Twitterverse, I need your help! I am going to be teaching math next year for the first time. 7th grade! I’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

I’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.

Most recently, they let me think about *time* with some second graders – the youngest kids I’ve ever taught and probably ever met, I can’t be sure – and especially how to tell time on an analog clock.

My goal in these experiences is always to find *areas of agreement* between the teaching of different age groups and different areas of math. Whether I’m learning about time with second graders or about polynomial operations with high schoolers or about teaching with math teachers, I’m asking myself, what’s going on here that crosses all of those boundaries, not one of which is ever drawn as sharply as I first think.

The odds of me stepping on a child go way up, for one.

For another, these students were *inexhaustible*. Their default orientation towards me and my ideas was *rapt engagement* and an earnest, selfless desire to improve my ideas with stories about their friends, their pets, and their families.

My tools for curriculum and instruction were forged by students who communicated to me that “none of this matters” and “I can’t do it even if it did.” Those tools seemed less necessary here. Instead, I needed tools for *harnessing* their energy and I learned lots of them from my friends at Beach Elementary – popsicle sticks for group formation, procedures for dismissing students *gradually* instead of *simultaneously*, *silent* signals for agreement instead of *loud* ones, etc.

Even still, with these second graders, I tried to *problematize* conventions for telling time, just as I would with high school students. I asked students to tell me what bad thing might happen if we didn’t know how to tell time, and they told me about being late, about missing important events, about not knowing when they should fall asleep and accidentally staying awake through the night!

I tried to elicit and build on their early language around time by playing a game of Polygraph: Clocks together. I told them I had picked a secret clock from that array and told them I would answer “yes” or “no” to any question they asked me. Then they played the game with each other on their computers.

One student asked if she could play the game at home, a question which my years of teaching high school students had not prepared me to hear.

I saw in second grade the students I would eventually teach in high school. Students who were anxious, who shrunk from my questions, either wishing to be invisible or having been invisibilized. Other students stretched their hands up on instinct at the end of every question, having decided already that the world is their friend.

Those students weren’t handed those identities in their ninth grade orientation packets. They and their teachers have been cultivating them for *years*!

Rochelle Gutierrez calls teachers “identity workers,” a role I understood better after just an hour teaching young students.

All mathematics teachers are identity workers, regardless of whether they consider themselves as such or not. They contribute to the identities students construct as well as constantly reproduce what mathematics is and how people might relate to it (or not).

I have wanted not to be an identity worker, to just be a *math* worker, because the stakes of identity work are *so high*. (Far better to step on a child’s foot than to step on their sense of their own value.) We wield that power so *poorly*, communicating to students with certain identities at astonishingly early ages – especially our students who identify as Latinx, Black, and Indigenous – that we didn’t construct school and math class for their success.

I have wanted to give up that power over student identities and just teach math, but as Gutierrez points out, students are always learning more than math in math class.

My team and I at Desmos are forging new tools for curriculum and instruction and we’re starting to evaluate our work not just by what those tools teach students about *mathematics* but also by what they teach students about *themselves*.

It isn’t enough for students to use our tools to discover the value of *mathematics*. We want them to discover and feel affirmed in their *own* value, the value of their *peers*, and the value of their *culture*.

We’ve enlisted consultants to support us in that work. We’re developing strategic collaborations with groups who are thoughtful about the intersection of race, identity, and mathematics. A subset of the company currently participates in a book club around Zaretta Hammond’s Culturally Responsive Teaching and the Brain.

Before undertaking that work, I’d tell you that my favorite part of teaching Polygraph with second graders is how deftly it reveals the power of *mathematical* language. Now I’ll tell you my favorite part is how it helps students understand the power of their *own* language.

“Is your clock a new hour?” a second-grade student asked me about my secret clock and before answering “yes” I made sure the class heard me tell that student that they had created something very special there, a very interesting question using language that was uniquely theirs, that was uniquely valuable.

]]>Why did you become a math teacher?

Perhaps you loved math. Perhaps you were good at math, good, at least at the thing you called math then. Friends and family would come to you for help with their homework or studying and you prided yourself not just on explaining the *how* of math’s operations but also the *why* and the *when*, helping others see the purpose and application behind the math.

Helping other people understand and love the math *you* understood and loved – perhaps that sounded like a good way to spend a few decades.

Or perhaps you loved kids. Perhaps even at a young age you were an effective caregiver, and you knew how to care for more than just another person’s tangible needs. You listened, and you made people feel *listened to*. You had an eye for a person’s value and power. You understood where people were in their lives and you understood how the right kind of question or observation could propel them to where they were going to *be*.

Spending a few decades helping people feel heard, helping them unleash and use their tremendous capacity – perhaps you thought that was a worthwhile way to spend what you thought would be the hours between 7AM and 4PM every day.

Or perhaps you loved both math and kids. It’s possible of course that neither of the two previous exemplar teachers will speak fully to the path that brought you to math teaching, although one of them speaks fully to mine. Yet, in my work with math teachers, I find they often draw their professional energy from one source or the other, from math’s ideas or its people.

It took me several frustrated years of math teaching – and years of work with other teachers – to realize that *each* of those energy sources is vital. Neither source is renewable without the other.

If you draw your energy only from mathematics, your students can become abstractions, and interchangeable. You can convince yourself it’s possible to influence *what they know* without care for *who they are*, that it’s possible to treat their *knowledge* as deficient and in need of fixing without risking negative consequences for their *identity*. But students know better. Most of them know what it feels like when the adult in the room positions herself as all-knowing and the students in the room as all-unknowing. A teacher’s love and understanding of mathematics won’t help when students have decided their teacher cares less about them than about numbers and variables, bar models and graphs, precise definitions and deductive arguments.

If you draw your energy only from students, then the day’s mathematics can become interchangeable with any other day’s. Some days it may feel like an act of care to skip students past mathematics they find frustrating, or to skip mathematics altogether some days. But the math you skip one day is foundational for the math another day or another year. Students will have to pay down their frustration later, only then with compound interest. Your love and care for students cannot protect them from the frustration that is often fundamental to learning.

I could tell you that the only solution to this problem of practice is to develop a love of students *and* a love of mathematics. I could relate any number of maxims and slogans that testify to that truth. I could perhaps convince some of you to believe me.

But the maxim I hold most closely right now is that we act ourselves into belief more often than we believe our way into action. So I encourage you more than anything right now to adopt a series of productive *actions* that can reshape your *beliefs*.

Here are five such actions: anticipate, monitor, select, sequence, and connect.

Those actions, initially proposed by Smith and Stein in 2011 and ably illustrated here with classroom videos, teacher testimony, and student work samples, can convert a teacher’s love for math into a love for students and vice versa, to act her way into a belief that math and students both matter.

For teachers who are motivated by a love of students, those five practices invite the teacher to learn more mathematics. The more math teachers know, the easier it is for them to find value in the ways their students think. Their mathematical knowledge enables them to monitor that thinking less for *correctness* and more for *interest*. Would presenting this student’s thinking provoke an *interesting* conversation with the class, whether the circled answer is correct or not? A teacher’s mathematical knowledge enables her to connect one student’s interesting idea to another’s. Her math knowledge helps her connect student thinking together and illustrate for the students the enormous value in their ideas.

For the teachers like me who are motivated by a love of mathematics, teachers who want students to love mathematics as well, those five practices give them a rationale for understanding their students as people. Students are not a blank screen onto which teachers can project and trace out their own knowledge. Meaning is made by the student. It isn’t *transferred* by the teacher. The more teachers love and want to protect interesting mathematical ideas, the more they should want to know the meaning students are making of those ideas. Those five practices have helped me connect student ideas to canonical mathematical ideas, helping students see the value of both.

Neither a love of students nor a love of mathematics can sustain the work of math education on its own. We work with “math students,” a composite of their mathematical ideas and their identities as people. The five practices for orchestrating productive mathematical discussions, and these ideas for putting those practices into practice, offer the actions that can develop and sustain the belief that both math and students matter.

You might think your path into teaching emanated from a love of mathematics, or from a love of students. But it’s the same path. It’s a wider path than you might have thought, one that offers passage to more people and more ideas than you originally thought possible. This book will help you and your students learn to walk it.

]]>It’s my annual tradition to scrape all the handouts and slides from the NCTM annual conference website and put them in a more usable form and location, including a single, massive download link [2.8 GB].

But this year I also found *recordings* from every session.

Not video, nope.

Twitter!

People were recording the sessions on Twitter. People were in sessions the entire conference tweeting key findings, memorable phrases, and impactful slides.

Those tweets were enjoyed in the moment by people who were near their computer during the session. But every session’s tweets were mixed up with every *other* session’s tweets from that same time period. Then they were lost immediately afterwards to a timeline that *doesn’t stop moving*.

**So I re-captured those tweets and attached them to every talk.**

I wrote a script that copied every speaker’s Twitter handle and the time of their session from the program book. Then the script searched Twitter for their handle and “#NCTMSD2019” and – here’s the thing! – captured the results *only* from the time period of their session. Not during their flight to NCTM. Not during the fantastic happy hour after their session either.

So at this site you’ll see the usual listing of speakers and sessions – but you’ll *also* see columns that let you know a) how many attachments those speakers made available and b) how much Twitter coverage their sessions received.

Here’s why this is a blast. Lauren Baucom gave a dynamite talk describing her work supporting the de-tracking of her high school. No handouts or video, but here are 34 tweets to help you connect with her ideas. The same is true for loads of other presenters.

I think a lot about Jere Confrey’s statement that “students are the most underutilized resource in our schools.” Students bring value to our classes – their identities, their aspirations, their early and developing understandings of mathematical ideas – that too often goes uncelebrated, unexplored, and underutilized.

Similarly, thousands of teachers traveled at great length and great expense to San Diego last week. Thousands of brains, bodies, and souls all together in community. What did we *make* of that experience? What do we have to *show* for it? Here’s one thing and I’d like to know more.

If you have been successful in math, by public consensus, you must be smart. If you have been successful in the humanities, you *may* also be smart but we cannot really be sure about that now can we, says public consensus.

In a world where our finest mathematical minds ruined the global economy and perpetuate unequal social outcomes, outcomes most ably critiqued by people trained in the humanities, public consensus is wrong.

This worksheet is worse.

This worksheet associates smartness with a certain *way* of doing math, diminishing *other* ways your students might develop to do the same math. Because there are lots of possible ways to tell time – some new, some old, and some not-yet-invented!

Worse, this worksheet associates smartness with a certain way of doing math that is *culturally defined*, diminishing entire cultures. For example, depending on your location in the world, “2/5/19” and “5/2/19” can refer to the same calendar date. Neither of those ways are “smart” or “dumb.” They work for communication or they don’t.

If I’d like students to learn a certain way of doing math – whether that’s adding numbers a certain way or solving equations a certain way – I need to understand the reasons why we invented those ways of doing math and put students in a position to *experience* those reasons. I also need to be excited – thrilled even! – if students create or adapt their own ways of doing math when they’re having those experiences. Anything less is to diminish their creativity.

If I want students to learn how to communicate mathematically, I need to ask them to *communicate*.

So in this Desmos activity, one student will choose a clock and another student will ask questions to narrow 16 clocks down to 1.

I have no idea what ways students will use, create, or adapt in order to tell time. I will be excited about all of them.

I will also be excited to share with them the ways that lots of cultures use to tell time. When I share those ways, I will be honest that those ways aren’t “smart” any more than they are “moral.” They are merely what one group of people agreed upon to help them get through their day.

So I’d also offer students *this* Desmos activity, which tells students the time using several different cultural conventions, including the one the worksheet calls “smart” above.

Students set the clock and then they see how easy or hard it was for the class to come to consensus using that convention.

Later, we invite students to set the clock themselves and name the time using three different conventions. They make two of them true, one of them a lie, and submit the whole package to the Class Gallery where their classmates try to determine the lie.

The words we use matter. “Real world” matters. “Mistakes” matter. “Smart” matters. Those words have the power to shape student experiences, to extend or withdraw opportunities to learn, to denigrate or elevate students, their cultures, and the ideas they bring to our classes.

Defining smartness narrowly is to define “dumbness” broadly. Instead, we should seek to find smartness as often as possible in as many students as possible.

shoot. i say five fifteen and five forty five routinely. i guess i'm not "smart"

— Ms.B (@MathIsNotScary) March 18, 2019

Write each time "my way."

— corey andreasen (@coreyandreasen) March 19, 2019

There's nothing wrong with familiarizing students with these phrases. How about "Write each time in words in at least two different ways. Tell which way is your favorite."

Re: time as culturally bound

— Idil A (@Idil_A_) March 30, 2019

growing up in Mogadishu, Somalia my mom said they used a 12 hour am/pm system, but it ran 6am-6pm.

Makes a lot of sense living on the equator, sunrise was 0 and sunset was 12.

@ddmeyer more cultural time context. I always have to ask what’s “half seven” and there’s more than one answer. https://t.co/daR81RCAUC

— Calley Connelly (@CalleyMath) March 30, 2019

There are lots of details here. She’s trying to focus on the ones that matter. The color of the parabola doesn’t seem relevant. They’re all blue. The window of the graph is the same for all the parabolas.

She focuses on the orientation of the graphs and she asks a question using the most precise words she can given her current understanding. “Is it like a hill?” she asks.

Geoff answers back “No” and Amare eliminates all the “hill” graphs from consideration. So far so good.

Amare is now at a loss. She knows that the graphs are different but she isn’t sure how to articulate those differences. “Is it wide?” she asks.

After a long pause, Geoff answers back “Yes.”

Amare eliminates several graphs, one of which happens to be Geoff’s graph. Their definitions of “wide” were different.

Their teacher brings the class together for a discussion of the features the students found useful in their exchanges. The teacher offers them some language mathematicians often use to describe the same graphs. Then they all return to the activity to play another round.

Here is a diagram the GAIMME report uses to describe mathematical modeling (p. 13):

I contend that Amare and Geoff participated in every one of those stages.

Here is GAIMME’s definition of mathematical modeling (p. 8):

Mathematical modeling is a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena.

I contend that Amare and Geoff satisfy that definition as well.

Many mathematical modelers would disagree, I suspect, given the reaction to my panel remarks last week.

Polygraph isn’t “real world.” They’re convinced it isn’t. When asked to describe how we know a student is working in the “real world” or not, though, they beg the question with adjectives like “legitimate,” authentic,” or “not mathematical” (essentially “not *not* ‘real world'”).

They can’t offer a definition of “real world” that categorizes the shapes that are *right in front of the student right now* as “not real.” They just know “real world” when they see it.

**The distinction between the “real” and “not real” world doesn’t exist and insisting on it makes everyone’s job harder.**

**It makes the teacher’s job harder**. She has to maintain two models for how students learn – one for ideas that exist in the “real world” and one for ideas that exist in the “not real world.” But they can unify those models! The tasks that mathematical modelers often enjoy and Polygraph should be taught the same way. That’d be great for teachers!

**It makes the mathematical modeler’s job harder.** The tasks mathematical modelers enjoy are not categorically different from Polygraph. The early ideas that teachers need to elicit, provoke, and develop in those tasks differ from Polygraph only in their degree of contextual complexity. Instead of telling teachers, “Here is how this task is similar to everything else you’ve done this year,” and benefiting from pedagogical coherence, they tell teachers, “This task is categorically different from everything else you’ve done this year and why aren’t you doing more of them?”

I’m trying to convince mathematical modelers that their process is the same one by which anyone learns *anything*, that they should spend much less time patrolling borders that don’t exist, and instead apply their processes to *every* area of the world, every last bit of which is “real.”

Sol Garfunkel, a co-panelist and personal hero, would later call my introductory remarks “completely wrong.” A university professor called them “dangerous.”

I mention those reviews not to marshal sympathy. I’m really happy with my remarks and I don’t think I was misunderstood! I’m mentioning them to acknowledge that my remarks caused a lot of anxiety among people who call themselves mathematical modelers. I’ll respond to some of those anxieties below.

(Here is the whole panel, if you’re interested. Or here is an excerpt of my five minutes and the Q&A period. Or skip down to my responses to questions and criticism.)

Hey folks, I’m Dan Meyer. I work at Desmos where my team makes modeling activities using digital technology.

I’m an optimist so I’m hopeful for modeling’s future even though I feel like it’s in a diminished state right now.

On the one hand, you have the folks who are defining modeling down, folks who will call any problem modeling for the sake of a good alignment score for their textbook.

On the other hand, you have organizations like the ones that authored the GAIMME report who are defining modeling *up*, who are placing modeling on a mountain that is far too high for any mortal teacher to climb.

First, the report is 200 pages long, which is a lot of pages. I’m trying to think back to my time in the classroom, wondering during which interval of time I’d read a report of that length.

Passing period? No.

Prep period? No.

Weekends? Gotta finish up True Detective Season 3.

Summer? Maybe.

Summer if I was on a grant-funded project led by university professors like yourself? Now we’re getting somewhere.

But beyond the length of the report, it depends heavily on adjectives like “messy,” “open,” “real-world,” and “genuine,” adjectives which have *no shared meaning*. None. The only way to know you’re doing modeling is to ask the authors of the GAIMME report if they think what you’re doing is messy, open, real-world, or genuine enough.

I want to challenge that narrow definition of modeling.

The first number in a sequence is 1. What might the next number be?

[Audience members call out different numbers.]

Maybe 2? Maybe you’re thinking about counting or cardinality. It’s 2. What might be next?

[More audience call-out. People call out 3 and 4.]

Maybe you’re thinking still about counting. Maybe you’re thinking about powers of two. It happens to be 4. What might be next?

[Audience members call out numbers. More convergence now. People are feeling good about 8.]

It happens to be 7. What might be next?

[Audience members are really converging on the pattern now.]

That’s right. That’s the sequence.

A statement I suspect very few people in this room will agree with is *that was mathematical modeling*.

But it was.

You took your early knowledge of the pattern. You put it to work for you. You found out something new.

You revised your model. It came into sharper focus. Suddenly you *did* know the sequence. Several pleasure centers in your brain lit up simultaneously. That is modeling.

It’s the same with learning anything – from short, abstract sequences of numbers to huge, abstract concepts like love, which you think you understand as a kid. It’s defined by your relationship to your parent or guardian. That’s what love is. Or love is everything but that.

You go out and put your understanding of love to work for you as a young adult.

You find out something new that reveals the limits of your ideas of love. You revise and sharpen your ideas.

You put *those* ideas out into the world until you have that first traumatic break-up and you realize your model for love is even fuzzier than it was originally!

All these experiences help you revise your model for love – never completely, never correctly, never *incorrectly*, and always in process.

That’s modeling.

We think it’s like this, that modeling is a subset of math learning. And that our goal is to make the subset as large as possible.

But to name that distinction is to undermine the goal.

We cannot tell teachers that some days are modeling days and some days are *not* modeling days.

That on some days, you should draw on students’ funds of knowledge and on other days you can ignore them.

That on some days, you should elicit early student ideas about math and on other days you can transfer mature ideas from your head to theirs.

That on some days, you should provoke students to refine their ideas about math and on other days you can treat their ideas as though they’re finished and ready for grading.

That’s too confused to work.

I think this is actually true, though it isn’t the entirety of what I’m trying to say.

What I’m saying is this: that all learning is modeling.

It’s true about love. It’s true about a sequence of numbers. It’s true about modeling itself. You came in here with a model in your head about modeling. You’ll test that model here at MSRI. Everything you hear and see and experience will change and strengthen your model for modeling.

We will all walk away with a different model for modeling than when we got here.

So let’s not trivialize modeling by defining it downwards. Let’s not define it upwards, out of reach of anyone outside of the academy.

*Let’s define it everywhere*.

Here are a few follow-up thoughts, mostly addressed to the people at #CIME2019 who felt strongly that “mathematical modeling” and “learning” are fundamentally different processes.

**You’re going to have to actually define the “real world” and the “non-real world.”**

In something of a rebuttal to my remarks, Sol Garfunkel said:

So we might as well start this fight now. I think Dan is completely wrong. The reason we wrote the GAIMME report was to put out a standard defintion of modeling. Now you could use another definition. But the definition of mathematical modeling in the report and the one all the people I know who work in the field agree on is that it begins with a real-world problem. [..] Most people would agree or at least – it’s not a question of “agree” – it’s a definition. As some math teacher of mine once said, defintions are neither right nor wrong, they’re either useful or useless.

If your definition of “real world” labels the US tax code as *real* and polygons as *non-real*, your definition is not useful. To most US K-12 students, the US tax code is very *non-real* and polygons are very *real*.

If you define “real-world” as a property that is binary rather than continuous, that is fixed across all cultures and time rather than relative and mutable, if your definition doesn’t account for the ways (per Freudenthal) that contexts *become* real in someone’s mind, it isn’t useful.

And if your distinction between “mathematical modeling” and “learning” depends on “real world,” a descriptor without a definition, it isn’t a meaningful distinction.

The distinction Garfunkel (and many modelers) are trying to draw here is very similar to Supreme Court Justice Potter Stewart’s definition of pornography: “I know it when I see it.”

That lack of definitional precision will undermine broad adoption and cost teachers and students dearly, as I’ll describe next.

**Teachers need fewer ideas about teaching.**

I was happy that Sol took a moment to respond to my remarks but I was disappointed that in doing so he fully ignored the audience member’s question, which I thought was extremely important:

What is gained and what is lost by lumping all learning under the umbrella term of “modeling”?

Other people can describe what is lost. As I’ve said, I’m very unconvinced we’ve lost a connection to the “real world.”

What’s gained is coherence. What’s gained is the opportunity to take all these pedagogical toolboxes teachers currently have on their shelves – toolboxes for “real world” and “non-real world,” toolboxes for “mathematical modeling” and “not mathematical modeling” – and replace them with one toolbox: modeling.

**Modelers: teachers still need you.**

The audience member who called my remarks “dangerous” seemed worried that after working so hard to convince teachers that there is a special thing called “mathematical modeling” and that teachers should work to integrate it deeper into their practice, I’d come along and say something like, “No, everything you’re doing is already that thing. You’re fine.”

But that isn’t what I said and it isn’t what I believe. Serious work *is* necessary here and people who understand modeling are well poised to lead it.

Modeling is the process whereby a learner tests out her early ideas, determines their limits, and develops those ideas further. That’s also called “learning.”

To help students learn anything, teachers need to *initiate* the modeling process, eliciting early ideas, provoking students to determine their limits, and helping students develop their ideas further.

All learning is modeling. But not all teaching initiates the modeling process.

People who call themselves mathematical modelers understand that process better than most. We just need them to drop this meaningless distinction between the real and non-real world and apply their skills across all of teaching.

My proposal here makes modelers *more* necessary, not less.

Here are a couple of items where the world needs your help and one way where the world (a/k/a Desmos, the New York Times Learning Network, and the American Statistical Association) is here to support you.

Here is a clip from *60 Minutes* in which Congresswoman Alexandria Ocasio-Cortez proposes a 70% top marginal tax rate.

And here is a ranking of the last five words of that sentence for “concreteness of meaning to average Americans”:

1. 70%

2. tax

3 [tie]. rate, top, marginal

A 2013 YouGov poll convincingly illustrates how little Americans understand our marginal tax rate system. YouGov posed the following scenario:

Suppose that your income put you at the very top of the 28% tax bracket and you earned one more dollar such that you were now in the 33% tax bracket.

They asked 818 respondents to choose between two options:

- My tax bill would go up a very small amount.
- My tax bill would go up substantially.

Only 52% of respondents correctly answered that their tax bill would go up a very small amount. Democrats answered that question correctly nearly twice as often as Republicans, a difference which is amply explained by all the Republican leaders who are smart enough to know how marginal tax rates work lying to people about how marginal tax rates work.

Explaining tax rates before Reagan to 5th graders: “Imagine if you did chores for your grandma and she gave you $10. When you got home, your parents took $7 from you.” The students said: “That’s not fair!” Even 5th graders get it.

— Scott Walker (@ScottWalker) January 15, 2019

We need to help our students – whatever their political inclinations – believe fewer lies.

There are lots of interesting questions our students (and their teachers!) should ask about taxes and the kind of society they want to live in, questions like “What are the upsides and downsides of making Dwayne ‘The Rock’ Johnson pay 70% of the money he makes *above* 10 million dollars in taxes?” But we *cannot* answer those interesting questions if we don’t have a basic command of the facts of our tax system, if we allow our leaders to *deceive* us about how marginal tax rates work.

Stats teachers: you have the power to make your students smarter about taxes in a day than fully *half* of Americans have been in their entire lives.

- Take a few representative individuals from the world – real or imagined, but definitely including The Rock.
- Hypothesize their income. Just make up numbers across an income distribution.
- Use our tax brackets to calculate their tax bill.
- Create a new set of brackets that are more just.

At Desmos, we’ve been enormously enthusiastic about the partnership between the New York Times Learning Network and the American Statistical Association they call What’s Going on in this Graph?

In that partnership, the NYT team selects a newsworthy and timely graph from their award-winning output. The ASA team hosts a conversation on the NYT website about what students notice and wonder about those graphs.

For the rest of the school year, my team at Desmos will be helping teachers host those conversations *in their classrooms* from our Activity Builder platform. A new graph and activity every week. Give them a try.

There’s no time like 2019 to start all of this.

**Ben Orlin** offers a useful lesson plan here:

One of my favorite parts of teaching Precalculus was a “Design Your Own Income Tax System” project, which went something like:

-Make up your own system (minimum: 5 brackets);

-Write a paragraph justifying your decisions;

-Make up 5 individuals (+ their professions) and compute how much tax each would pay;

-Express MTR as a function of pre-tax income (and graph it);

-Express tax owed as a function of pre-tax income (and graph it). (By far the hardest step.)

An unhappy email:

While the information about Americans misunderstanding taxes did not surprise me, your choice to bring politics into the situation did. How do you propose we use your information about the tax knowledge of Democrats vs. Republicans with our students? Should we poll the students on their political affiliation and spend twice as much teaching taxes to our Republican students as we do our Democrat students?

The difference in tax knowledge between Democrats and Republicans along with the propensity of Republican leaders to lie to their constituents about marginal tax rates is meant to *heighten the importance of good instruction here*. Whether students grow into Republicans or Democrats is irrelevant to me (in this particular moment). If they or their parents pay attention to Republican leaders or media, they are getting lied to about this particular issue and we can do something about that.

Another unhappy email:

You’ve made this particular mail also about politics. Roughly: “Dem/good Repub/bad” Does that serve your core goal (statistics pedagogy)? Thanks for this free service. I hope to read the next one.

FWIW I’m more inclined to believe “people/bad” than “Democrats/good” “Republicans/bad.” But it certainly seems true that Republican leaders are willing to lie about this issue and that Republican voters are believing those lies. Teachers can and should subvert that relationship. Not necessarily converting Republicans to Democrats but making sure Republicans disbelieve their leaders when they lie.

Another:

This sort of proselytizing does nothing but polarize. You’re taking advantage of those who follow you for your insights with math education to preach your opinion about complex, controversial political topics that require nuanced thought and discussion to resolve. Have you ever changed your mind due to other people smuggling in their unsolicited opinions about politics like this? If you’re looking to change people’s minds about politics, do it through an open, honest platform where that is your sole purpose. I hope you stick to math without trying to fit in other agendas that distract from this purpose in the future.

I get that it’s a drag to see a) Republican leaders lying to their constituents, and b) their constituents believing those lies, but those aren’t “opinions” of mine. They’re the facts of the situation, and teachers need to work an overtime shift countering that deception. I hope that directly before or after you emailed me you emailed a complaint to the Republican leaders who are making your job harder.

Another:

When I was at university, we would have been required to present various viewpoints on an issue. Our professors would have required us to present pros and cons for each side. For examples, [dopey climate denier webpage] would give an alternative for our students to consider.

Thanks for the comment. Some issues are controversial enough that they deserve a hearing of multiple sides. Many other issues deserve no such hearing. The fact that “global climate change caused by human activities is occurring now, and it is a growing threat to society” is a settled issue by climatologists, our government, the military, and even for=profit insurance agencies. Now we need to decide what to do about it. By contrast, whenever I write my post declaring The West Wing the most overrated show of the 20th century, I will be sure to allude to opposing points of view.

Reader Bill Rider sent in his account of a lesson from his class:

]]>I stated that this prompt came from a political discussion that revealed a lack of understanding about marginal tax rates among fellow politicians and pundits. Our learning about this idea would make us better informed that many adults. (

This enhanced their attention.)In true Dan style, I didn’t want to script this too much. Together, we wrote the seven marginal tax rates on the board.

I asked them how much tax one would pay (Individual) on a salary of $9,525. (Upper end of the lowest bracket.)

I then asked how much

extratax one would pay if they received a $10,000 raise. This allowed the kids to learn that some of the money is taxed at one rate and the rest at another.We then considered someone who earns $520,000 a year. How much of that salary would be taxed at 37%?

In their groups, they collaborated to determine the total tax bill on $520,000.

Collecting each of the seven tax bracket computations allowed us to see things like: .32| 157,501 – 200,000| = tax for that portion of income

We determined that effective tax rate. Each class had at least one student who asked “why can’t we just make life easy and use that effective rate for everyone?” Rich discussions ensued.

We discussed the many who fear moving into the next tax bracket and why such a fear was unfounded.

We discussed a representative’s proposal to move towards a higher upper bracket of 70% , historical higher rates, the number of folks that might be affected and whether one would pay 70% on all income as a high earner.

I started to type up a worksheet

butit seemed to scripted/guided… and it didn’t come from their questions and their desire to be better informed than an adult.

I spent most of the fall in eighth grade classrooms, watching lots of teachers enact the same set of Desmos lessons in different ways and in different contexts and with different results.

Some classes were high energy, some were low energy.

Some classes seemed to learn a lot, others learned less.

There are *lots* of important explanations for those differences, of course, many of which have nothing to do with the teachers or students themselves. But it was also interesting to sit in some high energy, high learning classes and palpably feel that **these teachers are really, really curious about their students**. Curious about them personally, sure, but curious about their

Students *feel* that curiosity – “My teacher wants to know what I’m thinking about.” – and I find it easy to attribute some significant amount of those classes’ high energy and high learning to that feeling.

Teachers expressed that curiosity using the snapshotting tool when students recorded their thinking in Desmos. When students recorded their thinking on *paper*, **teachers expressed their curiosity with their cameraphones**, taking photos of student work and projecting them up on the board.

You see this on Twitter *all the time*! Curious teachers share diverse student thinking with other curious teachers.

Making students scratch their heads with our notice and wonder today. It sparked some great conversations! #exceptional #iteachmath pic.twitter.com/uOkcta4ASm

— Teri Beth Shearer (@tbethshearer) January 11, 2019

And that practice creates no fewer than twelve virtuous cycles, a few of which I can quickly describe:

- When teachers express curiosity about diverse student thinking, students
*feel*that and feel license to express even*more*diverse kinds of thinking. - The more perspectives on an idea a teacher can help students connect, the more students learn about that idea.
- That all feels great so the teacher becomes
*more*curious about student thinking and consequently re-evaluates her curriculum and instruction to emphasize tasks and pedagogy that are*more likely*to elicit diverse thinking. - The teacher becomes interested in learning
*more mathematics*because the more math you know, the more you’re able to identify and connect diverse student thinking when you see it.

Run that cycle for a few months and you have a different class.

Run that cycle for a few years and you have a different teacher.

Run that cycle across a *department* and you have a different *school*.

It starts with your cameraphone.

**BTW**. If your students’ diverse thinking currently fills you with more anxiety than curiosity, I encourage you “act your way into belief” instead of the reverse. Take two minutes at the end of class to share “My Favorite Whoa,” a photo of student thinking during the day you thought was *so* interesting and why you thought it was interesting. That’s low commitment with a lot of upside.

**BTW**. If you *already* use your cameraphone to express curiosity about student thinking, head to the comments and let us know *how you do that*. Your colleagues want to know your workflow.

**Featured Comments**

Daniel Peter uses whiteboards:

Need to be able to put up multiple solutions at the same time so the teacher can use questions to help students create explicit connects between the solutions: similarities/differences, aha (unique, elegant, just plain interesting) and help students make connects to the underlying properties, principles of mathematics. The advantage of paper/vertical whiteboards (or old school individual slates) is I can create the congress or bansho to make those connections explicit through the organization.

Several people use Reflector. Here’s Gretchen Muller:

It turns my phone into a portable document camera. Multiple devices can be shown at a time so I can do compare and contrast between different pieces of work at the same time. I now use it in my work with educators. The first question I always get is “How did you do that?”. I use it both as a live camera so that students can explain from their desk or still pictures from my phone and iPad when I want to compare.

Allison Krasnow describes students using *their* cameraphones to take pictures of student work:

]]>I received three texts (I use remind.com) this evening with students sending photos of their homework showing where they got confused and asking for help. Them texting me photos of their homework when they are stuck and at home with no one to help them is incredibly powerful.

I do agree with what is written, but I am still wondering what I’m supposed to do with that information and the student’s copy.

**T**: Oh this is so interesting! You’ve actually answered a *different* question correctly. Check this out:

**T**: How does that help you come up with an answer to the original question? Talk about it. I’ll be back.

That’s a *script* right there. It works for *any* incorrect answer. The script is all-purpose and all-weather but it has two challenging requirements:

- You have to actually believe that student ideas are interesting, especially ones that don’t correctly answer the question you were trying to ask.
- You have to identify the question the student answered correctly.

This is why I want to learn more math and more math and more math.

The more math I know, the more power I have not *just* to show off at parties but also to appreciate student ideas and to identify the different interesting questions they’re answering correctly.

Barbara Pearl, via email:

Can you write about it briefly again in a simpler way so I can try and understand it? When students make a mistake or answer something incorrectly, you want to …

I want to teach in a way that honors the specific student and also the general ways people learn.

So in any interaction with students, I need to a) understand the sense they’re making of mathematics, b) celebrate that sense, saying loudly “I *see you* making sense!”, and then c) help them *develop* that sense, connecting the question they answered correctly to a question they *haven’t yet* answered correctly.

Rachel, in the comments:

So, if we don’t call it a mistake, then what do we call it?

I don’t have any problem saying a student’s answer is incorrect, that they didn’t correctly answer the question I was trying to ask. But my favorite mathematical questions defy categories like “correct” and “incorrect” entirely:

- So how would you describe the pattern?
- What do you think will happen next?
- Would a table, equation, or graph be more useful to you here?
- How are you thinking about the question right now?
- What extra information do you think would be helpful?

How can you call *any* answer to those questions a mistake or incorrect? What would that even *mean*? Those descriptions feel inadequate next to the complexity of the mathematical ideas contained in those answers, which I interpret as a signal that I’m asking questions that *matter*.

**Featured Comment**

Denise Gaskins quoting WW Sawyer:

Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.

“So, if we don’t call it a mistake, then what do we call it?”

THINKING

]]>I find that I have to keep insisting that they restate the question in their own words. The culture of “right answer” is filled with shame and shaming, and students will try repeatedly to just give me the “correct” answer to the original question. But this is a missed opportunity for developing understanding, in my view.