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.

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?”

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.

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

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

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 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.

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

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!”
• “More taller” is coming up a lot.
• These kids think that as they get older, they’ll get bigger and I’ll get smaller and turn into a baby.

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 and to protest school closures in my city, to create interesting mathematical activities and to 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.

Any tips around a young lad with ASD who cannot get his head around estimation? He just cannot see that it would be ‘nearly†or ‘around†something when he can clearly work the answer out. My gut says give him something else to do, but if anyoneâ€s come across this in the past….

— Ashley Booth (@MrBoothY6) September 11, 2019

I think it’s possible we should cut the student some slack here.

If the student has all the tools, information, and resources necessary to calculate an answer, we should be excited to see the student calculate it. Asking students to do anything less than calculate in that situation is to ask them to switch off parts of their brain, to use less than their full capacity as a thinker.

If we treated skills in other disciplines the way we often treat estimation in math …

… we’d ask students to spell words incorrectly before spelling them correctly.

… we’d ask students to recall historical facts incorrectly before recalling them correctly.

Estimation shouldn’t ask students to switch off parts of their brains or use less than their full capacity as thinkers. It should ask them to switch on new parts of their brains and expand their capacities as thinkers. Estimation tasks should broaden a student’s sense of what counts as math and who counts as a mathematician.

Estimation and calculation should also be mutually supportive in the same way that …

… knowing roughly the balance of yeast and sugar in bread supports you when you pour those ingredients exactly.

… knowing the general direction of your destination supports you when you drive with turn-by-turn directions.

… knowing the general order of your weekend schedule supports you when you carry out your precise itinerary.

Engaging in one aspect of mathematics makes the other easier and more interesting. That’s what Kasmer & Kim (2012) found was true about estimation. When students had a chance to first predict the relationship between two quantities it made their later precise operation on that relationship easier.

If we want students to develop their ability to estimate, we need to design experiences that don’t just ask them to calculate badly on purpose.

Create tasks where estimation is the most efficient possible method.

Take that worksheet above. Give students the same sums but ask them to order the sums from least to greatest.

Students may still calculate precisely but there is now a reward for students who estimate using place value as a guide.

Create tasks where estimation is the only possible method.

This is the foundation of my 3-Act Task design, where students experience the world in concrete form, without the information that word problems typically provide, without sufficient resources to calculate.

“Estimate the number of coins.” Estimation feels natural here because there isn’t enough information for calculation. Indeed, estimation is the only tool a student can use in this presentation of the context.

Meanwhile, in this presentation of the same task, there is enough information to calculate, which makes estimation feel like calculating badly on purpose.

Estimation isn’t a second-class intellectual citizen. It doesn’t need charity from calculation. It needs teachers who appreciate its value, who can create tasks that help students experience its benefits.

BTW

• Here’s a beautiful children’s book on exactly this topic. [via Julia McNamara]
• Contemplate then Calculate is an instructional routine that cleverly blocks calculation by only showing a mathematical structure for a limited time only.

Featured Comment

William Carey:

One thing I love about calculus is is proceeds from estimation to exact calculation, and there’s no way to justify the exact calculations without working through the estimation first. We often think of mathematics as a discipline that proceeds deductively from perfect truth to perfect truth, but there are whole swaths of mathematics where the best way forward is to work from an answer whose incorrectness we understand towards an answer whose correctness we don’t yet understand.

Mark Betnel:

I agree with you, but I think it’s interesting to turn your non-math examples into better activities that reflect what we’re trying to do with “good” math estimation tasks.

Mr. K references Fermi problems, which fall really nicely in the category of “tasks where estimation is the only possible method.”

Theresa Clifford:

At the beginning of the year, I fill four jars around the room. One with M&M’s, one with eraser caps, one with cotton balls, and one with paper clips. They are all allowed a guess for how many in each jar. They enter their answer and their name on a slip of paper and place it in a collection jar. Whenever we come to a question where I want them to estimate first, I remind them of what they did when they first looked at the jar. I don’t tell them how many in each until the winter break – the suspense is awesome. Then in January I start with four new jars.

Joel offers an example of this kind of estimation exercise.

• One of Idil Abdulkadir’s students asked her “Are there any other people of colour?” about a summer program she was attending. She describes what she thought before she responded, illustrating how much of the work of great, relational teaching takes place invisibly and nearly instantaneously.
• Sarah Schwartz writes in Education Week about the most interesting controversy in math curriculum right now: the publisher suing a parent for allegedly lying about their curriculum.
• Bob Janes describes his implementation of the five practices for orchestrating productive mathematics discussions with technology and without.
• Chatbot company Hubert recommends teachers regularly ask their students three questions to promote the development of their teaching practice.
• Jessica Wynne photographed mathematicians’ blackboards and the results do not disappoint.

Twitterverse, I need your help! I am going to be teaching math next year for the first time. 7th grade! Iâ€ve previously taught ELA & SS. Any teachers/gurus I should follow? Advice/tips for me? Resources you recommend? I know @Desmos is a good resource! #iteachmath #WEAreLakota

— Emily Ladrigan (@EmilyLadrigan) June 28, 2019

Iâ€ll be teaching 7th grade math next year. Switching from Social Studies. Any tips or helpful websites?? #iteachmath

— Ms. Nesmith (@ms_nesmith) July 24, 2019

I saw those tweets and had to sit back and collect myself.

That’s because I know how well I’m served by my knowledge of mathematics, how that knowledge helps me find value in early student thinking, how that knowledge helps me connect and build on thoughts from different students that, without that knowledge, might seem totally unrelated.

This isn’t a critique of those two newly drafted math teachers at all. Most of my horror here results from the thought of being drafted to teach history after a career teaching math. So what can they do?

You’ll find lots of people in those threads recommending resources and curricula. But resources and curricula are only as good as the teacher using them. A developing teacher can make a good resource bad and an expert teacher can make a bad resource good. (This is why John Mason prefers to talk about “rich teaching” instead of “rich tasks.”)

So my own advice is for these teachers trained in the humanities to focus on their teaching, not the resources or curricula.

Specifically, I hope they’ll resist the idea that math should be taught any differently than the humanities. I hope they’ll resist the idea that only the humanities deal in subjectivity, argumentation, and personal interpretation, while math represents objective, inarguable, abstract truth.

Math is only objective, inarguable, and abstract for questions defined so narrowly they’re almost useless to students, teachers, and the world itself.

In social studies, an analogous question might ask students to recall the date of the Louisiana Purchase or the name of the king who signed the Magna Carta — questions that are so abstracted from their context, so narrowly defined, and so objective that they make no contribution to a student’s ability to think historically.

The National Council for the Social Studies describes what’s necessary for students of social studies:

Students learn to assess the merits of competing arguments, and make reasoned decisions that include consideration of the values within alternative policy recommendations. [..] Through discussions, debates, the use of authentic documents, simulations, research, and other occasions for critical thinking and decision making, students learn to apply value-based reasoning when addressing problems and issues.

All of which rhymes perfectly with recommendations from the National Council of Teachers of Mathematics:

Teaching mathematics with high expectations for all students in mathematical reasoning, sense making, and problem solving invites students to learn to identify assumptions, develop arguments, and make connections within mathematical topics and to other contexts and disciplines.

Teaching math like the humanities asks us to:

• Broaden the scope of the problems we assign. We can always narrow the scope in collaboration with students but the opposite isn’t true. Students don’t have the opportunity to “identify assumptions,” for example, if we pre-assume every detail in the problem.
• Focus on mathematical ideas that are big enough to be understood in different ways. Ask students to make claims that demand to be argued and interpreted rather than evaluated by an authority for correctness.
• Celebrate novel student contributions to mathematics. History is made every day and so is mathematics. If our students leave our classes this year without understanding that they have had made unique and original contributions to how humans think mathematically, we have defined “mathematics” too narrowly. (For example, someone just decided to call this shape a “golygon.” If that person has the right to notice and name things, then so do your students.)

Instead of the worksheet above, show your students this video of a pallet of bricks and then immediately hide it.

“Does anybody have a guess about how many bricks we saw up there?”

“Did anybody notice any features about the bricks that might help us figure out exactly how many bricks we saw there?”

“Let’s look at the video again. Okay, what’s the most efficient way you can think to figure out the number of bricks.”

“How were you thinking about the number of bricks you figured out? What assumptions did you make?”

“Someone else got a different answer from you. How do you think they thinking about the number of bricks?”

“Here’s the number of bricks. What’s another question we could ask now?”

These questions rhyme with the kinds of questions you’d hear in a productive, engaging humanities classroom, questions which are no less possible in mathematics!

Humanizing math class means teaching like the humanities. And if you’re joining us from the humanities, please be generous with your pedagogy. We need all of it.

BTW: This is my contribution to the Virtual Conference on Humanizing Mathematics, a fantastic learning opportunity hosted by Hema Khodai and Sam Shah through the month of August 2019.