Is this a function?

This is clearly a function. Functions are patterns.

But there's more. This is a function:

This isn't:

Functions are like the top graph. Non-functions are like the bottom graph.

What's the difference between the top graph and the bottom graph? One difference, for sure, is that the bottom graph has loops. But why do the loops matter? Can you tell a story that matches the top graph? Can you tell a story that matches the bottom graph?

It's harder to tell a realistic story about the bottom graph. Try making other graphs with the string. What if you make a "C"? A "U"? Are these more like the top graph or the bottom graph?

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Here's another function:

Google Translate is a function.

But people's ears and minds are not functions. To us, the words "I love you, man" can mean an immense variety of things. A single utterance can mean "I love you" or "I don't love you" or "I think you're funny" or "Let's just stay friends," and it all depends on a million difference things like the tone of his voice, the tone of her voice, whether you're at a carnival with friends or on sitting alone on a couch.

Functions are automatic translators. Non-functions aren't always sure how to understand a sentence.

And, by the way, a confused boy, unsure how to interpret "I love you, man" is sort of like an impossible graph, no? He knows what time it is, but he just isn't sure how happy to be.

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But now we're getting a bit melodramatic. Let's tone it down a bit.

With credit to Christopher Danielson, this is a function:

But this isn't:

Functions are reliable machines. Non-functions are unpredictable machines. You always put in the same number of tokens, but tons of different things could happen as a result.

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Now, let t stand for time, and let h stand for happiness. Is t^2 = h a function? Is t = h^2? 

(In the comments, Gregory Taylor rightly points out that the question is whether time is a function of happiness, or happiness is a function of time.)

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And, after all that, is this a function?

If we tell our kids that functions are machines, then the only question a kid can ask himself is "Can you think of this as a machine?" But with a richer set of images to draw on, "Is this a function?" becomes connected to a series of more reliable and helpful questions:

• "Is this more like a slot machine or a change machine?"
• "Is this more like Google translate or a confused boy?"
• "Would its graph look more like a loopless graph or a graph with loops?"

The point here isn't to be precise. When we want precision, we'll use the formal definition. The point is to provide students with a set of images around that formal definition that guides their thinking in helpful ways.

Let's try to find a richer set of images for both functions and non-functions. Let's also be more intentional about bridging the gap between linear, quadratic and exponential things and the sort of semi-arbitrary pairings that we want students to recognize as functions.

Postscript:

This post constitutes my final project for Christopher Danielson's really wonderful functions course. He's going to offer stuff like this in the future, and it will definitely be worth your while. 

There's more to say, but I'll save it for another post or the comments. 

Update (3/31/13): After some helpful criticism on twitter, I edited the post for quality.