words from Persi Diaconis

This is a delightful article about Persi Diaconis,who is now working on the mathematics of "smooshing" cards. https://www.quantamagazine.org/20150414-for-persi-diaconis-next-magic-trick/ But the most memorable take-away for me is the following:

When Diaconis returned to college after a decade as a professional magician, his first three grades in advanced calculus were C, C and D. “I didn’t know you were supposed to study,” he said. His teacher told him that he should write down the proofs and practice them as if they were French verbs. “I said, ‘Oh, you’re allowed to do that?’” Diaconis said. “I thought you were just supposed to see it.”

Write down the proofs and practice them as if they were French verbs! I love that analogy. And it makes so much sense. Math is a language and a certain amount of rote practice is helpful for those of us who are learning it in later life. (Persi Diaconis had a somewhat unorthodox and nontraditional educational trajectory. I guess mine may be more unorthodox still.) The following paragraphs from the article were also fascinating:

When it came to smooshing, instead of just trying to “see it,” Diaconis devoured the literature on fluid mixing. “When we started talking about the connections between cards and fluid mixing, he read the whole 200 pages of my Ph.D. thesis,” said Emmanuelle Gouillart, a researcher who studies glass melting at Saint-Gobain, a glass and construction materials company founded in Paris in 1665. “I was really impressed.”

While Diaconis grew more conversant in fluid mechanics, Gouillart benefited from his unique insight into card shuffling. “It turned out that we were studying very similar systems, but with different descriptions and different tools,” Gouillart said. The collaboration led her to develop a better way to measure correlations between neighboring particles in the fluids she studies.

Diaconis, meanwhile, has developed a mathematical model for what he calls “the sound of one hand smooshing.” In his model, the cards are represented by points scattered in a square, and the “hand” is a small disk that moves around the square while rotating the points under it by random angles. (It would be easy, Diaconis noted, to extend this to a two-handed smooshing model, simply by adding a second disk.)

Diaconis has been able to show — not just for a 52-card deck but for any number of points — that if you run this smooshing model forever, the arrangement of points will eventually become random. This might seem obvious, but some shuffling methods fail to randomize a deck no matter how long you shuffle, and Diaconis worried that smooshing might be one of them. After all, he reasoned, some cards might get stuck at the edges of the table, in much the same way that, when you mix cake batter, a little flour inevitably gets stranded at the edges of the bowl and never mixes in. But by drawing on 50 years of mathematics on the behavior of random flows, Diaconis proved that if you smoosh long enough, even cards at the edge will get mixed in.

His theoretical result says that the smooshing model will eventually mix the cards, but doesn’t say how long it will take. The model does provide a framework for relating the size of the deck to the amount of mixing time needed, but pinning down this relationship precisely requires ideas from a mathematical field still in its infancy, called the quantitative theory of differential equations. “Most studies of differential equations focus on what happens if you run the equation for a long time,” Diaconis said. “People are just now starting to study how the equation behaves if you run it for, say, a tenth of a second. So I have some careful work to do.”

Diaconis is optimistic that the work will lead him not just to an answer to the smooshing question, but to deeper discoveries. “The other shuffles have led to very rich mathematical consequences, and maybe this one will too,” he said.

Avoid Einstellung (getting stuck)

On the recommendation of another older learner of mathematics, whom I discovered as a guest poster on the blog of Miles Kimball, a former student of mine, I have begun reading a quite wonderful book.

Despite the subtitle, the book is actually designed for "both math experts and mathphobes." I myself am somewhere in the middle of that spectrum. I am not a "math expert," since I have a lot to learn (and, in some cases relearn) in math, and I am not your typical "mathphobe" either. I love math. And I certainly never "flunked algebra". But at the same time, there are times I find learning new things scary--so I do understand what it is like to be a mathphobe.

And, in fact, I found myself feeling pretty phobic in my final exam in my graduate class this past Monday. I got impossibly bogged down on what was probably intended to be a simple problem #1 on the exam.

What happened to me on Monday was a classic example of the phenomenon Barbara Oakley refers to by the wonderful German word "Einstellung", which means getting stuck. As she writes:

[t]he Einstellung (pronounced EYE-nshtellung) effect. In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better idea or solution from being found.

I was so stuck on a particular approach to the problem (based on a misconception) that I couldn't get the perspective to see I was just wrong and had to abandon it for a different approach.

Part a of problem 1 was actually kind of fun, calculating the Fisher information for a random sample drawn from a normal distribution with unknown mean $\theta$ and variance 1. It looked kind of ugly at first (since it involved taking the second partial derivative of the log of the joint pdf of the random sample), i.e., it was:

$$I(\theta)=-\frac{\partial^2 \log(\prod_{i=1}^n\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}(x_i-\theta)^2})}{\partial \theta^2}$$

but it was kind of beautiful that the ugly looking expression above ultimately boiled down to 1, and as soon as I saw that the answer was 1, it made complete sense that 1 would be the answer (for reasons involving the Rao-Cramér lower bound, which I won't go into here.) And I thought,wow, what a great choice for a first problem, because it was one of those "lightbulb going off in my head moments.)

But then came part b of problem 1, where we had to compute the efficiency of an obviously inefficient estimator for the unknown parameter. That meant that I had to compute the variance of the estimator, which should have been an easy piece of cake.

The estimator was $Y=\frac{1}{2}X_1+\frac{1}{3}X_2+\frac{1}{6}X_3$. It was easy for me to prove that $E(X)=\theta$, so I knew the estimator would be unbiased. So then all I needed to do was to compute the variance of the estimator. And I was used to using the following clever trick to find the variance of statistics: Variance(Y)$=E(Y^2)-[E(Y)]^2$. So, I thought to myself, well, I already know $E(Y)=\theta$, so all I need to do is to find $E(Y^2)$. So I wrote:

$$E(Y^2)=\frac{1}{4}E(X^2)+\frac{1}{9}E(X^2) + \frac{1}{36}E(X^2) =\frac{7}{18}(1+\theta^2)$$

But looking at that answer, I realize that could lead to a negative variance if $\theta$ is sufficiently large and I knew THAT couldn't happen, so I must have done something wrong, but my brain was starting to go "deer in the headlights" and freeze up and I was so stuck on my approach to computing the variance as $E(X^2)-[E(X)]^2$ (which can often be a clever approach) that I couldn't see that my equation above was fundamentally flawed.

If my head had been clearer and I could have gotten unstuck, I could have made the problem trivial by writing:

$$Var(Y)=\frac{1}{4}Var(X)+\frac{1}{9}Var(X)+\frac{1}{36}Var(X)=\frac{7}{18}$$

which is the simple, clear, and straightforward approach to solving what was intended to be an easy softball problem. Somehow I had gotten my brain stuck on an unworkable and incorrect approach. Having done that on the first problem, I was then sufficiently rattled that I strongly suspect I messed up some later problems on the exam as well.

Oh well, live and learn. At my age and stage of life, I don't really care about grades, but I was disappointed because I thought the professor had been terrific and that I had learned a lot in the class and I would have loved to have done a better job of showing him what I had learned. Fortunately, there was also a takehome part of the final, which I felt much better about, plus I felt pretty good about all the homeworks I had turned in as well as the midterm.

But the whole thing was really a remarkable learning experience to me. As an economics professor, I often see students do incredibly silly things on exams and I beat myself up, thinking "Haven't I taught these folks *anything*? How could they be so confused." But having seen the incredibly silly thing that *I* did on the exam and how it didn't really reflect my understanding (but more my nervousness under time pressure), it reminded me again that tests are necessarily flawed measures of learning.

But I also realize that maybe if I knew how to study more efficiently and master a subject more deeply than I had, I might have done significantly better on that exam. And I like challenges, and I feel that following some of the ideas in this book may be helpful to me.

So feel free to stick around if you want and follow the future challenges of a sixty-something "Velveteen Mathematician" who wants to become a "real mathematician."

Struggling to learn math again in my 60s

My last post was written a little over two years ago, shortly before my world turned totally upside down when my husband died unexpectedly.

Ross loved many many things and math was a particular love we both shared. Both of us had undergraduate degrees in math, but we met at the beginning of graduate school in a PhD program in economics, not math. Why? Because in 1975, economics seemed like a vastly more practical choice. There were far more jobs for economists than mathematicians.

But both of us loved math anyway and did it just for fun in various ways. In fact, the year before he died we enjoyed a nightly ritual before bedtime where I would read aloud from a mathematically fun Japanese novel called Math Girls, which I learned about from Math Mama's blog.

And Ross was very supportive of my amateur mathematical adventures like founding and advising the Albany Area Math Circle, doing Guerilla Math Circle outreach activities and other such mathy stuff. In fact, just two days before he died, he was on one of his solo walks around the neighborhood on a fine spring day and a guy came up to him and said, "Excuse me? Are you Mary's husband?" and Ross said "Yes???" and then the guy hugged Ross and shook his hand and thanked him profusely, saying that his daughter was headed to MIT next year and that it never would have happened without our math circle. Anyway, Ross came home and hugged me and told me how good that made him feel.

And then, the next day (which turned out to be the day before he died, though of course we had no way of knowing that was coming), our math circle had its end of year picnic and the students wound up deciding to built a really large tetrahedron and they were so excited about it that they didn't want to take it apart at the end of the picnic, so one of the moms with a large minivan brought it back to our house and a couple of students managed to get it up the winding stairs from the garage (with a bit of minor disassembly and reassembly) and into our family room. And Ross loved seeing the students' excitement about the tetrahedron as they brought it into the house and then every time he walked by the tetrahedron on what turned out to be the last 24 hours or so of his life, the tetrahedron made him smile. He liked living in a house with things like a giant tetrahedron built by math-loving students in it.

Anyway, the next day he left this earth and since then I have been faced with the question of finding a reason to keep on living in as joyful a manner as possible.

I did not want to continue running the ocnsulting firm we had run together. Economics is known as the "dismal science" for a reason. I do feel our work did some good, but I couldn't continue it without him.

So...instead I decided to *seriously* undertake graduate studies in math. It hasn't been easy. I started out with a graduate class in real analysis in fall 2013, studying a subject for which I'd taken the prereq (undergrad real analysis) forty years earlier! There were more than a few cobwebs in my brain, but I loved the challenge. Then in spring 2014, I took graduate abstract algebra--more cobwebs. What they say about abstract algebra is that it is like doing a jigsaw puzzle in a completely dark room. But for me (given the number of years since I'd taken the undergraduate course prereq to the graduate class in abstract algebra), a better analogy is that I felt like I was doing a jigsaw puzzle in a completely dark room WHILE WEARING THICK MITTENS!

But I persevered and learned a ton--not only about math and the process of learning. This past year, I have been taking a two semester sequence in the theory of statistics. It's been challenging but illuminating. One difficult issue for this most recent class is that I really never had the undergrad prereq class, so unlike the previous year (when I was just dusting off cobwebs in my brain) I was largely working with a lot of completely alien theoretical concepts.

I have decided to use this blog to share more of my journey and my insights as I go.