Teaching Log: 0.9999… = 1

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.

SPMPS School Admissions, Google Trip, and More!

And now, my latest update e-mail about SPMPS.  If you want to receive SPMPS updates by e-mail, let me know.  Perhaps I should have a formal sign-up system for e-mails, but the truth is that I like the informality of a simple text e-mail that I send to our supporters.

Dear friends of the Summer Program in Mathematical Problem Solving,

I would like to take a quick break from preparations for summer 2013 to celebrate the achievements of our students from summer 2012.

Before I get to that: last Friday, we had a reunion at Google’s NYC headquarters.  Google engineers made some fantastic presentations about their work.  In one presentation, we learned about binary and RGB values to encode images on a computer.  In another, we learned about the math behind apps, and the geometry that underlies a tap, pinch, or drag.  In a third, we heard about one Googler’s path from a challenging background to success at Goldman Sachs and then Google.  Then students could ask their own questions in a panel about careers, and got a tour of the great facilities.  Thanks to Google for providing such a wonderful reunion day!

Now… drumroll please… congratulations to all of our students who were admitted to selective schools:

  • Amy, Ana, and Jahdel for gaining admission to Brooklyn Tech!
  • Jamila for gaining admission to Medgar Evers College Preparatory School!
  • Amy, John, Nicole, Quentin, and Salimatou for gaining admission to Bard High School Early College!
  • Emalee and Kiara J. for gaining admission to the Brooklyn Latin School!
  • Joel for gaining admission to Westminster School!
  • Taylor for gaining admission to Manhattan / Hunter Science High School!
  • Jeremy for gaining admission to NYC iSchool!
  • Nathaniel for gaining admission to Manhattan Village Academy, Cardinal Hayes High School, and Cardinal Spellman High School!

Ana was also admitted to the Center for Talented Youth’s Academic Explorations program.  We still have many more students left to hear from about their school admissions, so I hope to have more great news for you soon.

Thanks as ever for all your support!


I’d also like to add something to this e-mail: all of our students are absolutely incredible.  I’m tremendously proud of their admission to selective schools, but the truth is that all of them have bright futures.  I didn’t mention the students attending the NY math circle, or those applying to other summer programs.  And the tremendous intelligence and insight they will bring to whatever school they attend.

I just saw them a week ago, and I miss them already!

Beyond Motivation to Self-Identity

Why do people want to do things?

I’ve been asking myself this question a lot.  If you believe that teaching is about more than simply imparting knowledge and skills, that it is also about inspiring students to achieve greatness through academic study, then you have to ask yourself: how do you give students experiences that will make them want to learn your subject and then apply it in a future career?

For many educators, the answer is often  to show students that the subject is beautiful.  For others, it is to show students that the subject is useful.  Educators talk about motivation, and sometimes intrinsic vs. extrinsic motivation.  These are all very useful conversations.  I think they are also insufficient.

Let me give an example from my own life.  I love playwriting.  I wrote some plays that I consider decent while in college, and I would love to write some more plays and perhaps to see them performed.  Yet I “don’t have time” to keep writing plays.  Why not?  Well, for one thing, I have a job doing something I love—education—and that job probably pays better.  But I could still write plays as a hobby!  I don’t, because there are other things I can do for fun, and because most of my friends are math/science-oriented and so I don’t have a community of people to talk to about it and encourage me.  Instead, I go to see a lot of plays each year and content myself with occasionally imagining what plays I would write if I had the time.

My point is that even if someone likes doing something, that is not enough for them to dedicate their time to it.  Just as we recognize this fact in our own lives, we should think about the many factors that play into our students’ decisions, both consciously and unconsciously:

  • Do they see it as a productive/gainful use of their time?
  • Do they have a supportive peer group?  Will they be able to do it with others?
  • Will it give them social status?
  • Can they see themselves doing it in the future?  Is that future self someone they like and admire?  Do they have role models?
  • Is there an established pathway for doing it that they understand?
  • Do they expect success at it?

I believe that focuses on curriculum or study skills or content knowledge are all good steps, but they are insufficient to the task.  Just because students learn that mathematics is useful for building bridges does not mean that they see themselves doing it.  We need to consider the whole context surrounding a child, the whole environment that might encourage them to become a scientist (or not), and how we can make it more likely that they see math and science as a viable pathway.

Ultimately, I think we need to build their self-identity as scholars.  To me, self-identity goes much farther than just motivation, be it intrinsic or extrinsic.  Motivation plays a part, but self-identity is about how they see themselves.  Indeed, I claim that without a resilient self-identity, all of our efforts to teach knowledge and skills are less effective.

Of course, saying that we should accomplish all these things is a far cry from specific proposals to do so.  I hope to explore more about self-identity in future posts over the coming weeks: to try to give a better definition, and to give concrete thoughts for how to help students develop it.

Careers in Mathematics Education

You’re a well-qualified graduate in a STEM field.  You could make lots of money in Silicon Valley or in finance, doing interesting things, but you want to be in education.  What can you do that makes good use of your talents and maybe even lets you feed your family?

Too many people don’t understand that there are good career opportunities available.  There are curriculum development roles; education technology companies; programs that cater to more motivated students; and all kinds of exciting smaller initiatives.

Teaching can also be a tremendously rewarding career that mixes many different kinds of very interesting challenges.  You engage with academic material on a fundamental level, but you also have some very deep engagement with ideas in pedagogy and psychology.  You also are in a very social career, so you get to interact with many interesting people and see the impact that your work has on them. The pay might not be great, especially initially, but there are prestigious fellowships that can supplement your pay and ease your transition.

This post exists to share what I’ve learned about exceptional opportunities in education that can be part of a serious career.  I hope that it will be a resource for those who would want to pursue work that we so desperately need.

You want to…

  • Change the mathematics classroom: Consider Reasoning Mind, a 140+ employee company that develops math software for elementary instruction.  Their work produces great results and is based on serious mathematics.  Math specialists can be “Knowledge Engineers”, and they are very rigorous about who they hire; many of them have PhD’s.
  • Work with talented students: A natural place to go is Art of Problem Solving, which creates outstanding curricula for elementary through advanced high school level and teaches it online.  They have a huge online community of dedicated students and lots of innovative online tools for them to do math.
  • Work with talented, underserved students: My own program, the Summer Program in Mathematical Problem Solving, is now hiring a Director of Programs to take over leadership and expansion of our work.  During the summer, we also hire instructors, and while you’re in college, you can be a residential counselor/TA.  Over the coming years, there may also be year-round curriculum development work.
  • Develop material about exciting mathematics: The National Museum of Mathematics is developing curricula surrounding their exhibits and also gives you the opportunity to teach to students who come in for field trips.
  • Work with math circles: Josh Zucker makes his living as an “itinerant math teacher”, running and teaching at math circles as well as online at Art of Problem Solving.  My friend Japheth Wood, on the other hand, makes his living as Executive Director of the New York Math Circle.
  • Teach: Do you want to understand the subtlety and rich intellectual life that goes into teaching really well?  Take a look at Sameer Shah’s blog, or Dan Meyer’s blog, to see how really smart people approach mathematics teaching seriously.  If you’re worried about the pay, consider either the Math for America Fellowship or the Knowles Science Teaching Fellowship.  Another way that many people try this out (tends to produce a love-it or hate-it result) is through Teach for America.
  • Do technology: I recently had a great conversation with Zach Wissner-Gross who founded School Yourself which makes lovely interactive tools to study mathematics and is a startup you might want to look at.  You could work at edX or Coursera or Khan Academy.  If you want the kitchen sink, here’s Quora’s list of education technology startups, but beware that there may be a mismatch between many startups and what works in practice.
  • Science enrichment: Unfortunately the sequester has put something of a hold on this for now, but NASA has always produced excellent science outreach materials.
  • Do Research: A number of my colleagues, including Yvonne Lai and Nina White, have made the transition from mathematics to research in mathematics education as well.  Math ed research is in serious need of qualified mathematicians right now!

Even more exciting, this is not a comprehensive list of opportunities.  There are many, many other organizations out there, and you can always start your own.  In fact, just as many underserved students don’t know the landscape of colleges, enrichment programs, or selective schools, many of us in the math world don’t know the landscape of education organizations that would love to have more qualified mathematicians working with them.

If you want to pursue education, you should go for it.  There are great careers awaiting you if you’re very good at what you do.