Creative Thinking

The title of this post may not seem particularly related to Mathematics. Creativity, some might say, is limited to the world of Art. Mathematics, they may continue, is about using specific sets of tools, to solve problems, mechanically. Unfortunately, because of the way that arithmetic and Mathematics in general are taught in schools, this does seem to be the case. But, in reality, it could not be further from the truth.

Consider the following problem: 87 contestants are taking part in a knock-out tournament (perhaps in tennis, or chess; the specifics are irrelevant). The first round, will consist of 23 matches and 41 'byes'. That is, 41 contestants will move to the next round without a match and the remaining 46 will have their elimination matches (why 41 byes?). Then, the tournament will continue normally, with no more byes. How many matches will take place, before the champion can be declared?

This problem can be solved in a few different ways. The mechanical method would be to simply count the number of matches. The first round will have 23 matches. Then 64 players will remain. They will have a total of 32 matches in the second round. Continuing in this way, we find that the total number of matches needed is 23 + 32 + 16... + 1. But there is a more elegant and creative method to find the answer. There are 87 contestants. We can ignore the information on byes and so on. Each match will eliminate exactly one contestant. We need only one contestant to remain. So, 87 - 1 = 86 matches will be required. (This is also the answer we get by adding the number of matches in each round). This second method, is undeniably better. It is simpler, shorter and less prone to error from bad computation. That is where creativity enters the realm of Mathematics. It is often possible to solve a problem in many different ways. But some ways are clearly more creative than others.

The problem with this is that creativity cannot be taught. There is no way of knowing in advance which method will be most suitable to solving a specific problem. A good fundamental knowledge of many different methods can often give one a good idea of how to proceed, but again, this takes personal practice. A student needs to solve many problems, before they are fully familiar with a specific method. Unfortunately, this does not, generally, happen. School systems being what they are, do not create a setting whereby students are required to work through enough problems, to understand the concepts behind them. Worse, the students' natural curiosity for these things is not nurtured enough, for them to want to learn such things.