All my action is over at Substack for now.

]]>Students are receiving more feedback from computers this year than ever before. What does that feedback look like, and what does it teach students about mathematics and about themselves as mathematicians?

Here is a question we might ask math students: what is this coordinate?

Let’s say a student types in (5, 4), a very thoughtful wrong answer. (“Wrong and brilliant,” one might say.) Here are several ways a computer might react to that wrong answer.

**1. “You’re wrong.”**

This is the most common way computers respond to a student’s idea. But (4, 5) receives the same feedback as answers like (1000, 1000) or “idk,” even though (4, 5) arguably involves a lot more thought from the student and a lot more of their sense of themselves as a mathematician.

This feedback says all of those ideas are the same kind of wrong.

**2. “You’re wrong, but it’s okay.”**

The shortcoming of evaluative feedback (these binary judgments of “right” and “wrong”) isn’t *just* that it isn’t *nice* enough or that it neglects a student’s emotional state. It’s that *it doesn’t attach enough meaning to the student’s thinking*. The prime directive of feedback is, per Dylan Wiliam, to “cause more thinking.” Evaluative feedback fails that directive because it doesn’t attach sufficient meaning to a student’s thought to cause more thinking.

**3. “You’re wrong, and here’s why.”**

It’s tempting to write down a list of all possible reasons a student might have given different wrong answers, and then respond to each one conditionally. For example here, we might program the computer to say, “Did you switch your coordinates?”

Certainly, this makes an attempt at attaching meaning to a student’s thinking that the other examples so far have not. But the meaning is often an *expert’s* meaning and attaches only loosely to the novice’s. The student may have to work as hard to *understand* the feedback (the word “coordinate” may be new, for example) as to *use* it.

**4. “Let me see if I understand you here.”**

Alternately, we can ask computers to clear their throats a bit and say, “Let me see if I understand you here. Is *this* what you meant?”

We make no assumption that the student understands what the problem is asking, or that we understand why the student gave their answer. We just attach as much meaning as we can to the student’s thinking in a world that’s familiar to them.

**“How can I attach more meaning to a student’s thought?”**

This animation, for example, attaches the fact that the relationship to the origin has horizontal and vertical components. We trust students to make sense of what they’re seeing. Then we give them an an opportunity to use that new sense to try again.

This “interpretive” feedback is the kind we use most frequently in our Desmos curriculum, and it’s often easier to build than the evaluative feedback, which requires images, conditionality, and more programming.

Honestly, “programming” isn’t even the right word to describe what we’re doing here.

We’re *building worlds*. I’m not overstating the matter. Educators build worlds in the same way that game developers and storytellers build worlds.

That world here is called “the coordinate plane,” a world we built in a computer. But even more often, the world we build is a physical or a video classroom, and the question, “How can I attach *more* meaning to a student’s thought?” is a great question in each of those worlds. Whenever you receive a student’s thought and tell them what interests you about it, or what it makes you wonder, or you ask the class if anyone has any questions about that thought, or you connect it to another student’s thought, *you are attaching meaning to that student’s thinking*.

Every time you work to attach meaning to student thinking, you help students learn more math and you help them learn about themselves as mathematical thinkers. You help them understand, implicitly, that their thoughts are *valuable*. And if students become *habituated* to that feeling, they might just come to understand that they are valuable *themselves*, as students, as thinkers, and as people.

**BTW**. If you’d like to learn how to make this kind of feedback, check out this segment on last week’s #DesmosLive. it took four lines of programming using Computation Layer in Desmos Activity Builder.

**BTW**. I posted this in the form a question on Twitter where it started a lot of discussion. Two people made very popular suggestions for *different* ways to attach meaning to student thought here.

I wonder if there is option 6, that plots a diff point like, shows the coordinates, and asks if they want to revise their (4,5). This could actually be cool for Ss who plots it correctly the first time as a double check.

— Kristin Gray (@MathMinds) December 10, 2020

]]>Unpopular opinion (apparently) from someone who’s seen many Ss start switching coordinates AFTER they’ve learned slope. Since coordinates represent location, not movement, I’d prefer #4 or better yet, “the meeting of the x&y” pic.twitter.com/mxoz8gM6Sv

— Ms. (Lauren) Beitel (@ms_beitel) December 10, 2020

“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.

]]>When schools started closing months ago, we heard two loud requests from teachers in our community. They wanted:

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.

]]>The American Time Use Survey is a fantastic data set. You can find out how many more hours per day women spend on household activities than men. You can identify the time of day that the majority of Americans wake up.

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 people’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.

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

notthe highest priority of theiroryour 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.

]]>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**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

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:

What do we think of this Maths problem ... interestingly fuzzy or annoyingly vague https://t.co/aqeYmcz4X9

— Hull PGCE (@hullpgce) February 7, 2020

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

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:** 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.

]]>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

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.

]]>EdSurge invited me to review the last decade in math edtech.

Entrepreneurs had a mixed decade in K-16 math education. They accurately read the landscape in at least two ways: a) learning math is enormously challenging for most students, and b) computers are great at a lot of tasks. But they misunderstood why math is challenging to learn and put computers to work on the wrong task.

In a similar retrospective essay, Sal Khan wrote about the three assumptions he and his team got right at Khan Academy in the last decade. The first one was *extremely* surprising to me.

Teachers are the unwavering center of schooling and we should continue to learn from them every day.

Someone needs to hold my hand and help me understand how teachers are anywhere near the center of Khan Academy, a website that seems especially useful for people who do not have teachers.

Khan Academy tries to take from teachers the jobs of instruction (watch our videos) and assessment (complete our autograded items). It presumably leaves for teachers the job of monitoring and responding to assessment results but their dashboards run on a *ten-minute* delay, making that task *really* hard!

Teachers are very obviously peripheral, not central, to the work of Khan Academy and the same is true for *much* of math education technology in the 2010s. If entrepreneurs and founders are now alert to the unique value of teachers in a student’s math education, let’s hear them articulate that value and let’s see them re-design their tools to support it.

Five of my favorite articles from the last month.

**Daniel Willingham**argued in*the Los Angeles Times*that math specialists should teach children math in elementary school instead of elementary teachers. (A thread of unhappy math specialists and elementary teachers on Twitter.)**Ben Orlin**interviews**Matt Enlow**about his lined paper from another dimension, an art and math project that may transport you into a skewed version of your childhood in school.**Gallup**surveyed teachers and students on Creativity in Learning, revealing several disagreements about how often they think students memorize facts, apply their learning to the real world, and discuss topics with no right or wrong answer.**Maya Kosoff**writes Big Calculator: How Texas Instruments Monopolized Math Class, another critical examination of TI’s profit margins and their effect on working class families. (My thread about the article.) Very related: Desmos is now permitted on the Texas end-of-course exam.

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

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 anyone’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**

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.

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.”

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.

]]>Come hang out with me at California Math Council’s North and South conferences in November and December.

**CMC-South**. Palm Springs, CA. November 15-16. I’m going to describe how “rich tasks” and “bland tasks” both fail our students. And I’m going to do it with 15 students on a stage in a live lesson demonstration. Let’s gooo! [register]

**CMC-North**. Pacific Grove, CA. December 6-8. I’ll share some of the ways my colleagues and I at Desmos are designing for belonging in math class, specifically how we try to expand the list of who counts as a mathematician and what counts as mathematics. [register]

Five of my favorite articles from the last month.

- 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.

Economist Herb Stein’s quote ran through my head while I read The Hustle’s excellent analysis of the graphing calculator market. This cannot go on forever.

Every new school year, Twitter lights up with caregivers who can’t believe they have to buy their students a calculator that’s wildly underpowered and wildly overpriced relative to other consumer electronics.

The Hustle describes Texas Instruments as having “a near-monopoly on graphing calculators for nearly three decades.” That means that some of the students who purchased TI calculators as college students are now purchasing calculators for their *own* kids that look, feel, act and (crucially) cost largely the same. Imagine they were purchasing their kid’s first car and the available cars all looked, felt, acted, and cost largely the same as *their* first car. This cannot go on forever.

As the chief academic officer at Desmos, a competitor of Texas Instruments calculators, I was already familiar with many of The Hustle’s findings. Even still, they illuminated two surprising elements of the Texas Instruments business model.

First, the profit margins.

One analyst placed the cost to produce a TI-84 Plus at around $15-20, meaning TI sells it for a profit margin of nearly 50% — far above the electronics industry’s average margin of 6.7%.

Second, the lobbying.

According to Open Secrets and ProPublica data, Texas Instruments paid lobbyists to hound the Department of Education every year from 2005 to 2009 — right around the time when mobile technology and apps were becoming more of a threat.

Obviously the profits and lobbying are interdependent. Rent-seeking occurs when companies invest profits not into product development but into manipulating regulatory environments to protect market share.

I’m not mad for the sake of Desmos here. What Texas Instruments is doing isn’t sustainable. Consumer tech is getting so good and cheap and our free alternative is getting used so widely that regulations and consumer demand are changing quickly.

Another source told The Hustle that graphing calculator sales have seen a 15% YoY decline in recent years — a trend that free alternatives like Desmos may be at least partially responsible for.

You’ll find our calculators embedded in over half of state-level end-of-course exams in the United States, along with the International Baccalaureate MYP exam, the digital SAT and the digital ACT.

I *am* mad for the sake of kids and families like this, though.

“It basically sucks,” says Marcus Grant, an 11th grader currently taking a pre-calculus course. “It was really expensive for my family. There are cheaper alternatives available, but my teacher makes [the TI calculator] mandatory and there’s no other option.”

Teachers: it was one thing to require plastic graphing calculators calculators when better and cheaper alternatives weren’t available. But it should offend your conscience to see a private company suck 50% profit margins out of the pockets of struggling families for a product that is, by objective measurements, inferior to and more expensive than its competitors.

**BTW**. This is a Twitter-thread-turned-blog-post. If you want to know how teachers justified recommending plastic graphing calculators, you can read my mentions.