Reading the Comics, March 21, 2016: New Math And The NCAA Edition

Terri Libenson’s The Pajama Diaries for the 20th of March mentions, among “reasons for ice cream”, the stress of having “helped with New Math”. It’s a curious reference, to me. I expect it refers to the stress of how they teach arithmetic differently from how it was when you grew up. I expect that feeds any adult’s natural anxiety about having forgot, or never really been good at, arithmetic. Add to that the anxiety of not being able to help your child when you’re called on. And add to that the ever-present fear of looking like a fool. There’s plenty of reason to be anxious.

Terri Libenson’s The Pajama Diaries for the 20th of March, 2016.

Still, the reference to “New Math” is curious since, at least in the United States, that refers to a specific era. In the 1960s and 70s mathematics education saw a major revision, called the “New Math”. This revision tried many different approaches, but built around the theory that students should know why mathematics looks like it does. The hope was that in this way students wouldn’t just know what eight times seven was, but would agree that it made sense for this to be 56. The movement is, generally, regarded as a well-meant failure. The reasons are diverse, but many of them amount to it being very hard to explain why mathematics looks like it does. And it’s even harder to explain it to parents, who haven’t gone to school for years and aren’t going to go back to learn eight times seven. And it’s hard for many teachers, who often aren’t specialists in mathematics, to learn eight times seven in a new way either.

Still, the New Math was dead and buried in the United States by the 1980s. And more, Libenson is Canadian. I don’t know what educational fashions, and reform fashions, are like in Canada. I’m curious if Canadian parents or teachers could let me know, what is going on in reforming Canadian mathematics education? Is “New Math” a term of art in Canada now? Or did Libenson pick a term that would communicate efficiently “mathematics but not like I learned it”?

Rudolph Dirk’s The Katzenjammer Kids on the 20th reprinted the strip from the 5th of September, 1943. I mention it here because it contains an example of mathematics talk being used as signifier of great intelligence. The kids expound: “Now, der t’eory uf der twerpsicosis iss dot er sum uf circumvegetatium und der horizontal triggernometry iss equal to der … ” and that’s as far as it needs to go. It isn’t quite mathematics, but it is certainly using a painting of mathematics to make one look bright.

Rudolph Dirk’s The Katzenjammer Kids for the 5th of September, 1943, and rerun the 20th of March, 2016. I know it’s a lot of text to read; I’m sorry.

Mark Anderson’s Andertoons got its appearance in here the 20th. It’s got a student resisting the equivalent fractions idea. he kid notes that 1/2 might equal 2/4 and 4/8 and 8/16, but “the ones on the right feel like more bang for your buck”. The kid has a point. These are all the same number. It’s usually easiest to work with the smallest representation that means what you need. But they might convey their meanings differently. I get a different picture, at least, in speaking of “half the class not being done with the assignment” versus “16 of the 32 students aren’t done with the assignment”.

Charlie Podrebarac’s CowTown for the 20th of March claims Charlie could “literally paper the Earth” with losing NCAA brackets. As I make it out, he’s right. There are 2⁶³ possible NCAA brackets, because there are 63 matches in the college basketball tournament. All but one of these are losing. If each bracket fits on one sheet of paper — well, how big is a sheet of paper? If each bracket is on a sheet of A4-size paper, then, each page is 1/16th of a square meter. This is easy to work with. Unfortunately, if Charlie cares about the NCAA college basketball tournament, he’s probably in the United States. So he would print out on paper that’s 8 ½ inches by 11 inches. That’s not quite 1/16th of a square meter or any other convenient-to-work-with size. It’s 93.5 square inches but what good is that?

Well, I will pretend that the 8 ½ by 11 inch paper is close enough to A4. It’s going to turn out not to matter. 2⁶³ is 9,223,372,036,854,775,808. Subtract one and we have 9,223,372,036,854,775,807. Big difference. Multiply this by one-sixteenth of a square meter and we have about 576,460,752,000,000,000 square meters of paper. I’m rounding off because it is beyond ridiculous that I didn’t before. The surface area of the Earth is about 510,000,000,000,000 square meters. So if Bob picked every possible losing bracket he could indeed literally paper the Earth a thousand times over and have some paper to spare.

T Shepherd’s gentle and sweet Snow Sez for the 21st of March is a bit of humor about addition and the limits of what it can tell us.

Ruben Bolling’s Super-Fun-Pak Comix for the 21st of March is a Guy Walks Into A Bar that depends on non-base-ten arithmetic for its punch line. I’m amused. I learned about different bases as a kid, in the warm glow of the New Math. The different bases and how they changed what arithmetic looked like enchanted me. Today I know there’s not much need for bases besides ten (normal mathematics), two (used by computers), and sixteen (used by people dealing with computers). (Base sixteen converts easily to base two, so people can understand what the computer is actually doing, while being much more compact, so people don’t have to write out prodigiously long sequences of digits.) But for a while there you can play around with base five or base twelve or, as a horse might, base four. It can help you better appreciate how much thought there is behind something as straightforward as “10”.

What Are The Chances Of An Upset?

I’d wondered idly the other day if a number-16 seed had ever lost to a number-one seed in the NCAA Men’s Basketball tournament. This finally made me go and actually try looking it up; a page on statistics.about.com has what it claims are the first-round results from 1985 (when the current 64-team format was adopted) to 2012. This lets us work out roughly the probability of, for example, the number-three seed beating the number-14, at least by what’s termed the “frequentist” interpretation of probability. In that interpretation, the probability of something happening is roughly how many times the thing you’re interested in happens for the number of times it could happen. From 1985 to 2012 each of the various first-round possibilites was played 112 times (28 tournaments with four divisions each); if we make some plausible assumptions about games being independent events (how one seed did last year doesn’t affect how it does this year), we should have a decent rough idea of the probability of each seed winning.

According to its statistics, and remarkable to me, is that apparently the number-one seed has never been beaten by the number-16. I’m surprised; I’d have guessed the bottom team had at least a one percent chance of victory. I’m also surprised that the Internet seems to have only the one page that’s gathered explicitly how often the first rounds go to the various seeds, although perhaps I’m just not searching for the right terms.

From http://bracketodds.cs.illinois.edu I learn that Dr Sheldon Jacobson and Dr Douglas M King of the University of Illinois (Urbana) published an interesting paper “Seeding In The NCAA Men’s Basketball Tournament: When is A Higher Seed Better?” which runs a variety of statistical tests on the outcomes of March Madness tournaments and finds that the seeding does seem to correspond to the stronger team in the first few rounds, but that after the Elite Eight round there’s not the evidence that a higher seed is more likely to win than the lower; effectively, after the first few rounds you might as well make a random pick.

Jacobson and King, along with Dr Alexander Nikolaev at SUNY/Buffalo and Dr Adrian J Lee, Central Illinois Technology and Education Research Institute, also wrote “Seed Distributions for the NCAA Men’s Basketball Tournament” which tries to model the tournament’s outcomes as random variables, and compares how these random-variable projections compare to what actually happened between 1985 and 2010. This includes some interesting projections about how often we might expect the various seeds to appear in the Sweet Sixteen, Elite Eight, or Final Four. It brings out some surprises — which make sense when you look back at the brackets — such as that the number-eight or number-nine seed has a worse chance of getting to the Sweet Sixteen than the eleventh- or twelfth-seed does.

(The eighth or ninth seed, if they win, have to play whoever wins the sixteen-versus-one contest, which will be the number-one seed. The eleventh seed has to beat first the number-six seed, and then either the number-three or the number-14 seed, either one of which is more likely.)

Meanwhile, it turns out that in my brackets I had picked Connecticut to beat Villanova, which has me doing well in my group — we get bonus points for calling upsets — apart from the accusations of witchcraft.