# data[].attributes.attachment_link.description
Today’s math curriculum is teaching students to expect -- and excel at -- paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. Dan Meyer shows classroom-tested math exercises that prompt students to stop and think.

# data[].attributes.attachment_link.readable_text
Here are two representations of the horror of this pandemic.

First, a 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.

Bergamo daily newspaper pic.twitter.com/N3ECABz8dr
- David Carretta (@davcarretta) March 14, 2020

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?"

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.

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.

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.

I have so much more curiosity when my kid says something incorrect. I find it so fascinating that she decided to say that 1 + 9 = 30. Why?!?

I get so much more 1:1 time with her than with students in my classroom. I feel that spaciousness in a deep way.
- Bree Pickford-Murray (@btwnthenumbers) March 10, 2020

This is sort of alluded to in your already listed reasons, but maybe(?) another reason: You may feel as if you have more time to engage with the thinking of someone who [hopefully] will be in contact with you for the rest of your life. With students, time can feel [is?] shorter.
- Benjamin Dickman (@benjamindickman) March 10, 2020

I posted that image on Twitter last week, asking:

Which of these bushfire relief donors was the most generous? What's your ranking? What information matters here? What would your students say?

Some teachers quickly identified a connection to ratios.

@math6falcon here's a ratio activity for you!
- Matt Murray (@Mr__Murray) February 6, 2020

Meanwhile, Lee Melvin Peralta critiqued ratios as too limited to fully model generosity.

I think it goes beyond a ratio problem when one considered the absolutely messed up things BP and Bezos has done in the areas of labor, the environment, and local economies and culture (negative externalities are just the start) /n
- Lee Melvin Peralta (@melvinmperalta) February 7, 2020

I tend to side with George Box here, who wrote:

All models are wrong, but some are useful.

Anyone who thinks that proportional functions fully describe runners in a race, or that linear functions fully describe the height of a stack of cups, or that quadratic functions fully describe the height of objects under gravity, or that ratios fully describe generosity is, of course, kidding themselves.

But those models are all useful. Ratios are a useful way to think about generosity.

Emily Atkin originally stirred this question up for me in her fantastic climate change newsletter:

Chevron's donation is paltry, however, given its earnings and relative contribution to the climate crisis. Not only is Chevron the second-largest historical emitter of all the 90 companies, it also earned about $15 billion in 2018. So a $1 million donation amounts to about .00667 of its yearly earnings. To the average American, that donation would amount to about $3.96.

So Atkin is evaluating generosity as ratio of net worth/earnings to donation size. But then she also considers the donor's contribution to climate change.

The model is complex and grows more complex!

One teacher wanted to add fame and notoriety to our model, something Chris Hemsworth donates that Mariam might not. (Maybe she's a TikTok teen, though. We can only speculate.) I talked with someone who lives in Australia about this question, and she said Hemsworth is less generous than someone from another country donating the same amount because of his identity as an Australian citizen. Robyn V wondered how to evaluate time donations, and even the donation of one's life.

So ratios aren't a perfect model for generosity, but they do offer us an important insight that, under some circumstances, someone who donates $75 is more generous than someone who donates one million dollars, which one teacher noted is a quantity that is really hard for students to fathom!

One teacher preparation program asked the question:

https://twitter.com/jetpack/status/1225610304110972935

If the ambiguity of the original question strikes you as anything other than a feature, then please don't risk the conversation.

If you go into the conversation presupposing a model for generosity rather than admitting to yourself in advance that all the models are broken, you're likely to diminish students who suggest variables you had already excluded.

Okay, yes, um, 'whether or not someone lives in Australia.' Okay, that's one idea, but can I get some other ideas, please? Perhaps ones more related to the math we've been studying?

All of these models are complex. All of them are certainly broken. And all of them offer you the opportunity to celebrate and build on your students' curiosity and contextual knowledge, an experience that is all too rare for students in math class.

BTW: Shout out to Christelle Rocha for her observation that individual generosity is no way to solve the climate crisis.

Also, given that 3 out of the 4 pictured are people, are we implicitly positioning the entity as a person, or the people as entities? Another wondering: when individual generosity is compared, how might this distract from the responsibilities of agencies to preserve communities?
- Christelle Rocha (@Maestra_Rocha) February 8, 2020

If you knew me as a classroom teacher, you knew I was very, very cranky about the ways many math textbooks treated students and mathematics, how they failed to celebrate and build on student intuition about mathematical ideas, how their problems were posed in ways that hid their most interesting elements, how they were way too helpful.

So it's been a joy to get to do something more active about that problem than write cranky blog posts, to get to team up with some fantastic teachers, designers, engineers, and funders all continuously interrogating their assumptions about education, design, technology, math, and society, all to create what I think is ...

... the very best middle school math curriculum.

This is it.

Call off the search.

You found it.

Read more about the curriculum at the Des-blog, including details about our upcoming pilot.

[extremely Oprah voice] You get a debt of gratitude! You get a debt of gratitude! You get a debt of gratitude!

Aside from my enormous gratitude to the fantastic team I work with daily, I'm especially grateful to two groups:

The authoring / publishing team at Illustrative Mathematics / Open Up Resources who created and openly licensed a fantastic math curriculum, one which is the foundation of our own work. They dropped a massive gift on the math education community (or a hydrogen bomb from the perspective of the K-12 math publishing industry) and we were extremely happy to pick it up and build on it.

You. I'm talking about the folks who have been reading this blog, commenting on my posts, critiquing my ideas from day one. Your thoughts and mine are all tied together and run all the way through this curriculum.

This blog has been quieter over the last few years for reasons that are predictable - family, Twitter, the death of blogs, etc. - but also because, for the only time in my career, I haven't been able to write about my work.

That changes today and I'm very excited to collaborate with you folks once again on the work that matters to me most. It won't be at its best without you.

# data[].attributes.attachment_link.description
Here are two representations of the horror of this pandemic. First, a 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. Bergamo daily newspaper pic.twitter.com/N3ECABz8dr - David Carretta (@davcarretta) March 14, 2020 Both of these are only representations of this pandemic.