I’m trying to help my brother with his mathematics homework, and I find he’s struggling to understand what he should do. I try to nudge him in the right direction (without giving him the answer, of course), but he shrugs off my help. Frustrated because he knows I have the answer, he tells me, “Just give me the formula. That’s all I need.”
I try to maintain my composure (helping siblings can sometimes be a tricky experience), and say, “But you don’t need a formula. You just have to think about what you’re trying to find.” However, I can see that he’s having none of it, and just wants me to tell him the answer so that he can look around on his memory aid and “find” the path to the correct answer.
This is a problem that I see all too frequently. It’s the result of an incorrect mentality, one that is set upon “finding the answer” instead of understanding what the answer is. It’s a subtle difference, but the best way to shed light on this situation is to ask this question: what does the answer mean or represent?
If the person understands what they’re finding, they should have no trouble explaining it. For example, there’s a huge difference (in terms of understanding), when one describes the zeros of a function as the points where a function crosses the horizontal axis, or as the result of making the function equal to zero. Sure, those two explanations come to the same thing, but the former focuses on the result of the concept, while the latter only describes the procedure to get there.
Thus far, you may be thinking that there’s not really any difference between the two methods of explanation. However, the problem becomes evident when the sort of question that is asked is changed. For some questions, it is not evident how the procedure one is used to applies to the situation. For example, if you know how to do the same three steps for a problem on a specific subject, what happens when the question is slightly different, and the steps needed are changed? For some people, they’ll be out of luck, since their one-size-fits-all approach doesn’t work. And as most scientists can agree, the universe is a wealth of complicated and not-necessarily-intuitive problems. Therefore, we must be willing to adapt to the situation, which also means we need to understand what our end result should be, as opposed to just the result of a procedure or formula. Plus, knowing different paths to solve a certain problem gives us the freedom to look at problems at different angles, giving us inspiration for when we are stuck on other problems.
Instead of trying to figure out the formula which governs the situation, we should try and understand the situation. It also pays to remember that the formula comes as a result of the situation, not the other way around. Comprehension is the key, and procedures just aid us in quickening the process. We should never just bow to the formula without actually understanding what it means.
If you find yourself using a formula to generate a result that you don’t even understand, take it upon yourself to find out. Educate yourself.
And remember that being able to use a formula doesn’t necessarily indicate understanding of a situation.