In the last few podcasts, you may recall that we’ve been discussing the Collatz Conjecture, a famous unsolved problem that’s very easy to state. Just take any positive integer, and repeatedly perform the following operation: if it’s odd, triple it and add 1, but if it’s even, divide it by two. The conjecture is: with this process, will you always end up eventually back at 1? While the conjecture has been tested and is true for a huge range of values, no mathematician has yet been able to prove that it will always be true. Based on that, I was surprised at a result I got when googling references on this conjecture for my last podcast. It was an article by a Kentucky professor named Benjamin Braun, in which he talked about using the Collatz Conjecture as a homework problem for an undergraduate math class.
Now, when hearing this, you may be reminded of the strange story of statistician George Dantzig, which we have discussed in a previous podcast. When in college, Dantzig arrived late in class one day, and copied down some problems he saw on the board, assuming they were the day’s homework. He thought the homework was harder than usual, but finished it in a couple of days, and handed it in. Then he discovered that these had actually been famous unsolved problems, and because he thought they were homework, he had solved them, earning his Ph.D. in one day! This story became a staple of “power of positive thinking” lectures throughout the 20th century.
So, does Braun hope there is another Dantzig lurking out there in his classes, who could solve the Collatz Conjecture if approaching it with the right attitude? That would be nice, but that’s a longshot. Actually, Braun believes that unsolved problems can have a significant educational value as homework. Too many students share a misperception that math is all about using known formulas and procedures to get answers, even after completing many college math classes, and are never in a position to explore a truly interesting problem. When challenged with the Collatz Conjecture, students need to experiment with various hypotheses without knowing which ones will be true, and then can proceed to develop and prove some interesting insights. For example, maybe some students will write a program to graph the number of steps it takes for various numbers to get down to 1. This might lead them to hypothesize and prove an intermediate theorem, such as a relationship between a number’s base-2 logarithm and the minimum number of Collatz steps to 1. In the end, they probably won’t solve the famous conjecture, but they will learn a lot along the way.
Braun describes three main benefits of this style of assignment:
- “Students are forced to depart from the answer-getting mentality of mathematics.” As we have just discussed, they probably won’t completely solve the problem.
- “Students are forced to redefine success in learning as making sense and increasing depth of understanding.” Since they are free from the pressure to find a solution, they can relax and concentrate on what they are learning.
- “Students can work in a context where failure is normal.” As they examine the problem, students may come up with various hypotheses, and it’s fine if some of them are wrong. As Braun describes it, they will understand the “pervasive normality of small mistakes in the day-to-day lives of mathematicians and scientists”.
Naturally, you might wonder how students react to being given an unsolvable problem. Apparently this varies a bit: Braun mentions that while some students love the exercise, others are inclined to feel frustration and defeat. But I think there is little doubt that this exercise is very unique in an undergraduate curriculum. It’s a great path for students to understand that math isn’t all about replicating known algorithms, formulas, and proofs, and that there is still a huge unknown mathematical universe out there to explore.
And this has been your math mutation for today.