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Danny Brown:

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.

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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.

Test With Water

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.

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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.


There are lots of great reasons to use this task from NCTM’s Illuminations site, which asks students to derive an algebraic function from a problem situation. But one of those reasons isn’t “to show students why they should derive algebraic functions.”

It’s a real world problem, by most definitions of the term. But let’s not let that fact satisfy us. It’s possible for math to be real-world, but also unnecessary. For example, I can ask students to use trigonometry to calculate the height of a file cabinet. But that math isn’t necessary when a measuring tape would suffice.

The same is true here. I can find Stages 1 through 5 by multiplying by three successively. So why did we invent algebraic representations? Life would be so much easier for both the student and the teacher if we relaxed that condition.

But if we added the question, “How long would it take the entire world to experience a good deed?” we will have both a) identified the need for algebraic functions – to calculate outputs given any input, even distant inputs – and b) put students in a position to experience that need.


That’s a two-step process. With the line, “Describe a function that would model the Pay It Forward process at any stage,” the author satisfies the first step. He understands the value of algebraic functions, himself. But without our added question, that’s privileged knowledge and we’re hoping students infer it. Instead, let’s put them directly in the path of that knowledge.

Real world, and also necessary.


Six years ago, I released a lesson called Will It Hit The Hoop? that broke the math education Internet. (Not a big brag. It was a much smaller Internet back then.)

I think the core concept still works. First, students predict whether or not a shot goes in the hoop based on an image and intuition alone. Then they analyze the shot using quadratic modeling and update their prediction. Then they see the answer. For most students, quadratic modeling beats their intuition.


The technology was a chore, though. Teachers had to juggle two dozen different files and distribute some of them to students. I remember loading seven Geogebra files onto student laptops using a thumb drive. That was 2010, a more innocent time.


So here’s a version I made for the Desmos Activity Builder which you’re welcome to use. It preserves the core concept and streamlines the technology. All students need is a browser and a class code.

Six year older and maybe a couple of years wiser, I decided to add a new element. I wanted students to understand that linears are a powerful model but that power has limits. I wanted students to understand that the context dictates the model.

So I now ask students to model this data with a linear equation.


Then I show students where the data came from and ask them to describe the implications of their linear model. (A: Their linear ball goes onwards and upwards forever.)


And then we introduce parabolas.

Gas Station Ripoff

Here are three gas station pumps. Which ones are trying to rip you off? Can you tell just by looking?

After your students have that debate and share their reasons (expected: “the third is a ripoff because it’s moving faster”) invite your students to collect data for each pump and enter it at Desmos. Here we’re establishing a need for a graphical representation. It may reveal patterns that our eyes can’t detect.


The third act helps clarify the underlying trends. The third pump is spinning faster, but the price and the gas still exist in a proportional relationship. The first pump, meanwhile, pumps less gas per dollar the longer it runs.

I am indebted to William G. McGowan and Sean Berg, whose NCTM 2016 session description included the words “gas pumps have been hacked,” and there went my weekend.

Their description reminded me how important it is to expose students to counter-examples of the relationships they’re studying, protecting against over-generalization. (ie. “Everything is proportional. That’s the chapter we’re in!”) I’m becoming fascinated, in general, by problems that ask students to prove that a mathematical model is broken rather than just apply a model that works.

[Download the goods.]

Featured Comment

Scott Farrar:

I’ve written before about expanding teaching to the “neighborhood” of the special case. If we always show the highlight reel, students never get appreciation for how special and how powerful the ideas are. So I like that this lesson is about finding the non-proportional “ripoff” as it stands out in contrast to the “normal/expected” proportional relationships. (Ironically I would have said that proportionality is the special case, and nonproportionality is the ‘normal’– before thinking about what we expect as consumers. ‘normal’ is all subjective!)

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