[This is the text of a presentation I made on October 7, 2017 at the kick-off event for Global Math Week, held at the Courant Institute of Mathematical Sciences in New York City. Earlier in the day, James Tanton gave his usual brilliant presentation on Exploding Dots, so in my talk I was able to assume that the audience knew what Exploding Dots is about; they also recognized my riff on Tanton’s signature line “I’m going to tell you a story that isn’t true”, as well as the significance of the word “Kapow!” (and its variants) in the Exploding Dots story. You might want to visit YouTube and sample Tanton’s Exploding Dots videos to get a feel for what it’s all about. For the full Exploding Dots spiel, try https://vimeo.com/204368634. I hope to post a high-quality video of my own talk soon; for now, there’s https://www.youtube.com/watch?v=x8wcbONcGHk.]
I’m going to tell you a story that’s as true as I know how to make it. I don’t have a background in ethnomathematics, or the history of mathematics, or the history of math education, so please forgive any mistakes, omissions, distortions, or mispronunciations (and let me know about them, but not now!). The story I’m going to tell is as old as civilization, or at least as old as money — because as long as currency has existed in different denominations, there’s always been a need to make change, and to find systems for making change efficiently and accurately. The story I’m about to tell involves many parts of the world over the course of many centuries. And in many ways it’s a story about sand.
You and your computer have a fundamental disagreement about how to represent numbers. Your computer was designed to calculate in base two (binary), while you use base ten (decimal). But there is something that your decimal self and your binary computer can agree on: representing numbers in base three-halves is a damn fool thing to do. I mean, I haven’t even told you yet what “base three-halves” is, but you probably already guessed it’s one of those things mathematicians came up with not because anyone asked them to but simply because they can and because they think it’s fun.
When the path from a simple question to a simple answer leads you through swamps of computation, you can accept that some amount of tromping through swamps is unavoidable in math and in life, or you can think harder and try to find a different route. This is a story of someone who thought harder.
His name is Arthur Engel. Back in the 1970s this German mathematician was in Illinois, teaching probability theory and other topics to middle-school and high-school students. He taught kids in grades 7 and up how to answer questions like “If you roll a fair die, how long on average should you expect to wait until the die shows a three?” The questions are simple, and the answers also tend to be simple: whole numbers, or fractions with fairly small numerators and denominators. You can solve these problems using fraction arithmetic (in the simpler cases) or small systems of linear equations (for more complicated problems), and those are the methods that Engel taught his students up through the end of 1973. Continue reading
Last month I launched a venture similar to Mathematical Enchantments: a YouTube channel called Barefoot Math. The first few videos are about a game I invented called Swine in a Line. The rules are easy to state but the winning strategy is not easy to find, and the challenge I posed is to find that strategy. In the videos, and in this essay, I explain the strategy and I explain how a person might figure it out. It’s a story about how numbers, and the ways we represent them, can turn out to be relevant in surprising ways. I’ll also sketch how the game relates to a hot topic called the abelian sandpile model. Finally, I’ll connect the Swine in a Line game to James Tanton’s exciting way to make pre-college math vivid through his device of “Exploding Dots”, which will be the subject of Global Math Week later this year. Continue reading
My career as a serial extortionist was triggered by an act of theft — more specifically, by an honor student’s appropriation of another student’s words on a homework assignment.
To tell this story properly, I should back up a bit and describe my earlier, non-extorting self. When I was a naive young assistant professor, I was convinced that if I wrote up detailed solutions to the homework problems I assigned, students would eagerly read them and absorb not only habits of effective problem-solving but also habits of clear writing. I know for a fact that the students appreciated the extra effort I went to in writing up the solutions; they praised me for it in their end-of-term evaluations. There was only one problem: over the course of years, it became clear to me that hardly any of them actually read my solutions.
Today we’ll talk about some paradoxical things, like the logarithm of zero, and the maximum element of a set of real numbers that doesn’t contain any real numbers at all. More importantly, we’ll see how mathematicians try to wrap their heads around such enigmas.
All today’s logarithms will be base ten logarithms; so the logarithm of 100 is 2 (because 100 is 102) and the logarithm of 1/1000 is −3 (because 1/1000 is 10−3)). The logarithm of 0 would have to be an x that satisfies the equation 10x = 0. Since there’s no such number, we could just say “log 0 is undefined” and walk away, with our consciences clear and our complacency unruffled.
BUT WE AREN’T GOING TO DO THAT, ARE WE?
I thought my earlier essay on .999… did a pretty good of explaining why I (along with 99.999…% of mathematicians) say that it equals 1, until I asked some of my students what they got out of it; then I got a humbling jolt of pedagogical reality. The students agreed that .999… is the limit of the sequence .9, .99, .999, etc., and they also agreed that the limit of that sequence is 1. So you might think that they would have agreed that .999… equals 1, but no: they couldn’t swallow that conclusion.
I’ve decided that part of what’s going on is that my students arrive at college with a number-sense that’s so deeply grounded in their experience with terminating and non-terminating decimals, in worksheet after worksheet, that these concrete representations have taken on an independent reality for them. At that point, it does little good to tell them “.999… doesn’t mean anything till we assign it a meaning” or “We’re going to define .999… as a limit”, because they already “know” what .999… is: a dot followed by infinitely many 9s! No attempt to redefine .999… can shake loose their sense of what it already means to them.
So today I’m going to come at the problem of .999… from a totally different direction. Continue reading