Category: uncategorized

Total 492 Posts

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”.

2020 Resolutions

Meanwhile, Nepantla Teachers, a group of math educators focused on social justice in their work, asked several educators to contribute a resolution for the new year. Here’s mine:

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.

Estimation Isn’t Just Calculating Badly On Purpose

Here is a tweet I haven’t stopped thinking about for a couple of months.

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

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.