How to Ace Calculus: The Art of Doing Well in Technical CoursesNovember 14th, 2008 · 105 comments
Calculus is easy. Or at least, it can be. The key is how you digest the material. Here’s an example: when you’re first taught derivatives in calculus class, do you remember it like this…
Or do you intuit this image…
As I will argue in this post, for any technical course — be it calculus, physics, or microeconomics — the key between an ‘A’ and a struggle comes down to this distinction. Below I’ll explain exactly what I mean and reveal how top technical students use this realization to consistently ace their classes.
How Every Technical Class is Taught
Technical classes have a simple structure. In each lecture, the professor presents a series of concepts. Depending on the difficulty of the material, she may cover anywhere from one to more than a dozen. For each concept, the professor will derive the result from concepts you already know and/or provide an example of the concept in practice.
This simplicity is good. It will make it easier for us to develop a strategy to conquer the material…
The Magic of Insight
What do you do with the concepts being spewed by the professor? Most students dutifully copy them down along with their accompanying examples. For example, if it’s the first week of calculus, you might record the standard derivative equation I reproduced above.
This is fine, but it’s not enough….
In addition to capture, you need to develop insight.
What do I mean by insight? That click in your brain — the moment when the tumblers of your mental locks align, the door swings opens, and an intuitive sense of what and why come flooding out. Forget the equations you copied from the blackboard, I’m talking about developing an understanding deep down in your bones.
For our example of the derivative, this might mean having a solid mental grasp of this image:
A derivative at a given point is just the slope of the tangent line that kisses that point. Even more intuitively: it can be though of as the “steepness” of the graph at that point. That’s all. The complicated equation from above is just a way to calculate a specific number that describes this steepness.
If you understand this graph — really understand it — you understand the insight behind derivatives. If all you know is the equation from above, then you’re screwed.
I am now ready to reveal the big dark secret about technical class studying: If you want to do well in a technical class all you have to do is develop insight for every single concept covered in lecture.
That’s the whole ballgame.
That’s how every high-scoring technical student does it.
There’s no shortcut.
It’s the only way.
Here’s what I commonly observe: the students who struggle in technical courses are those who skip the insight-developing phase. They capture concepts in their notes and they study by reproducing their notes. Then, when they sit down for the exam and are faced with problems that apply the ideas in novel ways, they have no idea what to do. They panic. They do poorly. They proclaim that they are “not math people.” They switch to a philosophy major.
Without insight you can’t do well.
How to Develop Insight
Developing insight can be hard. (Though it gets easier with practice.) Especially when you’re given a dozen new concepts per lecture. The implication: you have to invest a lot of effort during the semester — not just right before the exam — to keep up with a technical course. Every one of those concepts described in lecture has to be translated from symbols on a blackboard to a shiver-inducing deep comprehension. It’s not easy, but at least the challenge is now well-defined.
Here are some tips that can help:
- If you have a hard time understanding the material as the professor presents it, prep the concepts before class by reading the textbook.
- Ask questions when the professor loses you. Often their answer can knock you back on track to insightful understanding.
- Ask the professor or TA for clarifications immediately following lecture.
- Try to review your notes as soon as possible after class to cement insights while the information is still fresh in your brain.
- Always go to office hours. But before you show up, spend time with the troublesome concepts trying to build insight. Figure out exactly where you get stuck. This will help the TA or professor give you targeted, useful advice. Never just say: I don’t get it.”
- Keep a running list of every concept taught so far in the semester. Mark the ones that you have an insight for and the ones you don’t understand. It helps to see clearly exactly what insights you still need.
The Practice Factor
Once you’ve developed an insight for every concept in a technical course, the final step before a test is to do a small number of practice problems for each to practice applying it. (This is where the mega-problem sets of Straight-A come into play.)
Here’s the crucial observation: if you skip the insight-generating phase, no amount of practice problems will help you side-step exam disaster. If it’s a week before the exam, and you lack insights on most of the concepts: you’re out of luck.
It’s Not Easy, But It’s Also Not Complicated
It’s hard to do well in technical courses. But it’s not complicated.
During the semester, you have to see yourself like a lone soldier trying to fight back the tide of encroaching concepts. Do everything you can to build insights in the heat of battle. Become obsessive about conquering concepts.
Once you’ve turned your attention to the real battle needed to do well in technical classes, you can invest your time and energy exactly where it’s needed.
And if not, there’s always philosophy…