First, John Rowe shows his students this image as a lead-in to counting problems, asking them, “Which state has more registered vehicles and how do you know?”

(Here’s an alternate version of that image that allows students to wonder “Where do you think these license plates are *from* and how do you know?” Because you can always add, but you can’t subtract.)

Second, Jenn Vadnais creates a stop-motion animation that does exactly what it wants to do, which is compose a cylinder into a sphere and then decompose the sphere into doughy little cubes. How close will math take you to the actual answer?

Go check out their posts, and keep up the great work, everybody.

]]>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.

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**

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.

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.

]]>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**:

]]>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.

]]>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**

]]>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!)

Yet on the level of thinking of our Silicon Valley overlords, aspects of my cognitive abilities that are absolutely central to my educational success are taken to have literally no value at all. In educational research, perhaps the greatest danger lies in thinking “that which I cannot measure is not real.” The disruption fetishists have amplified this danger, now evincing the attitude “teaching that cannot be said to lead to the immediate acquisition of rote, mechanical skills has no value.” But absolutely every aspect of my educational journey — as a student, as a teacher, and as a researcher — demonstrates the folly of this approach to learning.

I’ve said it many times, though people never seem to think I’m serious:

years studying literary analysis, now widely assumed to be a pointless and wasteful activity, have helped me immensely in acquiring the quantitative, monetizable skills that ed reformers say they want.

I applied to film school out of high school and spent a large fraction of my university math education reading screenplays and writing about movies. The coffin eventually closed on those aspirations, but my interest in narrative and storytelling has permeated *every* aspect of my teaching, research, and current work in education technology.

Freddie deBoer’s argument, both as I read it and experience it, isn’t that a liberal arts education makes a productive life in STEM *whole*. It’s that a liberal arts education makes a productive life in STEM *possible*.

**Featured Tweet**

]]>@ddmeyer https://t.co/tiZszn1l3Z "A liberal arts education makes a productive life in STEM possible." My experience as well. #MusicMajor

— David Wiley, PhD (@opencontent) March 11, 2016

Goetz worries that our collective interest in mathematical modeling risks granting the premise of the question, “When will we use this?” Math doesn’t *have* to be useful, argues Goetz. It’s beautiful on its own terms.

An emphasis on modeling—seeing mathematics as a tool to help us understand the real world—needs to be tempered by an approach that gives some prominence to the beauty that abounds in our subject. I want my students to understand how mathematics can explain the world—there is beauty in that notion itself—but also to see the inherent beauty and magic that is mathematics.

Agreed. But I no longer find *adjectives* helpful in planning classroom experiences, whether the adjective is “beautiful” or “useful,” “real” or “fake,” each of which is only in the eye of the beholder. Instead I focus on the *verbs*.

Mathematical modeling comprises a *huge* set of verbs that range from the very informal (noticing, questioning, estimating, comparing, describing the solution space, thinking about useful information, etc.) to the very formal (recalling, calculating, solving, validating, generalizing, etc.). One of the most productive realizations I’ve ever had in this job is that *all of those verbs are always available to us, whether we’re in the real world or the math world*.

**Existence Proofs**

“Math world” is the only adjective you could use to describe these experiences. When students find them interesting it’s because the verbs are varied and run the entire field from informal to formal.

- Which One Doesn’t Belong?. Students notice, name, argue, etc.
- Malcolm Swan’s Area v. Perimeter Problem. Students choose their own rectangle, draw it, graph it, imagine other solutions, imagine
*impossible*solutions, construct an argument about them, generalize to other shapes, etc. - Polygraph. Students notice, name, communicate, eliminate, etc.

Trick your brain into ignoring adjectives like “real-world” and “math-world.” Those adjectives may not be *completely* meaningless, but they’re close, and they mean so much less than the mental work your students *do* in those worlds. Focus on those verbs instead.

**Related Reading**

**Featured Comment**

]]>We shouldn’t overlook the usefulness of using

this part of mathto modelthat part of math. I see calculus as a way of describing and analyzing curves, including their curvature. I see analytical geometry as a way of representing “pure” geometry. I even see algebra as a way of modeling numerical patterns. Modeling is not just about the real world.

A small rectangular prism measures 7 inches x 2.3 inches x 4.6 inches. How many times could it fit in a larger rectangular prism with a volume of 39.3 cubic feet?

**Treatment #2**

Nissan is going to stuff the trunk of a Nissan Rogue full of boxes of Girl Scout cookies. Nissan lists the Rogue’s trunk space as 39.3 cubic feet. A box of cookies measures 7 inches x 2.3 inches x 4.6 inches. How many boxes will they fit in the trunk?

**Treatment #3**

Show this video.

- Ask for questions.
- Ask for wrong answers.
- Ask for estimates.
- Ask for important information.
- Ask for estimates of the capacity of the trunk and the dimensions of the box of cookies.
- Show the answer.
- Ask for reasons why our mathematical answer differs from the actual answer.

**Hypothesis**

Treatment #1 and Treatment #2 are as different from each other as Treatment #2 is from Treatment #3.

A layperson might claim that Treatment #2 has made Treatment #1 real world and relevant to student interests. But the real prize is Treatment #3, which doesn’t just add the world, but changes the *work* students do in that world, emphasizing formal *and* informal mathematisation.

“Real world” guarantees us very little if the work isn’t real also.

**Design Notes**

You can check out the original Act One and Act Three from Nissan.

I deleted this screen from Act One because I wanted students to think about the information that might be useful and to *estimate* that information. I can always *add* this information, but I can’t *subtract* it.

I added a ticker to the end of the video because that’s my house style.

I deleted a bunch of marketing copy because it was kind of corny and because it broke the flow of their awesome stop motion video.

I left the fine-print advisory that you should “never block your view while driving” because the youth are impressionable.

**The Goods**

[via whoever runs the Bismarck Schools’ Twitter account]

]]>Beginning in the 1960s psychologists began to find that delaying feedback could improve learning. An early lab experiment involved 3rd graders performing a task we can all remember doing: memorizing state capitols. The students were shown a state, and two possible capitols. One group was given feedback immediately after answering; the other group after a 10 second delay. When all students were tested a week later,

those who received delayed feedback had the highest scores.

Will Thalheimer has a useful review of the literature, beginning on page 14. One might object that whether immediate or delayed feedback is more effective turns on the goals of the study and the design of the experiment.

To which I’d respond, yes, exactly!

Feedback is complicated, but to hear 99% of edtech companies talk, it’s *simple*. To them, the virtues of immediate feedback are received wisdom. The more immediate the better! Make the feedback immediater!

Dan’s Corollary to Begle’s Second Law applies. If someone says it’s simple, they’re selling you something.

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