Let me preface what I'm saying by just expressing gratitude to 3b1b for making an amazing math animation library, getting tons of people excited about math(Including me), and producing some really great videos to give people intuition.
All that being said, I've found that 3b1b videos tend to bound the intiution you can get on a topic. Most of his videos tend to reframe an idea in some sort of interesting way that can be easier to grasp without rigirous definitions and theorems. If your goal is to just understand the ground level of the idea, then this intutition is really great.
I've found that this "visual" intuition becomes unhelpful as soon as you try to grapple with a more advanced version of the topic. Your understanding of the topic is often completely limited to the way he frames it, and becomes quite useless when you leave that limited framing. He doesn't teach you "fourier transform", he teaches you a very specific context of the fourier transform which doesn't generalize well.
Again, this is still super valuable if you just want a broad understanding, but not particularly helpful for a comprehensive education of a topic.
> Your understanding of the topic is often completely limited to the way he frames it, and becomes quite useless when you leave that limited framing.
I don't think the limited understanding is useless once you leave the simplified framing. As an example, linear algebra is far more than just 2/3D continuous space that he covers in "Essence of Linear Algebra", but I think having the solid visual intuition for those simple cases gives you a foundation to learn the less visualizable versions of linear algebra.
I think this is just another manifestation of the "office hours phenomenon":
1. You don't understand a concept
2. You go to your professor's office hours with a question and ask them to explain.
3. They draw some squiggles on the board and walk you through the issue
4. You say "Aha! I get it! Thank you for explaining!"
5. Two steps out the door you've completely lost the idea and need help again.
You haven't internalized the idea, you're still just attached to someone else's idea. To understand something fully, you have to really take it in and play with it; find the holes and the limitations of the model.
> Your understanding of the topic is often completely limited to the way he frames it, and becomes quite useless when you leave that limited framing.
Completely limited?
My framing of human learning goes roughly like this:
1. When we are motivated to solve a particular problem, we can draw upon a multitude of “frames” (one example being metaphor, another being an equation, another being a visualization) to help us.
2. People sometimes cannot shift frames of reference effectively (i.e. perhaps fixating on only an few approaches). This is not the fault of the particular frame(s); this is a limitation in how people use them.
3. To our brains, almost everything is a “model” … a way of conceptualizing, deciding what to pay attention, what to ignore, what questions to ask, and when to recognize we aren’t getting anywhere.
4. Some models / metaphors are better than others for certain people and contexts.
- Some are better connected to other models, allowing more fluid thinking.
This reminds of Richard Feyman's observation how some people count visually while others do it by sound.
You might be one of those people that can do math purely symbolically, by parsing and manipulating expressions, a bit like regular language and you don't need your brain to build a mental image for the things you work with.
I am intuition-first kind of person, and let me tell you, intuition is not a crutch, it can do things where symbol manipulations would take orders of magnitude more effort.
It is actually possible to imagine and manipulate highly-complex, multi-dimensional things that don't have equivalents in nature.
Imagination is a muscle used a lot when thinking intuitively, some of us have it developed to a ridiculous degree.
I think that's a good way to look at it. I'm definitely on the symbolic side of things for my understanding. I study pure math and the 1 applied math course I had to take(Applied Complex Analysis) was a nightmare because I was doing math without a formal framework.
I will say though that not every topic can have a sensible visual model to it, and being able to acquire symbolic intuition is probably essentiall to go far in any field.
> I've found that this "visual" intuition becomes unhelpful as soon as you try to grapple with a more advanced version of the topic. Your understanding of the topic is often completely limited to the way he frames it, and becomes quite useless when you leave that limited framing.
All that being said, I've found that 3b1b videos tend to bound the intiution you can get on a topic. Most of his videos tend to reframe an idea in some sort of interesting way that can be easier to grasp without rigirous definitions and theorems. If your goal is to just understand the ground level of the idea, then this intutition is really great.
I've found that this "visual" intuition becomes unhelpful as soon as you try to grapple with a more advanced version of the topic. Your understanding of the topic is often completely limited to the way he frames it, and becomes quite useless when you leave that limited framing. He doesn't teach you "fourier transform", he teaches you a very specific context of the fourier transform which doesn't generalize well.
Again, this is still super valuable if you just want a broad understanding, but not particularly helpful for a comprehensive education of a topic.