Lessons from Grant Sanderson

Grant Sanderson, the creator of the acclaimed YouTube channel 3Blue1Brown, has become a prominent voice in mathematics education, known for his visually-driven and intuitive explanations of complex topics. Through his videos, interviews, and writings, he has shared a wealth of insights into the nature of mathematics, the process of learning, and the art of explanation.

On the Philosophy and Beauty of Math

On the Nature of Math: "I think of math as being the study of abstractions over patterns and pure patterns in logic." [1]
The Interplay of Discovery and Invention: "I think there's a cycle at play where you discover things about the universe that tell you what math will be useful, and that math itself is invented in a sense, but of all the possible maths that you could have invented, it's discoveries about the world that tell you which ones are." [1]
The Unreasonable Effectiveness of Math: "History has this strange trajectory of things that seem to be useless coming back and being useful. But it's a real deep question. Why?" [2]
Math as an Art Form: Sanderson aims to "motivate math as an art form" by exploring "puzzles and problems with some intrinsic beauty, particularly those which seem quite challenging at first, but where some shift in perspective makes them both doable and beautiful." [3]
The Power of Abstraction: "Abstraction is the cost of generalization." [4]
Symmetry in Mathematics: "Math has a tendency to reward you if you respect its symmetries." [3][5]
The Emotional Side of Math: Instead of prioritizing usefulness, Sanderson emphasizes "emotion, wonder and imagination — similar to how one would engage with fiction works like Harry Potter." [6][7]
Math as Storytelling: "I think the thing not enough people talk about is what I'm just going to call story. And when I use that word I mean appeals to emotion i mean having comedy having some notion of characters that you care about i mean having a mystery you need to see resolved." [8]
The Universality of Math: When asked if alien mathematics would be different, Sanderson suggests that while the path of discovery might differ, the underlying logical truths would likely be the same, especially concepts rooted in physics. [9]
The "Aha!" Moment: The most beautiful ideas in mathematics are often the ones you have a little bit of understanding of, but not a complete one, leaving an element of mystery that makes them beautiful. [1]

On Learning and Problem-Solving

Active vs. Passive Learning: "You remember about 10% of what you read, you remember about 20% of what you listen to, you remember about 70% of what you actively interact with in some way, and then about 90% of what you teach." [10]
The Importance of Problem-Solving: "Your temptation will be to spend more time like watching [or] reading. Try to force yourself to do more problems than you naturally would." [10]
Struggling is Part of the Process: "Don't consider yourself done with the chapter until you've actually worked through a couple exercises." [10]
The Value of Reinvention: "The things that I've learned best and have the deepest ownership of are the ones that have some element of rediscovery." [11]
Acknowledge Your Gaps: "Math helped me realize I wasn't that smart." [12]
Look at Problems Before Reading the Chapter: "If you can take a little look through those questions at the end of the chapter before you read the chapter a lot of them won't make sense some of them might and those are the best ones to think about." [10]
Flexibility in Learning: "The people at the end who made it through weren't the ones who had more intuition. They were the ones who were flexible enough to try different ways of learning." [6]
Give Things Meaningful Names: A key problem-solving tip is to "Give things (meaningful) names." [13]
Leverage Symmetry: Another core problem-solving technique is to "Leverage symmetry." [13]
Ask a Simpler Version of the Problem: When faced with a difficult problem, a helpful strategy is to "Ask a simpler version of the problem." [13]
Draw a Picture: Visualizing the problem is a powerful tool: "Draw a picture (have some numbers? Make them coords!)." [13]
The Role of Programming in Learning Math: "I know a lot of people who didn't like math got into programming in some way and that's what turned them on math." [10]
Reflect on Solutions: "When you aren't able to solve something and you read a solution don't just read it and see if you understand it try to step back and say 'Okay what was the instinct that I didn't have that I could have potentially had that would lead me to that solution?'" [14]
The Power of Teaching to Learn: Doing something to teach or to actively try to explain things is huge for consolidating knowledge. [10]
Don't Be Afraid to Toss a Book Aside: "The most important step seems to be finding the book that meets me where I want it to, not too easy, not too hard, and written in a way that meshes with my goals." [15]

On Teaching and Explanation

The Goal of Explanation: "My goal is for you to come away from this series feeling like you could have invented calculus." [5]
Visuals First: "When we put the visuals first, then start to articulate the meaning, you can get a sense of ownership over it." [6]
Resist the Urge to Start with Abstractions: "Pretty much all textbooks and explanations do it the other way around [starting with the general abstract thing]. And it's because that's so tempting to do once you understand a topic." [13]
Education as "Educing": The etymology of "education" shares a root with "educe," meaning to bring out. The goal is to bring understanding out from within the student, not just push information in. [16]
Motivation Over Explanation: "The main issues with education and limiting factors for new students learning new things are problems of motivation not of explanation quality." [16]
The Role of the Educator: "The role of educators goes beyond simply providing explanations; it also involves facilitating projects, identifying students' interests, and curating appropriate resources." [14]
The Power of Storytelling in Teaching: Thinking of a math lesson as a story you're telling, even a technical one, can result in something that keeps people's attention for longer. [17]
Don't Just Give the Answer: "A meaningful part of the value to add is not just the technology but to give the story around it as well." [11]
Authenticity Over Simplification: "People don't want to feel like they're just being watered down or talked down to, instead having a sense that what they're staring at is the authentic material as if they were studying it and becoming an expert in it i think that's a very inspiring feeling." [18]
The "Pedagogical Curse": Mathematicians can sometimes fail to give pedagogical clarity the same importance as mathematical rigor, making math harder than it needs to be for learners. [19]

On Technology and the Future of Math Education

Manim's Origin: Manim (Mathematical Animation Engine) started as a personal programming project to create a graphics library in Python. [7][20]
The Purpose of Manim: "I wanted something that was flexible enough that you didn't feel constrained into a graphical environment." [21]
When to Use Programmatic Animation: "Try to find a workflow that distills down that which should be programmatic into manim and that which doesn't need to be into like other domains." [21]
The Role of Online Platforms: While online platforms have revolutionized math education, they cannot fully replace the personalized guidance and feedback of educators. [14]
AI in Math: Sanderson views AI as a powerful tool that is "better than most people at math" but not yet at the level of the best human mathematicians. He sees its development as more of a continuous process than a discrete jump to AGI. [12]
The Future of Learning with Technology: "Whenever it becomes the case that the right medium to do that [explore a math topic] lends itself to simulation and to programming and all that, that feels like it would get to the point where it shifts the way that you even think about stuff." [4]
YouTube as a Primary Source of Learning: Sanderson gives a "yes and no, emphasis on the no," highlighting that while videos can provide intuition, they are only a partial part of the learning process which must include active problem-solving. [16]
The Value of Community in a Digital Age: "The things that motivate us are probably first and foremost our peers." [16]
On Not Being an "Influencer": "I feel proudest of a video when its sole focus is the lesson at hand, and if whatever limited influence I have over the viewer is not sold to a third-party, but is directed towards making that viewer love math." [3]
The Importance of Open Source: Sanderson made Manim open source because so many people asked how he made his videos, and he figured sharing the code was the easiest answer. [3]

On Creativity and Personal Growth

Follow Your Curiosity: "I start with whatever sparks my own interest and curiosity. If I'm not genuinely fascinated by a topic, I doubt I could convey it in a way that excites others." [3]
The Value of Low-Stakes Beginnings: "There's a benefit to starting in a way that is low stakes like you're not banking on it growing." [20]
Embracing the "Nerd": "I think there's always this unappreciated pent up demand from nerds for content that doesn't just give a surface level description of what science is saying." [13]
The Power of a Personal Project: 3Blue1Brown began as a personal programming project, driven by a desire to explore and create. [3]
On Making an Impact: "I unequivocally want more people to self-identify as liking math." [12]

Learn more:

Lessons from Grant Sanderson

On the Philosophy and Beauty of Math

On Learning and Problem-Solving

On Teaching and Explanation

On Technology and the Future of Math Education

On Creativity and Personal Growth

Written by Antoine Buteau

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