tl;dr — “Real world” is tougher to measure than “interest” and less important overall. So rather than asking which of these three different versions of a word problem is more “real world” I asked a couple hundred people which is more interesting. Only the geometry treatment was significantly better than a coin flip at generating questions.

————————–

Here are some closing words about “real world” math, mostly distilled from your comments on the last post. As with previous investigations, I am indebted to the folks who stop by this blog to comment and make me smarter.

Real-World Math Is Hard To Define

What other conclusion can we draw from the dozen-or-so definitions of “real-world math” I found here and on Twitter?

• It depends on whether we’re talking about the world of procedures, concepts, or applications. [Karim Ani]
• A problem simulating a project / job / task being performed by someone performing their normal job duties, such as the example of the contractor building a pool to meet municipal code. [Ben Rimes]
• A problem involving objects or tasks that would be considered an experience students are likely to have in the “real world.” [Ben Rimes]
• If it came from a math teacher it’s basically not real-world, unless it was a math teacher doing something in the outside world leading to an interesting problem in mathematics. [Bowen Kerins]
• An observation / question that would be interesting to humans outside a math class. [Kate Nowak]
• Something is “real” to a student if it’s concrete, attainable, comprehensible. [Michael Pershan]
• Something is “real” to a student if it has a non-mathematical purpose. [Michael Pershan]

Real-World Math Doesn’t Guarantee Interest

David Taub argued the whole question confused “interest” with “real world.” M Ruppel listed other criteria for judging the value of a question, one of which was “kids want to solve it.”

Their arguments may seem obvious to you. They aren’t obvious to the three people who emailed me here or these presenters at the California STEM Symposium or Conrad Wolfram or the New York Times editorial board.

As Liz said, “Real world is just a means to an end. The goal is interest.” We should reject simple explanations of student interest.

So Which Version Is More Interesting?

Asking which of these three versions (candy, geometry, text) of the same math problem is more “real-world” was a pointless task since basically everyone has a different definition of “real-world” and it doesn’t guarantee interest anyway

So let’s ask about interest instead.

I used Mechanical Turk (and Evan Weinberg’s invaluable Internet skills) to show a random version of the problem to 99 people. I asked them if they had a question or not.

Of the three treatments, only the geometry treatment was statistically better than a coin flip at generating questions. (Here’s the experiment and the data.)

Then I showed another 80 people the same three treatments and asked how interested they were in the equal area question as measured on a Likert scale from -3 to 3, including 0. (This measured interest another way. Perhaps a question didn’t occur to you impulsively but once you heard it you were interested in it.)

Here again only the geometry treatment had an interest rating that was significantly different from “neutral.” (Experiment and data.)

Why the geometry treatment? I don’t know. It’s more abstract than the candy treatment, which features objects from outside the math classroom. 88% of the people I surveyed in the first experiment answered “within the last year” to the question “When did you last use math to solve a problem in life, work, or school?” That’s a math-friendly crowd. It’s possible that a class of elementary schoolers would find the candy treatment more interesting and that a coffee klatch of research mathematicians would tend towards the text treatment.

I don’t know. I’m just speculating here that real world is a pretty porous category. And for the sake of interesting your students in mathematics, it’s more important to know their world.

2014 Mar 26. Fawn Nguyen asked her eighth grade geometry students which version they preferred.

An ongoing question in this “fake world” series has been, “What is real anyway, man?”

Are hexagons less real-world to an eighth-grader than health insurance, for example? Certainly most eighth graders have spent more time thinking about hexagons than they have about health insurance. On the other hand, you’re more likely to encounter health insurance outside the walls of a classroom than inside them. Does that make health insurance more real?

I don’t know of anyone more qualified to answer these questions than our colleagues at Mathalicious who produce “real-world lessons” that are loved by educators I love.

I’m sure they can help me here. Here are three versions of the same question.

Version A

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle.

Find P such that the area of the square and circle are equal.

Version B

When do the circle and the square have equal area?

Version C

Where do the circle and square have the same number of candies?

Version D

[suggested by commenter Jeff P]

You and your friend will get candy but only if you find the spot where there’s the same number of candies in the square and the circle. Where should you cut the line?

Version E

[suggested by commenter Mr. Ixta]

Imagine you were a contractor and your building swimming pools for a hotel. Given that you only have a certain amount of area to work with, your client has asked you to build one square and one round swimming pool and in order to make the pools as large as possible (without violating certain municipal codes or whatever), you need to determine how these two swimming pools can have the same area.

Version F

[suggested by commenter Emily]

Farmer John has AB length of fencing and wants to create two pens for his animals, but is unsure if he wants to make them circular or square. To test relative dimensions (and have his farmhands compare the benefits of each), he cuts the fencing at point P such that the area of the circle = area of the square, and AP is the perimeter of the square pen and PB is the circumference of the circular pen.

Dear Mathalicious

Which of these is a “real world” math problem? Or is none of them a real-world math problem?

If anybody else has a strong conviction either way, you’re welcome to chip in also, of course.

Featured Mathalicians:

• Kate Nowak says “real world” means “interesting to humans outside a math class.”
• Ginny Stuckey asks if I’m trolling.
• Karim Ani is on vacay and promises a response when he gets back to business. [2013 Mar 22: He delivers.]
• Matt Lane asks, instead, if the task imbues people with joy for mathematics.
• Chris Lusto asks, “Is the question self-referential?”

Featured Comments

David Taub:

These are the two ideas that seem to be confused here are just “real world” and “interesting”. There seems to be an inherent assumption that they are somehow related when I doubt they even should be. Sometimes they will overlap, and sometimes not, but it is a bit random and quite personal in my opinion.

Kevin Polke:

I caught the maintenance staff at my high school using “real-world math.”

Ben Rimes:

The problem probably lies in the many ways that one could define “real world”.

What qualifies as “real world”?

• A problem simulating a project/job/task being performed by someone performing their normal job duties, such as the example of the contractor building a pool to meet municipal code?
• Is a problem involving objects or tasks that would be considered an experience students are likely to have “real world”(dividing up the M&Ms to be equal size)?
• Actual evidence of math in the “real world “(video or otherwise) being applied as a part of someone’s job (the maintenance staff example above from @Kevin Polke) could qualify, or perhaps the application of math to a problem that is foreseeable in that person’s job duties.

I think it’s best to take the more pluralistic viewpoint on this one, as it would be quite a task to attempt to statically define the exact nature of “real world” math, as there are always countless more examples lining up to disprove whatever narrow definition you choose.

Liz:

“Real world” is just a means to an end. The goal is interest.

M Ruppel:

What matters for a ‘good task’ is not whether it’s real, it’s whether
1) its meaning is clear right away
2) kids want to solve it
3) have the mathematical tools to solve it (even if not very sophisticated tools)

Bowen Kerins:

I’ll say again though: I don’t really care if a problem is real-world. There are so many great problems that aren’t, and so many terrible problems that are. I don’t think it carries huge added value. Everyone decides what’s “real” to them (as Jeff P said). Right now for my kid, 7-5 and 5-7 being related to one another is plenty real, even though there is no connection yet to physical objects or money or any of that.

Featured Tweets

@ddmeyer Intriguing matters to my students -Wonder if 'real world' imperative comes from assumption math is horrible http://t.co/WkgCG7lAmr

— Cathy Bruce (@drcathybruce) March 11, 2014

@ddmeyer a missing aspect in the "Real-world math debate? "Nothing ever becomes real 'til it is experienced." – John Keats

— Micah Hoyt (@MicahHoyt) March 11, 2014

2014 Mar 26. Fawn Nguyen asked her eighth grade geometry students which version they preferred.

a/k/a Worshipping The Real World

Here are three e-mails I received from three different people over the last three months. Spot the common theme.

November:

My co-teacher and I were puzzling over what kind of problem would create an intellectual need for systems. Do you have anything you could send, by chance?

December:

We are planning to launch a unit on systems of equations in early January (after December break) and wanted to try out your approach to create an intellectual “need.”

January:

Showing two straight lines on a piece of graph paper and finding points of intersection has very little significance to most people. I’m looking for a real-world problem that has an answer that is not self-evident, but which requires a little thinking and finding the intersections and is infinitely more productive and satisfying and will stay with them for the rest of their lives. That is what I am looking for.

I receive these questions on Twitter also. I find them almost impossible to answer because I don’t know what your class worships.

Here’s what I’m talking about:

Class #1

You start class by asking your students to write down two numbers that add to ten. They do. Most likely a bunch of positive integers result.

Then you ask them to write down two numbers that subtract and get ten. They do.

Then you ask them to write down two numbers that do both at the exact. same. time. “Is that even possible?” you ask.

Many of them think that’s totally impossible. You can’t take the same two numbers and get the same output with two operations that are natural enemies of each other. They’d maybe never phrase it that way but the whole setup seems totally screwy and counterintuitive.

Then someone finds the pair and it seems obvious in hindsight to most students. We’ve been puzzled and now unpuzzled. Then you ask, “Is that the only pair that works?” knowing full well it is, and the class is puzzled again.

You define systems of equations as “finding numbers that make statements true” and you spend the next week on statements that have only one solution, that have infinite solutions, and the disagreeable sort that don’t have any solutions.

Students learn to identify the kind of scenario they’re looking at and how to find its solutions quickly (if any exist) using strong new tools you offer them over the unit.

Class #2

The same lesson plays out but this time, after we’ve determined the pair of numbers that solve the system, a student pipes up and asks, “When will we ever use this in the real world.”

Worshipping the Real World

David Foster Wallace wrote about worship — the secular kind, the kind that applies to everybody, not just the devout, the kind that applies especially to us teachers in here:

If you worship money and things – if they are where you tap real meaning in life – then you will never have enough. Never feel you have enough.

If your students worship grades, they won’t complete assignments without knowing how many points it’s worth. If they worship stickers and candy, they won’t work without the promise of those prizes.

If you say a prayer to the “real world” every time you sit down to plan your math lessons, you and your students will never have enough real world, never feel you have enough connection to jobs and solar panels and trains leaving Chicago and things made of stuff.

If you instead say a prayer to the electric sensation of being puzzled and the catharsis that comes from being unpuzzled, you will never get enough of being puzzled and unpuzzled.

The first prayer limits me. The first prayer means my students will only be interested in something like The Slow Forty — a real world application of systems. The second prayer means my students will be interested in The Slow Forty (because it’s puzzling) but also the puzzling moments that arise when we throw numbers, symbols, and shapes against each other in interesting ways.

The second prayer expands me. Interested people grow more interested. Silvia writes, “Interest is self-propelling. It motivates people to learn thereby giving them the knowledge needed to be interested” (2008, p. 59). Once you give your students the experience of becoming puzzled and unpuzzled by numbers, shapes, and variables, they’re more likely to be puzzled by numbers, shapes, and variables later. That’s fortunate! Because some territories in mathematics are populated exclusively by numbers, shapes, and variables, in which cases your first prayer will be in vain.

That’s why I can’t tell you what to teach on Monday. Your classroom culture will beat any curriculum I can recommend. I need to know what you and your students worship first.

BTW

• Review assignment: Which prescriptions from our earlier review of curiosity research are evident in Class #1 above?
• Michael Pershan draws a similar distinction between pedagogy and curriculum.

References

Silvia, PJ. (2008). Interest – the curious emotion. Current Directions in Psychological Science, 17(1), 57—60. doi:10.1111/j.1467-8721.2008.00548.x