Probably the most efficient way to improve at mathematics is just to do a large number of exercises. But how do you choose the right problems to try? If the problems aren’t challenging then you don’t learn anything, but if they are too difficult they become frustrating or even unachievable. The ideal problem is one that forces you to put in serious effort, but which is still achievable. This is an idea that goes by the name of desirable difficulty in the education literature. But practically how do we actually choose problems of this type. A sensible approach is to use textbooks with graded problem sets, where you can choose the greatest difficulty where you (or your student) will still consistently solve the problems. While this is pragmatic it relies on the textbook having accurate grading, how can you actually tell how hard a problem is going to be?
There is another approach that is a bit more theoretical in nature and which is admittedly impractical for a classroom setting, but which is all the same interesting and perhaps still useful. This is the use of the Elo system. The Elo system was originally invented to compute the difference in skill between chess players, but it can also be used for our purposes. For chess, the Elo system gives players numerical scores, the higher the score the better the player. It then predicts the likelihood of each player winning based on the difference in scores and updates scores based on the actual outcome of the game. For our problem we could give each student and each math problem an Elo rating where we consider the students to be competing against the problems. If they get a problem right they win the game and their rating goes up while the problems rating goes down (and vice versa). The purpose of this is that Elo lets us find the probability the student will solve the problem. We can then give students problems they are likely but not guaranteed to solve, these would be exactly the problems that are desirably difficult.
