There were two particularly useful comments in response to this problem:

gasstationwithoutpumps:

The moment of inertia for rotating a I-beam about its long axis has no practical relevance in structural engineering. This is a fake-world problem, of no interest either mathematically or to engineers.

Even if this task did have practical interest for structural engineers, its presentation here will move the needle on student engagement only a fraction of a degree. The issue here isn’t the usefulness of the application to professionals but the tedious, pre-determined work students do.

Jane Taylor:

When I saw the two boards, I wanted to go get a board and try standing on it. How much weight could we put on the board in each position before it broke? That would be an engaging problem.

I don’t know. That might be an engaging problem.

There are 100 different directions that question can go in terms of the work students do in class and only a handful of them will actual leave kids mathematically powerful and capable.

Watch me ruin the problem:

The maximum load a board can hold before it snaps is given by the formula:

[formula involving cross-sectional area and mass]

Dan weighs 90 kilograms and the dimensions of the board are 2 inches by 4 inches by 70 inches. Will the board hold his weight?

I have no confidence this task will result in the sense of accomplishment and connection the editors of the NYT seem to think it will.

There are other ways to present this kind of task, though. Which is my point. The “real world”-ness or “job world”-ness of the task is one of its least important features.

In case this whole series seemed to you like a bit of a straw man (it did to Kate Nowak and Michael Pershan) here’s the New York Times Editorial Board:

A growing number of schools are helping students embrace STEM courses by linking them to potential employers and careers, taking math and science out of textbooks and into their lives. The high school in Brooklyn known as P-Tech, which President Obama recently visited, is a collaboration of the New York City public school system and the City University of New York with IBM. It prepares students for jobs like manufacturing technician and software specialist.

[..]

Though many of these efforts remain untested, they center around a practical and achievable goal: getting students excited about science and mathematics, the first step to improving their performance and helping them discover a career.

Pick any application of math to the job world and I promise you I can come up with 50 math problems about that application that students will hate. Get a little coffee in me and I’ll crank out 49 more. It’s that one problem, the one out of 100 that students might enjoy, that’s really tricky to create, and often times its “real world”-ness is its least important aspect.

Chris Hunter reminds me (via email) that the British Columbia Institute of Technology has made a similar bet on “real-world” math. Here’s an example:

Once again, we’re asking students to substitute given information for given variables and evaluate them in a given formula. Does anyone want to make the case that our unengaged students will find the nod to structural engineering persuasive?

The “real world” isn’t a guarantee of student engagement. Place your bet, instead, on cultivating a student’s capacity to puzzle and unpuzzle herself. Whether she ends up a poet or a software engineer (and who knows, really) she’ll be well-served by that capacity as an adult and engaged in its pursuit as a child.

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Chris Hartmann points out that these application of math to jobs often miss the math that’s most relevant to those jobs:

And, in the job world a lot of the mathematics isn’t done by human minds or hands anymore, with good reason. Faster, more accurate means are available using technology. What often remains is puzzling out the results.

Mr. K:

The telling thing is that the Times’s example of a real world problem that real world people can’t solve, that of calculating the cost of a carpet for a room, is pretty much a guaranteed loser for any math class that I have ever taught at any level.

On the other hand, yesterday I had a room full of third round algebra students engrossed in building rectangles with algebra tiles. That’s about as non real world as it gets.

gasstationwithoutpumps:

The moment of inertia for rotating a I-beam about its long axis has no practical relevance in structural engineering. This is a fake-world problem, of no interest either mathematically or to engineers.

There are real-world applications for moment of inertia problems, but this is not one of them.

nerdypoo:

This seems to be a perennial favorite. In 2011 the Times asked if we needed a new way to teach math, with this quote:

“A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. ”

I’m certain I could find an example of such an article from every few years …

Let’s just call them “theories of engagement” for now. Every teacher has them, these generalized ideas about what engages students in challenging mathematics. Here’s the theory of engagement I’m trying to pick on in this series:

This theory says, “For math to be engaging, it needs to be real. The fake stuff isn’t engaging. The real stuff is.” This theory argues that the engagingness of the task is directly related to its realness.

This is a limited, incomplete theory of engagement. There are loads of “real” tasks that students find boring. (You can find them in your textbook under the heading “Applications.”) There are loads of “fake” tasks that students enjoy. For instance:

• Tetris.
• Tic Tac Toe.
• Sudoko.
• Shikaku.

No context whatsoever in any of them. Perhaps the relationship actually looks more like this:

I’m being a little glib here but not a lot. Seriously, none of those tasks are “real-world” in the sense that we commonly use the term and yet they captivate people of all ages all around the world. Why? According to this theory of engagement, that shouldn’t happen.

Here are fake-world math tasks that students enjoy:

• Ihor Charischak’s Jinx Puzzle.
• David Masunaga’s Magic Octagon.
• Malcolm Swan’s Area v. Perimeter.
• NRICH’s Factors and Multiples Puzzle. (Megan Schmidt: “… I have one student in particular who is not particularly motivated by much … when I bust out a puzzle, he’s all in.”)
• Matt Vaudrey’s Magical Triangle Theorem.
• Andrew Stadel’s Weekly Puzzle.
• The meaning of the sequence 3, 3, 5, 4, 4, 3, … , which drove kids bananas the day I wrote it on the board at the end of a test.
• The proof that 2 = 1.
• 2013 Dec 3. The four fours puzzle.
• 2013 Dec 5. Timon Piccini’s Broken Calculator.
• 2013 Dec 10. Patrick Vennebush sends along “How much is your name worth?”

My point is that your theory of engagement might be limiting you. It might be leading you towards boring real-world tasks and away from engaging fake-world tasks.

We need a stronger theory of engagement than “real = fun / fake = boring.”

Homework Time!

Choose one:

• Write about a fake-world math task you personally enjoy. What makes it enjoyable for you? What can we learn from it?
• Write about an element that seems common to those enjoyable fake-world tasks above.

Hans Freudenthal changed the conversation from “real world” to “realistic world“:

The fantasy world of fairy tales and even the formal world of mathematics can provide suitable contexts for a problem, as long as they are real in the student’s mind.

This complicates our task. It’s easy to create real world tasks that aren’t real in the student’s mind. It’s harder to create realistic tasks.

Here’s one way to test if the context is “real in the student’s mind”:

Can they construct an argument about it?

From Jennifer Branch’s presentation handout at CMC-South [pdf], I’ve pulled a series of questions she calls “Eliminate It!”

None of these are “real” in the sense that most of us mean the word. But each of these groups is “real” to different students. Triangles are real. Pentagons are real. Diameters are real. We know they’re real because those students can construct an argument about which one doesn’t belong. That ability to argue proves their realness.

(Of course, the value of the task is that different arguments can be made for each member of the group.)

On the other hand, consider:

These elements are definitely “real.” They’re metals. But are they realistic? Are they real in your mind? Can you construct an argument about their substance?

If not, how is it in our best interests to promote a definition of “real” that admits “magnesium” but denies “pentagons”?

2013 Nov 26. Similarly, it’s “real” if they can sort it meaningfully.