Conceptual Pathfinding
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.[1] 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.
I’m reading your post as an example of an active learning strategy. At what point do you verify that your concept is correct? Do you stop when it feels “satisfying and like the concept has clicked into place”? Or do you test your concept, for example against textbook exercises?
Good question! I stop the “conceptual pathfinding” process when the concept has clicked into place.
Textbook exercises are a great way of testing your understanding, of course. But I think the signal of “this feels like it’s making sense” is important, and usually precedes being ready to work on problem sets. Textbook exercises are a way of going beyond the limits of feeling like a concept makes sense and probing the gaps in your knowledge.
Conceptual pathfinding functions as a second way to identify and fill in those gaps, one that’s complementary to exercises.
This so called “conceptual pathfinding” doesn’t seem like an intermediate approach. I don’t understand what the differences between “conceptual pathfinding” and “information/material bootstrapping” are.
You defined conceptual pathfinding as a process that uses textbook explanations and involves figuring out how to get from rough intuitions to detailed mathematical formalizations. Basically, starting from basic information and deriving to target information.
My interpretation of your post is that you are dividing “information bootstrapping” into multiple parts and each part focuses on one concept and then doing information bootstrapping or conceptual pathfinding on those parts. I don’t think there is any real difference between “conceptual pathfinding” and “information/material bootstrapping”.
Thank you for making me the recipient of your first LessWrong comment!
First, a response to your concrete question.
An example of what I regard as “bootstrapping” is a class that entails rederiving calculus from a small set of starting axioms. In case you didn’t know, this is a real thing that people do, including some here on LessWrong.
On the opposite extreme might be the act of memorizing a proof. I am not talking about empty memorization, where you can recite the words but don’t understand what they mean. Instead, I mean that as you rehearse the steps in the proof, you come to understand the proof, and at the end, you also have it memorized.
To me, “conceptual pathfinding” is an intermediate between these two.
Let’s say you were trying to understand what a t test is in statistics. To start with, it’s just words on the page, maybe a fuzzy mental picture of a bell curve, the idea of a comparison or of significance, the memory that you don’t know the “real” standard deviation.
You might start by saying something like, “a t test is a way of checking if there’s really a difference between two groups. There’s a t value and a p value… and if the p value is low enough, then the test is significant!”
Obviously, that’s an inadequate understanding, but it starts you going in the right direction. Based on your performance, you can then self-reflect and make additional comments or ask questions. You might think thoughts like “OK, and that’s what alpha is—a way of saying whether the p value is “low enough” for the test to be significant.” Or you might say, “how do we calculate a t value?” Or you might notice, “it’s important that all the data points are numeric for a t test.” Or you might say, “a t test assumes we’re dealing with roughly Normally distributed data.”
And then you might start over, saying something like “a t test models numeric data as coming from a Normal distribution, and asks whether two samples are significantly different. We choose an alpha value to say what our signficance threshold is, and then calculate a t value, which we use to get a p vlaue, and then compare it with the alpha value to decide if the test is significant or not.”
Again, still inadequate, but displaying a maturing understanding. And you can just keep repeating this process, looking up individual facts when you need to, until you’re satisfied. For me, this sort of process keeps me engaged. When I look up individual facts, I feel like I’m fitting them like a puzzle piece into a puzzle.
Responding to your point of view
If I’m understanding you correctly, you seem to think that what I’m calling “bootstrapping” is still what’s fundamentally going on in the above—that there’s no real difference between, say, rederiving calculus from first principles and going from one’s hazy impressions of the t test to iteratively building a more full and precise articulation of it. That may be true. There’s probably a lot of what I’m calling “conceptual pathfinding” that would go into an effort to rederive calculus—it’s just the scale of the challenge that’s different.
My underlying motivation here is that sometimes, when people are confused about topic X, advice is given that if they’d just commit to understanding topic X from first principles, rather than trying to learn it in the typical college textbook format, that they wouldn’t be so confused.
And in the struggle to learn, I have also found myself gravitating toward the very actionable, but not very helpful or meaningful activity of just trying to memorize stuff to make the ideas go in.
The point of this post is to articulate what I do find helpful. It may not be very good as a crisp formalization, but I stand by it as practical learning advice for people with brains like mine.
I don’t find much use in defining conceptual pathfinding as local information bootstrapping since you mentioned only the scale of the challenge is different. What I often experience is that studying one concept always lead to me studying another concept since it depends on other concepts. This means that I am going to end up deriving all of calculus as long as I have the time and will. And that implies that given enough time and will, a single conceptual pathfinding turns into a global information bootstrapping (a very holistic understanding).
Basically, what I found useful about this post were:
-Your algorithm
-The concept of bootstrapping
What I don’t like:
-Pointless categorization
-Algorithm does not state how to manage the information that will come out of information bootstrapping.
Thanks for the feedback, Duck Duck.
What I liked about your comments:
You clearly read and thought about my post
You honestly expressed your own experience, including disagreement, in enough depth to promote a discussion
What I didn’t like:
Sharp tone (“pointless” categorization)
Complaining (“Algorithm does not state how to manage the information that will come out of information bootstrapping,” “I don’t think there is any real difference between “conceptual pathfinding” and “information/material bootstrapping”)