In grad school, we referred to those magic steps handed down from on high as "proof by proctological extraction."

1:02:59 "How does he phrase it? He's so much more articulate than I am," said one of the most articulate guys I've ever seen on youtube.

I love how the lecture itself passes all three checks

When I was younger, I was into both mathematics and creative writing. The result is that, when I went on to teach mathematics, I would always try to structure my lessons like a story. Not to say that I'm *literally* up there telling a tale, but rather that I would use the concepts and frameworks that ensured a story was interesting and meaningful to create a lesson that (hopefully) had the same effect. Freytag's triangle (it looks like _/\_) is one particular tool that I would always use. The "intro" is the motivating question, the "rising action" is concrete examples with higher and higher levels of abstraction, the "climax" is the general or abstract idea I'm trying to convey in the lesson, the "falling action" is the re-interpretation of the concrete examples as instances of that general idea, and the "resolution" revisits the original motivating question and shows how the new concept answers it -- and sometimes sets up for the "sequel" (the next lesson) when there is an obvious follow-on question. I think it's really intriguing that the ideas you have are very similar to this storytelling approach, even if they aren't consciously or explicitly inspired by storytelling. That also includes the "discovery fiction" you mention as an answer to one of the questions. I find this concept to be especially helpful in motivating definitions, because it preemptively answers the most common question of abstract definitions in math: how/why would anyone come up with that? Thanks for giving this talk, it's a very inspiring lecture and shows how exciting of a time it is now to be both a student and an educator in mathematics.

I think pedagogical consideration in university math classes is often missed by professors who are just focused on the subject and not who they are teaching it to and how the students feel in their learning process. Great talk Grant, beautifully visualized and great live presentation skills! Awesome to see

The "I thought I liked math" part is really funny to me because I'm the exact opposite, I always saw math as something I didn't like at all but could do okay at if I worked at it, until I took linear algebra and absolutely loved it. And was shocked to see myself near the top of the class in it, after feeling mediocre to inadequate the previous semester in differential equations.

I used to use that x^2-y^2 trick all the time in non-math related classes I taught. People usually thought I just memorized a bunch of them. Then you let them in on the "secret" and they're actually more impressed than when it was just magic to them.

Also, what he's talking about around 45:00 is right on the money in terms of pedagogical theory, and is one of the things that drives me crazy about standard textbooks and the like. The pedagogical research is very clear that students learn better when you start with the specific (e.g. numerical examples) and move from there to the general. But the usual approach in traditional teaching is to *start* with the general, and present specific examples afterwards. This is especially true in the way math gets presented in courses and books, but is also true in other disciplines, such as my own, which is physics. It's a habit we need to get out of.

I said in another video comment that Grant deserves some new, yet-to-be invented prize that should be the equivalent of an Oscar for best computer generated imagery, an Emmy for outstanding narration / editing and a Nobel Prize in science for fostering interest in mathematics and science. I guess this is a start. He does amazing, inspired work. His videos visualizing the mechanics and mathematics of the propagation of viruses in a population may have saved many lives during COVID and could saved many more had they appeared in mainstream media.

Grant showed me how being thorough does not have to be tedious. Others I highly respect are: Penn, Parker, BPRP, Eisenbud, Grimes, and Copeland. All of these people have changed my view of mathematics and science in the last 5 years (I'm 57). I wonder how my life would be different if I had this resource in my teens and twenties when most are at their peak of whatever brilliance they're given.

~40:00, on the power of abstractions: I'm reminded of one of my favourite maths jokes. It's one of those jokes that has a certain element of truth in it: “It's in fact extremely easy to visualise a 13-dimentional Euclidean space. All you do is imagine an _N_ -dimensional Euclidean space, and then simply let _N_ =13.”

This man has saved a generation of students and will be a beacon in math for generations to come. As a young adult mathematician who is a visual learner, his work has inspired me many times over. To see math as a beautiful, creative, and curious field. Yes with rigor, but also with a capacity to maintain an open mind; to invite the self into the equation. I’m so thankful to him and to this organization for recognizing his work. He changed my life and rekindled the love and joy of math inside me ❤ I’m deeply thankful to him and his team

❤ My goal as a high school math teacher is to facilitate “a sense of how I might have discovered that myself”, and I agree that “the act of doing that really raises the ceiling” which at times causes uncertainty and discomfort not only for the students but also for me as the educator. I believe that’s the space in which one is truly learning how to be a mathematician. Thanks for sharing this. I needed a 4th quarter pick-me-up. Feeling inspired, continue sharing your message, especially your layers of abstraction. I know that my use of that sort of framework is the reason that many of my students have commented that they enjoy math again.

Such an interesting talk! 1, diagrams & visualization 2, concrete examples 3, motivating questions

Very often, I find myself procrastinating my homework on math, by watching grant talk about math. You're doing something right, and I pray math educators out there are paying attention. 🎉Congratulations, the award is well deserved. I hope you win many more.

Maybe not suitable for a strictly math audience, but another pedagogical check is "why does the audience care?" For engineers, physicists, economists, and others who don't relish in math for it's own sake, knowing where the current discussion applies to their field both better engages them and makes them more likely to use the math.

Grant. I am a recent graduate in applied math and data science. I can not stress enough how helpful your channel has been in my efforts to learn mathematics. I am so proud of you for your success and wish you the best of luck with your future endeavors, I will certainly be following them. Congratulations!

Concerning the checklist of "things you want to have for a pedagogical math text", I'd add that each definition needs 4 types of examples: 1) *Common examples*: ie, the things you'll want as the go-to 'simplifying' image in your mind for basic operations (eg, R2 and R3 for vector spaces). These help ground the student and gives them confidence. 2) *Less common, more advanced, but important examples*: ie, things which are the first "profound extension" you'll see of the abstract mathematical concept (eg, sequences N -> R and functions R -> R for vector spaces). These are often highly motivating, because when presented progressively as an extension of the common examples, they generally lead the student to have a much easier time with more complex material. 3) *Non-examples*: ie, things that are "almost valid" for the definition, but fail for one specific aspect (sometimes just one suffices, sometimes you'll want one non-examples per aspect of the definition) (in the case of K-vector spaces, R-modules, where you can't always divide by a scalar, can serve as such). These provide the technical nuance as to why the various aspect of the definition are important, and where some missing aspect will lead to different theorems. 4) *Exact examples*: ie, examples which are just precisely in the definition, but aren't an example of something more complex (ie, distinguishing neatly between a vector space without a dot product, and an inner product space; I tend to make a visualization of 3 parallel real lines for R3, like a free vector space over 3 elements, to distinguish from the usual cartesian 3D space which presupposes angles and a dot product, for example). These are often the crutch you want to rely on when you're unsure as to whether your common example is the right intuition to follow, and can provide both rigor and a sense of security. Having all these sorts of examples, I've found, is what most cements both the technical/rigorous aspect, AND the intuitive aspect, of any new concept in mathematics. It's the best checklist I've come up with when thinking about "making things as simple as possible, but not any simpler than that", and it works quite well (once you've built the right rapport with the students).

I've known about the difference of two squares formula for half a decade and never saw the visual proof, and that still feels incredible to find out! So many steps in math education get skipped because the context isn't viewed as important as the rules and conclusions!

I'm teaching Vector Calculus in my 1st year college students who just started to touch differentials and integrations last semester. They got the concepts like gradients, divergences, and curls better when I made a lot of visuals. Also I used numbers as examples than some abstraction. I'm still learning a lot when it comes to the better pedagogical approach in teaching physics and math. You are a gem, Sir Grant! Thank you that you are saving millions of people around the world by teaching mathematics clearly and interesting.

As a teacher, I have a lot of coworkers that would really like this video.

His point right at the beginning about the tensions and synchronicities of popularization vs. education vs. research presentation and how it is good to avoid watering down something because it is "just" a popularization is very apt. I think his Essence of Linear Algebra is a great example. Is it educational? Is it a popularization? Yes! Definitely both. It is beautiful and highly engaging. But it is also deep enough that even though I've been "doing linear algebra" for over 20 years, I learned things from watching that series of videos. I honestly think those videos are one of the best things on the internet.

Every math teacher should watch this. TOTALLY explains what our school teachers lack. This is a very inspiring lecture for me, as I explore how to rediscover the excitement in learning maths, for myself and others. Thank you again.

One of the hardest lessons I've had to learn in communication is that I need to tell my audience what I'm writing (or speaking) about before I dive into what I want to say. I've noticed this pattern in the writing of other mathematicians: they often forget that the audience needs to know what they're talking about. The reader is forced to piece together from context why these equations are important, useful, or necessary. This makes the process of reading math soooo much more difficult, and for someone who struggles with math to begin with, perhaps impossible.

As a chemist, i think a lot of math is very beautiful and satisfying, but also incredibly fussy/neurotic and some people can't really get past that.

34:49 WOW this example blew my mind. THIS is the purest example of the most common pitfall that math teachers fall into. It's not just about difference of squares it's about *everything*. My heart breaks for all the people in the world who fell out of math because they were a part of class B. I know I was several times, and I never fully understood why so many of my classmates *hated* math.

Grant has me accidentally clicking on a video and paying attention for an entire hour.

In high school, I found it absolutely impossible to memorize the quadratic formula, and so, in calculus or whatever (including when taking the BC calc AP exam - hope the proctor who graded it liked this) I would have to use a margin of my paper to derive the formula. I got fairly quick at it, but I also think it gave me a much deeper understanding of what the formula was doing.

Thank you Grant, for allowing us all tol be Class A students. Really, we're grateful!

This reminds me of an intro accounting class which, at least for me, worked so well because it was taught almost entirely with the framing of a history lesson. That is, how modern accounting developed (starting with Venice), and what problems each addition or alteration was aiming to address. The fact that the class was grounded in real people trying to solve real problems made a world of difference to what can otherwise be a very boring and abstract subject.

50:23 For future reference, I'll point out that the transfer scores are flipped (the paper reported that the group that first did problems and then had instruction had a transfer score of 5.4 +/- 1.5).

Great stuff as always, well deserved, you’ve helped me in my teaching (and learning) of many things. Fact check at 50:50 is the Transfer result a transposition error? You have the transfer figure a standard deviation WORSE for the right hand group!

A good example of this is watching Grants videos for years (examples), then this video shows the “formula”, i.e., oh yeah that’s why I like those videos so much. It’d be way different if you just watched this talk and said “yeah that makes sense” and then just left it at that

17:08 This, to me, is THE core problem I had with my classes in school. The insistence that I just trust that something works and refusing to elaboarate on *why* -- demanding _faith_ for something as exacting as mathematics -- is such a grandly miserable failure of educational praxis that it's distressing to me that you even feel like you need to gently walk an entire audience through why it's a problem. That said, I still believe _in general_ that the "original sin" of maths education is in hiding so much of what it can do for us. Mathematics is so incredibly vast as a discipline that pretty much any problem you run across in your actual non-mathematician life probably touches on it in some way. But students who aren't already extremely deep into it are unlikely to recognise that fact. Or, perhaps more frustratingly, recognise that there must be a mathematical approach to what they're looking at but not what that approach is or even have the language to articulate which subfield(s) of mathematics deals with it. I think a lot of people who say stuff like "I'm not a 'math person'" are victims of this poorly-conveyed mapping between concept and application.

It’s cool how closely your pedagogy checklist lines up with how I was taught to give an engaging research talk. Really speaks to the overlap you mentioned

5:37 This was me in my third year of my math program. I still like the concept of math, but I had to get off the academic train for my own sanity, as I continued to notice the increasingly wide gaps in my knowledge that was preventing me from learning things at the pace a degree wanted me to.

This is the magician showing how the trick is done, and yet it doesn't spoil the trick.

I personally would not have picked up on the pattern of those numbers, but I think that's because I'm very slow at basic arithmetic. You show me the visual grid and I pick up on the connection to the difference of squares formula almost instantly. I tend to be more visual and see math as symbols and relationships. With the pure algebra, the symbols and relationships are stated pretty explicitly. With the visual grid, the symbols are the physical rectangles themselves, and the relationships are how you can cut up and move around the pieces. You just show me a number and ask me to factor it though, and I won't notice that it's one away or n² away from another square without manually doing the calculation, or I end up with unsimplified square roots. Also, the algebraic form led me to realize that the squared modulus of a complex number is just a special case of the difference of squares.

Well deserved award. Congratulations Grant! His methods of exposition connect with my learning style more than any math instruction I've ever experienced. I believe that analogy and metaphor are requisite steps to ratcheting human brains up to the higher levels of abstractions.

The audience laughed at the worksheet, but that is how I was taught most math. Towards the end of college I was shown how to painstakingly take the "class B" style instruction and create my own examples, giving myself something kind of like "class A".

YES!!!! I adore that KZfaq channel! As a neurodivergent person, I really can’t express the amount of help, support, joy and hope that 3 blue 1 brown has provided for me and my education. I’m naturally a creative person so to put the visual creativity into demonstrating mathematical concepts really is something so close to my heart.

Something I have been saying for a long time is the importance of history, both in mathematics itself and in the pedagogy of mathematics. How does this mathematics connect to previous mathematics? What were people thinking when they developed it? Although who they were isn't so important, they'll be mentioned, of course. History ties into some of the other points you made, Grant, but it's worth including as something to be checked off.

Man, I love the An Opinionated History of Math call out, that man is great, always something deep to say about math and culture.

What a wonderful talk! Thank you Grant!

Congratulation Grant! Keep up the great work.

Well deserved, Grant! Looking forward to watching more of your content!

38:52 holly mollly, i knew a2 - b2 = (a+b)(a-b) all these years, but yeah, the Class A examples never crossed my mind _with such an impression_. Yeah, i had also went nearly that off by 1 thing (6*8 =~ 7*7) while thinking about approximation for logs, but i never sort of extended it to off by 4/9/25 etc ... and i am someone who _has_ (fortunately still) interest in maths (though, it's deterred many times in the due course, but i am a keeper lol)

yayyy, congratulations Grant, deserved recognition 🥳🥳

Got to see this in person, great talk!

I want to point out that the Poisson Summation Formula is nothing more than a mere restatement of the validity of a Fourier expansion. - This means, that if the Complex Analysis textbook already justified the Fourier expansion (for example, by proving the inversion formula for the Fourier transformation), then a much shorter proof could theoretically just hang off the Fourier expansion formula. - Alternatively, you can consider the Fourier expansion as a special case of Sturm Liouville problem - its eigenspace is dense, because the inverse operator (convolution with the Greens function) is a compact operator, and we could appeal to the functional analysis result. - Alternatively, you could observe that the Fourier expansion is a series of matrix elements of the compact Lie group U(1). And we could then appeal to the Peter-Weyl theorem (which in turn turn appeals to the Stone Weierstrass theorem).

Absolutely brilliant video!

I thought I don't particularly like math until... I kinda got a glimpse of "real math" and "real mathematicians" and how _they_ perceive math and think about math. Now I'm pretty much fascinated and mesmerized by maths. :)

43:07 Layers of Abstraction: Categories Vector Spaces Functions Algebra Numbers Quantities

This is a profoundly important talk, capturing with great clarity why mathematicians (or really any domain expert) are often such terrible teachers. I'm going to write a considered response to this video and post it as an article, and as a KZfaq video. Three points: 1. The Mathsjam folks in the UK do in fact have explicit training that teaches mathematicians how to present ideas clearly to a mass audience. 2. It's amazing to me that something about the contract that KZfaqrs have with their audience has cause math youtubers as a whole to produce explanations that are drastically clearer and more humane than anything that has come before. 3. As one of the commenters below mentions, it's all about storytelling. For a story to be meaningful, you have to experience the protagonist's struggle. A mathematician jumping straight to the formula is like J. K. Rowling condensing the Harry Potter series down to "Harry killed Voldemort", and discarding the rest of the series, or a Mario game being reduced to a single button on the screen that says "Rescue Princess", or a Golfer walking to the green and dropping the ball into the hole, as if it were logically equivalent to playing golf.

This reminds me of the "Crash Course Computer Science" series here on KZfaq. First they show you how to make logic gates with transistors, then how to make different compnents by combining thsese logic gates etc. It happens so often that they have "another level of abstraction" as a catchphrase.

You too are a personal hero for many, Grant and a very well deserved award indeed! Thanks for all your hard work over the years which has led many people to be able to look at mathematics in a fun, engaging and encouraging way. Especially in your empathetic voice!

Great lecture. Thank you very much🙏

I *really* agree about the re-discoverability of proofs, I remember in class feeling like : "you mortals check proofs that genuises have invented". And I found it extremely frustrating. Even if there is a strong truth into it, I think we need to make palatable the path of discovery of the proof, and tell kids that not everything needs a genius, and that the order in which we teach maths is not the historical order of discovery, so a lot of proofs are discoverable by mere mortals because we don't have to respect historical order.

Great lecture Grant!! On that checklist of yours, I find math history, as in the specific story of the persons discovery, is incredibly helpful and engaging. It often gives you a sort of "mnemonic" or trick for thinking about it. Many of Euler's stories come to mind. Much Love!! And Thanks For All Your Hard Work!!!!

Congratulations to my favorite Math teacher!

Great talk, really well presented!

Another thing is that, when we write proofs, we start from the conclusion and argue in favor of it. When we discover new shit, we explore the argument, and find the conclusion, by stumbling and bumping on related things around the problem

Very insightful talk, you're such an educator.

So great Grant, thank you

The shout out to opinionated history of math was unexpected. I love it so much. I'm glad Grant is aware of it and I hope he starts making some again.

Congratulations, great work !!!

Please bring the podcast series back

Great work Grant!

Great lecture! Thank you for helping inspire me to be a better teacher.

Amazing talk. Thank you

What a FANTASTIC talk.

Fantastic talk! Is that a typo in the slide at 50:45 -- the problem-solving-first class also had HIGHER transfer, right? But you switched the Transfer values.

Please continue the podcast series!!

One checkmark I would love to see is: "Is the way I think about and conceptualize the problem communicated and can/should the students think about it the same way just yet". In the mathy lectures I have been in that has been the key component. The professor motivates a definition in all its glorious abstractness which is great when you want to use it later. But only the good professors ever said something like: This can be x, y or z but it suffices to think about it like an A for now. This is also useful to communicate what level of understanding of the topic is required to follow the lesson further, leaving fewer people behind.

Great job! Have enjoyed your presentations in the past. This one should be shown to anyone teaching math grade nine or above before they start. In 50 years teaching math, I found stories and activities that steered to the topic very effective. In the levels I taught, rigor was a luxury topic.

This is an *amazing* lecture, thank you so, so much

I was really missing the ben ben and blue podcast just last week actually and this was great, if it could be on Spotify or other streaming platforms so I could listen on the bus that would be amazing but I appreciate this regardless (Edit: And the 3b1b podcast itself of course, the guests were always amazing)

Please sir, can we have some more! Love the discussion of the method and the reasoning as opposed to the practice, while also incorporating the practice as an example to motivate the methodology.

Really enjoyed this talk! I hope to potentially think back on these in order to explain things better myself!

So well deserved! Congrats Grant!

Thank you for what you said about honesty about the short-cuts. In college, the moment I "gave up" on physics for a while was Physics 3 when relativity was introduced and I found it greatly angering that literally everything I had been taught in Physics my whole life set me up to have the wrong intuitions about everything because it wasn't presented as leading up to advanced concepts, it was just teaching Newtonian physics because it's what was taught at that level a hundred years ago and the teaching was never updated. I felt lied to and frankly betrayed. How did I know I was being told the truth now, or was this just another thing to believe now that was convenient for my teachers? On the other hand, I vividly remember, including the placement on the page and the color of the call-out box, a blurb in my geometry text book in the section on parallel lines that said something like, "later discoveries like gravity and relativity have caused people to doubt this definition of parallel lines, and alternative definitions of this forms the basis of non-Euclidean geometry." I've never studied non-Euclidean geometry (yet!), but just knowing, "oh! There's something interesting here, just over the horizon, if I want to go in that direction." Just that quick paragraph kept me from the same frustration years and years later with mathematics that I got from physics. This "lies to children" approach named and excused lately by Ian Stewart and Jack Cohen I find so frustrating.

I thought I liked math until I learned about the axiom of choice, then I learned I like practical and computable math.

On the issue of applicability and motivation, I think there’s a general lesson that some students learn and others don’t that’s predictive of whether they’ll be motivated: the concept of a versatile tool. If you were packing a “survival kit,” you’d pack: -a sturdy, steel blade -flint (for fire) -rope(s) -hooks and pulleys Why would you pack those things without even knowing what you’ll use them for? It’s because they’re versatile tools. If you have to ask what, exactly, a versatile tool is “for,” then you’re missing the point entirely; instead, the right question is “what is its nature?”. A blade applies a large force to a small region; a fire is energetic; a rope is long, flexible, and strong; a mechanical advantage makes you stronger; mathematics makes you smarter.

Congratulations grant!

In allowing myself to indulge in my love of formalisation and metalinguistics. Edging the very pitfall our beloved Grant advises us against. I loved how his discourse about math exposition and it being pedagogically sound, is, itself, e pedagogically sound material. The delight that you can ludically learn about the Riemann Zeta function in his exampling of his own material of the subject. I wonder if he noticed himself the pedagogical nature of his talk about math pedagogy and rigor. Or simply is this naturally functorial about the subject. Chefs. Kiss.

Congratulations on the well-deserved award!

No-one would teach the use of any other "tool" without teaching what problems it solves and when to use it - except for math equations.

I wish I fell in love with math in my younger/schooling days...too bad there was no 3B1B back then. I had the wrong idea that math is such a boring subject but now all I can say is that math is the most interesting and very satisfying (and though often more challenging) thing one can learn and do. All I can do now is just teach myself and learn thru your wonderful videos. For that I am very grateful!

An idea just popped up in my head when you spoke about how starting a lesson with a motivating question is kind of hard in a video because people tend to be more passive and just watch the entire video: Split the video in 2 parts and publish the 2nd part after a couple of days as to gently nudge your viewers into really taking their time to "pause and ponder" :)

Well deserved. In my opinion one of the best educational communicators out there

I actually asked one of our professors at engineering school about an obscure notion in kinematics and when he explained it I was so shocked why nobody had told us about it before and what he said to me kind of justified why not everything is explained in class: he said "if we hand you everything the easy way, you'll just stop thinking"

Is it possible to have a link to the mentionned "Opinionated history of math"? I'm unable to find it on youtube in any way, either a channel (as mentionned) or a video. By the way, great speech Grant, still and always inspiring. I've been following your video's since before 3b1b and always have been charmed by the clarity and engaging way you present topics.

Congratulations for the award! I also appreciate your tokens of wisdom :)

This is brilliantly presented (as expected) and also super relevant to other subjects as well. My team is developing software for internal use in our company and some of the stuff in the pedagogical checklist is totally applicable to writing good documentation as well.

I really liked the part about the difference of squares. That result is really special to me, as I remember seeing those patterns for myself, and figuring out the connection, and that feeling of "discovery" was so amazing (although when eventually learning about it and seeing how easily it can be verified algebraically it took away some of the magic of it). If I ever were to teach it to someone, it would definitely be by showing them some examples, and help them see the patterns.

This was such an enjoyable talk! One framing for some of your points that you might enjoy considering is the importance of "play" behavior (by some loose definition of "play") in learning and memory. Play behaviors of different sorts are present across many (most?) animals, especially social ones, and seem to be intrinsically rewarding and a strong enabler of learning. The trick to motivating in many cases where it might be otherwise difficult seems to be in framing some intrinsically motivating "play" in a way that lets you cheat your point into the conversation. Well motivated examples, to me, feel like a specific instantiation of play behavior. It might be fun to consider what other reproducible categories of play could be helpful. This concept may also help in looking at why some play-like systems work for some people and not for others. Play is most fun when it's challenging but not too challenging. In a sort of inverted-U entropy style way, play is contextually optimal based on where each student is at that exact moment.

Brilliant video. 💯💯🙏🙏

Math was one of my fav topics back in school, but I never really went super deep into it back then. There are many topics in math I know very little about (for example most things related to statistics and probabilities). Either way, probably one of the biggest reason I love the 3blue1brown channel so much is exactly what you talked about there: the way it is being communicated. I think I've watched most videos, most of which I probably dont really remember anymore. But I can certainly say that for most videos, simply watching it was a joy. The wonder of discovering new things I didnt even know was possible has quite literally made me emotional at times, even if as I type this comment right now I dont really remember the meat of specific videos anymore. But that doesnt matter too much (also for the simple fact I can always go back, search for it and watch it again). Being a programmer I have a very technical brain that is fond of logic. Having things "click" and indeed following the path with the idea that I could have discovered that myself gives me so much joy. All that said, the main point here is what hooks me the most is that I consider your videos art, and im saying this completely unironically

Great talk, and I love every single 3b1b video

Totally agree with this. Would bring more people into the fascination that is math.