Case Study: How I Got the Highest Grade in my Discrete Math Class

November 25th, 2008 · 68 comments

A Hallway Encounter

During my sophomore year at Dartmouth I took a course in discrete mathematics. The tests were not calibrated to any standard scale, so it was difficult to judge how well you were doing. On the midterm, for example, scores around 50 to 60 out of 100 were at the top of the class, whereas for the final those would be failing.

Rewind, then, to the end of the winter quarter, and imagine my surprise in the following scenario. It’s the day after the final. I’m walking through a hallway when I encounter the TA:

“You…got the highest grade,” he said.

“On the final?” I asked, somewhat surprised.

“No, for the entire course.”

This was hard to believe. The course had 70 students. Three of them were from Eastern Europe where, educated in the old Soviet-style talent-tracking system, they had already studied this subject in high school!

I didn’t think of myself as a math person. Before this class, I had shown no particular talent for the subject. I was trying to just hang in there with a decent grade. My victory, as we like to say here on Study Hacks, was tactical.

In this post I will explain how I achieved this feat, and how following similar strategies can help you dominate even the most thorny technical courses…

No Tolerance For Lack of Insight

At the high-level, my strategy was exactly what I spelled out in my How to Ace Calculus post of two weeks ago: learn the insights. But I want to dive into the details of how I accomplished this goal for this specific class. Think of this as a case study of the insight method in action.

Here was my specific strategy:

Proof Obsession: Discrete math is about proofs. In lecture, the professor would write a proposition on the board — e.g., if n is a perfect square then it’s also odd — then walk through a proof. Proposition after proposition, proof after proof. As the class advanced, we learned increasingly advanced techniques for building these proofs. I soon developed a singular obsession: I wanted to be able to recreate, with pencil and paper, and no helper notes, every single proof presented in class. No exceptions. Lack of understanding of even one proof wouldn’t be tolerated.

My Obsession in Practice



Here’s how I learned every proof.

I bought a package of white printer paper. As the term progressed, I copied each proposition presented in class onto its own sheet of paper. I would write the problem as the top of the sheet and recreate the proof, from my notes, below. I tried to do this every week — copying the most recent material onto its own sheets — though I often got behind. While doing this work I would sometimes — okay, many times — realize I didn’t quite understand the proof I had copied in my notes. In these cases, I would break out the textbook, or do some web searching for the problem, to see if I could make sense of what I was writing down. This usually worked. In the worst case scenario, I would ask the professor or the TA for help. Not understanding the proof was not an option. I wasn’t practicing transcription; I knew I had to learn these. About two weeks before each exam I started scheduling sessions to aggressively review my “proof guides.” I always worked on the second floor of the Dana Biomedical Library on the outskirts of campus. (Think: dark, concrete-floored stacks, with desks tucked away at then end of long rows, each illuminated by a single, bright incandescent bulb…study heaven.) I did standard Quiz and Recall: splitting the proofs between those I could replicate from scratch and those that gave me trouble, and then, in the next round, focusing only on those that gave me trouble, and so on, until every sheet had been conquered.

By the day of the exam, you could give me any problem from the course and I could rattle off the proof, without mistake and without hesitation.

Lots of Work, but Not Hard Work

In retrospect, it’s not surprising I did well in this class. Most of the other students — even the Eastern European students — started studying for the exam 48 hours in advance, trying, frantically, to review as many of the high-level techniques as possible. Not surprisingly: a lot of details were missed. They knew the basics. But they lacked mastery.

Consider, by contrast, my approach. If you add up the time I spent copying out the proofs on the white paper, add in the time required to track down help for the proofs I didn’t understand, and then throw into the mix the time spent reviewing, the total is somewhat staggering. To try to do the same a few days before the exam would have been literally impossible.

This doesn’t mean, however, that my life was hell. If anything, this was a relaxing term. The secret was that I inlined my work throughout the term. I never spent more than 2 hours at a time working on these proofs. I never stayed up late. I never ground through material. I kept attacking it fresh, with high energy, time and time again.

There are two lessons I hope you take from this case study:

Conquering a technical class requires a massive amount of work. There is no short-cut. If you’re pulling high school bullshit and trying to wait until a few days before to learn everything you slept through in class, then you’re screwed. You need to grow up and leave that behavior in the past. Conquering a technical class doesn’t have to be painful. The key is to define your challenge — learn every insight — come up with a plan for winning the challenge — e.g., in my case, using proof guides to learn every single proof — and then putting the plan into motion with time to spare. No cramming necessary.

Know thy enemy and it becomes a lot less fearsome…

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