Let’s start by giving away that the answer turns out to be 50%.
I'm teaching Discrete Math this term - Math 3020. For the first time. And since I’m a philosopher/physicist and not a "real" mathematician, it's required a lot of work. For example, writing problems for a new course is not easy. It takes a lot of knowledge of a subject to be able to create problems that are (1) challenging, (2) not too challenging, (3) require the mastery of important learning goals of the course, and (4) are assessable. Also, knowing when to suggest a particular problem rather than some other, or knowing how to get a student unstuck without doing the work for them — those are black arts.
It's been said that all the best ideas are stolen ideas - And I don’t have a problem with that. So I troll twitter and blogs and other sources looking for interesting applications and ideas, while I also work through the textbook, grade papers, plan class activities, etc., etc.
So I found a tweet with the following:
ProbFact: The airplane boarding probability problem. http://ow.ly/9mqwx.
Go read it. It's a cool problem. Maybe try to figure it out. There’s an answer link in that post.
I was strong, and did not look at the answer for a long time. The answer was not obvious to me, so I started playing with it. Now I don’t know about you, but when I see an interesting problem that I don’t know how to answer, my first reaction is FEAR.
I feel very, very afraid that I won’t be able to figure it out. Or worse, that someone else will find out that I couldn’t figure it out. For a problem like this one that is simple to understand but for which the answer isn’t obvious, the fear is much worse. My impostor syndrome kicks into high gear. My students must not find out!
But I'm trying to get better, so I persevered. I spent about 3 days on the problem. Not continuously - I was at a conference that week, so whenever I had a spare moment between sessions, or on the train back and forth to Boston, I would pull out some paper and think on it some more. I was very proud of myself, to be honest. Maybe later I'll post some of the work I did, but I did all the things I tell my students to do. Play with it. Visualize it. Simulate it. Form conjectures. Test them. Iterate. Prove stuff. It was great exercise.
And I concluded that the answer was some expression involving n, the number of passengers. I checked the answer post, and found that my answer did not match the advertised 50%. Basically, I looked at the answers in the back of the book, and my answer was wrong!
So my fear factor shot up again. What could have gone wrong? Had I misunderstood the problem? Had I made a mistake? Had the problem originator made a mistake?
I checked my work. I repeated everything. I was CERTAIN that you get a different answer for different values of n. So I put it all together. Diagrams, simulations, explanations, and I finally decided — the original poster must have made a mistake.
So I sent him a note, explaining my answer. And breathlessly checked my email every 5 minutes for the next day.
He had not made a mistake. I had misinterpreted the problem statement. I had assumed that the very first passenger, the one that has lost their boarding pass, is not allowed to take their own seat in the random assignment — the originator had meant (and, in hindsight, the problem statement was clear on this) that the first passenger takes a random seat, which MIGHT INCLUDE THEIR OWN ASSIGNED SEAT.
That small difference is enough to change the result from one that depends on the number of passengers, to 50%, regardless of the number of passengers.
But I am strong. I am a math professor. I can handle this kind of thing. And laugh about it. And learn from it. I just worked out, in great detail, the complete analysis of TWO different problems! Huzzah!
But it really made me think about what I (and we) often put our students through. Our students who are a little more fragile, a little less experienced, and a little more afraid of the material to begin with.
How many times have I seen a student work VERY HARD on a problem, only to make some silly mistake somewhere in the huge chain of calculation and logic, ruining the whole thing? How many times have I seen a student NOT try very hard on the next problem? How often have my own instructions been not quite clear enough?
Most students would give up on that particular problem after about two minutes. But my goal is for them to learn to be patient problem solvers. To push through in spite of their initial ignorance and in spite of the difficulties along the way. Obviously I want them to get the right answer too — but how do I offer incentive to push through the way I did? How do I show examples of that kind of patience?
And the thing is, even now, saying that, I can’t stop myself from thinking “Don’t post this! Someone will find out! They’ll take your math professor license away! You should have been able to get that problem in two seconds — what is wrong with you! Hide your shame!”