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.
Thanks for sharing Freytag's triangle, it seems it will be a very practical tool for me and others. Grant does talk about storytelling in some of his other talks and videos.
Lovely read! I am constantly told by those around me that I should go teach, because the things I talk about are motivated by their questions, and I can just go off of their passions. But in general, you can't gage every students current interests, and I would find it hard to be engaged with half the class while the other half are scribbling or whatever, I just don't think I would handle a real classroom well.
When I was a math major, ~35 yrs ago, one of my profs suggested that I transfer to a journalism program 'cause of my "expository" way of talking about math. I think that the entrenched culture in math and physics seems to be that if you veer away from talking about math in an abstract sense, and try to take a more intuitive approach to it, you are "watering it down". With the advent of the internet, I'm so glad to see people like Grant use a multimedia approach to help motivate, and illustrate mathematical ideas. Of course, rigor and logic must prevail in formalizing proofs. But when teaching the concepts, images and verbal exposition help build intuition.
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
@@jessewolf7649 That's unfortunately untrue. People only have so much energy, and interpreting an unclear lecture takes a lot. Not only is that an issue for college students, as for many this is the hardest worked time in their lives, but this also closes the door on understanding for lay people and those with mental disabilities.
I have thought about this as well. But the situation is a bit complicated in an educational setting. Many students take a class because they are interested in it. But often, students take a class because it is a core course and/or they need it to complete their degree. On the other hand, professors have a time constraint because they have to complete a certain syllabus volume (for various reasons like grants, accreditation, etc.). Both these play a part in how much time and effort professors can spend on making a lecture. Also, the professor can be demotivated if students do not perform well in homework and tests.
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.
❤ 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.
My biggest problem with math has always been that I try to follow what is going on, but after the second step where I go "There is no way I would have thought about this" I lose track of what is going on. Combined with my terrible memory which means the "just memorize it" thing doesn't work, and I am not really surprised the only ways I have started to learn it are trough machine learning and optimization. Because both are fields where it is quite intuitive what you can do to get where you want to go, and in general the equations are not horrible or not to be solved by hand. The exact optimizations that make for "the best" solutions are still things you look up, but at least you understand what they do, where they come from and why it work. edit: maybe I should clarify that even for the things I do "understand" I basically just retrace the steps that made me understand it the first time I did, it takes 1 or 2 months working with it before I memorize enough to skip large steps and even then I usually end up making mistakes.
Slightly off-topic, but my partner is a high school English teacher, and I have to thank you for the hard work you do for our future generations. It's a thankless job that's getting harder every year and you guys aren't getting paid nearly what you deserve for the service you do for society.
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.
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.
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.
This is kind of me too! I was always pretty good at math, and liked it well enough until I took linear algebra. Then I started to really *love* math, so much so that I'm considering adding it as a second major in college.
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.
Speaking as a pure mathematics student, you also have to be careful of going the opposite route. My worst math course grade was differential equations and it's because I couldn't work my way through the bogged down applied problems that could've come straight from a physics textbook.
I think it's still important to explain why you should care even to mathematicians. For example, a typical Abstract Algebra class starts by defining a group as a set of elements and an operation where 1. The set is closed under the operation. 2. Every element has an inverse. 3. There is an identity element. 4. The operator is associative. Then, the math class just goes forward and proves a bunch of theorems about the properties of groups, but they don't really get to why people would care about groups besides it has some applications in like Robotics with Lie groups or something. Instead, you can motivate groups as "the simplest non-trivial algebraic structure in which you can solve equations". For example, let's say we need to solve x # a = b where # is the operation. To solve this equation, you add the inverse of a to both sides, which means you need an inverse for every element. Then, you need to be able to choose which # to do first so you can cancel out the a with its inverse, which gives you associativity. You then need the a and its inverse to produce an element that doesn't do anything, so you need an identity. Lastly, this process only works if # doesn't freak out and give you something not in the set, which means you need the set to be closed under #. See? Even if you never give any practical uses for groups, you'll still have given sufficient motivation for an entire field of math.
@@josephmellor7641 Nice point of view. I wish my algebra teacher would care to teach like this. Another problem I found is that Abstract Algebra is usually taught by discrete mathematicians, who are heavily influenced by the program to classify finite simple groups. Tha field-internal program is now competed, and a lot of the action with groups that interests me more has to do with group objects in categories of spaces, a way to get to invariants via algebra. With groupoids peeking out from beyond the connected component.
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.
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
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.
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!
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.
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).
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.
Thank you for seeking out videos like this one. My high school classes would've been infinitely less boring if the lessons started off with even the slightest pedagogical thought.
I remember it all started to come together for me when they drew a segment of a surface and described a vector pointing out from it as a value of a function representing the flux of a fluid through the surface at that point.
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.
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.
I agree with this deeply. Whatever the Nobel Prize for education would be, it’s his. I would nominate him and his team and would buy the plane ticket/tune-in to watch the ceremony live.
I think the guy has a unique take on maths and its exposition. He's a brilliant guy but he gets the whole idea of how abstractions and generality in maths are built from more concrete ideas. Not the other way around.
Absolutely. Old UK medic here. I quoted Grant's videos early on, whilst exploring the mathematics of particulate transfer. Instead, I was actually seen as nuts partially because of wearing a N95 under a discreet standard 'blue' - even though Mr Biden wore the same combo months later. If only Grant's videos were better known... Important man.
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).
This is perfectly on point, especially the one about quasi-examples. So often counterexamples chosen just aren't close enough to really show how the definition fails.
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!
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.
To borrow a little from the presentation, I think the "ceiling" on teaching is vastly higher than people realize. Great teachers have to master such a wide variety of skills that I think it's hard to expect someone like Grant to be the norm. But I also think there are basic guidelines that people can use to improve their practice, like those he outlined here, so that at the very least they're pushing toward their potential rather than plateauing because the way they teach is "good enough".
@@kymion Agreed. At least in the US, we have a teacher competency issue, caused in large part by a lack of motivation for intelligent people to become teachers and a shortage in educational labor in general. Many school districts around the country cannot afford to hire new teachers or cannot pay teachers enough to keep good teachers, and otherwise cannot find competency. Many teachers do not even have a teaching degree, nor is it required due to severe shortage. However, even though I think it is necessary to make teaching materials and content as accessible to teachers as possible as well as to students, the unfortunate truth is that competency and knowledge is required on the part of the teacher for the student to have a sufficient educational experience, particularly in math but in other subjects as well.
~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.”
@@Reashu Haha, didn't think of it! 😁Do you know of such a language? My mental combing through languages that I know a bit of got me a curious near hit close to home. The words _an_ and _one_ both come from the Old English "an," _one._ Alas, no cigar: "an" didn't sound like N :-( Its /a/ was, in both O.E. and Middle E., quite close to the /a/ in British RP "ask" or "after" (North-Am. dialects lack this sound). O.E. did not have the article; the article _an_ arose in M.E. from the usage “one [=certain unknown] smth./smb.” _One_ still sounds fine instead of _a_ in the same sense, “a certain, unknown until this time _X,”_ if only a little bit emphatic: “I met a guy at JMM'23 who taught me how to juggle ripe tomatoes” = “I met one guy at JMM'23 who...” The form _a_ has later developed from _an_ in M.E., very gradually. And _the_ is, of course, a late-O.E. form of _that._ Except in “the more, the better,” where it's not an article, and comes from _another_ case form of _that._ Aw yeah, _Englisc_ was quite different from English. :-)
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.
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
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
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.
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".
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.
I had a really similar experience. My teacher gave us the formula and I could not for the life of me remember it, we had a test and I nearly failed it. Then he showed us how to derive it using completing the square. As soon as I understood where all the parts come from it just made sense. x = (-b +- sqrt(b^2 - 4ac))/(2a)
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.
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 TH-cam 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 TH-camrs 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.
YES!!!! I adore that TH-cam 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.
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. :)
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!
The answer is in the description: "Correction at 50:24 - The measure of transferability score for Class B (3.1) and that for Class A (5.4) should be swapped."
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.
I thought I liked math until my first year of undergrad, and quickly changed major. I thought I disliked math until I discovered 3b1b and now wish I'd stuck with it 🤷🏻♂️
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.
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.
This reminds me of the "Crash Course Computer Science" series here on TH-cam. 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.
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"
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.
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" :)
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 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!!!!
16:48 this reminds me of the first time i was reviewing a lesson plan for my first calculus tutoring session. I opened the book to limits and finally understood the limit definition of a derivative because something about it just CLICKED. Also in college, my real analysis professor for math ed had our class experiment with definitions for a limit. I have never seen the delta-epsilon definition more clearly than that and it was magical because the way we got there was not through handed-down definitions but through a culmination of our definitions that we created through our own experimentation of our theories
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.
I think Grant made the point that people who have motivating examples remember things better. Considering topics rigorously and making connections teaches your brain how to think, but students often go about learning math without learning how to think better, just memorizing things and knowing where to apply them. In my college class(not a math major) I have a friend who literally studied definitions by heart, got to the epsilon-delta definition of a limit and did not think about what he was memorizing for a second - he thought that delta was a square root. And he only got his mistake after the exam. A lot of my colleagues are similarly uninterested in using this opportunity to think rigorously. Point is, knowing math doesn't make you smart, studying math properly, however, does.
The problem is when you're giving a knife to someone who's never seen one before, and has never cut anything (i.e. teaching an abstraction without concrete applications). And as the study at 50:00 shows, students have significantly better understanding and transfer learning when they start with concrete examples and progress to abstraction. (The image has a mistake that shows worse transfer but the video description corrects this).
@@petardraganov3716 I only mean that "math makes you smarter" in the same way that "a mechanical advantage makes you stronger." I'm not making the claim that math will raise your IQ, but it will certainly allow you to solve more problems with whatever cognitive faculties you have.
@@LowestofheDead You don't just hand somebody a knife--you show them what it does. I guess there are people who can genuinely understand the concept of applying ~10^5 PSI to a small region and not instantly be amazed and flush with imagination and wonder of seemingly infinite possibilities. I don't think that the solution, in this case, is to try to list the infinite possibilities--that's giving a man a fish, and you really ought to teach him how to fish. That is, tell them stories, show them how people in the world do things, and just show them how awesome and full of interesting challenges the modern world is. Once they recognize that--and they see how powerful mathematics is--they will be able to appreciate how valuable a knife is.
@@alexandersanchez9138 That argument doesn't conform to reality. The study of real-life students at 50:00 shows that they have significantly better understanding AND transfer when they start with concrete examples, not abstraction. Those students were given _one_ example - they did not need an "infinite" number. And they worked on the example themselves; this is nothing like 'giving a man a fish', it's a teacher doing their job. The less effective technique is done in both your argument and modern math education. Students are given abstract details of what a concept is, but not what it does or why it's used. In the example at 35:00, class B is taught the difference of two squares without ever factoring an actual number (from a real lesson plan). That's like giving someone a rod but not telling them about fish. Sure they could try _every single possible way_ of using a rod... but that's just poor teaching that wastes everyone's time and kills students' passion for no reason. The justification that 'some people are just worse at imagination' is just blaming the students for terrible teaching methods. I could draw an analogy to textbooks which leave out a "trivial" step as an exercise for the reader, which wastes everyone's time in trying every possible method. After going through so much unnecessary pain as a student (which drives away so many people) we convince ourselves that the pain was meaningful to separate us from the rest. So we inflict that pain on future students. This is why people say math is elitist, and honestly they have a point. This is why Grant Sanderson's work is so important.
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).
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.
I think I actually skipped a layer of abstraction. I loved the version with the physical quantities and could see how each step directly corresponds to each part of the algebraic version, but you show me the version with concrete numerals and I'm completely lost because I didn't pick up on the fact that the examples were one away from squares.
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)
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.
@@100c0c it's less about teaching relativity as just mentioning that there's more to learn. Teaching something as, "this is the way it is, and everything you learn will be built on this basis," when that's not true is what was upsetting. "This thing is longer than that thing," isn't even something you can rely on, when the relative speeds are high. Things are simultaneously longer than each other, and you can live in a whole different world just by traveling faster. If you believe in (Can't remember the name; Umbra Radiation? That's not right), then there's even matter that only exists when you are accelerating. My point was just that when I was learning math, it was at least mentioned that Euclidean geometry was just the simple version, and when I was taught Physics, it wasn't mentioned that it was the simple version; it was actually all known to be untrue. It could have at least been mentioned that, just over the horizon, there were other ways of looking at things.
I like to talk math and science with anyone around me that listens. I build off their interests, as much as I can, to draw them in and get them asking questions, and then I can lay down the real physics/math wizardry and blow their minds. Many people I have conversations with tell me I should be a teacher, but this only really works in a 1 on 1 environment, and I don't think it would go over as well with a class where half the kids don't care. Maybe I could be a professor if I had tried to go down that path, but I liked being out in the real world in practice, vs some kind of PHD track.
If only you could bottle the "path of rediscovery". I would say I largely bounced off most of my mathematics course work from text books but rediscovered much of it through engineering problems. Great talk!
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!
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.
Your layers of abstraction have a good precedent in the word under-standing itself, you have to know what stands under the learning outcome to be able to parse it.
I come from a background of math, computer science, and classroom teaching, and I've had the chance to see a huge variety of teachers and the diversity of their approaches. What I've found more often than not is that content mastery is inversely proportional to ability and/or motivation to create lessons that engage a broader audience than other content masters. In other words, the experts are only good at explaining in an expert-oriented way. I think there's a separate-but-correlated effect whereby most teachers of content came from a background of expert use of that content--e.g. a comp sci lecturer actually had a career as a software developer--rather than a teaching background. The combination of the two leads us to a situation where teaching of these "hard" topics becomes very exclusionary and leads many people who could otherwise learn the material to conclude that it's just "not for them" or they're not cut out for it. The reality is that the teachers are simply not doing what they can to reach more people. Grant not only articulated similar thoughts in his presentation, but has proven to be one of the exceptions to what I'm describing. He clearly has very deep understanding of the material, but also goes to great lengths to teach people in a much broader way than these kinds of topics would normally be presented. To say it's difficult to have the combination of skills to do this is a vast understatement, as illustrated by how few people seem to teach this way. I'm glad that he's found a platform to share his lessons, and I look forward to learning from his philosophy of teaching in my own endeavors!
Folks like Grant and Mark (Rober) are gifts to humanity and this is not hyperbole. Kids of all ages who are able to follow their work and be inspired are truly lucky indeed!
@@ster2600 in promoting learning and raising inquisitiveness in kids, I think they are alike; in fact there are many more on TH-cam who fall in that category of great teachers. It is such a joy to watch them unravel complex topics
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.
This was a great talk and the questions at the end were very insightful. I’m definitely going to take a look at the podcast you mentioned. Ps. I appreciate not having midroll ads during the talk
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
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.
The work you and the other great math communicators are doing will change lives! I wish I had been exposed to the sort of stuff you and folks like Matt Parker and the Numberphile presenters when I was in middle/high school.
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.
I would describe mathematical pedagogy more this way: Look I just painted the Mona Lisa; now you as should paint a Mona Lisa in the same style. Almost a paint by numbers exercise. Better is I have the tools and the skills to paint the Mona Lisa; now you should acquire those tools and skills and paint whatever beautiful thing that comes to mind.
I thought I loved math until... I looked in my University course catalog, and found that I wanted to take Physics and Electronics. It's the engineering and calculation I love. When I got into my Honors Calculus class at University, we did a lot of convergence and series. Finally, half-way through the quarter, we all asked "Will we Ever do some Calculus?" I DO love differential equations and Eigen Vectors, and solving things brute force.
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In grad school, we referred to those magic steps handed down from on high as "proof by proctological extraction."
“Pulled out of my ass” love it
I've also heard the expression "rectally sourced statistics".
Shhh! My career depends on this... \o/
Oh, is that why they call them proctors? :P
Working your logs out with a slide rule. perhaps??? 😁
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 mean, articulate people recognize other articulate people.
@@iantaakalla8180 True true
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.
Thanks for sharing Freytag's triangle, it seems it will be a very practical tool for me and others. Grant does talk about storytelling in some of his other talks and videos.
@@fbkintanar No problem. I'll have to check those out!
Amazing, would you be able to share your blog / writings / videos? You indeed have an amazing approach.
Lovely read! I am constantly told by those around me that I should go teach, because the things I talk about are motivated by their questions, and I can just go off of their passions. But in general, you can't gage every students current interests, and I would find it hard to be engaged with half the class while the other half are scribbling or whatever, I just don't think I would handle a real classroom well.
When I was a math major, ~35 yrs ago, one of my profs suggested that I transfer to a journalism program 'cause of my "expository" way of talking about math. I think that the entrenched culture in math and physics seems to be that if you veer away from talking about math in an abstract sense, and try to take a more intuitive approach to it, you are "watering it down". With the advent of the internet, I'm so glad to see people like Grant use a multimedia approach to help motivate, and illustrate mathematical ideas.
Of course, rigor and logic must prevail in formalizing proofs. But when teaching the concepts, images and verbal exposition help build intuition.
I love how the lecture itself passes all three checks
Pedagogically Recursive.
It's really staggering how many works about communication and motivation don't follow their own rules.
Autological
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
Math professors should “just focus” on the subject! If the students are not interested in the subject they should not be taking the course.
the problem with education in just one arrogant borderline comment.
@@jessewolf7649 That's unfortunately untrue. People only have so much energy, and interpreting an unclear lecture takes a lot.
Not only is that an issue for college students, as for many this is the hardest worked time in their lives, but this also closes the door on understanding for lay people and those with mental disabilities.
I have thought about this as well. But the situation is a bit complicated in an educational setting. Many students take a class because they are interested in it. But often, students take a class because it is a core course and/or they need it to complete their degree.
On the other hand, professors have a time constraint because they have to complete a certain syllabus volume (for various reasons like grants, accreditation, etc.). Both these play a part in how much time and effort professors can spend on making a lecture. Also, the professor can be demotivated if students do not perform well in homework and tests.
Students fellings? What is that?
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.
Any technology explained in sufficient detail is indistinguishable from magic lol
❤ 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.
My biggest problem with math has always been that I try to follow what is going on, but after the second step where I go "There is no way I would have thought about this" I lose track of what is going on. Combined with my terrible memory which means the "just memorize it" thing doesn't work, and I am not really surprised the only ways I have started to learn it are trough machine learning and optimization. Because both are fields where it is quite intuitive what you can do to get where you want to go, and in general the equations are not horrible or not to be solved by hand. The exact optimizations that make for "the best" solutions are still things you look up, but at least you understand what they do, where they come from and why it work.
edit: maybe I should clarify that even for the things I do "understand" I basically just retrace the steps that made me understand it the first time I did, it takes 1 or 2 months working with it before I memorize enough to skip large steps and even then I usually end up making mistakes.
Slightly off-topic, but my partner is a high school English teacher, and I have to thank you for the hard work you do for our future generations. It's a thankless job that's getting harder every year and you guys aren't getting paid nearly what you deserve for the service you do for society.
That rediscoverability is what Grant is all about. He's just a brilliant, inspiring guy.
Ur a W teacher for even looking at content like this on how to teach better. Wish I had you in high school.
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.
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.
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.
This is kind of me too! I was always pretty good at math, and liked it well enough until I took linear algebra. Then I started to really *love* math, so much so that I'm considering adding it as a second major in college.
I almost gave up on my physics degree because of linear algebra. Had to take it twice. Still don't understand a thing about any of the theory
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.
Engineers don't, Physicists do, Economists wish they did.
Speaking as a pure mathematics student, you also have to be careful of going the opposite route. My worst math course grade was differential equations and it's because I couldn't work my way through the bogged down applied problems that could've come straight from a physics textbook.
I think it's still important to explain why you should care even to mathematicians. For example, a typical Abstract Algebra class starts by defining a group as a set of elements and an operation where
1. The set is closed under the operation.
2. Every element has an inverse.
3. There is an identity element.
4. The operator is associative.
Then, the math class just goes forward and proves a bunch of theorems about the properties of groups, but they don't really get to why people would care about groups besides it has some applications in like Robotics with Lie groups or something. Instead, you can motivate groups as "the simplest non-trivial algebraic structure in which you can solve equations". For example, let's say we need to solve
x # a = b
where # is the operation. To solve this equation, you add the inverse of a to both sides, which means you need an inverse for every element. Then, you need to be able to choose which # to do first so you can cancel out the a with its inverse, which gives you associativity. You then need the a and its inverse to produce an element that doesn't do anything, so you need an identity. Lastly, this process only works if # doesn't freak out and give you something not in the set, which means you need the set to be closed under #.
See? Even if you never give any practical uses for groups, you'll still have given sufficient motivation for an entire field of math.
@@josephmellor7641 Nice point of view. I wish my algebra teacher would care to teach like this. Another problem I found is that Abstract Algebra is usually taught by discrete mathematicians, who are heavily influenced by the program to classify finite simple groups. Tha field-internal program is now competed, and a lot of the action with groups that interests me more has to do with group objects in categories of spaces, a way to get to invariants via algebra. With groupoids peeking out from beyond the connected component.
@@fbkintanar groupoids is a crazy word
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.
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
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.
Hi! If you were to give advice to a teen who has access to these tools, what would you say?
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!
Thanks Josh! Best of luck with your post-graduation pursuits.
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.
Such an interesting talk!
1, diagrams & visualization
2, concrete examples
3, motivating questions
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).
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.
Thank you for seeking out videos like this one. My high school classes would've been infinitely less boring if the lessons started off with even the slightest pedagogical thought.
I remember it all started to come together for me when they drew a segment of a surface and described a vector pointing out from it as a value of a function representing the flux of a fluid through the surface at that point.
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.
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.
I agree with this deeply. Whatever the Nobel Prize for education would be, it’s his. I would nominate him and his team and would buy the plane ticket/tune-in to watch the ceremony live.
I think the guy has a unique take on maths and its exposition. He's a brilliant guy but he gets the whole idea of how abstractions and generality in maths are built from more concrete ideas. Not the other way around.
Absolutely. Old UK medic here. I quoted Grant's videos early on, whilst exploring the mathematics of particulate transfer. Instead, I was actually seen as nuts partially because of wearing a N95 under a discreet standard 'blue' - even though Mr Biden wore the same combo months later. If only Grant's videos were better known... Important man.
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).
This is perfectly on point, especially the one about quasi-examples. So often counterexamples chosen just aren't close enough to really show how the definition fails.
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!
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.
To borrow a little from the presentation, I think the "ceiling" on teaching is vastly higher than people realize. Great teachers have to master such a wide variety of skills that I think it's hard to expect someone like Grant to be the norm. But I also think there are basic guidelines that people can use to improve their practice, like those he outlined here, so that at the very least they're pushing toward their potential rather than plateauing because the way they teach is "good enough".
@@kymion Agreed. At least in the US, we have a teacher competency issue, caused in large part by a lack of motivation for intelligent people to become teachers and a shortage in educational labor in general. Many school districts around the country cannot afford to hire new teachers or cannot pay teachers enough to keep good teachers, and otherwise cannot find competency. Many teachers do not even have a teaching degree, nor is it required due to severe shortage. However, even though I think it is necessary to make teaching materials and content as accessible to teachers as possible as well as to students, the unfortunate truth is that competency and knowledge is required on the part of the teacher for the student to have a sufficient educational experience, particularly in math but in other subjects as well.
Thank you Grant, for allowing us all tol be Class A students. Really, we're grateful!
~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.”
Works best in languages where "n-dimensional" and "one-dimensional" sound similar enough to be mistaken for each other.
@@Reashu Haha, didn't think of it! 😁Do you know of such a language?
My mental combing through languages that I know a bit of got me a curious near hit close to home. The words _an_ and _one_ both come from the Old English "an," _one._ Alas, no cigar: "an" didn't sound like N :-( Its /a/ was, in both O.E. and Middle E., quite close to the /a/ in British RP "ask" or "after" (North-Am. dialects lack this sound). O.E. did not have the article; the article _an_ arose in M.E. from the usage “one [=certain unknown] smth./smb.” _One_ still sounds fine instead of _a_ in the same sense, “a certain, unknown until this time _X,”_ if only a little bit emphatic: “I met a guy at JMM'23 who taught me how to juggle ripe tomatoes” = “I met one guy at JMM'23 who...” The form _a_ has later developed from _an_ in M.E., very gradually.
And _the_ is, of course, a late-O.E. form of _that._ Except in “the more, the better,” where it's not an article, and comes from _another_ case form of _that._ Aw yeah, _Englisc_ was quite different from English. :-)
@@cykkm Swedish is close - "ett en-dimensionellt rum" and "ett N-dimensionellt rum" differ only in how they are stressed
This is the magician showing how the trick is done, and yet it doesn't spoil the trick.
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.
My science teacher (Mr A. Carey) did that. It's called priming, or "tell 'em what you're going to tell "em".
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
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
As a teacher, I have a lot of coworkers that would really like this video.
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.
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".
Well deserved, Grant! Looking forward to watching more of your content!
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.
I had a really similar experience. My teacher gave us the formula and I could not for the life of me remember it, we had a test and I nearly failed it. Then he showed us how to derive it using completing the square. As soon as I understood where all the parts come from it just made sense. x = (-b +- sqrt(b^2 - 4ac))/(2a)
Same but for newton's law of universal gravitation in a 100-level astro course
Newton invented Calculus but couldn't use it in Principia because scholars didn't see calculus as a 'proof'.
Sam e for trig identities. I’d just go to Eulers equation.
@@DrDeuteron Maclaurin series are also nice for clarifying trig identities
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.
The Schrödinger eqs is beautiful for hydrogen, and gets rough when you attach it to some carbons.
@@DrDeuteronit is still beautiful, just non analytic :-)
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.
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 TH-cam 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 TH-camrs 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.
YES!!!! I adore that TH-cam 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.
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
I was hovering over the close button durring q&a but it was actually good too. nice
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!
The answer is in the description: "Correction at 50:24 - The measure of transferability score for Class B (3.1) and that for Class A (5.4) should be swapped."
@@nathanalday3062 cheers obviously missed that. One of the downsides of not being able to update/correct videos on YT..
yayyy, congratulations Grant, deserved recognition 🥳🥳
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.
Does he have a TH-cam channel?
@@zahraali5018 nah, only the podcast
Ok thanks!
I thought I liked math until my first year of undergrad, and quickly changed major. I thought I disliked math until I discovered 3b1b and now wish I'd stuck with it 🤷🏻♂️
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.
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.
This reminds me of the "Crash Course Computer Science" series here on TH-cam. 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.
Grant is so awesome they deserve all the recognition in the world!
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"
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.
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.
Grant has me accidentally clicking on a video and paying attention for an entire hour.
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" :)
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 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!!!!
16:48 this reminds me of the first time i was reviewing a lesson plan for my first calculus tutoring session. I opened the book to limits and finally understood the limit definition of a derivative because something about it just CLICKED.
Also in college, my real analysis professor for math ed had our class experiment with definitions for a limit. I have never seen the delta-epsilon definition more clearly than that and it was magical because the way we got there was not through handed-down definitions but through a culmination of our definitions that we created through our own experimentation of our theories
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.
I think Grant made the point that people who have motivating examples remember things better. Considering topics rigorously and making connections teaches your brain how to think, but students often go about learning math without learning how to think better, just memorizing things and knowing where to apply them. In my college class(not a math major) I have a friend who literally studied definitions by heart, got to the epsilon-delta definition of a limit and did not think about what he was memorizing for a second - he thought that delta was a square root. And he only got his mistake after the exam. A lot of my colleagues are similarly uninterested in using this opportunity to think rigorously.
Point is, knowing math doesn't make you smart, studying math properly, however, does.
The problem is when you're giving a knife to someone who's never seen one before, and has never cut anything (i.e. teaching an abstraction without concrete applications).
And as the study at 50:00 shows, students have significantly better understanding and transfer learning when they start with concrete examples and progress to abstraction.
(The image has a mistake that shows worse transfer but the video description corrects this).
@@petardraganov3716 I only mean that "math makes you smarter" in the same way that "a mechanical advantage makes you stronger." I'm not making the claim that math will raise your IQ, but it will certainly allow you to solve more problems with whatever cognitive faculties you have.
@@LowestofheDead You don't just hand somebody a knife--you show them what it does. I guess there are people who can genuinely understand the concept of applying ~10^5 PSI to a small region and not instantly be amazed and flush with imagination and wonder of seemingly infinite possibilities. I don't think that the solution, in this case, is to try to list the infinite possibilities--that's giving a man a fish, and you really ought to teach him how to fish. That is, tell them stories, show them how people in the world do things, and just show them how awesome and full of interesting challenges the modern world is. Once they recognize that--and they see how powerful mathematics is--they will be able to appreciate how valuable a knife is.
@@alexandersanchez9138 That argument doesn't conform to reality. The study of real-life students at 50:00 shows that they have significantly better understanding AND transfer when they start with concrete examples, not abstraction.
Those students were given _one_ example - they did not need an "infinite" number. And they worked on the example themselves; this is nothing like 'giving a man a fish', it's a teacher doing their job.
The less effective technique is done in both your argument and modern math education. Students are given abstract details of what a concept is, but not what it does or why it's used.
In the example at 35:00, class B is taught the difference of two squares without ever factoring an actual number (from a real lesson plan).
That's like giving someone a rod but not telling them about fish. Sure they could try _every single possible way_ of using a rod... but that's just poor teaching that wastes everyone's time and kills students' passion for no reason.
The justification that 'some people are just worse at imagination' is just blaming the students for terrible teaching methods.
I could draw an analogy to textbooks which leave out a "trivial" step as an exercise for the reader, which wastes everyone's time in trying every possible method.
After going through so much unnecessary pain as a student (which drives away so many people) we convince ourselves that the pain was meaningful to separate us from the rest. So we inflict that pain on future students.
This is why people say math is elitist, and honestly they have a point. This is why Grant Sanderson's work is so important.
Congratulations grant!
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).
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.
I think you had things a bit backwards. The fact that a lot of people know the formula without your beautiful insight is the sad thing.
I think I actually skipped a layer of abstraction. I loved the version with the physical quantities and could see how each step directly corresponds to each part of the algebraic version, but you show me the version with concrete numerals and I'm completely lost because I didn't pick up on the fact that the examples were one away from squares.
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)
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.
How would you even teach relativity before physics 3?
@@100c0c it's less about teaching relativity as just mentioning that there's more to learn. Teaching something as, "this is the way it is, and everything you learn will be built on this basis," when that's not true is what was upsetting. "This thing is longer than that thing," isn't even something you can rely on, when the relative speeds are high. Things are simultaneously longer than each other, and you can live in a whole different world just by traveling faster. If you believe in (Can't remember the name; Umbra Radiation? That's not right), then there's even matter that only exists when you are accelerating. My point was just that when I was learning math, it was at least mentioned that Euclidean geometry was just the simple version, and when I was taught Physics, it wasn't mentioned that it was the simple version; it was actually all known to be untrue. It could have at least been mentioned that, just over the horizon, there were other ways of looking at things.
I like to talk math and science with anyone around me that listens. I build off their interests, as much as I can, to draw them in and get them asking questions, and then I can lay down the real physics/math wizardry and blow their minds. Many people I have conversations with tell me I should be a teacher, but this only really works in a 1 on 1 environment, and I don't think it would go over as well with a class where half the kids don't care. Maybe I could be a professor if I had tried to go down that path, but I liked being out in the real world in practice, vs some kind of PHD track.
If only you could bottle the "path of rediscovery". I would say I largely bounced off most of my mathematics course work from text books but rediscovered much of it through engineering problems. Great talk!
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!
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.
Well deserved. In my opinion one of the best educational communicators out there
Your layers of abstraction have a good precedent in the word under-standing itself, you have to know what stands under the learning outcome to be able to parse it.
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 come from a background of math, computer science, and classroom teaching, and I've had the chance to see a huge variety of teachers and the diversity of their approaches. What I've found more often than not is that content mastery is inversely proportional to ability and/or motivation to create lessons that engage a broader audience than other content masters. In other words, the experts are only good at explaining in an expert-oriented way. I think there's a separate-but-correlated effect whereby most teachers of content came from a background of expert use of that content--e.g. a comp sci lecturer actually had a career as a software developer--rather than a teaching background. The combination of the two leads us to a situation where teaching of these "hard" topics becomes very exclusionary and leads many people who could otherwise learn the material to conclude that it's just "not for them" or they're not cut out for it. The reality is that the teachers are simply not doing what they can to reach more people.
Grant not only articulated similar thoughts in his presentation, but has proven to be one of the exceptions to what I'm describing. He clearly has very deep understanding of the material, but also goes to great lengths to teach people in a much broader way than these kinds of topics would normally be presented. To say it's difficult to have the combination of skills to do this is a vast understatement, as illustrated by how few people seem to teach this way. I'm glad that he's found a platform to share his lessons, and I look forward to learning from his philosophy of teaching in my own endeavors!
Got to see this in person, great talk!
Grant Sanderson is nailing his 95 theses to the door and I'm here for it.
What a FANTASTIC talk.
59 times 61, this does not happen in middle school... this is where you won my updote.
Please bring the podcast series back
Congratulations to my favorite Math teacher!
I factored 9,991 in my head!! What the heck!! That actually felt like magic! Truly I have earned the title Mathamgitian!
Great lecture! Thank you for helping inspire me to be a better teacher.
Congratulation Grant! Keep up the great work.
Very insightful talk, you're such an educator.
PREACH! Grant is a genius
Folks like Grant and Mark (Rober) are gifts to humanity and this is not hyperbole. Kids of all ages who are able to follow their work and be inspired are truly lucky indeed!
Mark rober is nothing like grant.
@@ster2600 in promoting learning and raising inquisitiveness in kids, I think they are alike; in fact there are many more on TH-cam who fall in that category of great teachers. It is such a joy to watch them unravel complex topics
I thought I liked math until I learned about the axiom of choice, then I learned I like practical and computable math.
Totally agree with this. Would bring more people into the fascination that is math.
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.
Please continue the podcast series!!
I always loved learned but hated school. I hope things change to engage and empower children.
This was a great talk and the questions at the end were very insightful. I’m definitely going to take a look at the podcast you mentioned.
Ps. I appreciate not having midroll ads during the talk
Thanks, I forgot to change TH-cam’s default there. It should have no midrolls now.
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
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.
The work you and the other great math communicators are doing will change lives! I wish I had been exposed to the sort of stuff you and folks like Matt Parker and the Numberphile presenters when I was in middle/high school.
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.
I would describe mathematical pedagogy more this way: Look I just painted the Mona Lisa; now you as should paint a Mona Lisa in the same style. Almost a paint by numbers exercise. Better is I have the tools and the skills to paint the Mona Lisa; now you should acquire those tools and skills and paint whatever beautiful thing that comes to mind.
I thought I loved math until... I looked in my University course catalog, and found that I wanted to take Physics and Electronics. It's the engineering and calculation I love. When I got into my Honors Calculus class at University, we did a lot of convergence and series. Finally, half-way through the quarter, we all asked "Will we Ever do some Calculus?" I DO love differential equations and Eigen Vectors, and solving things brute force.
17:00
This is exactly what I love and seek about learning.
But as much as I aked for it, my profesor don't recognize it as important...
Congratulations, great work !!!