This is an easy call. I need a question to carry me through my thirties and I can’t think of a better one than, “What does the math textbook of the future look like?”

I’ve known for awhile I need a certain set of collaborators for that project. I have worked with Eli, Eric, and Jenny for the last three years. We need each other. They need what I do (the math teaching stuff) and I need what they do (the computery stuff). They’re great at what they do and we get along great. Stay tuned.

The lesson asks for an imprecise sketch rather than a precise graph.

This is so rare. More often than not, our curricula rushes past lower, imprecise, informal, concrete rungs on the ladder of abstraction straight for the highest, most precise, most formal, most abstract ones. That’s a disservice to our learners and the process of learning.

You can always ask a student to move higher but it’s difficult to ask a student to move lower, forgetting what they’ve already seen. You can always ask for precisely plotted points of a model on a coordinate plane. But once you ask for them you can’t unask for them. You can’t then ask the question, “What might the model look like?” Because they’re looking at what the model looks like. So the Times asks you to sketch the relationship before showing you the precise graph.

Their reason is exactly right:

We asked you to take the trouble to draw a line because we think doing so makes you think carefully about the relationship, which, in turn, makes the realization that it’s a line all the more astonishing.

That isn’t just their intuition about learning. It’s Lisa Kasmer’s research. And it won’t happen in a print textbook. We eventually need students to see the answer graph and whereas the Times webpage can progressively disclose the answer graph, putting up a wall until you commit to a sketch, a paper textbook lacks a mechanism for preventing you from moving ahead and seeing the answer.

This isn’t just great digital pedagogy, it’s great pedagogy. You can and should ask students to sketch relationships without any technology at all. But the digital sketch offers some incredible advantages over the same sketch in pencil.

For instance:

The lesson builds your thinking into its instruction.

Once it has your guess — a sketch representing your best thinking about the relationship between income and college participation — it tailors its instruction to that sketch. (See the highlighted sentences.)

The lesson is the same but it is presented differently and responsively from student to student. All the highlighted material is tailored to my graph. I watched an adult experience this lesson yesterday, and while she read the personalized paragraph with interest, she only skimmed the later prefabricated paragraphs. It should go without saying that print textbooks are entirely prefabricated.

It makes your classmates’ thinking visible.

The lesson makes my classmates’ thinking visible in ways that print textbooks and flesh-and-blood teachers cannot. At the time of this posting, 70,000 people have sketched a graph. It’s interesting for me to know how much more accurate my sketch is than my classmates. It’s interesting to see the heatmap of their sketches. And it’s interesting to see the heatmap converge around the point that the lesson gave us for free, a point where there is much less doubt.

In a version of this article designed for the classroom, students would sketch their graphs and the textbook would adaptively pair one group of students up with another when their graph indicated disagreement. Debate it.

I’m not saying any of this is easy. (“Sure! Do that for factoring trinomials!”) But we aren’t exactly drowning in great examples of instruction enhanced by technology. Take a second and appreciate this one. Then let me know where else you think this kind of technology would be helpful to you in your teaching.

Featured Comment

Avery:

And as far as I know, even with Apple proclaiming “Textbooks that go beyond the printed page” since 2012?, there isn’t a single digital math textbook doing this yet.

tl;dr. I made another digital math lesson in collaboration with Christopher Danielson and our friends at Desmos. It’s called Central Park and you should check out the Walkthrough.

Here are two large problems with the transition from arithmetic to algebra:

Variables don’t make sense to students.

We give students variable expressions like the exponential one above, which they had no hand in developing, and ask them to evaluate the expression with a number. The student says, “Ohhh-kay,” and might do it but she doesn’t know what pianos have to do with exponential equations nor does she know where any of those parameters came from. She may regard the whole experience as one of those nonsensical rites of school math which she’ll forget about as soon as she’s legally allowed.

Variables don’t seem powerful to students.

In school, using variables is harder than using arithmetic. But what does that difficulty buy us, except a grade and our teacher’s approval? Meanwhile, in the world, variables are responsible for anything powerful you have ever done with a computer.

Students should experience some of that power.

One solution.

Our attempt at solving both of those problems is Central Park. It proceeds in three phases.

Guesses

We ask the students to drag parking lines into a lot to make four even spaces. Students have no trouble stepping over this bar. We are making sure the main task makes sense.

Numbers

We transition to calculation by asking the students “What measurements would you need to figure out the exact space between the dividers?” This question prepares them to use the numbers we give them next.

Now they use arithmetic to calculate the space width for a given lot. They do that three times, which means they get a sense of the parts of their arithmetic that change (the width of the lot, the width of the parking lines) and those that don’t (dividing by the four lots).

This will be very helpful as we take the next big leap.

Variables

We give students numbers and variables. They can calculate the space width arithmetically again but it’ll only work for one lot. When they make the leap to variable equations, it works for all of them.

It works for sixteen lots at once.

Variables should make sense and make students powerful. That’s our motto for Central Park.

2014 Jul 28. Here is Christopher Danielson’s post about Central Park on the Desmos blog.

Featured Comment

Grant Wiggins:

In thinking further about your complaint about “Write an expression” I think what is also going on in this app is a NEEDED slowing down of the learning process. The text (and too many teachers) are quick to jump to algorithms before the students understands their nature and value. Look how long it takes to get to the concept of an appropriate expression in the app: you build to it slowly and carefully. I think this is at the heart of the kind of induction needed for genuine understanding, where the learner is helped, by scaffolding, to draw thoughtful and evidence-based conclusions; test them in a transfer setting; and learn from the feedback — i.e. the essence of what we argue understanding is in UbD.

Kevin Hall:

One reason I like this activity so much is that it hits the sweet spot where “What can you do with it?” and “What does it mean?” overlap.

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.

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.

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.

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.

Digital

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.

Networked

“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.

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.

You Should Check Out

Des-man: a Desmos Labs Project.

What It Is

First, you had Fawn Nguyen’s assignment where students created a face using the Desmos graphing calculator.

Students reviewed conics and domain and range. That was a blast for a lot of reasons. Now Desmos has created a system where the teacher can quickly see the creation of the faces in real-time and use filters to sort quickly through student work in productive ways.

Why It’s Important

Des-man explores the potential of networked devices in math class.

You could download the Desmos iOS app, flip off your iPad’s Internet connection, and still have a good time creating your Des-man. If the experience of using a digital math curriculum doesn’t get any better when you turn on the Internet, it is wasting the Internet.

With Des-man, an Internet connection lets you see all your friends’ Des-men instantly, as they’re being drawn. It lets the teacher see the Des-men quickly too and then select and sequence them in productive ways.

We have here a math activity for networked devices that doesn’t waste the network. That shouldn’t be noteworthy, but it is.

This is it, the last entry in our summer series of #MakeoverMonday. Thanks for pitching in, everybody.

The Task

Click for the PDF.

What Desmos And I Did

Lower the literacy demand of the task. The authors rattle off hundreds of words to describe a visual modeling task.

Clarify the point of the task. A great way to lower the literacy demand is to convey the point of the task quickly, concisely, informally, and visually, and then formalize, expand, and verbalize that point as students make sense of it. Here, the point isn’t all that clear and the central question (“How can you fit a quadratic function to a set of data?”) is anything but informal.

Add intellectual need. The task poses modeling as its own end rather than a means to an end. Models are useful tools for lots of reasons. Their algebraic form sometimes tells us interesting things about what we’re modeling (like when we learn the average speed of a commercial aircraft in Air Travel by modeling timetable data). Models also let us predict data we can’t (or don’t want to) collect. We need to target one of those reasons.

The most concrete, intellectually needy question, the one we’re going to pin the entire task on, pops its head up 80% of the way down the page, in question #2, and even then it needs our help.

Lower the floor on the task. We’re going to delay a lot of these abstractions — tables, graphs, and formulas — until after students know the point of the task. We’re going to add intuition also and ask for some guesses.

Motivate the different abstractions. The task bounces the student from a table to a graph to a power function in five steps without a word at any point to describe why one abstraction is more useful than another. Students need to understand those differences. A table is great because it lets us forget about the physical pennies. A graph is great because it shows us the shape of the model. And the algebraic function is great because it lets us compute. If those advantages aren’t clear to students then they’re only moving between abstractions because grownups told them to.

Show the answer. We tell students that math models their world. We should prove it. The textbook does great work here, asking students in question #2 to “check your prediction by drawing a circle with a diameter of 6 inches and filling it with pennies.” Good move. But students have already drawn and filled circles with 1, 2, 3, 4, and 5-inch diameters. I’m guessing they would rather draw and fill a 6-inch circle than do all that math. The circles need to get huge to make the mathematics worth their while.

So here’s our new central question: how many pennies will fill a really big circle? We’re going to pose that question by showing someone filling a smaller circle, then cutting to the same person starting to fill a larger circle. It’ll be a video. It’ll take less than thirty seconds and zero words. It’ll look like this:

We’ll ask students to commit to a guess. We’ll ask for a number they know is too high, and too low, asking them early on to establish boundaries on a “reasonable” answer. The digital, networked platform here lets us quickly aggregate everybody’s guesses, pulling out the highs, lows, and the average.

Then we’ll talk about the process of modeling, of looking at little instances of a pattern to predict a larger instance. We’ll have them gather those little instances on their computers, drawing circles and filling them.

Those instances will be collected in a table which will then be aggregated across the entire class creating, a large, very useful set of data.

(Aside: of course a big question here is “should students be collecting that data live, on their desks, with real pennies?” Let’s not be simple about this. There are pros and cons and I think reasonable people can disagree. For my part, the pennies and circles are basically a two-dimensional experience anyway so we don’t lose a lot moving to a two-dimensional computer screen and we gain a much easier lesson implementation. However, if we were modeling the circumference of a balloon versus the breaths it took to blow it up, I wouldn’t want students pressing a “breathe” button in an online simulator. We’d lose a lot there.)

Next we’ll give students a chance to choose a model for the data, whereas the textbook task explicitly tells you to use a quadratic. (Selecting between linear, quadratic, and exponential models is work the CCSS specifically asks students to do.) So we’ll let students see that linears are kind of worthless. Sure a lot of students will choose a quadratic because we’re in the quadratics chapter, but something pretty fun happened when we piloted this task with a Bay Area math department: the entire department chose an exponential model.

Eric Berger, the CTO at Desmos, suspected that people decide between these models by asking themselves a series of yes-or-no questions. Are the data in a straight line? If yes, then choose a linear model. If not, do they curve up on one side of the graph (choose an exponential) or both sides of the graph (choose a quadratic)? That decision tree makes a lot of sense. But the domain here is only positive circle diameters so we don’t see the graph curve on both sides.

Interesting, right?

All this is to say, if you’re a little less helpful here, if you don’t gift-wrap answers like it’s math Christmas, students will show off some very interesting mathematical ideas for you to work with.

Once a student has selected a model, we’ll show her its implications. The exponential model will tell you the big circle holds millions of pennies. We’ll remind the student this is outside her own definition of “reasonable.” She can change or finish.

We’ll show the answer and ask some follow-up questions.

What We Didn’t Do

Change the context. It’s a totally fair point that packing pennies in a circle is a fairly pointless activity, one with no real vocational value. When I pose this task to teachers as an opportunity for task revision, they’ll often suggest changing the context from pennies and circles to a) pepperoni slices and pizza dough or b) cupcakes and circular platters or c) Oreos and circular plates, basically running a find-and-replace on the task, swapping one context for another.

I don’t think that does nothing but I don’t think it does a lot either. It’s adding a coat of varnish to a rotting shed. You’re still left with all the other issues I called out above.

We’ll talk about the teacher dashboard next.

What You Did

• Andrew Shauver wants students to work with the physical pennies.
• Megan Schmidt focuses on all the different ways the directions of the task think for the students.
• Jonathan Claydon makes over a different modeling task with a bunch of nice ideas and I push back in the comments.
• Good material in the preview post.