Getting To Your Destination

You can be the best driver in the world and still you can be driving for hours on end without ever reaching your destination.

Some students have an interesting reaction when they first overcome some difficulties. For example, some people are used to making mistakes at solving equations, because they don't know the rules or they don't know how to apply them. Meaning, they cancel identical terms out of a fraction without realizing it is an illegal move because those terms are just terms, not factors of the whole numerator and denominator. Or they somehow mess up other algebraic steps of solving an equation. They know they are making mistakes, they just don't know what those mistakes are. They are used to getting the wrong answer because they know they make mistakes then cannot even identify. Then they come to get some tutoring and they start finding out what their mistakes are. They learn the correct way to apply the operations to algebraic expressions. At first they seem unsure. Little by little they gain confidence. And then, when they finally stop making the mistakes they had been making before, something interesting happens. All of a sudden they seem confused. They seem surprised to see they are still not finding the solution to the equation, even now when they are not making mistakes any more! They seem to think: "Where is the solution?" They used to think the reason why they were not getting the right solution before was only because they were doing all those mistakes. So, why are they not finding the solution now? They are not making mistakes any more, so where is the solution? Now they seem to keep on going, correctly applying operation after operation to the equation, without getting anywhere near to a solution. Why is that? Well, this is the metaphor I use in this situation. When you drive a car, you have to learn the correct way to use all the controls, the brakes, the steering wheel, the accelerator and everything. If you don't use them correctly, you will crash the car, or you will break the transmission or you will overheat the engine or cause some other catastrophe like that. Once you learn how to correctly use all the controls you can drive safely. Still, that is not enough for you to get to your destination. It is a necessary condition, but by itself is not enough. You can be the best driver in the world but if you keep aimlessly driving in circles all around town when you have to go from Los Angeles to Las Vegas, you are never going to make it. You need to pay attention not only to the car's controls, but also to the road signals. You may even need a map. That is similar to what happens when you are solving equations. Applying each operation correctly and following the rules well at every step is the equivalent of driving safely and operating the car correctly. Now, actually being able to reduce the equation to its simplest form and finding a solution is the equivalent of getting to your destination. You can be correctly applying operations for hours on end, producing more and more equivalent equations like pop-corn in a microwave, but in order to get the final numerical solution, you have to choose what operations to apply and when to apply them. You have to arrange your algebraic steps so that the equation actually gets simpler and shorter with almost every step. That is another skill you have to master but you couldn't see the need for it while you were crashing the car all the time. It's like when you are climbing a mountain and after getting to the top of a hill, you discover another hill in front of you, which you couldn't see before because its view was obstructed by the previous hill you just conquered.