I can think of two ways of approaching the distinction between functions and non-functions:

**Directly:** Don’t build functions/non-functions on any sort of prior foundation. Don’t try to have the distinction emerge out of the attempt to solve a problem. Just do something, like, “Here’s a table: {(1,3), (2, 4), (2, 5), (3, 6)}. What’s f(2)?” This is fine, but my issue with this is that there’s no context for it. It’s not easy to come up with examples of non-functions that relate to the rest of the math that the kids do. And the examples of non-functions that I’d be throwing in front of kids would be unlike the sorts of tables and rules that they’ve seen so far. They’ll get the vocabulary, but will it mean anything to them?

**Impossible Inverses: **Another way to handle this would be to introduce non-functions in the context of inverses. Non-functions could then be a generalization of non-inverses. Suppose that kids can find inverses. Then ask them, “What’s , if ? Then we get to have the conversation about what’s going wrong with the inverse of the function, and abstract that into a distinction between good and bad functions, or functions and non-functions. In this way of doing things, non-functions are invented as a way to describe the sorts of malfunctioning inverses that we sometimes get.

What’s hard about that second approach is getting into inverses without having a clear way of talking about functions. What sort of machinery is necessary for get far enough into functions to have the sort of conversation that I’m imagining?

- We need comfort with function notation.
- We need to have some sort of notation for referring to . I’m imagining that might be a good way to start.
- We need to be able to discover the inverse of a given rule.

I think that second approach is doable. Is it advisable? Thoughts?

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Personally, I prefer to give students a solid foundation of sets before I even talk about functions. To me, a relation has two sets and rule of assignment. A function puts a restriction on that rule of assignment.

After the foundation is laid, we can talk about x’s being assigned to more than one y in various forms: tables, sets of ordered pairs, maps, & graphs. Graphs last because if you talk to them too quickly about the vertical line test they will be blinded by it.

To introduce functions using inverses to puts you in a circular situation, since they need to know what a function is to talk about inverses, and by your argument, they’ll need to know what an inverse is in order to identify a function. My gut tells me that approach is not advisable.

I hear that, Marshall. But increasingly I’m open to non-linear approaches to teaching. What I’m imagining is that we start with a rough understanding of functions, a decent understanding of inverses, then cycling back to a stronger understanding of functions and an even stronger and more precise understanding of inverses.

I always strongly prefer to start with something real and concrete. I tell ’em that a function has only one y for every x… like prices in a store. The bananas cost 39c a pound. There’s only one answer for that. They don’t cost something different at the same time. Of course, lots of things could also cost 39 cents a pound.

A little more abstractly, the idea is that if you’ve got a function, it’s a reliable way to get the *one* right answer for a situation. Relationships that aren’t functions have more than one possible right answer. x^2 + y^2 = 1… hey, that thing that squaring does to signs means you get different answers.

Most of my guys would so not get any meaning whatsoever out of inverses 😉

I think there’s something going on here with the language of concrete and abstract.

That’s concrete?

And what do you mean that your guys wouldn’t get any meaning out of inverses? I mean, don’t you have to teach them that too? So how do you make that concrete for them.

What I’m suggesting is that kids can get a good understanding of what an inverse relationship is between two rules or tables before they have a good understanding of what is or isn’t a function. I’m further suggesting — super tentatively — that it’s a good idea to build their understanding of what is or isn’t a function atop their understanding of what an inverse relationship is.

I’ll pitch out a couple things here… first, the way I tend to do functions is with a “call-response” kind of deal, involving the whole class. Like, “I say subject, you state it’s math,” subject-math-subject-math, “I say teacher, you state my name,” teacher-me-teacher-me, “I say name, you state your name” name-CHAOS. Okay, you’re all saying something different, this is hard for me to deal with. Now, would we have the same problem if I gave you “x” and asked for the response in these math examples? I’ve also done it in terms of what graphs one could create if we were always talking distance and time (since “walkable” graphs are functions).

With respect to doing it in context of inverses… is it advisable? I’m honestly not sure. I never thought about it that way. That said, I’m trying something different in my course this year which might make it at least feasible. I began with parabolas, as a hook from last year, and effectively said “I can use f(x) because these are functions when we’re dealing with x; we’ll define what it means to be a function later”. (I know, I know, it’s not good when they have to accept things like that on faith… but it let me familiarize them with notation and draw on prior knowledge before tossing in completely new curve families. This is a work in progress, but I point out it would also give you point #1 as referenced above.)

With that in place, we could then get into a discussion of “inverses” in terms of “if I give you the answer, what was the question” (or “if you get to class late and only hear the answer, what was the question”). Of course, with the parabola, you can never be sure. But with something linear (my favourite standby is the Fahrenheit/Celcius conversion), you could know. And so on from there. Again, no idea if it’s necessarily advisable – I’m not certain I could pull it off, at the least.

Gah, even as I hit ‘send’, I’m like “With the parabola you CAN be sure if it’s the vertex!”. I should never use words like never. 🙂 ^^;

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