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.

a/k/a Dave Major Goes Bananas

Shorter: Dave Major and I are experimenting again with what math textbooks could look like on devices that are digital and networked. Our most recent experiment is Ice Cream Stand.

Longer: Last September, Kate posted this image to Twitter attached to the tweet, “Worst geometry problem ever: can’t be solved until after you solve it.”

Clever bit, right? Classic Kate.

We could print that out and have students use a compass and straightedge to construct the circumcenter (the point that’s equidistant from all three coffee shops). That’d be a fine summative assessment. Very “real world,” etc.

But if you’d like to use Kate’s tweet to motivate the need for the circumcenter, to give students a reason to care about the circumcenter, we’ll need to start much lower on the ladder of abstraction. We’ll need to throw out formal vocabulary and formal operations for a few minutes. We’ll need to start with intuition.

So we changed the domain from coffee to ice cream. We changed the environment from a roadway (a complicated space) to a park (an open space). And we gave students a few easy choices. “Which ice cream stand would you pick, given where you’re standing right now?”

Students see that they’re basically painting the field one dot at a time.

So we ask them to extend that metaphor and paint the entire field so that someone else can see which stand is the closest no matter where they are in the park.

This is a task that a lot of students can complete regardless of their mathematical knowledge. It’s expensive, but not impossible, to provide this task on paper. It’s impossible to do on paper what comes next.

We combine the entire class’ park paintings.

That’s a composite from three dozen people on Twitter.

Dave and I then asked students for some preliminary thoughts about how we could calculate the right painting. But that’s where we finished. The point is, students now want to know, “Who’s right? Who’s closest?” And what’s weird is that our intuition validates the math to a degree.

That is to say, you can see areas where Twitter agreed with itself. You can see areas where Twitter disagreed with itself. When you construct the circumcenter from the perpendicular bisectors, you’ll find that they overlay rather neatly on the areas of disagreement.

That’s the ladder of abstraction. It isn’t impossible to climb it with print-based tasks, but a digital networked device makes it a lot easier.

Open Questions

• Q: Where does this activity go next? We could add some expository text about the circumcenter. We could leave that to the teacher. We could calculate which student took the best guess in her painting of the field. A huge open question throughout these projects is, “What role does the teacher play here?”
• Q: Another huge, open question is, “What happens to the first student who runs through this activity?” Her composite painting is just her own painting. Dave and I are developing activities that exploit the network effect. They get better and more interesting when more students use them. So again: what happens to the first student through?

BTW. Dave Major wrote his own post about this project.

Featured Comments

Alexandre Muniz:

The burning question I have after looking at this is, why is the average line a bit wrong? (Especially the blue/green line.)

Evan Weinberg:

The line of uncertainty shows where the intuitive power of the brain breaks down. This is where the power of mathematical tools can step in to hone in on a more precise answer. What strikes me here is that the mathematical tools don’t do that much better of a job.

Jason Dyer:

If you allow the first student through to see the picture as it gets revised (via a reload button or some auto-update), I don’t see a terrible problem (except for the usual classroom dilemma of what you do with any student that finishes fast).