“Real-World” Math Is Everywhere or It’s Nowhere

Amare is looking at these 16 parabolas. Her partner Geoff has chosen one and she has to figure out which one by asking yes-or-no questions.

all 16 of the parabolas

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 responding "no" to "is it like a hill?"

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

now only 9 parabolas left

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.

"oh no! you eliminated your partner's parabola!"

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):

the modeling cycle from GAIMME

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.”

All Learning Is Modeling: My Five-Minute Talk at #CIME2019 That Made Things Weird

I contributed to a panel on mathematical modeling panel at MSRI this week – five minutes of prepared remarks and then answers to a couple of questions.

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.

Stats Teachers: 2019 Is Your Year

Folks, if you teach stats in middle school or high school, 2019 is your time to shine.

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.

Marginal Tax Rates

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:

  1. My tax bill would go up a very small amount.
  2. 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.

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.

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

What’s Going on in This Graph?

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.

Featured Comments & Emails

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

The Best App for Your Teaching is Already on Your Smartphone

tl;dr – It’s the camera. And using it thoughtfully can change your teaching in substantial ways.

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 thinking in particular.

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.

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

  1. When teachers express curiosity about diverse student thinking, students feel that and feel license to express even more diverse kinds of thinking.
  2. The more perspectives on an idea a teacher can help students connect, the more students learn about that idea.
  3. 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.
  4. 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.

[Mailbag] What Do You Do with the Ideas You Used to Call “Mistakes”

Guillaume Paré, in the very interesting comments of my last post where I urged us to reconsider mistakes:

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:

  1. You have to actually believe that student ideas are interesting, especially ones that don’t correctly answer the question you were trying to ask.
  2. 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.

Daniel Peter:

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

THINKING

cheesemonkeysf:

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.