One of the most important things we must learn in mathematics is to overcome our fear of actually trying. It is very easy to simply stare at a blank piece of paper and feel overwhelmingly helpless. Very often, we may even think we are stupid for not being able to start a problem. We should never think this!
What many people don’t understand about mathematics is this: being good at math doesn’t at all depend on how “smart” we are. It simply depends on our willingness to work hard, and to fail, and to try again. This is how we master the subject; we just fearlessly keep trying until we get it!
In this week's Problem (and pretty much every other!) there are two competing ideas of “trying”. We will call them “random experimentation” and “plan of attack”.
When you are confronted by a problem, and not sure what to do, do something. In the case of the problem we were solving, we need to figure out how to dissect a square into 6, 7, 8, etc. squares. How can we even begin? Don’t worry about it. Just begin. Start by drawing. Experiment! Here, we call this “random” experimenting, because you don’t have to have a plan in mind, you don’t have to know exactly what you’re doing, and you don’t even need to know where you’re going with your line of thought. And, you don’t even have to be successful at first (or second, or third, or fourth) try! You just need to start thinking. The more you think, the more likely you are to come upon a useful idea or concept that will help you solve the problem. And the useful things you find might be deliberately found, because you’re being all clever (go you!), or you might just accidentally stumble upon them. But no matter how you find them, you first need to embark on the journey that will lead you to them, and that means you definitely need to start doing something.
The point of all this experimentation is to find your “plan of attack” for the problem. You will at some point need to know what you’re doing! Your random experimentation will lead you to your plan of attack, the thinking that will actually allow you to solve the problem. For this, you need to be patient. It may take a few minutes, or a few hours, or a few days. In the Squares problem, your plan of attack essentially involves realizing how to fit smaller squares around another square. This allows us to notice that we can use a similar approach in many parts of the question.
Many problems require several plans of attack. In Squares, to answer later questions, we need to understand how to use the dissections that our first plan of attack gave to us, to answer the more general question of cutting a square into any number of smaller squares. Thus our second plan of attack involves seeing the patterns that arise out of our first plan. So, here it’s important to realize that mathematics builds upon previous concepts: your plans of attack are not completely separate ideas. You do not want to start from scratch to answer the last part of the problem! You need to have all your thoughts from the whole course of the problem, in your pocket, ready to apply when needed.
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