This is nicely put. It reminds me a bit of David Perkins's [0] "whole ballgame" approach (i.e. the best way to learn how to play baseball is to play a few games or approximations thereof, e.g. tee-ball; you'll struggle mightily at first, but you'll ultimately benefit by having a sense of how the game as a whole works, and it provides valuable context and motivation for subsequently working on specific skills like throwing the ball, swinging the bat, etc. And the ultimate result is greater mastery than if you had started from the skills and worked your way up to playing the game piecewise.) Fast.ai's pedagogy is based in part on this principle, and (as someone inclined to go down rabbit holes) it's been a huge help to me as I go through the course and book. [1]
I'd add too, recently released personal knowledge base apps (Obsidian, Roam, et al) are very well suited to fractal learning. I partition my notes into content (summaries) and concepts (with some Zettelkasten-style ephemeral notes mixed in). When I come across new concepts, I create a stub; then I fill it in when necessary with the necessary degree of detail, adding new details, revising, and forcing myself to make my account of it coherent as I encounter different facets of the concept in different books, articles, etc.
I second everything in this. I recently switched my note taking to be organized around concept "hubs" with Zettelkasten-esque linking between them, and it's maybe the biggest improvement I've made in terms of organizing my learning. I sometimes cover material a bit slower, but I rarely have a single session now where I don't have some kind of aha moment. This is a direct product of having to explain what I'm learning in order to fit it into my system, a la the Feynman technique.
The starkest example to me of breadth first vs depth first learning are math geniuses. They typically ignore everything except mathematics to begin with and then hyper focus on a specific area of mathematics intensely. The most striking I know of is Peter Schulze who basically never attended an undergraduate class in mathematics and picked up Linear Algebra while he worked out the requirements to understand the Langlands Conjecture. He got awarded his bachelor by doing a 6 hour exam when he entered University.
> He got awarded his bachelor by doing a 6 hour exam when he entered University.
I wanted to read more about that (I like such stories) but I couldn't find that story online. On Wikipedia it says that bachelor took him 3 semesters. Does anyone have the source?
I might misremember it slightly, I got told this by a professor (he was one of his examiners) and people that knew him at Bonn in ~2010. He could also have been awarded the majority of the credit points towards the Bachelor at entrance.
This doesn't sound very practical advice to me. It seems to be saying: Do a bit of breadth-first mixed with some depth-first. But whatever you do don't do only depth-first.
But maybe that is an important point. Don't dig too much into details try to get the overall picture so you can decide what you should focus on most.
> This doesn't sound very practical advice to me. It seems to be saying: Do a bit of breadth-first mixed with some depth-first. But whatever you do don't do only depth-first.
It seems to be saying that to me too.
Given the tree and maze visualizations, I'm a bit surprised that there was no corresponding visualization of the recommended strategy of switching back and forth between depth-first and breadth-first search (at regular or perhaps random intervals).
Regarding the optimality of various strategies, there is a distinction to be made between the optimal strategy for an individual learner/searcher, and the optimal individual strategy for searching collectively.
Also, there is a difference between strategies for acquiring existing knowledge that is new to you, vs. expanding the boundary of existing knowledge.
Regarding the "expanding the boundary of knowledge collectively" problem space specifically, it is worth noting that a pure depth-first approach isn't really possible in practice. Even in fields that reward depth-first individual search strategies such as math, an individual searcher's search is bounded by the length of their career, and each new searcher starts semi-randomly somewhere along the then-current boundary of collective knowledge.
That sounds pretty reasonable to me. You don't want to dive straight into depth-first because then you disappear down a rabbit hole with no context to tell you whether it's the right rabbit hole.
Start out with breadth-first until you have a feel for the lay of the land, then pick a direction to journey in.
Right that sounds very reasonable. My issue is that it is so reasonable that it is not a great insight worthy of an article. It is good but quite basic advice.
It also must be dependent on the topic under examination largely therefore the advice can only very generic.
Let's think about an example topic, say "Mathematics". What would be the best strategy for learning mathematics?
In fact, I've thought a lot about how to generalize these techniques to a data structure (i.e. knowledge graph) so that anyone can zoom in or out to whatever level of abstraction they please:
It's an analogy on learning. Every other day there's some blog post on HN about some analogy on some self improvement topic like learning, project management or complexity or whatever...
I'll tell you what learning isn't. Learning isn't coming up with new analogies for concepts you already completely understand over and over again.
I and pretty much everyone on this planet already knows what learning is, and unless you have a new learning technique, what is the point of comparing learning to a fractal? There's nothing new here.
I'm not trying to reprimand what this person wrote this but when you see some flavor of this on HN all the time it gets old.
You're not a stupid person so you being not stupid knows that I and pretty much everyone on the face of this earth knows that we all are aware that no one understands ALL of learning. So if you're not stupid, and you already know this, why did you take the time to point out some pointless flaw in my wording?
I'll tell you why. Because you liked this analogy. You thought it was profound to compare learning to DFS and BFS and fractals and you were sort of offended when someone like me didn't find it profound. Not only do I not find it profound, I find it obvious and trivial. Comparing learning to fractals is like some offhand thought I can have for like 5 seconds than forget about because it's so trivial.
Too each his own man. If this article blew your mind great. It didn't do anything for me.
Here's another analogy for you.
Learning is like traveling down a path and encountering a several forks in the road. You can BFS the paths or you can DFS the paths
Replace Learning with "Life" and remove BFS. In life you can only choose one path every time you encounter a fork and the decision is permanent. No popping the stack like in BFS to try the other paths. It's just DFS all the way to the last final leaf node. You have one life so live it right.
Did that analogy blow your mind? Because I made it up 3 seconds ago. That's how trivial these analogies are.
I was impressed by the article and when I read your comments, I felt like this random person is right, it does sound profound but doesn't add anything substantial to the topic of learning.
It's nice to have another aspect or perspective on learning but it is certainly not a "new learning strategy". I agree.
And yet there is a merit in analogies, they can help you understand the underlying principles of something, in this case learning. I'm not sure people (including me) always get the full picture of something and an article can help with that showing learning from another angle.
In general I agree with you and yet the article adds something for some to their understanding of what learning is.
Careful. Analogies are a common form of deception. Used often in religious speeches, political speeches, blog posts and more.
The speaker offers zero substantial information yet manipulates the audience into feeling the speech was "good" through the use of analogies. There is sort of a catharsis when the person listening to the speech connects the dots between two unrelated concepts, and this subtle emotion is used as a form of manipulation often deliberately when the speaker is incredibly intelligent and often accidentally as is what I believe is the case here.
It's easiest to see the deliberate case in religious speeches or sermons if you're not religious yourself. Often you will find that religious principles or sayings have no proof and you will often find that the analogy is deceptively used in place of evidence or proof where no proof exists.
I sometimes find tremendous insight in a piece of literature that is analogous to real life difficulties. Provided that, that piece of writing, also offers some sort of a solution to the "problem".
Those problems I'm referring to are for example confidence, psyche (esp. C.G.Jung), getting over traumas etc.
Obviously, my formentioned cases are not extensive and everyone finds something else that they struggle with where a piece of literature might help.
2. Thank you for your insights. If your answer to question 1. is no, I'd like to know what kind of analogies you think are valuable and why?
1. No, I don't disregard analogies, I'm just able to recognize and isolate analogies from a topic or a point. For learning, I agree an analogy is very useful.
Analogies are tools for illustration, not for proving a point. To use a tool to illustrate a concept that is already well known (learning for example) is often pointless.
In terms of utility, much of the utility of analogies is similar to the utility of art, music or humor. You can enjoy humor, but humor itself doesn't offer any greater insight or a topic.
2. Analogies are useful for illustrating things that cannot be imagined by the human mind. The extrusion of a 3 dimensional cube into a 4 dimensional cube cannot be pictured by your minds eye. You can only visualize the analogy: The extrusion of a 2 dimensional square plane to a 3 dimensional cube. The entire field and existence of higher dimensional geometry is inferred from analogies.
Note that these are only good for things that are not known by you. Much of literature and speeches use analogies for things that are well known. For a list of examples where analogies are mostly useless see here:
Thank you for answering my questions and expanding more on the topic. You actually didn't have much convincing to do because it seemed so fundamentally true that it was hard to come up with an argument against it. I still believe analogies have more value than being mere approximations to an unimaginable world (numbers, states, etc). But like with all beliefs, I don't have any evidence to support my beliefs and rely on my intuition and experience.
> I still believe analogies have more value than being mere approximations to an unimaginable world
I didn't say this. Although my example is actually impossible to understand without an analogy, I am saying an analogy can help you understand something you currently don't understand.... it does not need to be impossible to understand without an analogy.
The quotes in my link are actually mostly things and concepts you completely understand. Nothing new is learned... the analogy is deception.
There's a quantum aspect to learning, if your learn speckles of everything, your 'energy' level doesn't go up. If you go breadth first, you have to cross some pragmatic, practical, real newness barriers, keep that in mind.
I'd add too, recently released personal knowledge base apps (Obsidian, Roam, et al) are very well suited to fractal learning. I partition my notes into content (summaries) and concepts (with some Zettelkasten-style ephemeral notes mixed in). When I come across new concepts, I create a stub; then I fill it in when necessary with the necessary degree of detail, adding new details, revising, and forcing myself to make my account of it coherent as I encounter different facets of the concept in different books, articles, etc.
[0]: http://www.pz.harvard.edu/resources/making-learning-whole-ho...
[1]: http://course.fast.ai/