Helping Teachers Do the Very Hard Thing

The very hard thing is inviting, celebrating, and developing student thinking however and wherever you find it.

I participated in a panel at the Future of Teacher Learning conference at MIT earlier this year.

Justin Reich and Rachel Slama offered me the opportunity to hang with some absolute heroes and also the opportunity to distill all my current ideas and questions about teacher learning into four pictures and five minutes.

Let me know if any of this connects to your own thinking about supporting teachers, or about being supported as a teacher. Let the record show that I often experience my largest leaps in understanding when strangers send me emails.

Here are my prepared remarks.

Curriculum is Teacher Learning

At Desmos, we’re convinced by a variety of research, personal experience, and trusted testimony that the most productive change a math teacher can make is to develop an asset orientation toward their students’ thinking.

From “What are this student’s misconceptions about an idea?” to “What are this student’s conceptions about the idea?”

From “Is this answer correct?” to “How is this answer useful?”

From “This student is broken—how can I fix them?” to “This student is brilliant—how can I invite, celebrate, and develop that brilliance?” borrowing from Dr. Danny Martin.

A lot of my colleagues and I spent years working with teachers on these changes, but we ran aground constantly on curriculum.

A curriculum will never expressly prohibit student brilliance, but lots of curricula only make room for expressions of grownup brilliance—the brilliance of the teacher or the textbook author, for example. 

So supporting teachers meant building our own curriculum, a fork of the excellent curriculum from Illustrative Mathematics, optimized for our technology, design, and pedagogy. 

Rich & Digital Curriculum is Rare

Our curriculum is in a really useful position for supporting teacher development: 

  1. It’s rich—by which I mean it invites a broad array of student brilliance: noticings, wonderings, sketches, verbal arguments, equations, graphs, and text responses to interesting questions. Ideas that we couldn’t and wouldn’t want to label “correct” or “incorrect.”

  2. It’s also digital. Students express a lot of that brilliance in a form that’s visible to teachers and the people who support teachers, including site-based coaches and coaches at Desmos. Teachers are able to enact some sophisticated routines through the platform like selecting student ideas, sequencing them, and presenting them. We have visibility into how teachers are taking up those practices.

Lots of curricula is one or the other. The IM curriculum is extremely rich but print-first. Lots of digital curricula only make room for student brilliance in the form of multiple choice responses and numbers—stuff that computers often grade as correct or incorrect.

Continuous, Low-Intensity Teacher Learning

The visibility we have into student ideas and teacher practices has expanded our options for teacher professional development.

We offer teachers a large kick-off session at the start of the school year to get us lined up on some of our hopes and dreams for the year and to get us excited for the work. This session is high intensity and one-time only, and we know the energy and learning dissipates quickly.

So we’re also developing and researching interventions that are lower intensity but continuous. Frequent booster shots if you’ll forgive the timing of that metaphor.

Spiraling the Same Three Ideas

For example, teachers are always in their email. Let’s use that as a medium for professional learning. We know what lesson a teacher taught yesterday, so we send that teacher an email previewing the next lesson through the lens of one of the three themes, which we spiral throughout the year: inviting student brilliance, celebrating it, and developing it, with pedagogical techniques attached.

We offer those ideas to teachers not in a one-off session abstracted from their daily work, but rather as close to the teacher’s practice as we can get: “On this screen, here’s a great moment to tell students, ‘Look, you can’t break mathematics.’ ” These preview emails give teachers essential and generalizable teaching strategies and give us email open rates that email marketers can only dream about.

Big questions now:

  • What are the mid-range interventions? Medium frequency. Medium intensity.

  • How do we use this digital platform for rich student thinking to nourish professional learning community meetings or support site-based coaches? Don’t know.

Also, we’re sitting on a nice pile of data. If you’re a researcher looking for interesting collaborators, let me know at

What else?

  • The bad arithmetic problem that’s designed to go viral went viral again this week. I’m not going to fault your average internet user for passing along a bit of math that’s designed to excite the world’s passion for symbolic pedantry. But if you think math is about more than symbolic pedantry, well, please don’t share the bad arithmetic problem.

  • I’m obsessed with broken graphs. Graphs that reflect some kind of discontinuity or break in the social order itself. Over at Luke O’Neill’s Substack, I saw this grade A broken graph that indicates Americans are diagnosed with cancer at much greater rates at age 65. Over on Twitter, I asked people why.

  • Enjoy this time-lapse video of the introduction and disappearance of the Blockbuster Video franchises in the United States.

  • Education Doesn’t Work is the provocative title of a long essay that invites you to ask yourself, “What do I hope education will do? Is there any evidence it can do that?” If you hope education will bring about economic equality, for example, the evidence invites you to invest your efforts elsewhere.

  • Here is a quote I haven’t gone a week without thinking about since I first read it. “Everyone is mathematically smart as a result of living in the world.

Against Mastering Mathematics

"This idea of mastery gets us into some trouble because it keeps young people from exploring their mathematical ideas."

In the most recent episode of Math Teacher Lounge, Bethany Lockhart Johnson and I interviewed UCLA math education professor Megan Franke. She shared some thoughts about “mastery” that you should really read.

[“Mastery” often means that] if you master addition and subtraction you can move on and do other mathematical work, but you can’t move on until you’ve mastered that. What that does is sort students between who’s mastered and who hasn’t, and it keeps young people like this young person who may have said “twenty-ten” from being able to do more interesting mathematics. We say, “Well, we’ve got to practice the number sequence first. We can’t let you solve any problems that go into the twenties. We can’t let you . . . ” when actually letting you go farther is going to let you explore the idea of the number system better than if I stopped you. This idea of mastery gets us into some trouble because it keeps young people from exploring their mathematical ideas.

Check out the rest of the conversation, which is short and amazing, at An idea that struck me by the end is that, for teachers, the most profound professional transformations require profound personal transformations as well. That raises the stakes on this work in ways I didn’t understand as a new teacher. As someone who has never successfully maintained any kind of separation between the personal and professional areas of my life, I wouldn’t have it any other way.

What Else

  • “Results show that essays have a stronger correlation to reported household income than SAT scores.” This is the study that everyone in education research is talking about right now.

  • “No tax exemptions for universities with selective admission processes,” is probably the take on the study I found most interesting.

  • Get a spot at the free Amplify STEM Forum next week. Lots of great speakers, and I’m excited to contribute a new talk called “Math Without Mistakes.”

  • If you like the Notice and Wonder math routine in principle but struggle to structure that work in practice, check out this modification from Yorkville High School math teacher Kellie Stilson.

  • Berkeley High School math student Veronika Price made a beautiful hummingbird out of hundreds of mathematical expressions (see the image at the top of this email!) and I made a fancam video breaking the whole graph down.

Curriculum From the Future

We brought it back to the present.

In Back to the Future II, a movie that is distressingly old, villain Biff Tannen steals a time machine, travels decades into the future, buys a sports almanac, and becomes a casino baron back in the present by placing sure bets using that almanac.

Our team at Desmos did something very similar this last month.

✅ Stole a time machine.

✅ Traveled decades into the future.

❌ Brought a sports almanac back to the present. (We are both too nerdy and not mercenary enough for this.)

✅ Instead, we brought back the best middle school math curriculum we could find in the future.

You can preview all of it right now, no hurdles to clear. Much of it is also available to try out for free with students. This curriculum builds on and enhances the skillful work of the Illustrative Mathematics curriculum team.

My current favorite sequence of lessons teaches eighth grade students about equations using the context of hangar diagrams. It starts by inviting students to play and experiment. (Try it!)

Students then learn that a point in the plane has meaning—that one point defines two pieces of information: the weight of the triangle and the weight of the circle. (Try it!)

Then students learn that a point's position on one line, two lines, or zero lines also has meaning. Every point in the plane represents a different pair of hangars. (Try it!)

I realize that interactive digital experiences in math class aren't all that rare. What's uncommon here is the room we've made for both students and teachers to be creative in their work.

For students, we ask questions that have more than one right answer and where even the wrong answers are interesting. We don't try to fix the wrong answers. They’re interesting, and worth talking about as a class.

So we pass all of those answers on to their teacher, who mixes the responses together in the giant mixing bowl we call the Teacher Dashboard, sprinkling in some of her own knowledge, and creating learning experiences that are far beyond the ability of computers to replicate both now and even in the future. (Take our word on this—we were there!)

For me, this is the team-est of team efforts and also the best embodiment of ideas that have excited me for close to 20 years now. It’s the best thing I’ve ever had a hand in making, and I hope you will check it out.

What Else

Computer Feedback That Helps Kids Learn About Math and About Themselves

Students are receiving more feedback from computers this year than ever before. What does that feedback look like, and what does it teach students about mathematics and about themselves as mathematicians?

Here is a question we might ask math students: what is this coordinate?

Let’s say a student types in (5, 4), a very thoughtful wrong answer. (“Wrong and brilliant,” one might say.) Here are several ways a computer might react to that wrong answer.

1. “You’re wrong.”

This is the most common way computers respond to a student’s idea. But (4, 5) receives the same feedback as answers like (1000, 1000) or “idk,” even though (4, 5) arguably involves a lot more thought from the student and a lot more of their sense of themselves as a mathematician.

This feedback says all of those ideas are the same kind of wrong.

2. “You’re wrong, but it’s okay.”

The shortcoming of evaluative feedback (these binary judgments of “right” and “wrong”) isn’t just that it isn’t nice enough or that it neglects a student’s emotional state. It’s that it doesn’t attach enough meaning to the student’s thinking. The prime directive of feedback is, per Dylan Wiliam, to “cause more thinking.” Evaluative feedback fails that directive because it doesn’t attach sufficient meaning to a student’s thought to cause more thinking.

3. “You’re wrong, and here’s why.”

It’s tempting to write down a list of all possible reasons a student might have given different wrong answers, and then respond to each one conditionally. For example here, we might program the computer to say, “Did you switch your coordinates?”

Certainly, this makes an attempt at attaching meaning to a student’s thinking that the other examples so far have not. But the meaning is often an expert’s meaning and attaches only loosely to the novice’s. The student may have to work as hard to understand the feedback (the word “coordinate” may be new, for example) as to use it.

Alternately, we can ask computers to clear their throats a bit and say, “Let me see if I understand you here. Is this what you meant?”

We make no assumption that the student understands what the problem is asking, or that we understand why the student gave their answer. We just attach as much meaning as we can to the student’s thinking in a world that’s familiar to them.

“How can I attach more meaning to a student’s thought?”

This animation, for example, attaches the fact that the relationship to the origin has horizontal and vertical components. We trust students to make sense of what they’re seeing. Then we give them an an opportunity to use that new sense to try again.

This “interpretive” feedback is the kind we use most frequently in our Desmos curriculum, and it’s often easier to build than the evaluative feedback, which requires images, conditionality, and more programming.

Honestly, “programming” isn’t even the right word to describe what we’re doing here.

We’re building worlds. I’m not overstating the matter. Educators build worlds in the same way that game developers and storytellers build worlds.

That world here is called “the coordinate plane,” a world we built in a computer. But even more often, the world we build is a physical or a video classroom, and the question, “How can I attach more meaning to a student’s thought?” is a great question in each of those worlds. Whenever you receive a student’s thought and tell them what interests you about it, or what it makes you wonder, or you ask the class if anyone has any questions about that thought, or you connect it to another student’s thought, you are attaching meaning to that student’s thinking.

Every time you work to attach meaning to student thinking, you help students learn more math and you help them learn about themselves as mathematical thinkers. You help them understand, implicitly, that their thoughts are valuable. And if students become habituated to that feeling, they might just come to understand that they are valuable themselves, as students, as thinkers, and as people.

BTW. If you’d like to learn how to make this kind of feedback, check out this segment on last week’s #DesmosLive. it took four lines of programming using Computation Layer in Desmos Activity Builder.

BTW. I posted this in the form a question on Twitter where it started a lot of discussion. Two people made very popular suggestions for different ways to attach meaning to student thought here.

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