# How mathematicians are trying to make NFL schedules fairer

ESPN had a nice article about applied mathematicians at the University of Buffalo who are working with the NFL to create fairer schedules. A few quotes:

“This is a field I’ve worked in for 46 years, including 43 as a professor,” Karwan said by phone last week. “I’ve worked on very difficult problems that take more than 12 hours on the supercomputer to solve. And this is by far the hardest any of us have ever seen.”

And:

In developing the schedule, NFL assigns “penalty points” to outcomes such as three-game road trips, games between teams with disparate rest, and road trips following a Monday night road game. In their final proof of concept in 2017 before receiving the grant, Karwan and Steever took the 2016 schedule and lowered the penalty total by 20 percent…

The first step is based in both math and reality. Before creating the schedule, the NFL identifies a small number of games — usually between 40 and 50 — to lock in. The league refers to this as “seeding.” It helps accommodate expectations from television partners for key games in certain time slots, as well as about 200 annual requests from owners who prefer their stadiums not be used in a given week because of concerts, baseball games, marathons and other potential complications…

At that point, the NFL asks its computers to run schedule simulations until it finds one that has an acceptable penalty total. Usually that means juggling the 40 to 50 pre-seeded games. Karwan and Steever believe the key to improving the schedule is to better choose those pre-seeded games, allowing the computer to see stronger schedules that would otherwise be blocked by the initial choices through a process known as integer programming.

Not surprisingly, this research was publicized by the MIT Sloan Sports Analytics Conference, an annual conference dedicated to the integration (insert rim shot) of mathematics and sports.

# Veteran teacher shows how achievement gaps in STEM classes can be eliminated

This press release from UC Santa Cruz definitely gave me food for thought about new things to try in my own classes. A few short snippets:

[Professor Tracy Larrabee] uses a three-pronged approach to support underrepresented students in her class.

“The first is that we have had a very diverse teaching staff,” she said. “We have one professor, four TAs and four MSI tutors, and during this time it just happened that of those people, half were female, we always had at least one African American, one Latinx, and one non-gender conforming tutor so that everyone could feel a connection to someone on the teaching staff.”

“Another technique I use is to emphasize failure as the appropriate path to learning,” she said. “Engineering is hard; it’s good to fail the first time you attempt a problem. People who fail at a problem the first time tend to retain things better than those who luck into the right answer.”

Her final tactic is to explicitly discuss stereotype threat. This is the risk that someone (i.e., from an underrepresented minority) might take routine negative experiences as confirmation that they are fundamentally unsuited for something like higher education.

“One of my African American MSI tutors—who are extremely high achieving students selected to provide supplemental tutoring to others—told me it was like having a light bulb go off for him,” Larrabee said. ”Until I discussed the issue in class, he felt like he didn’t belong in this major, but after we talked about stereotypes, he realized it wasn’t that he was unsuited for the material. It was hard for everyone!”

# My Favorite One-Liners: Part 118

A cheap but funny Dad joke.

# My Favorite One-Liners: Part 117

I absolutely love this joke. The integral looks diabolical but can be computed mentally.

For what it’s worth, while it was able to produce an answer to as many decimal places as needed, even Wolfram Alpha was unable to exactly compute this integral. Feel free to click the link if you’d like the (highly suggestive) answer.

# My Favorite One-Liners: Part 116

This awful pun is just in time for Valentine’s Day.

# My Favorite One-Liners: Part 115

I credit Math With Bad Drawings for this new weapon in my arsenal of awful mathematical puns.

# Fun with Proportions and Atoms

I came across this fun video on proportions, imagining how large some objects would be if atomic (and subatomic) length scales were magnified to the size of a tennis ball.

# The End of “Statistical Significance”

I’ve linked to a number of articles about the misuse of p-values. Recently, I read a nice article in the October/November 2019 issue of MAA Focus summarizing a conversation between the Executive Directors of the Mathematical Association of America and the American Statistical Association about the ASA’s call to eliminate the use of p-values. Per copyright, I can’t copy the entire article here, but let me quote the lead paragraph:

In March 2016, the American Statistical Association took the extraordinary step of issuing a Statement on p-Values and Statistical Significance. This spring, the association went even further, publishing a massive special issue of its journal The American Statistician entitled Statistical Inference in the 21st Century: A World Beyond p<0.05. The lead editorial in that special issue called for the end of the use of the concept of statistical significance.

It’s going to be a while before entrenched statistics textbooks catch up with this new standard of professional practice.

Here’s an NPR article on the issue: https://www.npr.org/sections/health-shots/2019/03/20/705191851/statisticians-call-to-arms-reject-significance-and-embrace-uncertainty

Other articles cited in the MAA Focus article:

# Adding by a Form of 0 (Part 4)

In my previous post, I wrote out a proof (that an even number is an odd number plus 1) that included the following counterintuitive steps:

$2k = (2k - 1) + 1 = ([2k - 1 - 1] + 1) + 1$

A common reaction that I get from students, who are taking their first steps in learning how to write mathematical proofs, is that they don’t think they could produce steps like these on their own without a lot of coaching and prompting. They understand that the steps are correct, and they eventually understand why the steps were necessary for this particular proof (for example, the conversion from $2k-1$ to $[2k - 1 -1]+1$ was necessary to show that $2k-1$ is odd).

Not all students initially struggle with this concept, but some do. I’ve found that the following illustration is psychologically reassuring to students struggling with this concept. I tell them that while they may not be comfortable with adding and subtracting the same number (net effect of adding by 0), they should be comfortable with multiplying and dividing by the same number because they do this every time that they add or subtract fractions with different denominators. For example:

$\displaystyle \frac{2}{3} + \frac{4}{5} = \displaystyle \frac{2}{3} \times 1 + \frac{4}{5} \times 1$

$= \displaystyle \frac{2}{3} \frac{5}{5} + \frac{4}{5} \times \frac{3}{3}$

$= \displaystyle \frac{10}{15} + \frac{12}{15}$

$= \displaystyle \frac{22}{15}$

In the same way, we’re permitted to change $2k-1$ to $2k-1 + 0$ to $2k -1 - 1 + 1$.

Hopefully, connecting this proof technique to this familiar operation from 5th or 6th grade mathematics — here in Texas, it appears in the 5th grade Texas Essential Knowledge and Skills under (3)(H) and (3)(K) — makes adding by a form of 0 in a proof somewhat less foreign to my students.