So I’m teaching this course this year. It’s for the math faculty of a high school. It’s called:
MA600 Algebra and Analysis with Connections to the K-12 Curriculum
I am unspeakably excited, and want to do the best job possible.
The class: 7 teachers, deeply committed to kids, serious, not real talkative, rightly protective about their time, which is in short supply, but eager to get sh*t done.
The content: Basically, all of mathematics, seen as a unified whole.
It’s met twice. The second class was last Thursday. I need to get my thoughts sorted out here. I’m expecting this to help me visualize the next moves more clearly, just by doing it, but I’d love your thoughts too.
I didn’t really know anything about the mathematical background of the group when I wrote the syllabus, so for the first class I gave them a getting-to-know-you problem set with a wide range of problems and just let them work the whole time. Magically the experience of watching folks work on the problems and then later looking at what they did on paper gave me just enough information to plan the direction of the class’ first unit. We’re beginning with analysis. My first goal: the – definition of the limit. (I.e., the definition of the limit, for the snobby among you.) My second: the completeness axiom.
The plan: generate the need to define the limit by working with 2 everyday concepts that are actually limits. Namely, infinite decimals, and instantaneous speed. My hope is that by pressing on these concepts, we’ll see that in spite of our familiarity with them, we don’t actually understand them unless we have a precise way to talk about limits. Then, develop the definition out of the need to fully understand the familiar. Then, develop the completeness axiom out of the desire to make sure infinite decimals have a limit.
Here’s what we did:
I opened class with a problem set designed to get them thinking about the meaning of decimals in particular, and various other contexts for the idea of limits. I shamelessly bit the format from PCMI. The problems span a wide range of skills and I didn’t leave enough time to do them all, so people could attack problems appropriate to their skill level. This is now my favorite way to differentiate problem sets, a propos of a) using it in some NYMC workshops last year, and b) hearing about how wonderful it was for everyone at PCMI.
Then, since we are all just getting to know each other, I did a short presentation on the mindset I wanted us to be in:
(Scribd did not handle slide 6 very well, which is too bad because I was proud of that slide. This is my first PowerPoint presentation ever. Actually I did it in Keynote.)
Then, we got to business. I put this up:
I asked them to talk about it with their tables. (I had them in 2 pairs and a group of three, in three tables in a horseshoe shape in front of the board. I like this and think I’ll keep it. Easy transitions from pair/group to whole-class; tables feel separate enough so you don’t feel like your conversation with your partner is in front of everybody; but everyone’s close enough so we can all talk. On day 1 I put us all around one table, for a sense of collegiality and common purpose, but it was too close; you couldn’t discreetly check in with your neighbor, for example.)
There was a widespread sense of mathematical discomfort, and rightly so. Infinite decimals enter most people’s math educations with no attention to the fact that they actually violate everything you’ve learned about math up to that point. You don’t get the full story until analysis, but unless you really get intimate with and own that content, you probably don’t connect what you learn there to what your teacher introduced without comment somewhere between 3rd and 7th grade, as though it weren’t a mind-boggling idea. “When you expand 1/3 as a decimal, the 3’s just keep going.” Or, “3.14159… It never ends or repeats.” Um, excuse me? It NEVER ENDS?
So it’s no surprise everybody has an underdeveloped idea of infinite decimals, and therefore that objects like 0.99999… cause some dissonance. This is very productive dissonance. I’m hoping it carries us all the way to the completeness axiom; we’ll see.
One of the three tables produced the standard argument that if x = 0.9999…, then 10x = 9.9999…, so 9x = 10x – x = 9, so x = 1; but even this table found this conclusion unsatisfying. I asked them why. The table that had produced the argument said, “usually this method gives you a fraction.” I asked for an example. They produced one from the problem set:
x = 1.363636…
100x = 136.363636…
99x = 100x – x = 135
x = 135/99 = 15/11
I asked how many folks found this argument convincing. 7 out of 7. (Well, one raised hand was kind of hesitant.)
I asked the same question about the same argument with .9999…. 4 out of 7. Then I dropped this:
How many people found this one convincing? 0 out of 7.
Then what’s the difference?
At first, they cast about a bit, but then one of them said, “1.363636… has a finite limit, but …9999.0 doesn’t.” Their ideas began to coalesce around this type of language. Another one said, “we can actually estimate 1.363636…, for example we know it’s between 1 and 2.”
From the point of view I am ultimately heading for, this is the rub. Infinite decimals suggest convergent series, and the standard way to give them meaning as real numbers is that they are equal to the limit of the convergent series they suggest. …9999.0 suggests a wildly divergent series, so it cannot become a real number in the same way. (To bring home that convergence is the heart of the matter: there is an alternative way to define distance between numbers, the 10-adic metric, according to which it is actually …9999.0 that has the convergent series, and in this alternative system the above proof is valid and it actually does equal -1.) What I’d like us to do is a) define limits precisely; b) use this to prove that when a series has a limit, you can do the above type of manipulations to find it; c) try to prove that the series suggested by an infinite decimal always has a limit; d) realize that we can’t prove this without articulating the completeness axiom; e) articulate the axiom; and f) prove from the axiom that any infinite decimal has a real number limit. (Somewhere along the line, produce an – proof that 0.9999… = 1.) Now, how to orchestrate this…
For next time I told them to try to craft a definition of the meaning of an infinite decimal 0.abcd… I gave them a few minutes just before the end to discuss this with their groups. I’m expecting to learn a lot about their thinking from what they come up with, but I’m not counting on anyone to have a mathematically satisfying answer. I’ll be pleased if somebody does though.
As I think about next class, here’s what’s on my mind:
1) When we develop the – limit, what I’m going for is for this definition to feel like a satisfying relief. I know how easy it is for this definition instead to feel like a horrible monstrosity designed to oppress analysis students. I think what I have to do is keep them thinking about the reasons why anything less than this definition is too vague, which means I need to keep coming up with objects and problems that throw monkey wrenches into whatever more naive definitions they go for. (Of course, if they come up with something equally precise as the – limit but different, that would be amazing.) I feel like we’re off to a good start on this, but I want a fuller catalogue of head-scratchers (like …9999.0 = -1) to push the level of precision higher.
2) Relatedly, I sense a danger that the “real answers” will be unsatisfying because it’ll feel like “wait, I already said that.” For example, the participant who said that the difference between 1.3636… = 15/11 and …9999.0 = -1 is that “the first one has a finite limit”… I mean this is basically the answer. But it’s not based on a precise definition of limit yet, so it’s not what I want yet. I’m afraid of a “what was the big deal?” moment when we’ve got the real sh*t up there. I think the way to avoid this lies in that catalogue of head-scratchers I need to develop, so that nothing less than the real thing is satisfying. What do you think?
3) Where to go immediately next. Basically the question is: stick with decimals? Or change gears completely and press on the notion of instantaneous speed? Most (not all, I think) of the teachers have had a calculus course, but think at most 1 or 2 of them have internalized the philosophical lesson that instantaneous speed needs to be defined as a limit in order for us to even access it. I’m attracted to the idea of switching gears because I’m drawn to the connection between the disparate realms: two highly familiar, but totally different, objects – infinite decimals and speed in a moment – both getting pressed on to the point where you realize you never fully understood either one, and then you realize that the missing idea you need is the same thing in the two cases. (A precise way to talk about what number some varying quantity is “heading toward.”)
Actually as I write this out, it seems clear to me that switching gears is the way to go. I think it’ll give us a clearer understanding of what we’re missing with the decimals. Also, it’ll allow us to access all this rich historical stuff around the development of calculus. For example, maybe I’ll share with them some choice quotes from Bishop Berkeley’s The Analyst, to help articulate why the 18th century definition of the derivative was inadequate.
Anyway. Very excited about all this. Will definitely keep you posted.
10 thoughts on “0.99999…”
While I don’t, of course, really understand the math at all, I just want you to know that I think your approach is great. You are using math to talk to math teachers about math. Exactly right. It’s so natural that you don’t even know it’s smart. But it is, and I bet it works.
I was just about to say that Berkeley is a gold mine for this stuff.
Also of interest might be statements like this one:
“in the later 18th century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function.” (from the Stanford philosophical dictionary)
You can take plausible sounding definitions like those and use them to motivate discussions about why the quantifiers must be nested precisely as the epsilon delta definition has them.
I’d suspect that building other machinery on top of faulty limits or infinitesimal concepts is the best way to make rigor seem necessary. History suggests that you should postpone epsilon delta as long as possible.
I’ll be interested to see how you handle the cases where someone knows what the answer should be (e.g. “Take the derivative to find the instantaneous velocity”) but doesn’t know where that idea comes from or what it means. It sounds like your students will recognize the need for understanding and won’t be satisfied with that answer.
In my college freshman calculus classes, I always have a few students who took calculus high school and remember just enough to jump in with the correct answer when I pose a question to the class. (“How can we find the derivative of ln(x)?” “Isn’t it just 1/x?”) But when I question them on how they got such a wacky answer and why they know it’s true, they have no idea and it never seems to bother them that they don’t really understand.
Regarding Scribd slideshows in WordPress.com blogs: In the scribd shortcode, change
BTW, on the very first day of my first math class in college (Calc II), the professor started class by writing “0.99999… = 1” on the board. Totally blew my mind.
@ Steve – thanks! It’s sort of a natural path for me since I’ve spent the last 3 years studying math pretty intensively and the opportunity to be a math teacher and learner at the same time has really shifted how I see math education.
@ Gilbert – thanks for helping me think about this. I think the overall architecture of the course is going to force me to do epsilon and delta sooner than later, but you’re right.
@ Susan – because they’re teachers, I think they’re all pretty concerned with the why of it. Nonetheless, I’m planning with this case in mind, since much of the course is content that they’ve seen before but not fully owned. My challenge is how to make the deeper understanding feel desirable.
@ Frank – thanks, just what I needed!
If you haven’t already, you should check out Elementary Mathematics from an Advanced Standpoint by Felix Klein – dated, yet relevant 🙂
Totally bought it precisely to help plan this class! I also got Courant & Robbins’ What is Mathematics? with the same idea. (And I also want Zalman Usiskin’s Mathematics for High School Teachers: An Advanced Perspective, but this is not available in an inexpensive paperback version…) Now I’d love to spend a summer (or a 3 year period!) reading and working through it all before even starting to plan the class! Oh well, next time…