Bootstrapping a company means starting a business with your own resources, without external funding. When learning from a textbook, we’re not completely bootstrapping our understanding because we’re not re-deriving the subject matter from scratch. While striving for conceptual understanding is common advice, bootstrapping the material by re-deriving it ourselves can be extreme. An intermediate approach is “conceptual pathfinding,” which uses textbook explanations but requires figuring out how to get from rough intuitions to detailed mathematical formalizations.
Conceptual pathfinding is a process that can help you build a deeper understanding of a concept. Here’s how it works:
Start by stating the basic concept in simple, rough terms. Don’t worry about being precise or mathematical, just try to capture the general idea.
Try to make your statement more specific, pushing yourself to see how far you can go. When you hit a dead end and can’t see how to make your description more formal or solvable, move on to step 3.
Look up the description in a textbook to figure out the solution to your puzzlement. This can provide genuine insight and help you see how the formalization of the concept is a beautiful way to operationalize your intuition.
Clear your mind and repeat the process, starting from your rough intuition and working your way forward. Try not to draw on cached bits of information or intermediate results, and focus on figuring things out rather than straining to remember.
When you can get from your blank slate intuition to the final, detailed result, it should feel satisfying and like the concept has clicked into place. It’s like moving to a new town and finally learning to navigate without a map.
By following these steps, you can deepen your understanding of a concept and build your intuition in a more organic and meaningful way.
As an engineering student, I find textbooks essential, but the prose descriptions often fall short. After skimming the text, I focus on hard theorems, proofs, equations, and diagrams to build a better understanding. However, this approach can make it difficult to retain information and re-derive math on my own. That’s why I find it helpful to build intuition from the math and then use conceptual pathfinding to move from the core conceptual understanding back to the math.
For learning new concepts, I start with rough intuitions and work to make them more specific. If I hit a dead end, I look to clarify where the gap is in my ability to connect intuition to the math. I’ve developed an algorithm for power calculations using this approach.
I’ve realized that starting with intuitions and working to make them more specific and accurate can train my mind to think deeply and constructively about a topic, leading to a deeper understanding. When learning new ideas verbally, I often rephrase the explanation in my own words to solidify my understanding.
Starting with a blank slate and working forward from there is essential to the intuitive-to-math approach. While it may seem obvious to some, it can be bewildering to others. However, starting with intuition, seeing how far you can get towards the answer, and examining what caused the block can make re-deriving proofs and concepts more natural, relaxing, and enjoyable.
For instance, when learning about statistical power, I might begin with the understanding that it’s “how good a study is at detecting an effect of a certain size.” From there, I work to make the concept more specific by defining terms like “how good” and “an effect of a certain size.”
The first time I tried this approach for statistical power, I hit a dead end and struggled to find my way back to the key insight that there’s a chance of rejecting the null hypothesis even when it’s true. However, this obstacle helped me to clarify where the gap was in my ability to connect my intuition to the math and develop a more effective algorithm for power calculations.
Now, if I were to work my way from intuition about power calculations to math, it might look something like this:
“Power is a measure of how good a study is at detecting an effect of a certain size. The more samples you have, the smaller the standard deviation, and the farther apart the null and alternative hypotheses are, the easier it will be to detect the effect. It’s crucial to remember that you always have a chance of rejecting the null hypothesis, even when it’s true. To start, calculate what your sample mean has to be to reject the null hypothesis. It’s easier to obtain a value at least that extreme if the sample mean is not zero, but is actually your alternative hypothesis. Imagine dropping a bell curve on top of your alternative hypothesis. What would be the z-score of the sample mean that allows you to reject the null hypothesis? That z-score corresponds to the probability of rejecting the null hypothesis if the alternative hypothesis is true, and that probability is your study’s power.”
This level of conceptual detail is sufficient for me to figure out what the math ought to be.
Though some very impressive people do things like rederiving all of calculus from first principles in advanced math courses. I’ve never done that myself. My focus here is on the ordinary challenges people face in typical college and graduate-level coursework.