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Functionary: Learning to Communicate Mathematically in Online Environments

Bloggy Abstract

I took a collection of recommendations from researchers in the fields of online education and mathematics education and asked our friends at Desmos to tie them all together in a digital middle-school math lesson. These recommendations had never been synthesized before. We piloted and iterated that lesson for a year. I then tested that Desmos lesson against a typical online math lesson (lecture-based instruction followed by recall exercises) in a pretest-posttest design. Both conditions learned. The Desmos lesson learned more. (Read the technical abstract.)

Mixed Media

You’re welcome to watch this 90-second summary, watch my defense, read it if you have a few minutes, or eventually use it with your students.

Process Notes

True story: I wrote it with you, the reader of math blogs, in mind.

That is to say, it’s awfully tempting in grad school to lard up your writing with jargon as some kind of shield against criticism. (If your critics can’t understand your writing, they probably can’t criticize it and if you’re lucky they’ll think that’s their fault.) Instead I tried to write as conversationally as possible with as much precision and clarity as I could manage. This didn’t always work. Occasionally, my advisers would chide me for being “too chatty.” That was helpful. Then I stocked my committee with four of my favorite writers from Stanford’s Graduate School of Education and let the chips fall.

Everything from my methods section and beyond gets fairly technical, but if you’re looking for a review of online education and the language of mathematics, I think the early chapters offer a readable summary of important research.

I raved for a minute on Twitter last week about this New York Times article. You should read it (play it? experience it?) and then come back so I can explain why it’s what math curriculum could and should become.

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


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.

Shira Helft and Rick Barlow gave one of the most stellar talks at a recent, stellar math education conference. Like many teachers of ELL populations, they struggled to help their students speak mathematically. They described their transition from blunt prompts like “justify your answer” to precise participation structures they developed with their colleagues. They talked about two of those structures in their session – “math fights” and “middle bits” – which are simultaneously accessible and demanding. We could have kept that session running for another few days for all I cared.

They gave that talk to a room of maybe 50 teachers. It should have been 500, or however many math teachers there are in the world. (I assume around 500.) I hassled the Global Math Department to set them up with a forum, so here they are.

The live show is Tuesday, May 5, 6:00 PM Pacific Time. The recording will live on forever. Do yourself and your students a favor and watch it.

2015 May 4. I’m informed there’s a 100-person cap on live attendance, which means 400 of the world’s 500 math teachers will have to watch the recording. I’ll be sure to add that here once it’s available.

Every Handout From NCTM

Some people on Twitter were grousing about the inconvenience of clicking every single session link in the NCTM directory to find out if the speakers uploaded handouts. NCTM also mentioned that the handouts would only be available for about a month. For both of those reasons, I wrote a script to crawl all 814 talks and yank out the 208 with file uploads.

So here’s that directory. Or you can download all of the handouts in one big swipe [360MB] if you’d rather.

If you find anything interesting, do let us all know in the comments. It’s almost like you were there!

BTW: My slides don’t make a bit of sense without my voice attached so I don’t tend to upload them or send them around. I will be editing together video from my talk and posting it later this summer, though.

2015 Apr 26. In the comments, Eric Henry asked if my script could be easily tweaked to download all of the NCTM Research Conference’s handouts. It was easy! Man, code is cool. Directory + giant ball of handouts.

NCTM 2015 Schedule

Poking my head up briefly to let you know where I’ll be speaking at NCTM in Boston this week:

I really think we should spend Thursday evening together.

From 3:30 on you have a) the Ignite sessions, which are always fun, then you have b) ShadowCon, which is basically the future of NCTM conferences, after which c) your two favorite math education companies want to buy you a drink.

If you’re looking for help with the rest of your schedule, have a look at this list of Internet-enabled presenters as well as any of the names from my list last year.

For my part, if I can only make it to one session, it’s going to be this one.

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