific ways.


If you print out each page of the digital networked curriculum, is it any different?


The answer here is "sort of."

When I look at iBooks in the iBookstore from Pearson and McGraw-Hill or when I see HMH publish their Algebra Fuse curriculum in the App Store, I see lots of features and, yes, they require a digital medium. They have a) interactive slider-type demonstrations, b) slideshows that walk students through worked examples, c) stock video in the margins instead of stock photography, d) graded multiple-choice quizzes, e) videos of Edward Burger explaining math concepts and f) probably other items I'm forgetting. None of those features would survive the downgrade to paper.

So the question becomes, "Is it different enough?"

Are these offerings different enough to justify the enormous expense in hardware, software, and bandwidth? Do they take full advantage of their digital birthright?

I don't think so.


"Is it any different?" here means "if you were hundreds of feet below the surface of the Earth, in a concrete bunker without any kind of Internet access, is the curriculum any different?"


Here, in September 2013, the answer is "no," which is a shocking waste of very expensive, very powerful device.

Look at the apps you have on the home screen of your smartphone and ask yourself "how many of these are better because they have a large network of people using them?" Me, I have 12 apps on my homescreen and eight of them – Tweetbot, Messages, Instapaper, Instagram, Phone, Mail, Safari, Spotify – are so much better because of the crowd of people that use them with me. When I switch off my phone's network connection, they get so much worse. Those are the apps I care most about also, the ones that enrich my life, the ones that justify the expense of a smartphone.

When you switch off the network connection, most curriculum stays exactly the same. It doesn't suffer at all, which means it isn't taking advantage of the network connection when it's on.

More Different

Digital devices should allow you to:

  • Pose more interesting problems using more diverse media types and fewer words. (eg. three-act-style tasks).
  • Replace your textbooks' corny illustrations of mathematical contexts with illustrations from their own lives. Students: find a trapezoid from your own life. Take a photo. Tap upload. Now it's in your textbook.
  • Progressively disclose tasks over multiple screens so students don't have to look at pages full of questions and information like this [pdf] and can instead start with a brief video and single sentence.

Networked devices should allow you to:

  • See all your friends' illustrations from their own lives. The teacher should be able to see that gallery of trapezoids, promote certain illustrations, and offer comments on others that are visible to everybody.
  • Start lessons with integrated, formative polling. I'm talking about Riley Lark's ActivePrompt software built right into the textbook.
  • Create student conversations. Use student data to find students who disagree with each other, pair them up, and have them work out their differences. All of that should happen without the teacher having to facilitate it because the device is smart.
  • Combine student data for better, more accurate modeling. (eg. Pennies, where each student collects a few data points which are then instantly collected into a much larger class data set.)

There are other possibilities, of course, some of which we'll only start to realize as these tools are developed. But don't just sit around and wait for an industry as reactive as textbook publishing to start making those tools for you. Publishers and their shareholders react to their market and that's you. As long as they can still profit by repurposing existing print curriculum they will. It's on you to tell your publishing reps that the curriculum they're selling doesn't do enough justice to the powerful, digital networked devices they're putting them on. It isn't different enough.

2013 Sep 27. And here's LA Unified buying a billion dollars worth of iPads and then wasting the network that might make that investment worthwhile:

By Tuesday afternoon, L.A. Unified officials were weighing potential solutions. One would limit the tablets, when taken home, to curricular materials from the Pearson corporation, which are already installed. All other applications and Internet access would be turned off, according to a district "action plan."

Featured Comment

Elizabeth Statmore:

This is always a problem in the early stages of a new technology. The “Technology Adoption Life Cycle” has proven itself over and over for the last 20 years to be the gold standard in analyzing tech markets.

The “innovators” adopt a technology because they need to be the first kids on their block to have whatever it is. The “early adopters” see strategic advantages and uses for it — and they are willing to put up with what they perceive as minor inconveniences like limited optimized uses in order to gain the advantages they seek.

That moment of “crossing the chasm” into the mainstream is that moment when a technology catches fire because vendors have figured out a way to reach beyond the techno-enthusiastic “early adopters” who have sustained their businesses to the techno-unimpressed “early majority” customers who are the major “show-me” skeptics. These skeptics form the first mass market for a technology, followed only later — and reluctantly — by a “late majority.”

Seems to me that we are still very much in an “early adopter” market in the race for digital textbooks. No one knows the “killer app” for digital curriculum is going to look like, but we do know it might bear some slight resemblance to the analog textbook. But this will not

As Steve Jobs always used to say, the “killer app” for the iPhone was making a phone call. But it was all the supporting infrastructure tht was built in (seamlessly integrated contacts, e-mail, texting, reminders, calendar, notes, & management of the technology) that transformed the act of making a phone call.

[Future Text] Math Cache

a/k/a Great Moments in Digital Networked Math Curricula

You Should Check Out

Math Caching and Immediately Useful Teaching Data from Evan Weinberg.

What It Is

Evan has his students working on some practice exercises. As they complete their exercises, they use their Macbooks to submit a) an answer (which is nothing new in a world driven by quantitative machine-graded data) but also b) a photo of their work.

The images are titled with their answers and then start populating a folder on Evan's computer.


Why It's Important

Mistakes are valuable. Student work is valuable. This collects both quickly.

Mistakes are valuable for starting conversations, for prompting to students to construct and justify arguments, for asking students, "What different question does this work correctly answer?"

Most machine-graded systems hold back students with wrong answers and let them advance once they've corrected their errors. But this essentially sweeps clear the brambly trail that led to that correct answer when there's so much value in the brambles. Those systems don't tell you why the student had those incorrect answers. They don't allow the teacher to sequence and select incorrect student work for productive discussions later. Math Cache does.

Here's Evan:

I didn’t need to throw out the tragically predictable ‘who wants to share their work’ to a class of students that don’t tend to want to share for all sorts of valid reasons. I didn’t have to cold call a student to reluctantly show what he or she did for the problem. I had their work and could hand pick what I wanted to share with the class while maintaining their anonymity. We could quickly look at multiple students’ work and talk about the positive aspects of each one, while highlighting ways to make it even better.

Somewhat Related:

Nicora Placa:

A main assumption that I work with when doing these [student] interviews is that children do what makes sense to them even if it seems like nonsense to me. My job is to figure out what makes sense to them and why.

2013 Oct 2. Pearson's research blog picks up this post and argues that I'm too pessimistic about machine-graded data.

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