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“Feel free to answer like a seventh grader,” I told teachers as I led them through one of the lessons from our Middle School Math Curriculum.
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.
I live adjacent to the Northern California wine country, which makes wine tasting a fairly affordable and semi-regular kind of outing. (Pre-quar, of course.) But wine tasting makes me anxious and sweaty in ways that help me relate to students who hate math class.
- 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.
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?”
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
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:
- 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 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.