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Bootcamp 2012 — Day 5: How to understand (and ace) really hard subjects

Some of you may have noticed a problem with the memory technique I shared on Day 2. It assumes you already understand the subject decently. Trying to create metaphors or simplify a topic is extremely hard when you have a broken understanding.

Today I’m going to share a method to understand the extremely hard ideas. These are methods I’ve used on graduate classes in mathematics, hard ideas in physics and tough concepts in philosophy and logic.

The Best Method

The best approach is to have a one-on-one tutor or teacher to help you get the hard ideas. This is definitely easier than trying to bootstrap the knowledge yourself and most of the research I’ve seen points to one-on-one tutoring as being one of the best learning environments, in general.

That said, often you don’t have access to a tutor. I certainly didn’t when working through the MIT Challenge, so I needed an effective way to teach myself. Even if you have access to help, learning this method is useful since it makes any guided education more efficient.

The First Step to Teach Yourself the Hardest Concepts

The most important step before learning anything is to have the right attitude. I can talk about specific tactics all day, but I’ve seen too many students fail from the beginning because they have a terrible attitude.

The correct attitude is that everything is learnable.

Many people have adopted the utterly false and damaging belief that they can’t learn certain things. That they’re not a “math” person, that they can’t become a good writer or presenter, or that a certain subject is off-limits to them.

This is nonsense.

Now it’s probably true that some people will learn faster than others. Sometimes this is an innate talent, other times it’s simply past exposure (if you’ve already done a lot of math, physics won’t seem so hard). But speed just determines how long it takes, it doesn’t preclude you from learning anything.

I emphasize this attitude because many students facing really hard subjects believe that they cannot truly understand the ideas, so they’re best off just memorizing facts or solutions. I see this belief expressed implicitly in the thousands of emails I’ve gotten from students complaining about a hard class.

Unfortunately this attitude strikes “smart” people far more often than anyone else. Because they breezed through tough subjects before, they don’t build the intellectual muscle of struggling through ideas they don’t understand. So, when they finally encounter an idea that requires a bit of grit, they give up and memorize.

That’s all I’ll say on attitude, but that doesn’t diminish it’s importance. Learning hard ideas takes perseverance more than intelligence, so don’t adopt an attitude that will cause you to fail before you start.

The Specific Steps to Learning Any Tough Idea

When you don’t understand an idea it’s because you have holes in your knowledge. These holes can be uncrossable chasms which prevent you from understanding an idea. The key to learning any hard idea is to fill these holes.

Step One: Recross the source material carefully

The first step is to filling those chasms is to rewalk the source material carefully. Follow the chain of reasoning that is presented carefully. This may not always lead to an understanding, for several reason I’ll indicate, but it’s an important first step.

My approach when doing this is to carefully reiterate everything said or written, on a piece of paper, in my own words. This slows down the process to a crawl, so you’re meticulously covering every detail. You’re not aiming to have original insights, just attempting to follow the insights presented.

Step Two: Note missing pieces

Let’s say the problem was a mathematical proof. During the first step, I may follow every step of the proof closely, but still notice gaps in the chain of reasoning. I may have understood how they went from step 12 to step 13, but step 15 to 16 is unclear to me.

This is almost always because there is an underlying assumption you’re missing. The author is taking for granted that you understand something, which you do not. In our mathematical proof, it could be a technique or idea that’s unfamiliar to you (or you’ve forgotten).

From here your mission becomes to fill that gap. If you have taken the prerequisites for a particular course, then the missing gap will likely lie in an old idea you had covered, but have now forgotten. My approach was to look through old notes and make a highly specific search for what might be missing.

This process isn’t as tedious as it sounds. Generally a missing insight can only be of a particular type, so you aren’t searching blindly.

Repeat this process until you can follow the main chain of reasoning presented without gaps.

Step Three: Expand Your References

Now that you can follow one source, you want to tackle the same idea, viewed from different angles. Go online and find as many explanations or descriptions as you can and follow them as well. Generally it’s only the first that requires meticulous preparation, adding new explanations is easier once you’ve already followed one.

When following the MIT Challenge, my first resource was Wikipedia, after that I’d suggest searching for the concept on YouTube and then on Google to find resources. If it’s a common idea, KhanAcademy or PatrickJMT might cover it (however I found these resources quite sparse once you’re out of the early undergraduate level).

You should be able to collect several resources and individually walk through their explanations.

Step Four: Hypothesis Generation and Testing

The final step is very similar to the memory technique I taught you on day two. Your goal now is to try to create an analogy or simplification of the idea to better understand it.

The difference with this phase is that you may be wrong. When tackling tough ideas, you may need to go through many different pictures before you settle on one that is accurate. For each idea, you want to think through the implications of the analogy or simplification and see how it matches up to the source explanations you had been given.

This is a process best understood by seeing examples, so I’ll share a few quick examples of how I used this to learn real topics from the MIT Challenge.

Example: Voltage

One idea which I found tricky on my first exposure was the concept of voltage. I understood how the equations worked, but I couldn’t wrap my head around what voltage was.

My first attempt was thinking of voltage as the quantity of electrons, maybe it was like volume if current was like speed. However, quickly checking the source material showed me this was incorrect. Charge quantity was a different unit, separate from voltage. Shoot.

My second attempt was to think of it like a force. Maybe voltage was like a pushing strength flowing through the wires. Not right again. Forces are different kind of thing in physics, which are expressed in Newtons, voltage wasn’t.

Finally I saw a connection between electric force and gravitational force. The two forces are fairly similar in equations, so what would the gravitational analogy be to voltage? The answer, which surprised me initially, was height. Voltage is electric potential energy, just as the height of an object lets us calculate its gravitational potential energy.

From this point I could revisualize batteries like elevators and resistors as staircases and voltage levels of nodes as being a kind of ‘height’ that they rested upon.

Example: Laplace Transform

This idea really stumped me the first time I was exposed to it. Especially since it was described to me as being “the continuous analog to the Fourier Series”, which is an explanation that to me is only comprehensible if you already have a good idea what the Laplace Transform is.

Once again, I went through the source material, and tried several failed analogies for understanding it. First I thought of it as possibly a compression, but reflecting on the material, that seemed to miss what was really going on. Maybe another way of thinking about it was filtering, taking a function and filtering out the parts that didn’t match.

After several attempts, I finally hit on an analogy that it was about rebuilding the function using waves of different wavelengths (and also waves growing and dying out). Rebuilding, it turns out, was a much better analogy than compressing or filtering.

ACTION STEP: Day Five

Ideally you would be practicing this technique on a really hard idea you need to learn. But you may not have a really hard idea right now that you’re working through, or you might not have enough experience with the method to use it well. So for this bootcamp, feel free to pick any idea you’re learning right now to practice on (hard or not).

This process can also take more time and patience than the other methods I’ve shared with you. As a result, I’m only going to have you complete the first two steps as part of this bootcamp, but you’re free to finish the method in its entirety if you wish.

Here’s the steps, again:

1. Find the idea or concept you want to understand better. The more specific the better, so instead of “calculus” or “Java”, pick something like “L’Hopital’s rule” or “the Visitor design pattern”
2. Begin by recreating the explanation, in detail, on paper. When you encounter a missing link in the chain of reasoning, look for explanations to fill that detail.
3. Now gather up at LEAST 3 other explanations/descriptions of the idea. Wikipedia is a good starting place, but a Google search should turn up many others.
4. Read through these other explanations, so that you can follow them. (If an explanation is too difficult, try finding another one)
5. Finally repeat the technique from DAY 2 and try to simplify your understanding into a possible metaphor, analogy or even just a simplification. This may take several attempts.

For this bootcamp, just complete the first two steps: find an idea and try to follow, meticulously, the chain of reasoning explained by the source. If you want, continue the remaining 5 steps to try to build up your intuition to the next level.

Please note that some hard ideas will take more than one iteration of this method to understand deeply. Each iteration merely allows you to understand the idea from one angle. Holistic learning means having connections from many angles, and many different insights, each building your confidence.

That’s all for today!

Best,
-Scott