These are two times I have used math in my life in the last few days, beyond math I do for fun, or to teach students.
(I work at a school that takes kids on a few wilderness expeditions each semester. We’re going to the canyons of southeastern Utah in the beginning of February.)
We have 26 boys and 22 girls who we need to break into 5 trips, with 3-4 instructors on each trip. Two of our permits have a limit of 12 people per trip, the other three have a limit of 15 people. Each trip is broken into smaller, single-gender groups of 3-4 boys or girls who will camp and cook together (with the instructors acting as one more group). The goal is to balance genders and keep group sizes consistent to the extent possible.
This problem was passed along to me, as the woman at my school who makes the groups couldn’t figure out a way to do it without a trip of 8 students, which she wanted to avoid. I won’t reveal a solution here, but it wasn’t trivial, and involved some fascinating thinking about constraints.
I just finished the Level 1 course from the American Institute for Avalanche Research and Education, “Decision Making in Avalanche Terrain”.
We did this:
After a long ski in, digging a pit to examine the snow pack, and an earlier run, we descended for our final turns of the day:
It was a great run. Except that, if you look closely, on the top right just below the steepest part of the slope, you can see the outline of a small avalanche that I triggered and was caught in.
It was extremely small, maybe 10 feet by 10 feet, and less than a foot deep. I wasn’t buried at all, and skied away, but watching the slab break and slide beneath me was one of the most terrifying things I’ve ever experienced.
Interestingly, the vast majority of serious injuries and deaths from avalanches in the backcountry are triggered by the most experienced skiers. It’s a simple misunderstanding of two issues in probability. First, even small risks become amplified if they are repeated over and over again. Second, just because you take a risk and get lucky doesn’t mean that you will get lucky next time — too many backcountry skiers make a poor decision, don’t have any negative consequences, and get a “false positive” — their feedback is that they made the right choice, even though the odds were against them.
I consider myself lucky that I was able to learn this lesson in a pretty harmless way. I am not an experienced backcountry skier, but I hope to continue touring in the mountains around Leadville for many years to come, and to do so safely — hopefully never triggering a slide again. But to do that, I have to think really carefully about the risks I’m taking — are they risks that will eventually catch up to me, and am I objectively evaluating the risks without putting too much weight on false positives?
Real World Math
I’m not sure exactly how one would categorize these two problems. The first might be called “logistics”, and the second lives somewhere in probability. Neither is a topic that is particularly emphasized in secondary math education, yet I would argue that the underlying principles are some of the most commonly used in that place people call the “real world”.
Three questions. What mathematical experiences have prepared me to do this type of thinking? And are those experiences that we should be prioritizing for our students? Are we?