There was definitely a clearly thought out pedagogical reason for having all of my students walking around the classroom marked with post its indicating whether or not they would “die” of an infectious disease. It was either that or geometric proofs with algebraic properties for an entire 65 minutes. I know it’s traditional but I have my suspicions that two column proofs were a bastardized way to present proof to high schoolers when they decided to move the conceptually difficult topic from the college curriculum. And appeals to tradition bother me more than any other logical fallacy.
I also know that being able to prove cool things in math requires a lot of time spent on the details. I once endeavored to prove that odd + odd = even and other seemingly trivial facts. At least the proofs of the Mean Value Theorem(If you go from below sea level to above sea level on a walking trip you must have been at sea level at some point) was pretty cool from what I remember of it when I learned it the final year of college in Advanced Calculus.
I’m trying really hard to figure out why we do two column proof, how I feel about talking about axiomatic systems explicitly and just how wrong a proof is if the kids forget the angle addition postulate.
This brings us to the point of proof. To convince others? To practice logical argument? Why do we learn proof in high school? I love it. I know it’s important but which aspects of it are important and why? What kind of nonsense can I gloss over and what is really important. Mathematics is too beautiful of a discipline to only show them the tedium of really rigorous arguments. To what extent must they experience this tedium to be able to prove the awesome stuff I did later? Or perhaps they don’t see it as tedium. I find myself a bit flummoxed when trying to explain why we need the angle addition postulate. (If you have two angles that share a ray the sum of the angles equals the big angle formed when you put the two together.) If we are teaching them to argue this point is so boring as to be unnecessary. And I’d guess a professional mathematician doesn’t put steps like this in. Of course he could if he wanted to and maybe that is the rub with this argument. Perhaps I do need to have my kids focusing on these details. I know that they will never encounter a question like this on a standardized test and that while logic will be important in all fields, the use of the annoying postulates that seem super obvious will only be helpful if they take a class on proof in college. (Of course I’m sure the all will since through the wonder of Wikipedia I’ve shown them how much cool math is out there.)
What does this have to do with monks? A boy I had a huge crush on once sent me a text message with the following problem:
“There is a colony of monks who can’t communicate in any way and who cannot look in any sort of reflective surface. No exceptions. Someone comes to their monastery one day and says to them at the meeting that one or more of them are infected with a disease. The only symptom they will ever have is a red dot on their forehead that they cannot see or feel or touch. He tells them that if they have the disease they must commit suicide so that they do not infect the rest of the monks. The infection will spread in a month or so but for the moment the number of monks with the disease is fixed.
The next day all of the monks return to morning meeting and no one has committed suicide. The stranger repeats the message and the monks listen and look around.
The next day it happens again and still no one has committed suicide.
On the morning of the fourth day the monks with the disease have killed themselves and everyone who was well is at the meeting.
Given that every monk follows perfect deductive logic all the time, how many monks are in the monastery?”
There is a perfectly satisfying and logical answer to this question(I promise the answer is not, they used sign language, or communicated telepathically, or used God to help them or anything silly of that sort.)
By getting the kids to put their heads down and marking some of them with a disease I was able to get some of the kids to come to the answer. They all wandered around looking at everyone’s post it and got excited about conjecturing and trying to figure out how to decide. Everyone was really good at not revealing who was going to die as well which was surprising given how fun it might be for high school students to laugh about their friends being the marked monks. Even once those kids had explained and demonstrated with the post it note infection notices some of the kids were still extremely confused about the answer. It was interesting to see that the ones who were capable of geometric proof were similar to the ones capable of deductive logic in this problem as well.
The of course was evidence for my claim of this being a meaningful mathematical task for a logic and proof class even if I did ignore some stuff on Geometric proof while I was at it.
After we finished I said “Do you guys want to do some easier proofs now?” and most of the class happily worked on seemingly “obvious” geometric proofs. A few kids said “Can we please just keep working on harder problems instead?” So adorable. At the end of class kids talked about how much they liked wandering around class and being active problem solvers. I told them that if they helped me lesson plan we could do more things like that but that it was hard to find problems that interesting on a daily basis. Hearing them this happy was sure good motivation to try. And thank god I don’t have a standardized test I’m trying to get them to pass. And thank god no one is really going to know just how many times they practiced two column proofs. And maybe this will even help them.