That kid's proof of the second problem was brilliant.

Thank you for this beautiful and important video. I used to think of myself as "not a maths person", and more of a creative person- assuming these things were opposites. I struggled through school maths, having been told I was not good at math by placement tests that I believed had accurately judged me. In university though I stumbled into an abstract algebra class and I remember having to prove that 0+0=0, and falling in love. Maths, it turns out, isn't an about memorising algorithms, it's about getting to the heart of why things are true. There's no recipe for that, and there's no comparison to the feeling of discovering a mathematical truth. It's such a pity most students never get to have that experience.

"Good, he did not have enough imagination to become a mathematician". - Hilbert's response upon hearing that one of his students had dropped out to study poetry

As a young artist, I had the impression that Math was the opposite of art. Now that I've gotten a bit older, I wish that people didn't see a separation between the two. I've been taking small steps of teaching myself what I missed out on.

Not gonna lie, these videos are much better that most documentaries.

This video is heart-touching - from a Asian high school student who is obsessively interested in Maths but be totured by the maths class in school.

20 years after my high school observation "It is not math nor science at this class, It's just following the pattern" I found you. Thank you for this

The biggest thing I noticed about maths teaching was that the kids who had no trouble with maths were assigned the best teachers. The other kids were actively encouraged to do as little maths as possible. They had the worst teachers; teachers for whom maths was second or third fiddle at best. It suddenly struck me that this was the exact OPPOSITE of education: to NOT teach those who need teaching.

You have no reason to feel called out. You're one of only two people, in my three decades of life, who's managed to get me interested in these things. Back in school it was cold dead memorization. I remember none of that, but I'll be thinking about that pyramid in a box for a while.

I’m studying math at MIT, and I’ve just now thought about what math truly is to me. It actually makes me a bit sad to think that I’ve looked at math as the “opposite” of arts for my life, when that isn’t true at all.

I can relate to this. Here in the Far East with the culture of respect and obedience, education is more along memorization. Schedules are really tight and teachers need to cram into the student's brains. Meanwhile, students don't really care as long as they can get the correct answer and pass their exams. They memorized the formulas and theories and just practise, practise, practise. Back in high school I felt like an outlier as I liked to ask questions like this video. If I get stuck on a certain question, or get something wrong all the time, I'd try to draw it out, sketch it out, and try to understand it deeper. I often get reprimanded by it, saying don't be weird, or, you're wasting your time. But for some reason this inquisitive nature hasn't really left me. I brought it along with me when I teach other things like maths, history and engineering. The students are still affected by the culture and they most often don't care, as usual. A few do have the initiative to ask questions. I just want the youngsters to learn something about the world that could be useful to them, and in a fun way.

"It's a sad race to nowhere" was the line that I felt called out on the most.

Mathematics the music of reason. Music the mathematics of emotion

I'm an engineering student, when i study i always treat the demonstration as the most important thing-unless i'm REALLY late on an exam- and it is in fact the most fascinating part, making even a simple demonstration is so satisfying.

“Fantasy remains a human right: we make in our measure and in our derivative mode, because we are made: and not only made, but made in the image and likeness of a Maker.”-J.R.R. Tolkien

As a teenager who is extremely passionate about mathematics, this video really hit home for me, and almost exactly describes my feelings towards math education. Incredible job!

I really started getting excited about maths when I started looking into the assumptions that we just made (for no apparent reason) before university maths, like "why is distance defined as c²=a²+b²", and then playing around with the idea of distance, seeing different weird shapes of voronoi cells that emerge if you don't use an L2 norm distance or whatever it's called, and then getting invested into this sort of idea of "I've been assuming thing X is always like that, what else can I find to try and break an assumption or play around with", that was one of those moments where my interest in maths really skyrocketed

Seriously, youtube needs more videos like this one.

Strangely featured on my homepage, KZfaq is doing strange things again.

This answers the question as to why I was failing math in school but was pretty good at programming in my spare time, often times unknowingly using mathematical concepts that were much more advanced than what I was learning. It clicked that programming was math when I got into machine learning, eventually read the book "A Programmer's Introduction to Mathematics". A program is a proof, and usually one you have full creative control over. And just like a proof, different brains will write them differently, if it works as intended (expected output for given input) there is no "wrong".

That translates much of the feelings I had in highschool, and still have as a physics undergrad. In my case, a have to be very much self aware that "physics is not that" to keep the passion alive. Always hated mindless decorating rules and formulas.

As a History of Art student, this is just beautiful.

Denying self-exploration in maths is similar to parents saying to their child: "Don't ever travel the world, it's dangerous! Here at home you can just watch series all day long in which you can see anything about the world you'd like to see!"

I've been fascinated by math since grade school when we started learning algebra. But I have been put down by the idea that math is only for certain people or only for those who truly "smart". However, I am slowly getting myself back into the love of it. I have taken Calculus 1-3 in the last year and it has given me so much happiness! I may not be great at math per se, but I do love learning it! I hope all future generations will come to love math and not discriminate for who can and cannot do math. Math is for everyone!!!

Me: thinking you're you're solving complex mathematical equations. You : drawing music icons.

When you forget to add +C at the end of integral

I hated Maths,until I met a Teacher who truly brings out the beauty of maths in his class in ways I didn’t even thought to be possible. Now it’s my favourite Subject!

Students are taught some checkpoints along the complete line of reasoning that built math theory in the first place. It's like you teach a child to walk by carrying it and letting it touch the ground in a few spots along the way. It learnt where to go, but not how to get there without support.

I can listen to her whole day long. Just because of her voice but the fact that she is usually talking about physics or maths makes it even better lol.

_And a special shout out to today's Patreon Cat Of The Day - a lawyer from West Texas._

I can palpably and manifestly discern that you genuinely care about mathematics and its educational dissemination. We need more profound, enlightened, and masterful professors like you. Thank. You. For. You.

I grew up loving math, and i was good at it, and i got a big rush from solving math problems. by 6th grade i was being accused of cheating on tests all the time, my folks would have to come in and defend me. I did a lot of mental math, and had very sparse notes. in highschool they created a class called 'worldly math' and i think it was meant to teach balancing check books and stuff. but this teacher decided to teach the class the "correct" way to do mental math. He even showed us the "correct" way to SHOW YOUR MENTAL MATH WORK. After i failed this test, they held their ground. I then had to unlearn my natural math skills, and learn this teacher's "mental" math and put it on paper. at 16 years old this basically broke my brain, and I've never been good at math again.

This video hit me to my core... thank you

I like how you chose Debussy's as background music. Debussy's compositions contain a lot of mathematics.

Im a maths teacher and have experienced the math curriculum in Canada, South Africa, Bulgaria, Dubai and Australia. I believe the Australian system is the worst at just having kids memorise, excluding creativity & explorative thought (from this list). In Canada, I was taught way more conceptually, where I got to question, be creative and experience the beauty in math. Its very hard to carry that across though, especially to kids who have been taught to be little robots from the very beginning. I've had some great moments though, its always more interesting and natural. I describe the problem and let the students explain their attempts at solutions. Also, another challenge in teaching this way is that math is associated with the importance of success because it has been given a higher status in our society than the arts, and thus, you often don't have time in class to explore this way, because they have a list of formulas and techniques they need to know for their standardised test.

10:45 If you feel inadequate as a teacher, you are most likely a very good teacher.

Lockhart's lament is one of my favorites. I've long had the idea of making an animation and read along. Great job.

Everytime a Mathematician forgot to add units to their answers: Mathematician: Video Thumbnail

This video made my heart grow 10 times its size, and tears well up in my eyes. I connected so deeply with the phrase that "The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such." All of the joy I had when painting as a kindergartener with my fingertips, and tooting with a recorder... you mean I could have had that with mathematics too? I totally feel like that was stolen from me, and so many of my classmates. To all of us teaching mathematics (professionally or not) - we have to make sure that we never let the dogma of "curriculum" stomp on the freewheeling-ness of uninformed creativity. Thanks for the video :)

You go through KZfaq searching for a good video. I don’t usually ever get satisfied, but this particular video is different. I feel like this video is what I have been searching for this whole time, and now I have finally completed my journey here.

OMG! Today I gave up studying for the math test I have in 2 weeks bcz it became so frustrating and I keep painting all day to distract myself but I had this realization after getting tired that math and art are not so different! And then you posted this video! What an amazing coincidence!!!✨ I'll get back to math now! :)

This made me cry a little... I love your videos, always so on point, honest, and inspirational.

You just described elegantly the beauty of mathematics. The why behind loving math all my life and working as an artist. Thank you!

As I watch this video, I realize that so much of what was being said, here, was "inflicted" upon me. Particularly, the idea of being convinced that I was not good at math because, rather than memorizing formulae, I was better at conceptualizing ideas, and figuring out those formulas as needed, kept me for years from realizing that I am, in fact, a "math person," and that I enjoy math. That said, as a university-level music educator, I can confirm that most of what Lockhart wrote about attitudes toward the learning of music is actually true. Rather than being a "nightmare scenario" showing the differences between attitudes toward the learning of music and math, I recognized in it the actual attitudes of my non-musician students. In particular, the statement that, "most of them couldn't care less about how important music is in today's world; they just want to take the minimum number of music courses and be done with it," rings too true. The one biggest difference, though, is that music education is NOT mandatory.

Wow, this is so good, this gives me that feeling of beauty and wonder that I only used to get from 3b1b videos.

You must make a series "Beauty in mathematics" .Take it serious you may be Carl Sagan Of Mathematics !

I'm studying to be a philosophy teacher, and everything mentioned in this video about math is true for my field as well. What is engaging and worthwhile in philosophy is exactly the joy and beauty of well-thought out argumentation. It is about developing an examinative and understanding approach, so that we may be better, fuller people. Studying is not supposed to be practical. Schools need to be refuges, places where students can come to see and learn the beauty and joy that is to be found in the world, in the knowledge we collectively have acquired, in doing activities for their own sake and most importantly, in themselves.

When that which one loves to do, becomes that which one has to do, it is such that one cannot bear to do it any longer

I became entangled in this mess... Totally relatable... I cleared high school and pre university easily thanks to my memory... But during the lock down I came across many such videos... I did a grave mistake relying on the teachers in high school and pre uni... It sucks now... I'm in a UG uni now and it's the same... I feel terrible... Keep going Tibees.. I got inspired by your result sheet video and I particularly liked how you deducted the rapunzel dilemma... The van de graaf video showed me the concept in a few minutes which my teacher took a month to cover and still didn't teach it well... Sucks... But it's life. Not everyones got the same starting line.. It's every man for himself. Let's move on.. Now this might be an irrelevant comment.. But there must be people out there who would find an inspiration in you. Keep going. Good luck

This book has profoundly affected the way that I tutor math.

this is lovely, thank you. this can really be extended to most fields taught in school. pushing through the absence of creative spark with routine is misery.

I love your channel so much, and this is one of your best videos. I taught math for 40 years, and maybe occasionally managed to do what you're describing here. I love that proof about the triangle and the circle! So beautiful.

Your content is so interesting and your voice is so calming! Thank you! P.S.: the proof of the last theorem was really elegant!

To me, this channel is what Paul Lockhart was looking for, you're an excellent teacher!

As someone who has a B.S. in Math - abstract math. This hit home - I was good at math because I could not remember a formula unless I understood it - if then. I had to rederive spherical integration on a final exam. But because I understood it, I could teach it to someone I was tutoring thirty years later. I had one good math teacher before tenth grade - among other things, he taught us how to calculate square roots when I was eight. The other thing that completely messes up math learning is that kids (and adults) arrive at different levels of abstraction at different rates, and real math is abstraction. If a kid isn't ready for a concept, they are left behind, and it's very difficult for them to catch up. One thing that illustrated this clearly was that one of my math profs taught a section of the basic course required for university graduation every semester. From virtually every one of those classes, at least one person became a math major. Often they were older students - in their late twenties or thirties - who had thought they were bad at math from some point in school. My belief is that their brains matured to the point where they could think more complex, more abstract ways - but it took someone who made concepts come alive for them to light the flame. Something in me dies when I hear a person who is perfectly good at thinking about abstract issues say "I've never been good at math."

Lockhart's lament is my lament. So many ideas that I try to communicate as a math teacher are expressed here. I deeply resonate with the part of it where he extols the virtues of playing with the math, just because. I believe in my heart of hearts that the greatest key to any success that I have had as a mathematician has been that I play with it.

Why do I become very relaxed and infinitely interested in whatever topic you do?!

I love these videos, I always learn so much from you, thank you.

It’s very beautiful how you’ve described the “ art “ of mathematics , the inspired creativity that it is at its heart and how it can be / must be taught especially at those thirsty young ages . I’m a musician with a B.A. in music and after over 40 years - since my College years and being now 68 - of reflecting on how school , my parents etc. hadn’t seen what my truest loves and talents were , I too often lament , more deeply each day , at how much of my own artistic potentials were / remain untapped ... ever more so with mathematics given my passionate love of Physics 💔 Thank you for the inspiration .

You are so inspiring, when ever I get frustrated in my math classes I sit and watch a few of your videos and feel motivated. Thank you.

Books respectively on the shelf, KZfaq reward on the floor. Got ma Respekt

"logic takes you from A-B imagination takes you everywhere"

This was both moving and inspiring. You breathed a lot of new life into a nearly 20 year old work. Thank you!

I was immediately caught by the title and had to watch. This is so true of the education system, that seeks to demand knowledge and supresses creativity as that is the anthesis of knowledge. I recall in a Maths lesson someone asking the teacher "What was the point of differentiation, they didn't need it to buy a Mars bar". We were being taught a process (to follow a pattern) before the relevance or time to consider why there was even a need to differentiate. In the end I recall the process therefore seemed onerous and ultimately the answers were meaningless.

This says a lot about society.

I rarely comment on youtube videos. But I'm making an exception here because this video really captures what I have found so special about mathematics. I'm sure I will share this video with many of my students. So many of them suffered so much in their grade school and high school math classes and have rejected the whole idea of mathematics being beautiful or exciting or enlightening. This is so sad.

Well done! You do so well as an educator. Your way of explaining math through creative means is a wonderful talent. Please do not doubt yourself.

This article was an eye opener for me back in my school days. It is probably my inspiration to keep a for a better explaination and learn from it.

I loved this video, Toby. You reminded me of how much passion I feel about teaching and math. :)

Thanks for making this iconic article more widely known. I had known the earlier versions and also bought the book when it came out a few years later. The connection with art and creativity and especially music, or poetry is often no more appreciated. Schools these days often are pressured `to force more applications" or ``build in messy data analysis parts" to give math more more purpose. Especially the competition with applied math or stats puts pressure on remaining relevant. It is important to see it also as an art. Since art is closely related to creativity, this can also make it harder to teach and motivate. It is important to point out that many mathematical topics which were done entirely void of applications like properties about prime numbers later became very applied like for cryptology. But it is in the nature of pure mathematics not to worry too much about applications at first. And that makes pure math so pure! By the way, the Thales theorem round minute 9 in your movie is much more interesting when the base of the triangle is not the diameter. The angle stillremains constant when moving on the circular arc. This brings in a moment of surprise and makes the theorem much more interesting. The diameter case can be proven as you illustrated with symmetry. It needs a bit more creativity to find a proof of the general case. I teach this theorem since many years in a course and still have always seen that the class was able to figure out a proof in the more general case (despite that most have not seen the general case). I myself do no more agree with the pessimism of Lockart. Especially if you look what is around also on youtube (like also your channel) makes some aspects of math more accessible. S lot has happened since 2002 when the article was written. I'm sure that if Lockart would rewrite the essay today, he would agree. The last two decades have also produced msny fantastic math books for more general audiences as well as quite many movies and documentaries illustrating the beauty of the subject. Certainly the pamphlet of Lockart had a good deal to do with it.

Although I never really had much trouble with school math. It didn't catch my interest beyond what I was required to do. Fortunately, in high school, I had a group of friends that created a sort of additional class powered by one enthusiastic teacher of ours which was dedicated to learn real maths. We were given complex problems and had time to think about them, discuss them and present our answers if we were able to formulate one given everything we did know so far. This experience really showed me what math is about, the simplicity of reasoning and creative thinking in tackling self-proposed idealized fantasies. I am very grateful to have had this opportunity and thankful for this because basically, it opened a whole other domain of creative exploration that previously was locked behind stiff bars. Currently, I do study a lot of maths just to get more ability in addressing some ideas/fantasies of mine concerning GRT and thus require a lot of algebraic geometry. There are moments when I prefer the simplicity of algebraic reasoning but the more profound ideas I have had insights that were purely geometric in nature which were just beautiful. Thanks for this wonderful video to remind me of the art that math really is.

I am completely amazed at both the wisdom in the content and the quality of the presentation hats off Toby hats off i just adore ur channel

This one was very deep, dear Tibees. I like when you talk about "philosophical" (sorry, I cannot find another word xD) aspects of mathematics, in this case about the teaching

all my exams got over today and ive decided to binge watch Tibees today 😊

One of the most wonderful educational videos in recent times I've come across when dealing with Mathematics, thank you

Tibees, your channel is truly outstanding. Nobody had ever got me interested in maths. I'm discovering thanks to you.

This is so insightful and eye-opening! Thank you, Toby!

3:56 Now this is Avengers-level discovery

Hi Toby, I'm no mathematician, but I find your videos are a delight. I particularly like it when you dip into philosophy and life. You are so good at teasing out the details in your unique and charming way in videos that are entirely enjoyable. Thank you!

Good idea to watch these videos a few times over a period of time. Rather like looking at your favorite painting, you gain more understanding and can enjoy the video each time.

most calm voice i've ever heard

Ur video is more informative than my whole educational carrier

The only reason people like Paul Lockhart have ever been blessed to see the beauty of math is the creative teaching methods of people like you. His lament is all about wishing there were more teachers like you. Teachers who make that vital connection between the rigorous rules of math and the exciting discoveries that fill the hearts of students with passion about the subject. Unfortunately, such gifted teachers are naturally very rare, even rarer than the gifted mathematicians.

Everything amazes me . Starting from human body to the universe...every abstract,every profound thing amazes me....every creation, every history amazes me...list goes on..but what amazes me most is how some things give such perspectives which literally changes everything...hey Tibees you are one of my perspective alternators..thanks for living.. : )

This is so true!!! I really really love Math and Physics but what they teach us in school bores me. I still want to learn more but I can’t because our entire education system is just based on rote learning and full marks🥺

Your videos are always inspiring and helps to enhance imagination.Thank you dear. Lots of respect from India.

I don't really watch your videos much, but I had to click on this one because Lockhart's views on mathematics education changed me forever. Your rendition of his words was very touching. Thanks for that!

Very cool. Something that is not immediately obvious about rectangle is that the "second" diameter actually is a diameter (i.e., that it runs through the center of the circle) BUT that is remedied if you are more careful about the rotation by pinning the triangle (or a copy of it) at the center and the rotating, that shows vertexes have to stay on the circle and that the 2nd diameter is truly a diameter.

I feel like Tibees has solved Lockhart’s issue through her amazing videos

The 'are you talented or you Are you good at memorizing' had me thinking 🧐

As a former math teacher, I think you touch on a systemic problem in math education. Math teachers who love the subject are often pressured by their administrators to "teach to the test", and subsequently feel like they don't have time to share their love of the subject with their students, especially within the confines of a 45-50 minute class. I tried to push back against these pressures in my own classes, to show my students that there is often more than one way to solve a problem, which is where the creativity part comes in, and that making mistakes is not an indication of failure or stupidity, but rather an essential component of the learning process.

I cried watching this. I'm a Math major myself, but often I found myself fumbling to answer questions from the past syllabus. I'm only great at answering of what I currently learning and remember.

THANK U SO MUCH FOR MAKING THIS...The moment you showed the difference between how fun and beautiful math is and how it is introduced to us in schools, it made me so happy, that there are a lot of people with pure minds who want to correct what's wrong, cuz I've been tallied as a "wierdo" for worrying about what's wrong with the education.........I hope we will see a better and changed world some day 😁✌️

0:44 wow I see where this is going and SO TRUE :( ❤️🤜🤛

You do a good job exploring the boundaries of careers, schools and ideas. You investigate those different worlds very honestly. Most people are thrown into these worlds, and there's a lot to sort out - some seem to just give up. I majored in mathematics in college because I saw it as an escape from having a more dogmatic, socially structured major. I felt that if I learned a few proofs and some set theory, people would leave me alone with my own thoughts and wouldn't try to impose a bunch of beliefs on me that I was supposed to ultimately impose on myself and to parrot. But I did find that learning abstract mathematics very indirectly gave me great thinking skills that made the professional, material and engineering world much more manageable somehow. I feel that music, drawing and Math are all great places to develop one's own non verbal thinking. Of course, I imagine there are many other endeavors that can be like that, but those are three great ones to develop yourself. Language is the main genius of humans - something that defines our society and makes it possible in the first place, but these other areas are good to explore away from society's common, sometimes very restrictive structures to make a space for yourself.

Thanks Tibees for bringing this up .Bring more

It's really a masterpiece Toby 😀😀😀

I really think the situation is improving, mainly due to all the Math-KZfaqrs out there! Keep it up!

loved this essay years ago in high school. thank you for covering it!

Her voice calms me down. Kinda helps me focus too.