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
I have trouble with three-dimensional space. Yes, I do live in it, and I get by without hurting myself too badly too often, but honestly, I miss a lot of what’s going on. There’s no reason for you to care about my problem, except that it touches on the issue of “What does it take to be a mathematician?”, and the details of my limitations might serve as a useful antidote to the idea that math ability is always linked to spatial intuition. But I’ve got an ulterior motive for these confessions: I need some help, and I’m hoping one of you can provide it.
BERKELEY, CALIFORNIA, 1983
I remember my zeroth day of grad school, the day before the start of classes. I was attending a workshop on how to be a good teacher — something the university required all incoming Ph.D. students to do, since most of us would serve as Teaching Assistants for at least one semester. The seasoned TA leading our orientation wanted us to experience the sort of Aha! moment that good teachers instigate, so she led us through the famous “handcuffs puzzle” in which two people tied together by cords must extricate themselves from one another. Continue reading
Pi, that most celebrated of mathematical constants, leads a curiously double life. On the one hand, we have numerical formulas for pi, like Leibniz’s formula π = 4 × (1/1 − 1/3 + 1/5 − 1/7 + …); imagining a world in which this expression converges to a value other than 3.14… is as hard as imagining a world in which 2+2 doesn’t equal 4. On the other hand, we have a geometric definition of pi as the ratio of the circumference of a circle to that circle’s diameter, and this definition of pi lets us imagine that pi is a physical constant like the speed of light — that it could have a different value in an alternative universe that’s built using a different kind of geometry. Could there be worlds in which geometrical pi equals 3.24…, say, and in which the more open-minded scientists and mathematicians speculate about other worlds in which pi has some crazy value like 3.14…? Continue reading