For the past two weeks or so, I’ve been battling with my 6th graders over the issue of infinity. More specifically, we’ve been engaged in two debates: the first about whether or not .9 repeating is equal to 1, and the second about whether or not I will reach the door if I walk halfway to the door, cover half of the remaining distance, then cover half of remaining distance, then… you know the deal.
These issues are really difficult. In the small amount of polling that I’ve done, it seems that the majority of regular, everyday, non-math-teacher people will say, when asked, that .9 repeating definitely does not equal 1. They will also be very confused about the walking to the door question – mostly because they know that you’re supposed to be able to reach the door, but they still can’t help but believe that you won’t.
I’ve been putting off writing a post about this because I wanted to be able to say, “Here’s what I did to convince my 6th graders that .9 repeating equals 1 and that they’ll reach that door – and it worked!” But, despite all that I’ve tried, there’s still some confusion. No complete success! In fact, I think I’ve been going about it all wrong!
I had the idea that I’d been going about it all wrong after I read this post. You should read it, but, basically, in it the author of the blog, Chris Lusto, argues that thinking about limits as processes is confusing, because the process behind a limit never ends – yet the limit itself is an actual object. Chris writes of limits,
Could the object be the result of the process? If the answer is no (which my gut believes it to be in the infinite case), then how can we reasonably talk about it as both a process and an object? Does the duality break down?
After reading this, I realized that the process-versus-object distinction was the root of my infinite problems. For my 6th graders, “1” is an object. It’s a number. It’s tangible. But .9 repeating is a process. It’s a series, .9+.09+.009+.0009+…. And not only is it a process, but it’s a process that never produces an object that can be associated with it, because it never ends.
In class, I’d touched on a number (tangible!) of proofs that .9 repeating equals 1. I won’t describe them all here, but if you want to see them, check out this excellent Vi Hart video. (The best thing I did, actually, was to show my students this Vi Hart video. We spent an entire class watching it again and again and again, correcting it, until the kids burst out laughing every time she made a ridiculous mathematical mistake. Oh, math humor!)
But none of them got to the heart of the matter, because… all of the ways that I’d been trying to get them to see that .9 repeating equals 1 presumed that .9 repeating is an object. The arguments mostly rest on the fact that since .9 repeating can be used like 1, it must be 1. But my students didn’t believe that you could use .9 repeating at all! Because (according to them) it isn’t an object! It’s a never-ending, totally intangible process!
The problem here, I think (at the moment, at least), is that a limit is a mathematical idea that works just fine, but a philosophical (or linguistic?) idea that doesn’t sit well. When we say that the limit of some infinite process is a particular number, we mean this: that, in doing the process, we can get as close as we want to the object of our desire by narrowing the window of possibility. Why do we do this? I think so that we can use that thing that’s always just about to be produced as an object. (Am I wrong about this? Please correct me or add nuance if I am.) That doesn’t really say “IS” – that says, “It’s as close as we want, so we’re going to say, so that we can use it as an object, that it IS.” In treating .9 repeating as a process, we have to do what we’ve agreed to do with all processes like it – say that the process of getting close enough works to produce an object.
I like what Christopher Danielson says to Chris Lusto in response to the statement I quoted before:
But I do think the transition from process to object is at least in part one involving imagination. I have to imagine the object into being in mathematics precisely because mathematical objects are abstract.
And when I’m struggling to understand a new object (say a limit), it is often helpful to imagine the process that produced it. But I don’t have to see the process through to the end.
There’s no genuine, tangible end to the .9 repeating process. Yet, as an object, it acts like 1. So, how do I get my students to imagine that the .9 repeating process actually produces 1?
I don’t really know what to do about this, but, as I have to do something tomorrow and can’t just drop the issue, I’m trying the best thing I could come up with. On Friday we folded hyperbolic paraboloids, paper mathematical objects that you make by repeatedly dividing sections of a piece of paper in half. For homework over the weekend, I gave my 6th graders the chapter from The Number Devil: A Mathematical Adventure, by Hans Magnus Enzensberger, about infinite series. I asked them to think about this: how does the subject of this chapter relate to folding hyperbolic paraboloids? The hope is that in thinking about the limiting process backwards – starting with the single sheet of paper (1) and breaking it into halves again and again and again – the result of the limit process will be a bit more tangible.
Any suggestions for what I can do to help my students (and myself) understand this?