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.

Modeling

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.)

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.

Christopher Danielson:

Your quest for the perfect image that will get 100% of viewers on board with the same mathematical question may be a bit quixotic …

Avery Pickford:

In [my ideal] world, I imagine spending a greater amount of time talking about the aesthetics of what makes for an interesting math problem and much less time cajoling students to ask the “right” question.

John Scammell

It’s unfortunate that we are so curriculum driven that we have to trick them into inventing the question we want them to come up with.

Here’s the thing: nobody watches Jaws and feels cajoled into wondering the question, “Won’t anybody stop that shark?!” No one watches Star Wars and feels tricked into wondering, “Will the rebels defeat the Galactic Empire?!” Those questions are irresistible, not on account of any deception on the part of the cast or crew, but because the cast and crew evoked the central conflict of their story skillfully.

This isn’t to say those questions are irresistible to everybody. Some people lack the cultural prerequisites to care about Star Wars. Some people possess the prerequisites and simply don’t care. Not everyone is interested in every movie, however skillfully it creates a narrative.

The point of the #anyqs challenge is to evoke a perplexing situation so skillfully that the majority of your students will wonder the same question (whatever that is) and the rest of the class won’t find that question unnatural or uninteresting, even if it wasn’t the first question that struck them.

I recognize that I am probably not going to persuade you (or most of your readers) on this point. But these kinds of strategies have never been tried before in a math software package, and if they do work, then the developers stand to make a lot of money, and it could help a lot of students. I can’t be sure how effective these strategies would be until they’re tried, but I have a lot of reasons (which I tried to explain in the article) to think that they have the potential to make a big difference. That’s why I’d like to see a publisher or software company invest a few million dollars to produce a really high-quality software product.

I respond in the comments.

The one nagging question I continue to have about WCYDWT is…what exactly does it accomplish?

Yes, it appeals tremendously to our intuition. Students are looking at the world and designing questions about curiosities. This certainly appears to be the kind of skill we want to teach. But research is at best divided about the kinds of gains similar projects have had in the past.

Dan has pointed out that his students out-performed his entire department, that they showed real and true gains according to every measure we currently use.

But still I wonder. How much of those gains were attributable to the actual WCYDWT lessons? How much of it was attributable to his skill and highly developed craftsmanship as a manager, questioner, evaluator? WCYDWT doesn’t strike me as terribly efficient. And I know that’s not the point, but where is the research showing that these kinds of problems are effective?

I think Dan also mentioned that he would do these kind of lessons bi-weekly (about 1 in 10 days). To which I say, fair enough. He was efficient and skilled enough in those other nine days to experiment with something like this.

I don’t know, though. These problems appear to make students conceptually flexible, which is brilliant. To make them procedurally flexible- arguably more important and more difficult to teach- is probably what Dan was doing the other 90% of the time.

I know the focus of this blog is WCYDWT and debunking conventional textbook wisdom. But when there’s no coke or sprite, escalators, three-pointers, or cheese, how do you do the other stuff?

I can certainly accept working definitions that require clarification, but the Car Talk problem confuses the issue [of pseudocontext] (at least for me). I’ve only done a little tweaking to the Car Talk problem:

“The fuel gauge of an 18-wheeler is broken, so the driver decides to check the gas level of his cylindrical gas tank with a dipstick. When the level of the gas measures 20 inches high, the tank is completely full. What will the dipstick measurement be when the gas tank is one-quarter full?”

Based on the working definition of pseudocontext, this problem fails on both counts. It completely ignores reality: Why wouldn’t you just fix the gas gauge? Then the problem asks for an irrelevant measurement: Why would we need to know that the tank is one-quarter full?

I assumed the trucker wanted to know one quarter of a tank (rather than four fifths) so he’d know when he had to refuel. An arbitrary number maybe (why not one fifth of a tank?) but not irrelevant. As for ignoring reality, I know more about mid-century Russian architecture than I do about long-haul trucking, but it seemed plausible to me that the trucker couldn’t waste time fixing the gauge in the middle of a run. In both of these cases, I deferred to the authority of the radio hosts. If either of Mike’s complaints were valid, why wouldn’t the hosts have echoed them?

Mike has also misquoted the definition of pseudocontext in small but crucial ways.

Mike’s: “It completely ignores reality.”
Mine: “Context that is flatly untrue.”

Mike’s: “Problems that ask for an irrelevant measurement.”
Mine: “Operations that have nothing to do with the given context.”

Mike:

Personally, I find the Car Talk problem kind of boring and not very mathematically rich.

Once again we find that a problem’s basis in either pseudocontext or context has nothing to do with how much anyone enjoys or profits from it. (Seems only to fair to mention, though, that Alex’s class had the opposite reaction.)

Mike:

Another word of caution: Mathematics is part utility and part artistry. By limiting mathematical study to problems related to genuine physical phenomena can only serve to retard the growth of mathematics.

It’s worth clarifying my total agreement here. My blog covers math applications pretty much exclusively not because I think those are the only problems worth studying but because those problems are the easiest to create and teach poorly.