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Several months ago, I asked you, “You’re about to plan a lesson on concept [x] and you’d like students to find it interesting. What questions do you ask yourself as you plan?”

There were nearly 100 responses and they said a great deal about the theories of learning and motivation that hum beneath everything we do, whether or not we’d call them “theories,” or call them anything at all.

  • “How can [x] help them to see math in the world around them?”
  • “How can I connect [x] to something they already know?”
  • “How can I explain [x] clearly?”
  • “What has led up to [x] and where does [x] lead?”

You can throw a rock in the math edublogosphere and hit ten lessons teaching [x]. They might all be great but I’d bet against even one of them describing some larger theory about learning or mathematics or describing how the lesson enacts that theory.

Without that theory, you’re left with one (maybe) great lesson you found online. Add theory, though, and you start to notice other lessons that fit and don’t fit that theory. When great lessons don’t fit your theory about what makes lessons great, you modify your theory or construct another one. The wide world of lesson plans starts to shrink. It becomes easier to find great lessons and avoid not great ones. It becomes easier to create great ones. Your flywheel starts spinning and you miss your highway exit because you’re mentally constructing a great lesson.

Here is the most satisfying question I’ve asked about great lessons in the last year. It has led to some bonkers experiences with students and I want more.

  • “If [x] is aspirin, then how do I create the headache?”

I’d like you to think of yourself for a moment not as a teacher or as an explainer or a caregiver though you are doubtlessly all of those things. Think of yourself as someone who sells aspirin. And realize that the best customer for your aspirin is someone who is in pain. Not a lot of pain. Not a migraine. Just a little.

Piaget called that pain “disequilibrium.” Neo-Piagetians call it “cognitive conflict.” Guershon Harel calls it “intellectual need.” I’m calling it a headache. I’m obviously not originating this idea but I’d like to advance it some more.

One of the worst things you can do is force people who don’t feel pain to take your aspirin. They may oblige you if you have some particular kind of authority in their lives but that aspirin will feel pointless. It’ll undermine their respect for medicine in general.

Math shouldn’t feel pointless. Math isn’t pointless. It may not have a point in job [y] or [z] but math has a point in math. We invented new math to resolve the limitations of old math. My challenge to all of us here is, before you offer students the new, more powerful math, put them in a place to experience the limitations of the older, less powerful math.

I’m going to take the summer and work out this theory, once per week, with ten skills in math that are a poor fit for other theories of interest and motivation. As with everything I have ever done in math education, your comments, questions, and criticism will push this project farther than I could push it on my own.

The first skill I’ll look at it is factoring trinomials with integer roots, ie. turning x2 + 7x + 10 into (x + 5)(x + 2). All real world applications of this skill are a lie. So if your theory is “math is interesting iff it’s real world,” your theory will struggle for relevance here.

Instead, ask yourself, “Why did mathematicians think this skill was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?”

I’m signing on with Desmos as their Chief Academic Officer. Job one is producing the best digital math curriculum in the world. We’ve started that project already.

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.

This is the post I’ll re-read when I want to remember my five years at Stanford’s Graduate School of Education.

Why Grad School

My first year at Stanford was almost my last. A talk I had given right before I arrived at Stanford rolled past one million views. That opened up a lot of opportunities outside of Stanford, very few of which I declined. During what was supposed to be a perfunctory first-year review, my advisers invited me, with as much grace as I could expect of them, to leave Stanford, to return when I had more focus. I stuck around but I think all of us knew then I wasn’t really cut out for R1 university work. Still, I figured I’d work with teachers in a preservice program somewhere and a doctorate wouldn’t hurt my employment odds.

Two years later, just before my dissertation proposal was due, I received a job offer that was really too perfect to pass up, from people who didn’t care whether or not I had a graduate degree. They were nice enough to allow me to defer that offer until this summer.

All of this is to say, I had every incentive to walk, to join the ranks of the ABD. Here’s why I stayed, why I’d do it again even though my new employers don’t care about the letters after my name, and why I’d recommend graduate study to anybody who can make the logistics work: developing, proposing, studying, analyzing, and writing a dissertation works every single mental muscle you have and forces you to develop a dozen new ones. It’s the academic centathlon. I know how to ask more precise questions and how to better interrogate my prior assumptions about those questions. I know many more techniques for collecting data and statistical techniques for answering questions about those data. I know how to automate aspects of that data analysis through scripting. My writing is stronger now. My presentation skills are more polished. My thinking about mathematics education is more developed now, though still a work in progress.

It’s certainly possible to develop all of those muscles separately, without the extra overhead of a dissertation. (Michael Pershan seems to be making a go of it on Twitter, with Ilana Horn as his principle adviser.) But tying them all together in the service of this enormous project was uniquely satisfying.


If you’re thinking about grad school, take advantage of your tools:

Papers to manage references. Dropbox to sync them across machines. iAnnotate PDF to read and mark them up on an iPad. Google Scholar for everything. Scrivener for writing anything with more than five headings. Google Docs for writing anything else. I couldn’t survive grad school without those six tools.

Google Tasks for scheduling to-do’s. Boomerang for scheduling emails. I couldn’t survive professional life without those two tools.

The Last Five Years

  • Wrote two books.
  • Foster parented three kids.
  • Buried my dad.
  • Traveled around the world with my wife.
  • Learned from the best.
  • Collaborated with great people on interesting projects.
  • Traveled to a bunch of states and several countries, meeting basically all of you at one point or another.
  • Keynoted a couple of big-ish conferences.

Opportunity Cost

  • Never presented at AERA, PME, or ICMI.
  • Never attended AERA, PME, or ICMI.
  • Never gave a poster talk.
  • Never gave an academic presentation of any kind until my dissertation defense.
  • Never taught a course.
  • Never TA’d a course.
  • Never supervised any of the promising new teachers in Stanford’s teacher prep program.
  • Never connected with the people in my research group as much or as often as I would have liked.
  • Attended only a small fraction of the lunch talks and job talks and colloquia and dissertation defenses from the great thinkers passing through Stanford.
  • Heard “Oh – do you still go here?” way too often.


  • You guys. I thanked you all in my dissertation’s front matter and I’ll thank you here. The difference between a happy and sad graduate school experience often cuts on whether or not you like to write. In our conversations here, you guys made me, if not a great writer, someone who likes to write. As much as some of you drive me crazy, our back-and-forths made my arguments sharper and easier to defend in the dissertation. There was also that time that I asked on Twitter for help piloting assessment items in your classes and dozens of you helped me out. You have no idea what that kind of support is worth to a grad student around here.
  • I never got sick of my dissertation. I didn’t enjoy some of the logistics of its data collection. I didn’t always have the time I wanted to work on it. But I never got tired of it, which is some kind of gift.
  • Michael Pershan. My codes needed interrater reliability, the stuff that says, “Someone else can reliably see the world how I see the world, whether or not that’s the right way to see the world.” I hired Pershan onto my research team (doubling the size of my research team) when my time was crunched. He coded a bunch of data as fast as I needed and also changed “how I see the world” in some important ways.
  • Desmos. I had some of the area’s best computer engineers and designers building my dissertation intervention. I got very lucky there.
  • Jo Boaler. Jo was my principal adviser for all but my first year of grad school. There were a lot of great reasons to ask for her mentorship, but one of the best is that I never had to hide from her my lack of ambition for a tenure-track research job. As those ambitions faded, a lot of advisers in her position would have waitlisted me, focusing their efforts (rationally) on students who stood a chance to carry their research agenda forward. She invested more in my work than I had any reason to expect and I’ll always be grateful for that.

So that’s that. On to the next thing.


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.

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