Have an assignment due tomorrow, and have no idea where to even start? In office hours or class and so lost that you don't even know what your question is? No matter the context of your confusion, you're not alone!
First, let’s understand what exactly these situations have in common: They stem from the classic problem of “not knowing what you don’t know.” It’s important as a student to understand how to get out of this state so that you can start asking questions that one-shot your confusions instead of “How do I get started?” or “How do I solve this problem?” that yield generic answers. Here are two main ways to get you out of the fog!
Follow the breadcrumbs
If you don’t understand the topic at hand, it’s likely because you didn’t understand something before this. Especially in fields like math and physics (my personal areas of focus), advanced courses may be filled with jargon, but they still rest on logic and fundamental concepts. To ask a directed question, you need to identify what actually confuses you.
Let’s take a look at introductory mechanics, for example:
A cannon-launcher with an adjustable angle fires a cannonball at 20 m/s straight out of the barrel. What’s the farthest from the launcher that the cannonball can land?
This deceptively simple question has many concepts at play. Let’s start from the question itself, and at every step, we’ll ask ourselves: “How do you do that?”
You want to find the farthest distance that the cannonball goes. In other words, your goal is to maximize the distance that the ball goes. Before you can answer this, you’ll need to know how to calculate the distance that the cannonball goes. This calculation now depends on the angle of the launch. To find this distance, you must use trigonometry to break the problem into x and y components, then use the kinematic equations of classical mechanics in 2D…
You can see how fully tracing the “how to” in a problem leads to many fundamental concepts and tools, as illustrated in the reverse flowchart. If you find yourself not knowing how to answer “How do you do that?” at any given step, that means you’ve found the question you need to ask!
In this example, say I can trace back how to do everything until I need to optimize the distance traveled by the cannonball with respect to the angle. Then, the question I should raise to my friends, teacher, or tutor is “How do I maximize a function, again?”
Learning shortcuts
Sometimes, you can trace back the “how to” diagram in theory, but actually getting to the answer feels insurmountable. In this case, you effectively have a data analysis problem — you need to efficiently put together your given information to get an answer. Let’s take the following example, adapted from Challenging Problems in Algebra by A. Posamentier and C. Salkind.:
To conserve the contents of a 16 oz. bottle of tonic, a castaway adopts the following procedure. On the first day, he drinks 1 oz of tonic, then refills the bottle with water; on the second day, he drinks 2 oz. of the mixture, then refills the bottle with water; on the third day, he drinks 3 oz. of the mixture, then refills the bottle with water. The procedure is continued for succeeding days until the bottle is empty. How many ounces of water does he drink in total?
In this example, the question appears at the end: “How many ounces of water does [the castaway] drink in total?” We’re given the pattern that the castaway uses to drink water, so we actually know exactly how much water he drinks on a given day. With that information, we can solve the problem!
At this point, it’s possible to take a few different paths towards getting the final answer. The most straightforward way is to calculate the ratio of water to tonic every day, then figure out how much water was consumed that day. This method is tedious though, so you can imagine that a small adjustment to the problem — take a 100 oz. bottle and the same drinking procedure — becomes quickly intractable.
In this case, your question should not be about the “how to” steps that you’ve thought of, but rather about a way to circumvent the steps you’ve come up with. A well-constructed question would start off with how you could solve the problem but end with “Is there an alternative approach?” By asking this type of question, you can easily match a shortcut with a calculation type, and eventually, you’ll be able to come up with the clever method all on your own!
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