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 hereâ€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:

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 one idea, but can I get some other ideas, 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

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.)

Prepared Remarks

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.

Responses to Questions and Criticism

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.

Some of the other online modelling resources, such as Dan Meyer’s blog, don’t really fit what I would class as meaningful modelling, and can feel contrived, or of little relevance/import to students’ lives; if I am going to spend the time bringing modelling situations to my classroom, I want to address matters of importance, socially or politically.

Geoff Wake:

Yes, I’m interested see how Dan Meyer promotes a sort of pseudo-modelling that seems to be quite popular among certain teachers. I think one aspect that appeals is that he suggests a narrative that is immediately accessible. On the other hand some of the questions are not particularly meaningfully tackled using mathematics seriously.

I see two tacit questions.

One, should math be important?

And by “important,” I’m using Danny’s definition: relevant to a student’s life, either socially or politically.

See, there isn’t any one agreed-upon definition of “mathematics.” They’re all arbitrary, personal, and cultural. And given finite hours in a school year to spend learning math, they’re all political. They create winners and losers. Class time spent how you’d prefer is time not spent how someone else prefers.

So I help students learn math for one reason alone, and it doesn’t have to be your reason also. I want to help students learn to puzzle and unpuzzle themselves. Math offers us the opportunity not just to solve puzzles, but to generate them from scratch — just you and your brain and maybe something to write with.

Those puzzles may have sociopolitical importance, but that’s a higher standard than I choose to set for myself. So it’d make more sense for Geoff and Danny to criticize my standard than to assume I’m aiming at theirs and missing. I’m not.

Two, should modeling be important?

I suspect Danny, Geoff, and I would agree more about the point of mathematics than the point of modeling. Their criticisms specifically concern modeling, and the fact that I ask questions like “How many pennies are in the pyramid?” and “How long will it take the water tank to fill?” rather than questions like (I’m guessing here) “Is capital punishment sentencing just or unjust?” or “How should California manage its water supply?”

But there is much more consensus around the definition of “modeling” than “mathematics,” and that definition doesn’t specify culture, context, or importance. Modeling is mental work, work of a certain character, work that I think we’d all agree is uncommon in many classrooms and unfamiliar to many students.

Modeling asks questions about a context. It works to make those questions more precise and tractable. It nourishes those questions with data where none exists. It sets reasonable bounds on an answer before finding a solution. It solves questions mathematically and then tests those answers against the world’s answer.

Basically, “modeling” is a verb and it doesn’t help our understanding of the verb to attach it a priori to adjectives (like “important” and “relevant”) or to nouns (like “capital punishment” and “water supply”). If you want to understand modeling, ignore the adjectives and the nouns. Watch the verbs.

Featured Comments

Chris Evans:

Additionally, we have to remember (as math teachers) that we are not the only teachers and courses these students encounter. I teach mostly 11th and 12th graders, and they frequently tell me about the political conversations they are having in government class or the serious social topic they are writing about in English. I have observed that, although students seem to appreciate these connections to real-world problems, these topics are heavy, and at times students appreciate engaging in “lighter” application problems and activities.

Nick Hershman:

Except that when you watch students engaging with a task that they are motivated to understand they are doing all sorts of things that relate to the “way they view their place as a member of society“. I can’t imagine a situation in which a student isn’t both learning something about their place in society and simultaneously asserting some version of their belief about their place in society. It’s happening all the time.

John Mason:

So working on socially relevant issues is valuable. But ‘relevance to me’ means, ‘real to me’, and as the RMP project has shown, well as it has confirmed what has been known for ever such a long time, what can be real to someone has to do with what they can imagine, can grow to imagine, and is not confined to what they already do every day.

In a cooking mag: a 9×2 round pan holds the same as an 8×2 square pan. True? Test with math? Test with water? pic.twitter.com/LDfDheoQyZ

— Marilyn Burns (@mburnsmath) March 30, 2016

On Twitter, I remarked that Marilyn had summarized the entire modeling cycle in a single tweet. But the part of that cycle she summarizes best is the last: validating your answer.

With mathematical modeling, you don’t have to be the answer key. The world is.

If you have total faith in the perfect accuracy of your mathematical models, testing with water may sound unnecessary. For the 99% of your students who wonder if math has any power outside of their textbooks, test with water.

Featured Comment:

Bryan Anderson:

It was one thing to manually figure that out [if some glue could hold up a human], and then another to try the same thing with a bowling ball experiment modeling the same thing. We were able to see if our answer actually held up in that situation, it was a moment that will stay with me forever.