# Metaphors for learning

This post will be both ironic and trite. Ironic, because I am going to use a team sports metaphor in relation to learning, when I in fact have never been a team sports guy (I am a cardio and yoga guy). Trite, because thousands of people before me have used team sports metaphors in relation to learning.

Nonetheless, indulge me.

Here at Macalester College, our semester starts this week. As many faculty do every term, I am thinking about how to strike the right opening chord with my students. While I believe that teaching them math is important, my experience at a liberal arts college has convinced me that my even greater charge (a superset, if you will) is to teach them something about how to learn. This semester, I decided that to add on to my previous efforts, I should show an incredibly clear visual image on the first day of class that encapsulates the most important messages I have for my students. Here is my attempt.

A colleague of mine recently coined the brilliant term "anectotally," meaning, essentially, "with vehement conviction based on my completely anecdotal evidence." So anectotally speaking, most students begin my class subscribing to a metaphor for learning like the lefthand image in the link above. (Anectotally, and perhaps without realizing it, some faculty subscribe to this as well.) The professor is the bottle of water, each student is a glass, and learning takes place by the professor pouring knowledge (typically, via lecture) into the students. If learning is unsuccessful, it is either because the bottle isn't full enough or the glass isn't big enough.

Thanks to learning science research, we now know that this is not really an appropriate metaphor. The components of a successful learning environment are not a mystery. A wonderful, free book from the National Academies takes a scientific look at how people learn from multiple disciplinary perspectives. A wonderful chapter in that book discusses the design of learning environments. In the future, I'll post in much more detail about this topic. But for now, let me move on to the right-hand image on my slide.

I think a more effective metaphor -- and one that is anectotally surprising to many students -- is that of an athletics team. In the pouring water metaphor, the professor is the main attraction, providing the knowledge. In the team athletics metaphor, the student team is the main attraction. The professor is a coach who is not even always visible. In the pouring water metaphor, the student acquires knowledge passively. In the team athletics metaphor, the students are in action, sometimes as individuals and often as a cooperative group. The team is often on its own, only occasionally receiving guidance from the coach. The coach scaffolds an athletic experience, but the players go through the training regimen and play the game themselves. It is the players' hard work and dedication that are arguably the determining factors for success.

There are certainly many subtleties to this metaphor, and it should not be used too casually. But I am trying to go for simplicity-in-messaging on the first day of class.

Faculty and students, based on your anectotal evidence, what do you think is an apt metaphor for learning?

# Launching the Diversity Nominating Collective

Following up on this blog post and this one, both related to diversity and STEM, I am launching the Diversity Nominating Collective.

# Am I crazypants?

I'm writing from 30,000 feet, en route back home from the SIAM Annual Meeting. And more specifically, I'm writing to follow up on this post.

I had the opportunity to speak with a number of people at the conference... some involved in the selection of SIAM Fellows, some not. Regardless of whom I was talking to, I was really struck by how unpopular my own particular ideas about diversity seem to be... even with those folks who are active about such issues.

Here are two things I experienced.

1. Most people I spoke with feel that the solution to the problem of insufficient diversity amongst the Fellows is to get SIAM members to nominate more women and minorities. I somewhat disagree. I think this is an important part of the solution, but not the whole solution. I would like to see SIAM (the SIAM governance, that is) work actively to increase diversity rather than just say "we just need to encourage more members," which strikes me as punting. Many people pushed back directly against my contention, though.

2. Most people I spoke with believe (as communicated through implicit or explicit statements) that the make-up of the Fellows should proportionally represent the field at large. I stringently disagree. I want to see my professional society's Fellows have a more diverse make-up than the field. This way, the field at large gets sent a message. The choice of fellows is aspirational, from a diversity standpoint. In short: SIAM should lead, not follow. When I said this to people, I frequently got the pushback "we can't lower the standards for Fellows" which I find to be extremely offensive. I believe there are way more people qualified to be Fellows than there are slots for Fellows. We should choose a subset of the qualified people in such a way as to represent diversity.

Am I crazypants? Please tell me. I am asking because I really want to know.

# Professional societies and diversity

Today's rant -- a letter to my main professional society -- is posted below and needs little introduction. I am grateful to the people who have inspired me to speak up about diversity issues. For those who haven't yet spoken up, I hope you will consider doing so.

Dear President [removed], Executive Director [removed], and SIAM Fellows Selection Committee,

As I am so fond of telling my colleagues, I love SIAM. As a fifteen-year SIAM member, I've attended SIAM conferences; judged SIAM poster competitions; read, refereed for, published in, and edited SIAM journals; and, this year, won a SIAM best paper prize. Obviously, I have benefitted substantively from the excellent opportunities SIAM offers. But because I love SIAM so much, I find it necessary to ask some difficult questions related to SIAM Fellows selection and issues of diversity.

Just like I do every summer, this summer I am advising a cohort of undergraduate research students. It so happens that this summer, all four of my students are women. To boot, they are talented, hardworking, mathematically promising women working on advanced topics such as nonlocal PDE, correlated random walks, and more. These are women who have the interest and the ability to have future careers as applied mathematicians.

Besides working with my students on research, I expose them to other aspects of life as an applied mathematician. Yesterday, I found myself in the awkward position of trying to explain to them the 2013 SIAM Fellows list disseminated in the "SIAM Unwrapped" email I received. As we looked through the list of Fellows, we noted that merely two of the 33 new Fellows are women. One of my students commented "that's pretty grim" -- an assessment met with concurrence from the rest of the room.

Previous years are nearly equally grim. Based on my quick scan of data on the SIAM website, here is a tally of the representation of women amongst SIAM fellows:

2009: 194 fellows, 16 women (8%)

2010: 34 fellows, 3 women (9%)

2011: 34 fellows, 2 women (6%)

2012: 35 fellows, 7 women (20%)

2013: 33 fellows, 2 women (6%)

Overall: 330 fellows, 30 women (9%)

What am I to tell my students? What message are they meant to receive from looking at the Fellows roster? Is SIAM's contention that the dismal percentages simply are not a problem? Is the contention that in 2013, there were really only two women applied mathematicians qualified to be named Fellows? Or that because fellowship depends on a nomination process, there's just nothing SIAM can do if women aren't nominated?

One might have idealistically hoped that the Fellows program would strive to provide a little remedy to the underrepresentation of women in the field at large. It appears this is not the case. Furthermore, while I recognize that it is a dicey proposition to judge diversity based on inferences from a list of names and photos, the SIAM Fellows roster appears to include exceedingly few members from other groups (besides women) that are also traditionally considered underrepresented.

Please know that I am tremendously grateful for the hard work you do on behalf of SIAM. Also, I am in an odd position writing this email as an approaching-middle-aged white guy who has undoubtedly benefitted from the social privilege conferred by those demographics. But still, on behalf of young women and minority students, and with concern for the scientific potential that might be lost when some of them leave a field that appears to pay insufficient attention to diversity, I worry.

In case any of you share my worries and feel like discussing the issues I've raised here, I'll be present at the SIAM Annual Meeting in [location] on [date] and free for most of the day (except lunch).

Respectfully,

Chad Topaz

Associate Professor of Mathematics

Macalester College

http://www.chadtopaz.com

# Empathy, humility, and women in science

This post will eventually come around to the topic of women in science -- a topic about which I am wholly unqualified to write. But before getting to my main points, I will have to tell you about the time I had to tell a bunch of administrators why empathy is bad, I will have to explain why preschool graduation makes me angry, and I will have to have a talk just between us guys (but you ladies can listen in if you want).

For a time, I served on my institution's Student Learning Committee, which was charged with writing this Statement of Student Learning. One of the stated outcomes is that students will "demonstrate empathy by acting in a supportive manner that recognizes the feelings and perspectives of another cultural group." I think this is a nonsensical goal. Maybe my objection boils down to semantics, but to me, the basis of empathy is not "support." Empathy has to do with shared experience and emotion, and perhaps a partial dissolution of the boundary between oneself and others.

For instance, take a person from a cultural group other than my own -- perhaps a latino person. I can try to recognize the feelings and perspectives of a latino person, and I can try to act in a supportive manner, and I can even search for some of my own experiences that might approximate certain aspects of the latino person's, but in the end, I cannot deeply empathize about being latino because I have not had the experience of being a latino person in our country.

I laugh at myself when I recall that committee meeting, in which I found myself arguing passionately against having empathy as a goal for our students (at least, with the phrasing that was proposed and adopted). The people who were in that room must think I am real prince. But I felt -- and still do feel -- that there is a great deal of arrogance at the root of that goal.

I think that what we need is not empathy, but humility. Yes, we need to seek what experiences and feelings we might have in common, but at the same time, we need the humility to say "you know what? I can't know exactly what it is like to be you." If we don't have that humility, there is a real danger that our attempts to provide "support" will go off the rails. After all, I am sure that the imperialist missionaries to Africa thought they were being "supportive."

This is all one long preface to disclaim the rest of this post, in which I will write about women in science. I recognize that I am not a woman in science, and therefore, as I said, I am in some respects unqualified to write about this topic. But it is a topic of concern to me nonetheless.

If you can easily picture me in an academic conference room railing against empathy, then you will also be able to picture me getting irate at a preschool graduation. Recently, my daughter's preschool had its annual graduation for the kids who will move on to kindergarten. My daughter has one year left in preschool, but like most families at the school, we attended the ceremony anyway. During part of the ceremony, each graduating kid had a chance to go on stage, stand in front of a microphone, and say what they want to be when they grow up. Let's just for a moment suspend the overarching absurdity of asking five year olds to state a future profession. I didn't record data (I should have) but to the best of my recollection, there were 12 kids graduating, and about half were girls. Of those estimated six girls, four of them said that they wanted to be ballerinas when they grow up.

Before the ballerina lobby starts sending me hate mail, let me say that I think being a ballerina is a perfectly fine career choice. I have a lot of respect for ballerinas, who work incredibly hard to produce a beautiful art form. But I strongly suspect that the 66% of graduating preschool girls at my daughter's school didn't choose ballerina because they admire the grace and hard work. I'll conjecture that they chose ballerina because society tells them "little girls should want to be ballerinas."

I thought about discussing this with the school, but I can already hear their response in my head: "why are you complaining to us? It is important to let little kids dream about being whatever they want to be." Yes and no. Yes it is important to let them dream. But they are not dreaming in a vacuum. They are dreaming in the real world, where all sorts of messages -- good and bad -- get thrown at them. To me, one of the school's roles should be to provide a sensible and somewhat unbiased palate of options to the kids.

This is why my husband and I have a rule in our household that our daughter is not allowed simply to be "a princess," which society also loves to tell her is her life's calling. "You have to be a productive member of society," we tell her. "So you can be a princess poet, or a princess builder, or a princess teacher, or lots of other things. But you don't just get to sit around and be a princess." She has decided that she is a "princess scientist."

Speaking of princess scientists, I recently met a graduate student who is midway through a Ph.D. at one of the top applied math programs in the country. She reminisced to me that in her youth, her dad had once said something to the effect of "I always wanted to have a kid who was good at math but then I found out I was having a girl."

On that note, now it is time for me to talk to the guys. Ladies, feel free to stick around if you want. Guys, I know you are scientists, but I assume you still watch Sports Center. Please put Sports Center on pause and put down the remote. You can keep drinking your beer if you want, but please focus on what I am saying. **We have a women in science problem in this country. **Though this amazing historical anecdote isn't directly about the hard sciences, check it out. Sadly, this kind of bs is equally likely to happen today, though the sexism might be slightly less overt. I hate to be a predictable gay man quoting Madonna as the source of all wisdom, but she sure nailed it -- or at least, part of it -- in the song "(Do You Know) What it Feels Like for a Girl," when she said "When you're trying hard to be your best / could you be a little less?"

Dudes -- when you look around your science departments, do you notice that there are far fewer women faculty? When you look around your science classes, do you notice that there are far fewer women students? Do you ever wonder why studies show that there is a gender gap in scientific publishing with a tantalizingly-nearly-causal link to disparities in financial support, and why female scientific job candidates with equal credentials get evaluated less favorably than male counterparts?

Do you give a crap about this? You should. Because first of all, sexism and gender inequality are ethically wrong. And because selfishly, and closer to home, it might be your friend or your mother or your sister or your daughter who is one day penalized by our unfair system. And because if you care about the progress of science, you should want all the best scientists to be players in field without being held back by systemic bias against their gender.

Guys, you should all be concerned about women in science. So do something about it. If you need a suggestion, start modestly. When you write your next grant proposal, don't look at the broader impacts requirement and groan and think of it as some hurdle that has to be jumped over. Think of it as an opportunity to actually do some freaking good. But before you do some good, talk to every single woman in science you can find and try to learn from them about the challenges faced by women in science, and women-who-might-like-to-be-in-science. And have a little humility, because you don't know what it is like to be them.

# Research: It doesn't take a genius

What message do we want to send our young students about scientific research? I find it troubling that many students (at least, before they have performed any research) have the impression that it is something they cannot do. They envision research as consisting of Einsteinian leaps of knowledge attainable by only a few geniuses. I don't even like to use the word genius (even though I will in this post). I disagree with the fundamental framing it creates, and more practically, I think it's clear that genius is subjective and contextual. Think of all of the people who, over the course of history, have made the transition from crackpot to genius or, in reverse, genius to crackpot.

Regardless, I am not exactly sure of why many students have this view of research, but I can conjecture that it arises from 1) natural fear of the unknown, and 2) the attitudes of other people, including -- and perhaps especially -- students and professors in the academic world. I find many parts of academia have a somewhat elitist view of research. My opinion, to paraphrase the Chef Gusteau character in Ratatouille, is that "anyone can do research."

I think the view that research necessitates genius is a counterproductive and inaccurate. I worry that some students who might make meaningful contributions to the world through research (and of course, there are many other equally valuable ways besides research) are turned off by research-fear before they even start.

**Here's what I think we need to tell students**: 99.9% of research progress consists of teeny, tiny steps in knowledge rather than Einsteinian leaps. Even to take these tiny steps we must stand on the shoulders of the many people who have taken many tiny steps before us. And tiny steps are worthwhile. And research is a community effort, and it is satisfying to be part of a great community of past and future scholars who will take tiny steps to move forward our understanding of and appreciation for the universe. And still, even though we are taking only tiny steps, it's really hard. But very much worth it. And you can do it.

I advise a cohort of undergraduate research students every summer. I am lucky that I have well-prepared, motivated students. That said, my cohort this year has especially knocked my socks off. For example (and forgive the tech speak here) two of them have gone (in the span of two weeks) from not knowing what a partial differential equation (PDE) is to now: understanding how a particular PDE is derived as a model for biological swarming, solving the steady-state problem in 1-d, classifying the compactly-supported swarms that form, calculating the contact angle in the large-mass limit, and solving the full PDE on a computer. In case the tech speak means nothing to you, I can say: this is a lot for two inexperienced undergraduates in the span of two weeks.

In mathematics, we drastically overemphasize being really, really good at math. Being a math genius is neither necessary nor sufficient for doing meaningful mathematical research. I should know, because I am not a math genius and I do math research all the time. Some of it is even decent. I've been reflecting on why my students this summer have been such admirable and productive researchers (an outcome for which I take little or no credit). There are a few critical, basic attitudes and habits that they exhibit:

1. They believe they can make progress. For some reason, they lack the aforementioned research-fear. They have self-efficacy for the task. They are not put off by difficulty or failure. Sometimes they get stuck, and this does not seem to bother them, or make them think that they will not succeed in the end.

2. They put in the time to make progress. I see them in the lab for at least eight hours a day, and they are not on email or Facebook for most of that time. They are working. Still, they do take breaks to keep their heads from exploding, and this is very important too.

3. They are organized. They keep research logs in GoogleDocs and directories of files in Dropbox and "readme" files in those directories in order to organize everything, and to be able to retrace their steps days later when they forget what they did a few days in the past. Whenever they get new results, they write them up nicely in a manuscript file so that they retain in excruciating detail the most successful and relevant parts of the work they've done.

4. They collaborate. Each research project I am running this summer has two students on it. On a given team, sometimes both students do a task and check each other's results. Sometimes they sit down in front of the computer and code together. Sometimes they divide and conquer, and update each other on what has been done. I am convinced their collaborative habits have more than doubled the total productivity they would have as individuals working in isolation.

5. They take ownership. They act like the project is theirs, and not that it is some really hard homework assignment I have given them that they are required to complete. They understand that I don't have the answer to their problem.

6. Still, they are not afraid to ask me for help, guidance, and opinions.

I am sure there are more skills and attitudes I should mention, but this is what has jumped out at me thus far. They have been, in a word, inspirational.

# Learning is more than knowledge

This post runs a bit on the long side. If you are someone interested in my blog because of my recent writing about MOOCs, I promise that I will come back around to MOOCs by the end of this post.

If you only read one thing about human learning, read How People Learn, Chapter 6: The Design of Learning Environments. In case you are unfamiliar with this book, it was published in 2000 by the National Research Council of the U.S. National Academies. The book gives an overview of what is known about human learning from multiple disciplinary perspectives, including neuroscience, psychology, sociology, and so forth. Even though it should go without saying, I will point out that the National Academies are a credible source. This is solid work.

The whole book is useful, but Chapter 6 is a real gem. First, it gives some useful historical context for educational goals, outlining how these are (and always have been) in flux. But the main attraction of the chapter is a discussion of four characteristics that successful learning environments have. These learning environments are

- knowledge-centered
- learner-centered
- assessment-centered, and
- community-centered.

**Knowledge centeredness** refers to attending to the bodies of knowledge that we want students to have. Generic "thinking skills" are not enough. For example, a successful learning environment in the field of history will pay a decent amount of attention to, well, history.

I can feel you rolling your eyes at me because this is so obvious, but it is worth articulating. What is much less obvious -- and much more important to articulate -- is that **knowledge centeredness is not enough to constitute a successful learning environment**. This means that if you are a brilliant historian, if you are at the forefront of historical theory and knowledge, even if you have a better command of both the facts and ideas of history than anyone else in the world, even if you are the most outgoing, charming, and engaging lecturer in the world who weaves course lectures with meaningful examples and entertaining yet relevant anecdotes, if what your course consists of is you talking at students for 60 minutes three times a week, you have not come close to designing the most effective learning environment possible.

There is so much more to worry about besides knowledge. To my eyes, most of the higher education system still puts knowledge at the center, largely to the exclusion of other factors. In doing so, the system embraces a now-antiquated understanding of human learning, envisioning the endeavor of education as an expert pouring knowledge into the brain of a novice. We now know that this is a totally inaccurate model. We must pay attention to other aspects of learning.

**Learner centeredness** means paying attention to the "knowledge, skills, attitudes, and beliefs" that learners bring to their experience. In short, learners all differ from each other in ways that impact their educational experience, and you have to pay attention to these differences. One size does not fit all.

**Assessment centeredness** means giving students ongoing opportunities for formative feedback which they might use to impact their own learning. For example, courses that have only one or two assignments (say, a midterm and a final paper) are far from being well-designed. What feedback do students receive on their learning during the intervening weeks?

**Community centeredness** has several aspects, including creating the classroom environment as a community, and also connecting what happens in the classroom to the larger communities of students' lives: their neighborhoods, states, countries, and so forth. Courses that don't create a classroom community and/or don't connect the course to something bigger fall short of providing the most effective educational experience.

These aren't my opinions. This is what the National Academies think, based on their aggregation of evidence from numerous sources who investigated learning through a variety of disciplinary perspectives.

Why am I thinking about all of this today? I am prompted by discussion around MOOCs, which I have written about in this post and this post. In brief, most (but not all) MOOCs are almost exclusively knowledge centered. This means they won't provide the most effective educational experience. Some MOOC boosters are thinking about the economics of MOOCs and not considering the poor experience being provided. (Again, the experience doesn't have to be poor, but so far, it usually is.) At the same time, some of those faculty decrying MOOCs seem to fear that a MOOC will displace them. A message to these faculty: the more you address the four characteristics addressed above, the harder it is going to be for a MOOC to replace you. You will be providing a vastly superior experience.

Relatedly, and somewhat more broadly, someone recently made a comment to me that "active learning (and perhaps even some of the online media that you want to distinguish from MOOCs) transform the instructor's role," and if I understood correctly, the implication was that this was bad, and constitutes a threat to higher education. I, however, think this is as it should be. We know more than we ever have known about how human learning takes place. We also have better technology than we ever have. These changes suggest why it no longer makes sense to send kids to school with horn books.

In short, it is entirely appropriate that the role of the instructor change to keep up with our expanding technological capabilities, and even more importantly, our expanding understanding of human learning.

# Shocking developments

The semester must be over, because suddenly I find myself blogging again.

For those interested, I have migrated my old blog and other content (at www.integralmathsolutions.com) over to this site, so all of my postings will appear in one place.

This past week's news contained the story of a scientific study that found that electroshocking the brain might improve certain mathematical abilities. And of course, as for a response to this story, the Onion nailed it, as usual.

# Death, math, coincidence, and Chargers football

You may have read the sad news that Junior Seau, former linebacker for the San Diego Chargers, has died of an apparent suicide. Notably, Seau was a member of the Chargers' Superbowl team in 1994. Many articles reporting on his death have commented on an apparent coincidence:

Seau, who played in the NFL for parts of 20 seasons, is the eighth member of San Diego's lone Super Bowl team who has died, all before the age of 45. Lew Bush, Shawn Lee, David Griggs, Rodney Culver, Doug Miller, Curtis Whitley and Chris Mims are the others. Causes of death ranged from heart attacks to a plane crash to a lightning strike.

Without diminishing the tragedy of Seau's death -- or the deaths of his teammates -- let's analyze how coincidental this situation really is. On the way to sussing out the question of coincidence, I'll describe in layperson's terms some basic notions of probability, counting, and survival analysis.

I proceed with caution here, as I am neither a probabilist, a statistician, an actuary, a demographer... or even, really, a sports fan. I am confident that someone with more relevant mathematical training could do a better job than I do below. I welcome comments, corrections, and improvements.

Let's get going. As is often necessary with real-world problems, we will make some simplifying assumptions for tractability. According to Business Week, the average age of an NFL player in 2011 was 27 years old. We'll assume this was true in 1994. And to simplify further, we'll assume that *all* of the Chargers were 27 that year. Right now, it is 2012, so those players would be 45 years old.

Now we turn to life tables, clever little tools used by actuaries and demographers to calculate survival probabilities and life expectancies. There are two kinds of life tables. A *period* life table give the probability of death for people of different ages in a given calendar year. So for instance, a 2012 period life table would give the probability of a 90 year old dying this year, an 89 year old dying this year, and so on. A *cohort* life table shows the probability of death of people born in a particular year over the course of future years. So for instance, for people born in 1974 (my birth year), the cohort table would give the probability that they would die in 1975, 1976, and so on.

Since we are interested in people who were 27 in 1994, the best tool for us to use would be a cohort table for 1967 (since 1994 - 27 = 1967) but I couldn't find one. Instead, I looked at this historical period table for 1967 from the Centers for Disease Control. A period table and a cohort table are not the same thing, but nonetheless, we'll use the period table data to get a rough solution to our problem.

Take a peek at Table 5-3 (page 5-9) and focus on the first and third columns. The first column contains ages. The third column tells the story of 100,000 males. Specifically, it gives the number of the original 100,000 surviving until the age given in the first column. Of the original 100,000, 94,600 survive until age 27. These 94,600 live 27 year olds are our base population. Now we want to know the probability of living (at least) to age 45. Looking in the third column for the entry corresponding to 45, we find 89,456. This means that for those alive at 27, the probability of living to 45 would be 89456/94600 = 0.94562. In case you are unfamiliar with calculating probabilities this way, here's another example. Say a bag contains 10 balls, 3 of which are red. We reach in without looking and grab one. The probability that we choose a red ball is 3/10.

If the probability of an individual alive at age 27 in 1994 living to age 45 in 2012 is 0.94562, the probability of them dying before age 45 would be 1 - 0.94562 = 0.05438. That's because the probabilities have to sum to one (you can either live or die).

Now we have to consider a team. An NFL team has a roster limit of 53, so we'll take that as our team size. What is the probability that 8 players alive at age 27 in 1994, would have died by age 45 in 2012? This is a binomial probability question.

Let's work up to this idea. One basic idea used is that to get the probability of multiple things happening (different players living/dying) you multiply together their probabilities, so long as whether each person lives/dies is independent of the other people. For example, suppose that each day, the weather can be either rainy or sunny. And also suppose that the weather on one day is independent of weather on other days. If the probability of rain each day is 1/3, the probability that Monday and Tuesday will be rainy but Wednesday will be sunny next week would be 1/3 * 1/3 * 2/3 = 2/27. The 2/3 value is the probability of sunny weather, which is 1 - the probability of rain, or 1 - 1/3 = 2/3.

Back to our football problem. We want to know the probability of 8 players dying and the other 53 - 8 = 45 players living. With being the probability of one of the 1994 Chargers living until 45, this is something like

[p^{45} (1-p)^8 = 0.94562^{45} cdot 0.04538^8 = 6.1745 times 10^{-12}].

That's minuscule!

But wait, there's something we haven't accounted for. This is where the second idea related to binomial probabilities comes in. It's not a *specific* 45 players that we require to live, it's *any* 45 players. And similarly, it could be *any* 8 players who die. To account for this, we need to multiply by the number of different ways there are to select 45 live players and 8 dead players from a team of 53. This is

[frac{53!}{45!8!} = 886322710]

where the exclamation point is the *factorial* function. In case you don't know this function,

begin{eqnarray*}

1! & = & 1

2! & = & 2 cdot 1 = 2

3! & = & 3 cdot 2 cdot 1 = 3

end{eqnarray*}

and so on. So to get the overall probability that of the team of 53 Chargers in 1994, exactly 8 of them would die by 2012, we take

[6.1745 times 10^{-12} times 886322710 = 5.4726 times 10^{-3}]

or about one-half of one percent. This is small, but not negligible.

We can go even one step further. We've thus far been focused on the Chargers Superbowl team. Instead, we could ask about the probability that there would be exactly 8 deaths on *any* NFL team. We can figure this out with another binomial calculation, first noting that in 1994 there were 28 NFL teams. The probability that one of them would have exactly 8 deaths, as we have said, is which we'll call . The binomial calculation for exactly one team of 28 having exactly 8 deaths is

[r^1 (1-r)^{27} frac{28!}{1!27!} = 0.13213.]

To recap, that's a 13% probability that of all the NFL teams in 1994, exactly one of them would have exactly 8 players die by 2012 (with lots of simplifying assumptions thrown in). 13% is no guarantee, but neither is it so remote that we should be really surprised by it. By the way, note that when we say that one team has exactly 8 players die, this says nothing about the other teams. They could have fewer or more than 8 players die.

Let me finish with some inspirational words. As disclaimed above, I am not a probabilist. My only training in probability came when I was 11 years old. The calculation above is a very rough (an assuredly flawed) estimate based on data gathered from Googling and a few elementary notions of life tables that I picked up from the Internet. No fancy math was involved. I encourage you -- like I encourage all my students -- not to be afraid of trying to estimate things. It's do-able, it's interesting, and it's an excellent way to apply your powers of critical thinking to what's going on in the world.

**Addendum**: My excellent colleague Victor Addona has chimed in on this. He is a real pro who a) has a degree in statistics, b) publishes in the field of survival analysis, and c) does interesting and innovative work related to sports statistics. Victor sagely prefers a refined phrasing of my question, namely "of the 28 teams, what is the probability that the one with the most deaths would have at least 8 deaths?" He answered this question with a simulation conducted in the statistical package R, and his answer is about 19%. My very rough estimate is of 13% more like a lower bound to the answer to Victor's question. In short, the eight deaths are even slightly less coincidental than my post suggests.

[This post has been modified. Thanks to Louisa Bradtmiller for pointing out a factual error in the original post, in which I stated that the Chargers had *won* the Superbowl. Thanks to Victor Addona for pointing out a typo when I originally wrote down 1 - p incorrectly.]