The question about those images was, “What stays the same? What changes?” And people did *not* answer like seventh graders.

Instead, there was lots of discussion around proportionality, congruency, ratios, and other attributes of the shapes that are going to be one million miles from the minds of seventh graders in school right now.

But several teachers took me up on my offer and answered a little bit like children. I snapshotted them, paused the class, and presented them.

Things they told me that stay the same:

- The shape, the angles, the color, the orientation
- The color and the angle of the vertices
- The color and the paper size are the same
- The shape and the color
- Shape, color, orientation, centered on paper

“I love that you folks are finding patterns, noticing similarities, deciding what varies and doesn’t vary—including *color*!—using your eyes, your vision, your senses. That’s math!”

I read them an excerpt from Rochelle Gutierrez which is on my mind a lot these days.

A more rehumanized mathematics would depart from a purely logical perspective and invite students to draw upon other parts of themselves (e.g., voice, vision, touch, intuition).

**By naming those responses “mathematics,” I turned them into money.**

As a society, we decided long ago that certain pieces of paper had value—that they’re money. In much the same way, you are the central bank of your own classroom and you decide which student ideas are money. You decide which of them have value and, by extension, you influence a student’s sense of their *own* value.

I’m not hypothesizing here! Watch what happened with the teachers. On the very next screen in our lesson, we ask students to describe how this printer is *broken*.

Teachers clearly received my signal about what kind of mathematics was valuable.

They brought metaphors, imagery, and analogies that I don’t think they would have brought if I only praised deductive, formal, and precise definitions.

- My shape is drunk
- The lines do not stay straight…they are wobbly
- My pacman lines are no longer straight. The new figure looks droopy and sad.
- It got curvy, kind of sexy looking

The ability to decide what’s money is a lot of power! In this time of distance teaching, you have fewer ways to broadcast value to students than you would if you were in the same room together. But I’m so encouraged to see teachers using chat rooms, breakout groups, video responses, written feedback, snapshot summaries, whatever they can, to enrich as many students in their classes as possible.

]]>- There’s a sharp division between who is considered an expert and a novice, and an obsession with status (there are four levels of sommelier!) that’s only exceeded by some religious orders.
- Experts seem to have very little interest in the intuitions and evolving understandings that novices bring to the tasting room. (What you’re supposed to be experiencing – the answer key – is written right there on the tasting menu!)
- The whole thing is arbitrary in ways that we’re all supposed to pretend we don’t notice. (In math: the order of operations, the names of concepts, the y-axis is vertical, etc. In wine: the relationship between price and appreciation.)

I basically only enjoy tasting with a friend of mine, Michael Kanbergs, who is the man at Mt. Tabor Fine Wines in Portland, OR, if you’re local. He has expert-level knowledge about wine and enthusiasm to match but is allergic to most ordering forces in the world, including the expert / novice distinction. So he wants to share with you his favorite wines but he’s hesitant to offer his own perception too early because that’d undermine his curiosity about how *you’re* perceiving the wine.

I’m grateful to Michael for modeling good teaching, and grateful to other wine experts for helping me empathize a little better with math students who might find me and my habits alienating in similar ways.

]]>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:

- 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.

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.

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.

My concerns:

- 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? - 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.

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.

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*.

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.”

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.

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*.

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.**

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.

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.

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 peopleâ€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**

]]>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.

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 pandemicâ€”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 representationsâ€”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.

This is what I saw. Blistering heat. Folks in line since 7pm the night before. To get food. Hundreds of volunteers busting it to serve, so families could go home (probably to pass some out to their neighbors too) & get the nourishment they need.

— Robert R. Fike (@robfike) April 9, 2020

This is the COVID-19 Crisis. pic.twitter.com/CL8Be0wNwI

I have worked for the Star Tribune for nearly 29 years and have never seen 11 pages of paid obituaries in our Sunday paper. Stunning.

— Scott Gillespie (@stribgillespie) May 3, 2020

**2020 May 25**

The front page of The New York Times for May 24, 2020 pic.twitter.com/d14JhFp4CP

— The New York Times (@nytimes) May 24, 2020

]]>Sympathy card section in Walgreens today. pic.twitter.com/XfGo5bO1g9

— Victoria Weinstein (@peacebang) May 25, 2020

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

notthe highest priority of theiroryour life right now. Release yourself from high expectations right now, because thatâ€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:

**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.)**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.

I’m convinced that when teachers and students find the other side of this, it wonâ€t be because edtech companies offered junk for free, itâ€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 Naâ€ilah Suad Nasir and Megan Bang said in an open letter this weekend:

It may be that social distancing isnâ€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.

]]>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**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.*looks*like me**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**, communities that help me dissolve my sense of separateness from you. We are not separate.*my*kid is more valuable or special than*your*kid - I am working with a team to
**develop experiences in math class that lead to student answers that are**, or ones that at least lead to lots of*really*hard to call right or wrong*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?!?

— Bree Pickford-Murray (@btwnthenumbers) March 10, 2020

I get so much more 1:1 time with her than with students in my classroom. I feel that spaciousness in a deep way.

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

**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!”

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 hereâ€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:

Chevronâ€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/1225610304110972935If 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

oneidea, but can I get someotherideas, 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:**Â 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

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 â€“ family, Twitter, the death of blogs, etc. â€“ 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.

]]>I'm resolving to spend as much time next year thinking about student lives outside of school as I do their lives inside of school. Teaching and curriculum have enormous influence on student learning but the influence of those in-school factors is dwarfed by out-of-school factors like housing and food security. So I'm resolving to practice humanizing pedagogies

andto protest school closures in my city, to create interesting mathematical activitiesandto urge my representatives to protect and expand social programs. I'm resolving to ignore the distinction between educator and citizen.

Click through to read resolutions from thoughtful people like Carl Oliver, Hema Khodai, Idil Abdulkadir, Marian Dingle, Makeda Brome, and Tyrone Martinez-Black.

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