I wrote a message for one of our members, but I’d like to make it available to all our members and any student of math who might be interested, so I adapted it into this blog post.
For everybody, there comes a time when mathematics becomes hard…really hard. There are math problems that no mathematician has yet been able to solve. Right this moment, mathematicians all over the world are working on problems that they do not know how to solve and that they find tremendously challenging even to make progress on.
To be a mathematical or scientific researcher, one must not be easily discouraged in the face of a challenging problem. After all, the whole point of research is to resolve problems that are as yet unsolved. An important part of the process of becoming a researcher is learning how to handle situations where one feels incapable of solving a problem. Nobody can solve everything; therefore, solving everything is not what is important.
This psychological aspect of doing mathematics is something that we teach at Girls’ Angle through a gradual and careful process.
Confidence should not be based on whether or not one can solve everything, such as the problems on a math competition. I have seen contest superstars struggle when they try to earn their doctoral degrees in mathematics. For the first time, they met problems that they couldn’t resolve within time scales similar to those found in math competitions. Some lose their confidence and fail to complete their graduate programs. Instead of test or contest performance, students of math should focus on increasing mathematical understanding and what mathematicians call “mathematical maturity.”
It’s fine to take part in math competitions if you enjoy that sort of thing, but it would be unwise to attach any importance to the results, be they good or bad. It is a terrible mistake to become discouraged about doing mathematics because of a bad contest result. It is equally a mistake to gain confidence on the basis of good contest results, because such confidence is fragile.
Mathematics offers a rich arena of wondrous concepts and objects to ponder. Focus on enjoying the experience of learning about these marvelous ideas and concentrate on achieving deeper understanding. If you do that, you will find that contest problems you once thought were hard are now quite easy. Yet you may not even be able to pinpoint exactly why they’ve become easy. What will happen is that you’ll simply be thinking at a higher level than before and seeing the problems with greater perspective and context.
For those of you who like to use metaphors, here is one that may be useful. Think of a person trying to catch fish. Each fish is like a math problem. The person sees a fish and tries to grab it. The fish is fast. The fish is slippery. The person keeps missing, and the few times the person actually does get his hands on a fish, the fish slips away. It gets frustrating. The person thinks, “If I can just be quicker, I will catch the fish.” Or, “If I can just hold on tighter, I will catch the fish.” That is like a person who focuses solely on math contest problems as a way of learning math. Then, another person comes by, wades into the water with a big net, puts the net in, patiently waits a while, hauls in the net, and walks away with dozens of fish.
Focus more on building the net than catching the fish. To do that, you’ll have to go away from the river. Building the net will take time and patience, but when you’ve got your net, you’ll catch many fish.
One thing we do at Girls’ Angle is try to get members to shift their gaze from the fish to the net…to try to see beyond the problem to the larger landscape of mathematics. We work on helping members build a toolkit that can be used to more effectively learn mathematics. And we try to help members learn how to follow their instincts so that they can deduce the implications of their own observations.
Math is full of fundamental concepts which may seem very simple at first, but when fully grasped, provide the key to many problems. We work on helping students see how a seemingly simple observation can have far-reaching implications. As one studies mathematics and makes more and more of these notions a part of one’s intuition, their implications will crisscross like waves building upon each other and washing away more and more problems.
So don’t be concerned with artificial milestones. Concentrate on building strong fundamentals. Secure a foundation for yourself from which you can soar, and you will!