Dan Meyer

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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

The #1 Most Requested Desmos Feature Right Now, and What We Could Do Instead

When schools started closing months ago, we heard two loud requests from teachers in our community. They wanted:

  1. Written feedback for students.
  2. Co-teacher access to student data.

Those sounded like unambiguously good ideas, whether schools were closed or not. Good pedagogy. Good technology. Good math. We made both.

Here is the new loudest request:

  1. Self-checking activities. Especially card sorts.

hey @Desmos – is there a simple way for students to see their accuracy for a matching graph/eqn card sort? thank you!

Is there a way to make a @Desmos card sort self checking? #MTBoS #iteachmath #remotelearning

@Desmos to help with virtual learning, is there a way to make it that students cannot advance to the next slide until their cardsort is completed correctly?

Let’s say you have students working on a card sort like this, matching graphs of web traffic pre- and post-coronavirus to the correct websites.

Linked card sort activity.

What kind of feedback would be most helpful for students here?

Feedback is supposed to change thinking. That’s its job. Ideally it develops student thinking, but some feedback diminishes it. For example, Kluger and DeNisi (1996) found that one-third of feedback interventions decreased performance.

Butler (1986) found that grades were less effective feedback than comments at developing both student thinking and intrinsic motivation. When the feedback came in the form of grades and comments, the results were the same as if the teacher had returned grades alone. Grades tend to catch and keep student attention.

So we could give students a button that tells them they’re right or wrong.

Resourceful teachers in our community have put together screens like this. Students press a button and see if their card sort is right or wrong.

Feedback that the student has less than half correct.

My concerns:

  1. If students find out that they’re right, will they simply stop thinking about the card sort, even if they could benefit from more thinking?
  2. If students find out that they’re wrong, do they have enough information related to the task to help them do more than guess and check their way to their next answer?

For example, in this video, you can see a student move between a card sort and the self-check screen three times in 11 seconds. Is the student having three separate mathematical realizations during that interval . . . or just guessing and checking?

On another card sort, students click the “Check Work” button up to 10 times.

https://www.desmos.com/calculator/axlhe3shwg

Instead we could tell students which card is the hardest for the class.

Our teacher dashboard will show teachers which card is hardest for students. I used the web traffic card sort last week when I taught Wendy Baty’s eighth grade class online. After a few minutes of early work, I told the students that “Netflix” had been the hardest card for them to correctly group and then invited them to think about their sort again.

I suspect that students gave the Netflix card some extra thought (e.g., “How should I think about the maximum y-value in these cards? Is Netflix more popular than YouTube or the other way around?”) even if they had matched the card correctly. I suspect this revelation helped every student develop their thinking more than if we simply told them their sort was right or wrong.

We could also make it easier for students to see and comment on each other’s card sorts.

In this video, you can see Julie Reulbach and Christopher Danielson talking about their different sorts. I paired them up specifically because I knew their card sorts were different.

Christopher’s sort is wrong, and I suspect he benefited more from their conversation than he would from hearing a computer tell him he’s wrong.

Julie’s sort is right, and I suspect she benefited more from explaining and defending her sort than she would from hearing a computer tell her she’s right.

I suspect that conversations like theirs will also benefit students well beyond this particular card sort, helping them understand that “correctness” is something that’s determined and justified by people, not just answer keys, and that mathematical authority is endowed in students, not just in adults and computers.

Teachers could create reaction videos.

In this video, Johanna Langill doesn’t respond to every student’s idea individually. Instead, she looks for themes in student thinking, celebrates them, then connects and responds to those themes.

I suspect that students will learn more from Johanna’s holistic analysis of student work than they would an individualized grade of “right” or “wrong.”

Our values are in conflict.

We want to build tools and curriculum for classes that actually exist, not for the classes of our imaginations or dreams. That’s why we field test our work relentlessly. It’s why we constantly shrink the amount of bandwidth our activities and tools require. It’s why we lead our field in accessibility.

We also want students to know that there are lots of interesting ways to be right in math class, and that wrong answers are useful for learning. That’s why we ask students to estimate, argue, notice, and wonder. It’s why we have built so many tools for facilitating conversations in math class. It’s also why we don’t generally give students immediate feedback that their answers are “right” or “wrong.” That kind of feedback often ends productive conversations before they begin.

But the classes that exist right now are hostile to the kinds of interactions we’d all like students to have with their teachers, with their classmates, and with math. Students are separated from one another by distance and time. Resources like attention, time, and technology are stretched. Mathematical conversations that were common in September are now impossible in May.

Our values are in conflict. It isn’t clear to me how we’ll resolve that conflict. Perhaps we’ll decide the best feedback we can offer students is a computer telling them they’re right or wrong, but I wanted to explore the alternatives first.

2020 May 25. The conversation continues at the Computation Layer Discourse Forum.

The American Time Use Survey Is “Poetry, in Data.”

The American Time Use Survey is a fantastic data set. You can find out how many more hours per day women spend on household activities than men. You can identify the time of day that the majority of Americans wake up.

You can also determine the amount of time we spend with certain groups of people in our lives from childhood to late adulthood. For example, here are graphs of the amount of time we spend with friends and with co-workers.

graphs of time spent with friends and co-workers

Fantastic graphs, right? But will students think they’re fantastic? Will they learn from the graphs? How can you effectively introduce your students to the American Time Use Survey?

I use three strategies every time. You can read about them below and experience them in this new free activity from me and my colleagues at Desmos.

First, a meta-strategy:

I don’t allow myself to rest for a second in the false comfort that this is a “real world” context, and per se, interesting to students. Contexts are never “real” or “unreal.” They don’t exist in a vacuum. Contexts become real when teachers invite their students to interact with them in concrete and personal ways.

Here are three invitations I extend to students basically any time I’d like them to experience a graph as real.

1. I invite students to contribute their own data.

A table asking students to describe their OWN time usage.

The graph represents a group of people’s concrete and personal experiences: time spent with friends, co-workers, and partners. I ask students to contribute their own data so the quantities and relationships become more concrete for them as well.

2. I invite students to sketch their own graph before seeing the actual graph.

This invites students to share their own knowledge about the quantities and relationships. Students have ideas about how many hours people spend with friends throughout their lives. We should invite them to express those ideas with a graph.

The student's sketch and the actual answer.

I also place their own data from (1) on the graph. This extends an even more personal invitation to students and gives them an anchor for their graphing.

“That’s you on there, friend. Do you think American 15-year-olds spend more or less time with their friends than you? Okay, graph it!”

3. I invite students to reflect.

Jim Coudal called these graphs “Poetry, in data.” So I ask students to tell us which graph is most poetic and why. We’ve built up a lot of steam in the activity, and this question helps release it. It allows us to elicit from students the personal observations that haven’t yet found a home in our activity.

I posted this activity on Twitter and the majority of people said this was the most interesting graph to them.
Graph of time spent alone. It increases sharply towards the end of life.

People wrote:

It is sad to me that once we are old enough to have free time to spend with friends, we spend more time alone.

I wonder if the loneliness is by choice.

Alarming lack of social opportunities for seniors.

There is so much interesting research coming out about the impact of loneliness on people’s health.

How can we change this?

So consider the invitations you extend to students. In many curricula, those invitations are impersonal and abstract. “What is the value of the co-workers graph for a 75-year-old?” That’s a question that invites students to reflect on an adult’s knowledge of graphs and the context.

“What would your data look like? What do you think the graph looks like? Why?” These are questions that invite students to interact with the graph in personal ways, to inhabit the graph as if it were their own.

Featured Comment

Leigh Ann Mahaffie:

As I’m feeling mighty alone personally (even though there are folks in the house) and professionally (electronically just isn’t the same) during the current “Stay at Home” situation, this data definitely evokes some poetry for me.

Math Has Prepared Me Poorly for This Pandemic

Here are two representations of the horror of this pandemic.

First, a graph of coronavirus deaths in Italy.

Graph of Coronavirus deaths in Italy.

Second, the obituary page of a newspaper in the Italian city of Bergamo, first from February 9 and later from March 13.

Both of these are only representations of this pandemic. They point at its horror, but they aren’t the horror itself. They reveal and conceal different aspects of the horror.

For example, I can take the second derivative of the graph of deaths and notice that while the deaths are increasing every day, the rate of increase is decreasing. The situation is getting worse, but the getting worse-ness is slowing down.

I cannot take the second derivative of an obituary page.

But the graph anesthetizes me to the horror of this pandemic in a way that the obituaries do not. The graph takes individual people and turns them into groups of people and turns those groups of people and their suffering into columns on a screen or page.

Meanwhile, the obituaries put in the foreground the people, their suffering, and their bereaved.

Math has prepared me poorly for this pandemic—or at least a particular kind of math, the kind that sees mass death as an opportunity to work with graphs and derivatives.

For students, it has never been more necessary to move flexibly and quickly between concrete and abstract representations—to acquire the power of the graph without becoming anesthetized to the horror that’s represented much more poignantly by the obituaries.

For teachers, there has never been a more important time to look at points, graphs, tables, equations, and numbers, and to ask students, “What does this mean?” and particularly now, “Who is this?”

BTW

Two relevant quotes here.

  • “A single death is a tragedy; a million deaths is a statistic.” Commonly attributed to Joseph Stalin.
  • “Statistics are human beings with the tears wiped off.” Paul Brodeur, quoted in Mukherjee’s Emperor of all Maladies.

2020 Apr 10

Another example. It’s one thing to see a graph of unemployment, and another to see the lines for the food bank.

2020 May 25

We’re Only Getting Out of This Together

Desmos closed its San Francisco office on March 9, about a week before the surrounding county issued a “shelter-in-place” warning. When it became clear that our local school systems were going to close, we assembled a small team of people from across our company to figure out how we could support educators during a period of school closure that has no precedent in our lifetimes.

I ran webinars for teachers on Saturday and Sunday. (Check out the recording.) Approximately 600 people showed up and all of us were clearly looking for more than tips, tricks, or resources for distance teaching.

I told the attendees I figured that, because they were attending a webinar on the weekend, they were probably teachers who held their teaching to a very high standard. But now isn’t the time for high standards for teaching, I said. I referred to Rebecca Barrett-Fox’s fantastic essay, “Please do a bad job of putting your courses online.

… your class is not the highest priority of their or your life right now. Release yourself from high expectations right now, because that’s the best way to help your students learn.

I also mentioned Barrett-Fox’s admonition not to pick up new tools right now:

Also: If you are getting sucked into the pedagogy of online learning or just now discovering that there are some pretty awesome tools out there to support student online, stop. Stop now. Ask yourself: Do I really care about this?

You and I are likely receiving the same emails from ed-tech companies, ones that cloak in generosity their excitement to expand their user base, offering services for free they’ll charge for later. In our webinar I explicitly released the group from any expectation that they would learn Desmos as a beginner right now. Now is likely not the time. (It’s probably also worth pointing out that we’ve committed to never charging later for anything we make free now.)

But I told the attendees I had two hopes for their teaching during this time. That they would:

  1. Give students something interesting to think about. Hopefully mathematical, but maybe not. Hopefully towards grade-level objectives, but let’s be realistic about the stresses faced by students, teachers, and parents here. (Remembering also how many people cross more than one of those categories.)
  2. Make connections. I encouraged the group to make connections from teacher to student, from student to student, and from student ideas to other interesting ideas.

As an example, Johanna Langill, a teacher in my hometown of Oakland, CA, assigned her students our Turtle Time Trials activity. Students completed it on their own time, and then she recorded a review of their work, celebrating their early ideas, connecting those ideas to each other, and connecting those ideas to other interesting ideas.

In the week since that webinar, my team has had hundreds of conversations across every digital medium except maybe TikTok. We set up an email address and a hotline where teachers can ask for support, ask questions, or just vent omnidirectionally about how awful their situation is right now.

Our Facebook community is geared full-time towards supporting teachers in school closure. We are running webinars and drop-in office hours every day. We’re delivering new features and new activities specifically supporting distance teaching. We’re collecting all of these efforts at learn.desmos.com/coronavirus.

We’re trying to help teachers adapt to distance teaching, yes, but that’s really a secondary goal. Mainly, we’re trying to sustain community. Everything we’ve built or offered during this last horrible week has been an effort at preserving community between teachers and students, teachers and each other, and if I’ll confess to any selfish motive here, it’s that we’re trying to sustain our own community as well.

I’m convinced that when teachers and students find the other side of this, it won’t be because edtech companies offered junk for free, it’ll be through community, through solidarity across all of our usual divisions and now across divisions of time and space as well.

Like the Spencer Foundation’s Na’ilah Suad Nasir and Megan Bang said in an open letter this weekend:

It may be that social distancing isn’t quite the right frame for what we need right now. We certainly need physical distancing. But we also need to imagine and act from places of social closeness and care.

Teachers are our community and right now we intend to stay as close to them as possible.

But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers?

I have very small children which means my life is measured by little games and distractions stretched across the day. “What’s that called?” is one of those games. Point at a thing and ask for its name. Do that for another thing. Hey – it’s almost nap time!

So recently we pointed at an artichoke. “What’s that called?”

“Pinecone,” one of the kids says.

a drawing of a pinecone and an artichoke

That’s a factually incorrect answer, which is the same as lots of student answers in math class. But when my kid calls a pinecone an artichoke, I have a very different emotional, physical, and pedagogical response than when a student says something factually incorrect in math class.

With my kid, I am fine with the error. Delighted, even. I am quick to point out all the ways that answer is correct. “Oh! I see why you’d say that. They both have the kind of leafy-looking things. They both have the same-ish shape.”

I find it easy to build connections from their answer to the correct answer. “But an artichoke is greener, larger, and softer. People often eat it and people don’t often eat pinecones.”

However, if I’m teaching a math lesson and a student answers a question about math incorrectly, my reflex is to become …

… evaluative … “What did I just hear? Is it right or wrong?”

… anxious … “Oh no it’s wrong. What do I do now?”

… corrective … “How do I fix this answer and this student?”

I find it much harder to celebrate and build from a student’s incorrect answer in math class than I do an incorrect answer from my kids about artichokes. The net result is that my kids feel valued in ways that the students don’t and my kids have a more productive learning experience than the students.

I can give lots of reasons for my different responses but I’m not sure any of them are any good.

  • This is my kid so I feel warmer towards his early ideas than I do towards ideas from kids I see for only a small part of the day.
  • This kid looks like me so I’m more inclined to think of him as smart and brilliant and wonderful than I am a student with a different race, ethnicity, or gender.
  • The stakes are smaller. What’s the worst consequence of my kid referring to an artichoke as a pinecone? That he doesn’t get invited back to the Governor’s Ball? Who cares. This will work out. I’m not preparing him for an end-of-course exam in thistle-looking stuff.
  • I know the content better. I can build conceptually from a pinecone to an artichoke much more easily than I can build from early math ideas to mature math ideas.

But I find that every aspect of my professional and personal life improves when I try to neutralize those excuses.

  • I am a member of faith and educator communities that help me dissolve my conviction that my kid is more valuable or special than your kid, communities that help me dissolve my sense of separateness from you. We are not separate.
  • I am working with a team to develop experiences in math class that lead to student answers that are really hard to call right or wrong, or ones that at least lead to lots of interesting ways to be right or wrong. I am learning that it’s more helpful to ask a question like, “How are you thinking about this question right now?” than “What is your answer to this question?” because the first question has no wrong answer.
  • I am trying to develop pedagogical tools that make use of differences between student answers to replace ones that try to reconcile or flatten them. Tools like “How are these answers the same and different?” or “For what question would this answer be correct?”
  • I am trying to learn more math more deeply so I can make connections between a student’s early ideas and the later ones they might develop.

I am thinking about this idea from Rochelle Gutierrez more often:

All teaching is identity work, regardless of whether we think about it in that way. We are constantly contributing to the identities that students construct for themselves …

Whether my kid calls an artichoke a pinecone or a student offers an early idea about multiplication, they’re offering something of themselves just as much as they’re offering a fact or a claim. My goal is to celebrate those early ideas and build from them so that students will learn better math, but also so they’ll learn better about themselves.

Featured Comments

Several people mention that we have more time to enjoy our kids and their thinking than we do students in math class.

2020 Jun 13. Other examples of early ideas about language from around my home.

  • “Getting tangled out” a/k/a “getting untangled.”
  • “Yesterday” as a placeholder word for any time in the past.
  • “Foots” and “Gooses” as the plural for “Feet” and “Geese”.
  • Them: What do cows eat? Me: Hay, I think. Them: No, horses eat hay.
  • 6 looks a lot like a lowercase “g”.
  • “After” is any time in the future. Me [beleaguered]: “We’ll do that later, kids.” Kids [combative]: “AFTER!”