Month: March 2016

Total 7 Posts

Ignore The Adjectives. Watch The Verbs.

Last spring, Mathematics Teacher published my paper on mathematical modeling. In this month’s issue, they’ve published a response from Albert Goetz [$].

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.

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

Real Work v. Real World

Featured Comment

Howard Phillips:

We shouldn’t overlook the usefulness of using this part of math to model that 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.

[3ACTS] Nissan Girl Scout Cookies

Treatment #1

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.

  1. Ask for questions.
  2. Ask for wrong answers.
  3. Ask for estimates.
  4. Ask for important information.
  5. Ask for estimates of the capacity of the trunk and the dimensions of the box of cookies.
  6. Show the answer.
  7. Ask for reasons why our mathematical answer differs from the actual answer.


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

Download the goods.

[via whoever runs the Bismarck Schools’ Twitter account]