Math's pedagogical curse – Grant Sanderson [video]

**TL;DR: Grant Sanderson, founder of 3Blue1Brown, used the "touch of King Midas" as a metaphor in his JPBM communication award lecture, arguing that mathematics' precision is both a gift and a "pedagogical curse"—it makes rigor easy to measure but leaves teaching clarity difficult to verify. He therefore advocates "raising the ceiling and lowering the floor of mathematical exposition."** --- ## Opening Introduction: The Award and Its Background The lecture was hosted by Michael Pearson of the Mathematical Association of America (MAA). As part of JPBM (Joint Policy Board for Mathematics), the organizers presented this year's communication award to Grant Sanderson—creator and founder of 3Blue1Brown. The project is dedicated to bringing visualization to mathematics, and it is also a very easy-to-find YouTube channel. Pearson noted in particular that he is a harmonic analyst who has spent years thinking about convolution, convolution kernels, and related problems. So when he saw how Grant handled the topic of convolution, he was immediately captivated. He pointed out that the German word for convolution is "Faltung" (folding), and that Grant's video truly brought that word to life, making it clear why "folding" is an excellent word for describing convolution. Pearson also introduced Grant's background: he graduated from Stanford University in 2015 with a double major in mathematics and computer science; he worked at Khan Academy, producing videos and educational materials; and he has since dedicated himself entirely to 3Blue1Brown. ## The Acceptance Speech: Thanks to Those Who Inspired Him After taking the stage, Grant first thanked the members of the Joint Policy Board for conferring the award on him. He admitted that, having just received a communication award, being immediately asked to address a room full of people he greatly respects was a somewhat nerve-wracking experience. He did, however, take the opportunity to pay tribute to past winners—Steve Strogatz, James Gleick, and Matt Parker were the first names to come to mind, though he noted there were "many, many more." These were the heroes of his early days who had given him a great deal of inspiration. ## Threads of Mathematical Communication: Where Education, Research, and Outreach Intersect Grant explained that he wanted to discuss mathematical communication from a broader perspective, along with some of his thoughts on the subject. But he also pointed out that the topic can be divided into many different categories, and thinking about it as a whole is quite difficult—because one major component is education; and even within education, categorizing it as a single whole is inappropriate, since pre-college education and undergraduate-and-beyond mathematics education differ greatly in nature: people will consider which methods work best and what levels of student engagement to expect, and these are entirely different from other forms of communication, such as presenting one's research or popularizing and outreach. If he had to pick the most fitting category, he said he would belong to the latter. Grant did, however, believe that no matter which category one occupies, it is helpful to see these threads as intersecting one another and to recognize that most of their value comes from that intersection. Some of these directions are fairly obvious: - If you work in **education**, treat it as an attempt at **popularization and outreach**. - If you work in **outreach**, try to make it more engaging and attempt to connect with things people are already interested in. The same approach works perfectly well in the classroom as well. ## A Counterintuitive Lesson: People Come to Learn, Not to Be Entertained On the other hand, as a popularizer—or at least someone attempting to be—Grant frequently found that **going straight at it wasn't the right approach**: don't directly think, "What can I do to reach the most people, with the most eye-catching title possible?" What he noticed was this: **people simply want to learn.** What tends to make something more popular and gives it greater reach is often the content he made that felt more like "classroom supplementary material." The reason: people don't want to feel that they are being oversimplified or talked down to; they want to feel that what they're seeing is authentic, uncut material—as though they are learning it themselves and becoming experts in the subject. Grant believes that is a very exhilarating feeling. ## The Title: Raise the Ceiling, Lower the Floor On this basis, Grant titled the lecture **"Raising the Ceiling and Lowering the Floor of Mathematical Exposition"**—referring to a fact he believes: the popularization efforts made to bring mathematics to more people can actually make it more interesting even to those who are already experts. If he wanted to be a bit more provocative, he could easily describe what he had in mind as **"mathematics' pedagogical curse"**—because he does believe there is something strange about mathematics in a certain respect, especially compared to other fields, when it comes to the public's perception of it. ## That Absurd Opening Line Grant described an experience almost everyone has had: chatting with someone at a cocktail party, you mention that your work involves mathematics, and the other person says—and they seem to say it with no social taboo whatsoever—"Oh, I'm actually not very good at math." This statement in itself isn't absurd, after all—not everyone is going to love the discipline we study. What's truly absurd is: **you wouldn't say something like that about other fields.** And in Grant's view, there's an even more worrying variant of the statement: > "I used to think I liked math, until…" He recalled his experience as a high school student visiting various universities. He had always been the kind of student who knew from a young age that he wanted to study mathematics, and he had already fallen in love with the subject. When college students asked him what he wanted to major in, he answered, "I know I want to major in mathematics," and the response was: > "Oh yeah, yeah—I used to think I liked math, even wanted to major in it, and then I took a set theory course and realized I didn't really like math that much." Over the years, Grant had heard variations of this statement in various contexts: "I used to think I liked math, and then I took a linear algebra course." At some point, someone had become disillusioned with mathematics for one reason or another. ## So What Did We Get Wrong? Grant pointed out that sparking interest from the very beginning is extremely difficult. And even if the more important group to reach is the broader public who don't consider themselves "math people," we can equally ask: what did we do wrong, such that there are people who thought they liked math, until… One might say: math is simply harder than other fields; it takes rolling up your sleeves and working hard to master it, and not everyone is willing to do that. Grant believes this argument is unfair, because it effectively sweeps the problem under the rug. We must instead ask: **does math really have to be that hard? Have we done certain things that make it harder than it needs to be?** ## The Touch of Midas: Mathematics' Gift and Its Price On the question of why mathematics differs from other fields, Grant proposed a small theory, first asking the audience's indulgence as he illustrated it with a story about **King Midas**. In Greek mythology—or rather, a story woven together with the Greek historical accounts told by Aristotle and others, existing somewhere between fact and legend—there was a king named Midas who lived around the second millennium BCE. He performed some good deeds that caught the attention of Dionysus, the god of wine. Dionysus was willing to grant him a wish: name what you want, and I will give it to you. Midas said what he wanted was this: **anything he touched could turn into gold.** Dionysus said, "So be it." Midas touched a branch, and it instantly turned to gold. Overjoyed, he returned to his estate and touched all the roses in his garden; the roses turned into beautiful, metallic-gilded gold. He fantasized about all the wealth he would have, and the wealth his entire kingdom would have. Then, at a feast that evening, he touched the chicken leg in front of him—it turned to gold, completely inedible. And in another version of the story, when Midas realized that this gift might be a bit off, his daughter came to his side, he touched her, she turned completely to gold, and he lost her forever. What follows in the story is a story of redemption. ### Mathematics' "Gold": Precision Grant's analogy unfolds from here: **mathematics has a genuine gift—a very special kind of precision.** The pleasure we get is the ability to be 100% certain whether a conclusion follows from its premises. Of course, not absolutely 100%, since we might be misled by certain fallacies in logic, or overlook gaps in an argument. But at the greatest degree of precision one can hope for, mathematics truly possesses this precision, and many other fields can only dream of having such a thing. Even hard sciences like physics and computer science, which come quite close in this regard, are still affected by disputes—people can understand the same thing in different ways, and even the best experiments are not 100% precise. Soft sciences are further from this precision still; most forms of writing and discourse are far more openly subjective (such as art criticism—it's not without value, but it lacks this precision). The benefit this precision brings is: **it helps you advance a field.** When there is room for ambiguity and misunderstanding, you don't get mired in the kinds of disputes that might otherwise arise. Grant even suspects that the very fact that something is called "mathematical" may imply that it possesses this degree of precision. He also mentioned a friend in biology who, whenever they talked, lamented that biology needed to become a more mathematical science—how much progress humanity could make if biology could become a mathematical science. Grant understood that what she meant was a situation in which you can derive conclusions from premises with rigor comparable to physics: watertight, rock-solid. ### A Compiler-Style "Done" Check Grant further noted that another very good thing precision gives you is: **it provides a completion check that other fields simply don't have.** We can have a clear sense of concept: it isn't done yet; and at some point, there is a feeling that it is complete. You can go through it from start to finish, making sure every definition is entirely unambiguous—we can have definitions free of ambiguity. Other fields don't get that luxury. You can check whether every object in the theorems you cite has been previously defined. > It's like a compiler checking your code. You can walk through it end to end, and the answer is either a definitive "yes" or a definitive "no." And most importantly: **for every assertion you make, you can ask—has it been fully proven?** And this, in a sense, constitutes the very notion of "explanation." It gives you a clear milestone: before this point, the paper wasn't finished, the lecture wasn't finished; after this point, you can tie a little bow on it and declare it done. ### The Price: Everything You Touch Is Gold, But You Can't Eat It Grant said that he does consider this a gift, because it helps us keep advancing in ways other fields simply cannot. But this gift also comes with a price. Just as everything Midas touched turned to gold, but he **could not eat his own food**: mathematicians can touch anything and endow it with this rigor completely free of ambiguity—but sometimes that makes the content inaccessible to general readers, or harder for ordinary students to digest than it needs to be. Grant stressed that **this is not an inevitable consequence**. He didn't want anyone to get the impression that he was implying that "anything rigorous is therefore unclear"—drawing such a contrast from the Midas analogy would be a bit of an overstatement. But he suspects the following may be the case: the seductive pull of this ideal of precision—the fact that we can genuinely say "it's done" in some sense—means that **rigor can sometimes become a distraction**. ## Rigor Is an Endpoint; Clarity Isn't Grant acknowledged that there are certainly many excellent authors who have managed to be both rigorous and clear; in fact, the whole point of rigor often is to add more clarity. But he suspects that what sometimes happens is this: **because rigor gives you something very clear and beautiful to think about, while thinking about pedagogical clarity is far messier**—it's harder to define, you can't tie a bow on it, there's no clear milestone from "not finished" to "fully finished." So, once the itch of "knowing that something has been written out with complete rigor" has been scratched, it becomes very easy to declare a work finished—**and perhaps we shouldn't be so quick to declare it done.** Grant's proposal: in certain contexts this might still be acceptable, such as in research papers; but in content that is more explicitly aimed at teaching—like textbooks—perhaps we could introduce into that culture a focus similar to the one we currently place on rigor, but this time centered on **pedagogical clarity**. ## An Example: The Poisson Summation Formula and a Textbook Proof Grant offered an example, drawn essentially from a book that happened to be on his desk while he was assembling his slides. He emphasized that he actually really liked that book, that it was genuinely excellent—and the reason he dared to highlight it on air and poke fun at a part of it was precisely because it was so strong overall, one of the books he would recommend without hesitation. What he was thinking about was: **how deeply do authors usually consider pedagogical clarity beyond the concern for rigor?** The specific chapter he focused on covers the **Poisson summation formula**, which many in the audience may well be familiar with. It essentially says: for a sufficiently nice function, if you sum the values of that function at all integers, the result equals the sum of the values of the function's **Fourier transform** at all integers. It's a somewhat "crazy" formula that is extremely useful in many different contexts. The proof given in the book is actually very beautiful, and Grant liked it a lot. It opens by describing a function that seems a bit abrupt, but on closer inspection you realize the function has **poles** at all integers, and the **residues** at those poles are precisely the values the formula wants to sum on the left-hand side. Essentially, what the book does is: after providing this construction, after describing a **contour** along which you might integrate (with the background assumption that the reader is familiar with complex analysis up to that point in the book, knowing about contour integrals and the like), it walks through how, by interpreting what this integral means, you arrive at a very elegant derivation of the formula, step by step transforming it into the summation on the right. So it is very beautiful, involves many clever ideas, and all the pieces fit together perfectly— (The lecture fragment ends here.) --- **Source:** Math's pedagogical curse – Grant Sanderson (https://www.youtube.com/watch?v=UOuxo6SA8Uc)