God Plays Dice

In class today, I said approximately this:

So people decide whether to have children by flipping a coin, and if it comes up tails they have a kid, and if it comes up heads they don't. They repeat this until it comes up heads. This is probably not a good model of how people decide whether or not to have children, but maybe it's good in the aggregate. And anyway this isn't a class about how people decide whether to have kids.

Then there are two kinds of children, girls and boys -- well, not always, but this isn't a class about that -- and each child is equally likely to be a boy or a girl -- well, wait, that's not exactly true, but it's not a horrible assumption about how reproduction works on a cellular level, but this isn't a class about that either.

And people's decisions to stop having kids is independent of the sex of the children they've had -- which says this isn't China, because people do interesting things under the one-child policy -- but this isn't a class about that.

(Then I actually did some math -- namely, assume that the number of children a random family has is geometrically distributed with some parameter p, and assume that all children are equally likely to be male or female and that their genders are independent of the gender of any other children or the number of children in the family. Pick a random family with no boys. What is the distribution of the number of children they have?)

From Jones, Game Theory: Mathematical Models of Conflict (link goes to Google Books), in the preface:

"Some teachers may be displeased with me for including fairly detailed solutions to the problems, but I remain unrepentant [...] In my view any author of a mathematical textbook should be required by the editor to produce detailed solutions for all the problems set, and these should be included where space permits."

By the way, Jones was writing this in 1979; presumably if space does not permit, in the present day solutions can be posted on the author's web site. (This will pose a problem if websites move, though; perhaps an arXiv-like electronic repository of solutions would be a good idea?) A reviewer at Amazon points out that the inclusion of solutions to problems might be an issue for those choosing to assign the textbook in a course where homework is collected and graded. Jones has a PhD from Cambridge and as far I can tell was at Imperial College, London at the time of writing; the willingness to include solutions may have something to do with the difference between the British and American educational systems.

I've seen frustration about the lack of provided solutions in textbooks on the part of my more conscientious students. (This isn't with regard to this text - I'm not currently teaching game theory - but with regard to other texts I've used in other courses.) They want to do as many problems as they can, which is good. This practice of leaving out the solutions is perhaps aimed at the median student - in my experience the median student does all of the homework problems but would never consider trying anything that's not explicitly assigned. (And although I don't know for sure, the student who goes out of their way to get a bootleg solutions manual is probably not the conscientious student I'm referring to.)

a six year old solves the subset sum problem: a post on an operations research science fair project.

Roll five dice. Can you make a subset of the numbers appearing on them add up to 10? to 12? Laura McLay's six-year-old daughter did this experiment as a science fair project, solving the problem visually with bars of length 1 inch to 6 inch fitting in a 10-inch or 12-inch frame.

This is both the cutest and most awesome thing I have seen so far this morning. (But be aware that I have only been awake for half an hour.) There should be more things like it.

It's also what I'd do if I were seeing this problem for the first time. (Although, being a Grownup, I'd skip the manipulatives. Which would probably make it less fun.)

An example of a bad word problem, from Frank Quinn's article The Nature of Contemporary Core Mathematics, who is at Virginia Tech:

Bubba has a still that produces 700 gallons of alcohol per
week. If the tax on alcohol is 1ドル.50 per gallon, how much tax will Bubba pay in amonth? [Set up and analyze a model, then discuss applicability of the model.]

I have given an example with obvious cultural bias because I am not sure I could successfully avoid it. At any rate students in my area in rural Virginia would think this problem is hilarious. We have a long tradition of illegal distilleries and they would know that Bubba has no intention of ever paying any tax.

Did you know that there are actually things to say about whether zero is even or odd (from Wikipedia)? Obviously it is, but the math-ed folks have seriously looked at this.

I found this via a comment by John Thacker at The Volokh Conspiracy. There's a poll there; right now 2% of people have said 0 is odd, 51% even, 43% both, 4% neither. I can kind of understand what's going on with people saying "neither" (perhaps they're getting this from some elementary-school notions), but how is 0 odd?

My answer: yes, zero is even, because it's twice an integer.

(Or because the identity permutation on n letters is an element of the alternating group A_n -- I've been thinking about permutations a lot lately. But if you understand that, you probably are like me, think zero is even, and didn't even think there was anything to discuss.)

Incidentally, sometime recently -- I forget the context -- I saw something that referred to the Gaussian integer a+bi as "uneven" if and only if a and b had different parity.

Here's a fascinating article on what math is good for in biology: The "Gift" Of Mathematics in the Era of Biology, by Lynn Arthur Steen. Steen gives lots of examples about what math is good for in biology. Somewhat surprisingly to me, he doesn't really mention one of the first things that came to mind, namely the use of combinatorial techniques to study the genome, which is nothing but a word on a four-letter alphabet. It's possible that he subsumes this in "statistics", though; to take a simple example, one might want to know how many times a certain sequence of bases would appear in a "random" genome in order to determine whether the fact that such a pattern appears often is signal or noise. Still, he makes the point that while the traditional mathematics curriculum (with lots of calculus and differential equations) takes its scientific inspiration from physics, biology is ascending.

A shorter version of this article is available at The Chronicle of Higher Education.

(How did I find this? Steen was one of the authors of Counterexamples in Topology, which I mentioned yesterday, so I went over to his web site.)

I came across an article about a student who got a perfect score on both the ACT and the SAT. (These are the two standardized tests used for university admissions in the US; generally schools on the coasts use the SAT and schools in the interior of the country use the ACT, although this is a vast generalization. The geographical separation seems to be a function of where the tests originated, in Iowa and New Jersey respectively.

This article (which I'm not linking to because I found it by googling a student, and the student is probably already not happy that this is all over the Internet) points out that less than 1 percent of students get a perfect score on each of these tests. (As you'll see below, this is quite an understatement.) I think we're supposed to come to the conclusion that less than 1 in 10000 students would get a perfect score on both.

But of course scores on these tests are positively correlated! So the probability of getting a perfect score on both tests is much higher than the product of the probability of getting a perfect score on each. (I don't think knowing that would help you on the SAT. But it's been a while. In my day they were out of 1600.)

This article indicates that 294 of the high school seniors graduating in 2008 got a perfect score on the SAT, and 514 out of 1.4 million got a perfect score on the ACT. Wikipedia puts the number of SAT takers at 1.5 million per year; let's knock this down to 1 million since some people take the test more than once and we're talking about the total number of students. So the probability that a random student who takes both tests gets a perfect score on both is something like (294/1000000) (514/1400000), which is about one in 1.3 million. The number of students taking both tests is less than this (many people only take one of the two), so assuming independence there should be less than one student per year who gets a perfect score on both tests.

But a quick glance at the Google results will convince you that there are a few students per year who pull this off.

Counterexamples in Probability And Statistics (Joseph P. Romano and A. F. Siegel) and Counterexamples in Probability and Real Analysis (Gary L. Wise and Eric B. Hall) both seem to be books in the tradition of Counterexamples in Analysis (Bernard Gelbaum and John Olmsted) and Counterexamples in Topology (Lynn Arthur Steen and J. Arthur Seebach. These are books that collect the examples just "outside" the boundaries of the various standard theorems, the point being to explain why one needs the seemingly strange collections of hypotheses that seem to begin every analytic theorem. (Hence the tags "education" and "teaching"; I've often seen these counterexample books described as "anti-textbooks", and as being complementary to standard textbooks which often spend most of their time telling you what's true.)

It seems that these books are concentrated on the analytic end of mathematics; I couldn't find, for example, books of counterexamples in algebra, combinatorics, or number theory. There is, however, Theorems and Counterexamples in Mathematics. My sense is that the nonexistence of these books is connected to the fact that those fields don't seem quite as rife with theorems where all the work is hidden in the definitions.

An interesting statistic from this article from the Madison (Wisconsin) Times: in fall 2008, the average grade in social work courses was 3.70 on a 4.0 scale, and the average grade in mathematics courses was 2.79. (The article doesn't indicate why these two departments were chosen, but I suspect they're at the extremes of the distribution.) This is a fact offered in an article about how students feel more "entitled" to high grades than in the past.

I don't want to comment on how students may or may not feel entitled to high grades. Most of the information I've seen indicates that this sort of entitlement is more common now than in the past; I haven't been teaching long enough to feel like I can comment intelligently on historical trends. (And I wouldn't want to include data from when I was taking classes, because my friends and I may or may not have been a good sample.)

From 3σ → left.

I'm currently teaching a course "Ideas in Mathematics" in our summer session. This is a course generally taken by students not in technical fields; quickly speaking, my syllabus is some basic number theory, different notions of infinity, some bits of geometry (polyhedra, letting them know that there is such a thing as non-Euclidean geometry, etc.), fractals and chaos, and a smattering of probability. This is a course that's not a prerequisite for anything and the students aren't going into fields where they'll need math, so I, like a lot of other people teaching this class, take the approach of showing them that "math is beautiful" rather than that "math is useful".

So today I'm showing my students that the rationals are countable, first by the standard proof and then by the superior Calkin-Wilf proof. I find the Calkin-Wilf proof aesthetically superior because the "standard" proof, in my opinion, is "really" a proof that the set of pairs of natural numbers is countable; we then just cross off the pairs which aren't in lowest terms as a sort of afterthought. As a result, it's difficult to answer questions like "what's the 1000th rational number in the `standard' enumeration?". Then I will show them that the reals are uncountable, using Cantor's diagonalization argument.

While preparing today's class, I realized that I don't know when I learned that the rationals are countable and the reals are uncountable. Is this even part of the "standard" curriculum for math majors? These feel like facts that I have always known; presumably I picked them up from some popular mathematics book at an early age. Do any of you remember when you learned this?

I just wondered -- what is the typical age of a PhD recipient? A bit of Googling turned up this table from Inside Higher Ed, which conveniently sorts by discipline; it reports on an NSF brief. Mathematics and physics are tied for second lowest median age at 30.3; chemistry is the only discipline that's lower, at 29.6.

The table I linked to also gives the median time from getting the bachelor's degree to getting the PhD; by subtraction one can get some number that is a "typical" age of bachelor's degree receipt for students who eventually get a PhD. The median time from bachelor's degree to PhD in mathematics is 7.9 years. Subtraction, 30.3 - 7.9, gives 22.4 as a "typical" age (the difference of medians, which isn't really meaningful) for students getting a bachelor's degree who eventually go on to get a PhD in math. (The highest typical age at bachelor's degree is 25.3, for people getting PhD's in education.) This is the minimum among all eighteen disciplines covered here. It's hard to imagine a median much lower than that given the age at which students typically enter formal education and the number of years it takes.

I interpret this as saying that students who get PhD's in mathematics are less likely to take time away from formal education between high school and college or to take longer than the traditional four years to graduate from college. I'd be interested to see if this is because students who spend time away from formal education "lose" whatever mathematics they knew and have trouble picking it back up again; it's a popular conception that mathematics is more "hierarchical" and so this is more of a problem there than in other fields. (Not having much experience with other fields, I can't say.)

Also, chemistry has a median registered time to degree (time from entering a doctoral program to receiving the PhD) of 6.0 years; the next lowest is mathematics at 6.8. Why is chemistry such an outlier?

At Uncertain Principles and Unqualified Offerings there has been talk about how students in disciplines where the numbers come with units seem to be conditioned to expect numbers of order unity. (And so are professional scientists, who deal with this by defining appropriate units.)

Mathematicians, of course, don't have this luxury. But we are conditioned to think, perhaps, that rational numbers are better than irrational ones, that algebraic numbers are better than transcendental ones (except maybe π and e), and so on. In my corner of mathematics (discrete probability/analysis of algorithms/combinatorics), often sequences of integers a(1), a(2), ... arise and you want to know approximately how large the nth term is. For example, the nth Fibonacci number is approximately φⁿ/5^1/2, where φ = (1+5^1/2)/2 is the "golden ratio". The nth Catalan number (another sequence that arises often) is approximately 4ⁿ/(π n³)^1/2. In general, "many" sequences turn out to satisfy something like

a(n) ~ p qⁿ (log n)^r n^s

where p, q, r, and s are constants. There are deep reasons for this that can't fully be explained in a blog post, but have to do with the fact that a(n) often has a generating function of a certain type. What's surprising is that while p and q are often irrational, r and s are almost never irrational, at least for sequences that arise in the "real world". Furthermore, they usually tend to be "simple" rational numbers -- 3/2, not 26/17. If you told me some sequence of numbers grows like πⁿ I'd be interested. If you told me some sequence of numbers grows like n^π. I'd assume I misheard you. Of course, there's the possibility of sampling bias -- I think that the exponents tend to be rational because if they weren't rational I wouldn't know what to do! They do occur -- for example, consider the Hardy-Ramanujan asymptotic formula for the number of partitions p(n) of an integer n:

p(n) ~ exp(π (2n/3)^1/2)/(4n √3)).

I know this exists, but it still just looks weird.

(This is an extended version of a comment I left at Uncertain principles.)

I will let a more eloquent man speak for me.

We will build the roads and bridges, the electric grids and digital lines that feed our commerce and bind us together. We will restore science to its rightful place, and wield technology's wonders to raise health care's quality and lower its cost. We will harness the sun and the winds and the soil to fuel our cars and run our factories. And we will transform our schools and colleges and universities to meet the demands of a new age. All this we can do. And all this we will do.

- Barack Obama, from his inaugural address.

While at the laundromat today, I saw an episode of Mathematics Illuminated, about game theory. This is a series of 13 half-hour episodes on "major themes in the field of mathematics"; the game theory episode covered Nash equilibria, the prisoner's dilemma, evolutionarily stable strategies, etc. (I may be leaving out some things, because there was laundry-machine noise.) Their intended audience seems to be high school teachers and interested but perhaps mathematically unsophisticated adult learners.

It appears you can watch the whole series online. The main mathematician involved is Dan Rockmore of Dartmouth.

(And no, I don't know what channel it was on. Like I said, it wasn't my TV.)

Probability lessons may teach children how to weigh life’s odds and be winners, from The Times (London), January 5. David Spiegelhalter, of Cambridge University, claims that people need a better understanding of probability and statistics -- not in order to do mathematics, but to help them out in everyday life. He describes what needs to be taught as "risk literacy".

I agree. Much of the curriculum in US schools (the article is about British schools, but from what I can tell this is true there as well) basically seems to be set up so that students will be prepared to learn calculus either at the end of high school or right at the beginning of college. But there's so much more than calculus. And so many of the numbers people run into every day are probabilities or statistics of some sort.

Via X=Why?, I found an article on how a criminal investigation lab in California is inviting students to come in and showing them that math is useful for solving crimes. (In the CSI way -- figuring out how blood splatters, say -- not in the Numb3rs way.) This is certainly a good thing to do.

But can you make sense of this?

Craig Ogino, the department's crime lab director, started the event by offering a prize of 10ドル to the student who could use trigonometry to determine the number in gallons of a mixture used to make methamphetamine, based on his sketch.

I'm assuming that trigonometry was actually used for something else -- like, say, the aforementioned blood splattering analysis, seen later in the article -- and that the reporter made a mistake. But I'm not totally sure. Any thoughts?

A Russian teacher in America, by Andrei Toom.

It's what it sounds like. Toom repeats the familiar litany that in America, students learn for grades, and only incidentally for learning; it's an interesting read, mostly because of the perspective that he's able to bring to it as somebody who didn't grow up within the American system.

Mr. K points out that the Etch-a-Sketch toy helps students understand graphing. You turn one knob and the x-coordinate changes; you turn the other knob and the y-coordinate changes.

That's a good point -- but I was surprised to learn that eighth graders (that's who Mr. K teaches) are familiar with the Etch-a-Sketch.

Michael Alison Chandler, an education reporter for the Washington Post, is currently taking a high school "Algebra II" class. As she said in her first post, she's "not a math person" and yet math increasingly comes up in what she's reporting on, so she wanted to see what math education in the schools is like these days.

In today's post, she writes:

It's difficult to describe how or why math works. It's easier to just write the formula and say, "Do this." Several readers have commented on this blog that what's often missing from math education is more of a focus on why certain applications work. I agree. It's harder to remember what to do, if you don't have some sense of why it works.

This is something that many mathematicians should remember. But there's a balance to be struck; if you spend too much time on the theory ("why" it works) and never apply it that's silly too.

As you may have guesed, I primarily view mathematics as a tool for solving various interesting problems; I'm often not interested in the theory for its own sake, but I do like that knowing "theoretical" things makes lots of problems more tractable. Often such problems involve lots of calculation, and as many of you know it's easy to get lost in a calculation if you don't understand why you're doing it. But if you never calculate, and you only prove general results, I feel like you're ignoring why this subject exists in the first place. (Mathematicians of a more theoretical inclination may, of course, disagree.)

The Onion exposes poor math teaching in the schools.