It worries me that as I move into leadership roles, I get fewer and fewer opportunities to teach. Hence, I was excited to give a talk at the Sonya Kovalevsky Day at Barnard College. The entire 9th grade of the Urban Assembly Institute of Math and Science for Young Women was at the event, and about a quarter were in my class. It was a challenging talk for a number of reasons. First, the students had not chosen to be there, and had no particular interest in math. Moreover, I felt that it was important to do more than give a fun math talk. I wanted the students to really learn something that would help them in their mathematics in school.
The topic I chose was “Does 0.9999… = 1?” I did this for a few reasons:
- I could pose an interesting question at the beginning and let the students think about it and come up with ideas.
- I could vary the level of the discussion based on student background. At one end I could always retreat to 1/3 = 0.3333…, multiply both sides by 3, and get 1 = 0.9999… (ignoring issues of how you multiply an infinite decimal); at the other end I could talk about infinite geometric series and convergence (although that was very unlikely). In the middle, I could talk about how you prove things in general, give the overall idea of a series, and talk about what convergence means.
- I would be able to emphasize ideas of “math makes sense”—that different pieces of math fit together and work in the same way—and I would also be able to reinforce basic mathematics skills. This is as opposed to a topic like, say, combinatorial game theory, where I could talk about logical reasoning but wouldn’t be able to tie it into what they’ve already seen or build their ability to understand numbers.
I began by introducing myself and having all of the girls tell me their name. I asked about their subway ride to Barnard and generally tried to bond a bit and be friendly. Then I put up the question “Does 0.9999… = 1?” I took some questions (“what does the bar over the 9 mean?”) and took a poll. A bunch thought no, a bunch thought yes, several said they were unsure. Great!
I next asked students to explain why they were or were not equal. I got a few ideas but nothing very deep. I knew that underneath, students were struggling with what this question even means, so I asked them straight up, what does this question mean? I got some not terribly enlightening answers.
Now it was time to actually help them understand the question. “What does = mean?” I asked (and wrote on the board). We discussed the notion of equality for a while, and gave examples of things that were equal (1 = 1, 1/4 = 0.25, 2 + 3 = 1 + 4); part of my goal was to emphasize that = means “are the same as”, not “is the outcome of an operation”. I asked them if x + 3 = 2x, and we had some debate before deciding that it is only true if x = 3 (I’m still not sure if they got this); then we had some more debate about if x + x = 2x before deciding yes. Although I had hoped to doubly address the meaning of = and the hidden quantifiers in most algebra problems, I realize in retrospect that this probably muddled the picture more than it helped.
Once I felt like they understood =, I asked what 1 means (briefly, although I wish I’d had time to talk about it more) and then I asked what 0.9999… means. I ended up getting drawn off-topic in the discussion, which is OK, but I wish I’d gotten more of a bead on “it’s 9/10 + 9/100 + 9/1000 + …”.
Instead, we ended up talking about what 0.3333… is. None of them recognized it as 1/3, which took me rather by surprise. OK, time to talk about 0.3333…
We spent some time discussing how to turn decimals into fractions in general, but quickly concluded that you couldn’t straight-up turn 0.3333… into a fraction; 333333…/100000… makes no sense. So instead I suggested looking at 0.3, 0.33, 0.333, 0.3333, and so forth. I was again a bit surprised: they seemed to have no intuition that these numbers are actually very close together. So it was time to take out a number line and draw them.
Where, I asked, is 0.3 on the number line? Only one girl knew how to put it up; she knew that 0.1, 0.2, …, 0.9 were equally spaced and so she found 0.3. But when asked about 0.33, she didn’t know. So we spent some time talking about how 0.3 is 3/10, and how to find 3/10 (divide the number line into 10ths; there’s 1/10, now we want three of them, so we go over here). Then we decided that 0.33 is 33/100, and we talked about dividing the number line into 100 pieces. Then we saw that 0.333 is 333/1000 and we talked about dividing the number line into 1000 pieces. (I actually liked this part a lot, because it required a certain amount of abstraction to visualize dividing the number line into so many pieces!)
Then I asked, “OK, where is 0.3333333333 on the number line?” The students really had no idea that it would be right by the other numbers. Some thought it would be very far out indeed! Others thought it would be close to 0; an original hypothesis had been that as you add 3s, the number gets closer and closer to 0. Eventually I got them to write it out as a fraction: 3,333,333,333/10,000,000,000. I asked them where this fraction was on the number line, and again, no idea. (Although they were amused, perhaps even impressed, at my ability to rattle off “three billion, three hundred and thirty-three million, three hundred and thirty-three thousand, three hundred and thirty-three.”) Eventually we realized that this fraction was close to the others, although I’m still not sure all of them were convinced. I finally explained that if I offered to give you that many dollars, I’d give you 34 cents and then you’d owe me some money! This seemed to help put it in perspective for them, and we briefly discussed how you might scam someone with this “trick”. Note to my future self: give the example of 5,000,000,000/10,000,000,000; it should be much easier to see that it is just 1/2 despite the large numbers.
At some point we noted that these were getting close to 1/3, although many of them didn’t have an intuition for where 1/3 is on the number line. We talked then about how to convert 1/3 into a decimal and I divided 3 into 1, at which point some of them realized that they had seen this before. That said, I realize in retrospect that I didn’t emphasize that fractions are division and so something of a learning opportunity was lost.
We’d also gotten that 1/9 = 0.1111… in this discussion, but still no insight on 0.9999…. I had them discover the relationship between 1/9 = 0.1111… and 1/3 = 0.3333…, that in both cases you can multiply by 3 to go from one to the other. Then we saw that 1/3 times 3 is 1, and that 0.3333… times 3 is 0.9999…. At this point I tried not to hint further, and despite having everything right there, the class went off on a wild tangent, thinking they could “add” something to 0.9999… to get 1. (A perfectly good theory, if they were different, and I wish I’d emphasized that more. We spent a while talking about what happens if you add 1, if you add 1/2, if you add 0.1, and 0.01, and 0.001, and so forth; I wish I’d had more time to allow them to do this themselves.) Somewhat flummoxed, I did a poll. Nearly everyone was now convinced that 0.9999… is not equal to 1!
Eventually I brought us back on track, and someone made the magical connection, and a student realized that since 1/3 times 3 is both 1 and 0.9999…, they must be equal. This part could have used more discussion, relating it back to equality, but we were really quite short on time. I ended the class by hinting at a graphical argument, and tantalizing them with the sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + …, and then we were done.
In the end, the class was engaged and had very high energy, but had such a deficit of basic facts that they really struggled. I don’t think I managed to bring everyone with me as we went forward, nor was the class nearly as student-led as I’d hoped because it was hard to get them moving in this context. I’m actually happy with the presentation itself, and I hope to continue to refine it and find ways to make it accessible to everyone.
I am reminded just how much students in schools lack, however. The insistence on moving students forward to algebra and beyond when they don’t fully have the number sense to understand how things go on the number line seems to be doing them a great disservice, and I wish we could find a way to enable teachers to address these basic challenges with them. Otherwise, they’ll be continually forced to memorize procedures and they’ll never understand why they do what they do.
Consider Learning 