Saturday, March 30, 2013

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?


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.


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:

Slot Machine

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.


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.)


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.


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.


  1. I like the connections. I particularly like the translation analogy. Though as you say, language is subject to interpretation, and playing devil's advocate (as I do), we need to be careful of assumptions.

    1) "Functions are graphs that tell possible stories. Non-functions are impossible graphs." We're assuming the model with time there. What if money is my independent variable, and happiness is dependent on money?

    'I began unhappy. As I got more money, I became happier. Then I gave a lot of it away to a needy charity, and became happier still.' (Is that not a possible story and graph? But is it a function?)

    We're also assuming there that we're not graphing an episode of "Doctor Who".

    2) "Is t^2 = h a function? Is t = h^2?"

    Both are... depending on whether you think time is dependent on your happiness ('Time flies when you're having fun') or happiness is dependent on your time ('The longer this goes on, the happier I get').

    Which is possibly what you were going for there, but so often we stick time on the horizontal axis owing to our perverse need to control it. Just some extra thoughts.

    1. Lots of good points throughout, but I want to pick up on this one:

      We're also assuming there that we're not graphing an episode of "Doctor Who".

      I'm actually OK with that. Kids say, "Time travel is possible." Fine with me. Forget impossible graphs. Let's keep track of "graphs that imply time travel." Let's change the domain. What do these graphs say now?

      I care way more about the identification of a certain class of graphs than on the name that we give it.

  2. Lines I couldn't say better if I tried:
    "Functions are automatic translators."
    "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."
    I like the use of string to explore functions or non-functions. Students can quickly manipulate it. Keep up the great work man and thanks for being so candid in the course. You helped me a lot.