The way math should be taught

[@StephenBoothUK]

The problem I’ve found with how Maths is taught, and to a lesser extent other STEM subjects, is fundamental to how Maths is structured. In Maths each topic builds on the ones that came before. You need to understand counting before you can understand addition and subtraction, you need to understand addition before you can understand multiplication, you need to understand multiplication before you can understand division, you need to understand all 4 of those basic operations before you can understand geometry and algebra, you need to understand geometry and algebra to understand trigonometry, you need to understand trigonometry to understand calculus &c. You don’t find that to the same degree in most other subjects. You don’t need a thorough understanding of the 100 years war to understand Tudor England, whilst a decent understanding of Tudor England may help understanding of the English Civil war you can get by without it. In Geography knowledge of how glaciers shaped Northern European isn’t needed to understand the principle exports of South Africa or what the different sorts of clouds are. Maths is like a wall built of bricks whilst many other subjects are similar bricks scattered about. In Maths if you are missing a brick (maybe you missed that class due to illness or changing schools) or is damaged (maybe your teacher didn’t explain something in a way that clicked with you and didn’t give proper feedback when you got answers wrong in tests/homework, then had barrelled on to the next topic) then every brick (topic) that relies on that is weaker. At first you can get by, but as the damaged bricks and holes accumulate you find it harder and harder to understand each topic. A common experience for people who struggle with maths is that it all made sense up to a point, then it stopped making sense. That point often correlates to a life change such as changing school of a period of sickness.

[@barbh1]

When you get to that missing brick, you need help.

[@StephenBoothUK]

@@barbh1 it can be hard to identify which brick you’re missing. You can struggle through so by the time the struggle gets insurmountable you’re well beyond where the missing brick is. Imagine a Jenga tower where you’re having problems at the top, 10 or more layers from the base, but the root cause is on layer 3 because you weren’t taught something that wasn’t really needed by now. I recall one Electronics lecture where the lecturer was working through a derivation and suddenly point to part of an equation and said “And that of course is the complex plane definition of sine”. I had never been taught any definition of sine that didn’t involve the opposite and hypotenuse of a triangle. The name Euler had never passed the lips of any maths teacher in any class I had. No-one had ever said that sin z = ((e^(iz)) - (e^(-iz)))/2i. Apparently this should have been covered 5 years previously as part of O-Level Maths, unfortunately, due to an administrative error at school, I did CSE Maths which was supposed to be equivalent but didn’t cover quite the same material. I’d gone 5 years not knowing that I didn’t know that or might ever need to know that.

[@decare696]

That's not really that true. For example, a lot of geometry can be done without knowing about numbers. Going a bit further, you don't need topology to understand linear algebra, tho that's not really a level taught in school, so it might be irrelevant.

[@meropale]

To be good at virtually anything you need a solid foundation. With math the barriers to entry are more significant. It really matters, likely more so than in most fields.

[@jorianweststrate2580]

@@decare696 "you do not need topology to understand linear algebra" Functional analysis:

[@lars14701]

I was not familiar with this book prior to your video. Thank you!

[@umarus2]

Once, I covered an art class and saw how students interact. The students even invited me to join them. Although my artistic skills are almost non-existent, I painted a piece of paper motivated by their encouragement. Since then I have been considering implementing this approach (I believe this was the case in Ancient Greece and the Middle East.). However, one of the biggest obstacles to enjoying mathematics classes is the amount of content and limited time

[@JesusPlsSaveMe]

Jesus is the way, the truth and the life. Turn to him and repent from your sins today!

[@ClackerJack]

@@JesusPlsSaveMe Indeed, but what does that have to do with the price of fish?

[@andrewyork3869]

Drawing and rough graphing is way underrated in math.

[@sharonjuniorchess]

@@andrewyork3869 By starting off with arithmetic it automatically weeds out the spatial thinkers. Only later does maths need the spatial thinkers (who have all gone elsewhere).

[@frogandspanner]

Into GR!

[@jakobklug4978]

i had an excellent math teacher.... he started explaining that most math won't be necessary in most of our futures - but that it helps train complex logical patterns

[@fosheimdet]

Well, I disagree. Its usefulness depends mostly on the proficiency of the user. It's just that most people choose to (unknowingly or not) avoid the tools that make use of it. For example, when getting a mortgage, most people just use an online dedicated calculator, instead of working with the actual formula and a python script. And so they get no understanding or intuition of how their interest will accrue. They cannot plot it, and cannot play around to fill gaps in their understanding. If they had gotten to understand this formula, they could have also applied it to understand how their savings will grow. Instead, they know two websites: One for mortgage calculator, one for savings calculator. Statistics for example is something that can be applied basically everywhere you look, and will be useful throughout your life. From calculating probabilities in card games to analyzing your last month's spending or updating your beliefs.

[@RK-252]

​@@fosheimdet"updating your beliefs". I would give you 30 thumbs up for those three words. (As a side note: I agree with BOTH the broadest interpretation of the OPs post, AND the more technical interpretation of your own. My two cents below are tangential to this.) IMHO, over the years I have found statistical literacy (which is preconditioned on both numerical and logical literacy), to be one of the strongest tools I have to sanity check the tidal wave of information (and importantly disinformation), opinions, biases, prejudices and beliefs I am fed and absorb, consciously or subconsciously, on a daily basis. Statistics tends to remove ego from the equation, and in doing so helps me regularly reevaluate my own. And regardless of the many differing views on statistical literacy a year ago, a week ago, or even today - I think every human, bot, and machine learning algorithm reading this post will agree that mental tools to filter out misinformation are very very quickly becoming very very important.

[@adrianwright8685]

"Won't be necessary". - You could probably say that about most stuff learned at school!! But maths at least has the possibility of allowing you to check your bills, mortgage, pension, bank interest etc.

[@2bfrank657]

I think it's one of those situations where you don't know how useful a mathematical concept is until you've actually learned it. The more math you know, the more uses you find for it.

[@Gerii]

I always thought "I will never need all this" then I started a blue collar job and suddenly realized you can use it everywhere, especially the "basics" and it's very important for my daily work. even some more complex topics cross my way so I had to get into it.

[@jeremyholbrook2094]

Your calm disposition makes it less stressful for sure 😊

[@wraith8323]

In the future I would imagine AI teachers based -with permission- on certain personalities. I'm going to split my education between a Tibee personality and Morgan Freeman :)

[@jeremyholbrook2094]

@wraith8323 lol not bad !!

[@ferdievanschalkwyk1669]

Classical math teaching, is both abstract and prescriptive. This approach gets rid of the prescriptive portion, but the abstract portion is still there. That is the portion many people struggle with. Maths only really "clicked" for me, when it was taught the way it was discovered. First starting with ways to approximate with fair accuracy and then learning more sophisticated methods as increased accuracy is required. Even now, in computing, we use approximations to a large degree for greater efficiency, so that computational power can be expended on more important tasks.

[@JesusPlsSaveMe]

To everyone in this chat, Jesus is calling you today. Come to him, repent from your sins, bear his cross and live the victorious life

[@jaymathew]

Teaching the way its discovered is a great way to put it. I also began to understand geometry better using that technique and I only started doing that when I started to build things that I needed to be plum and level.

[@witokija]

@@JesusPlsSaveMe don't shove your stuff down peoples throat mate

[@darthbumblebee7310]

I always thought math is taught in a way pretty close to how it was discovered. What are some things you would do differently?

[@imacmill]

​@@JesusPlsSaveMeLOL.

[@trayne5151]

I remember taking calculus in high school, my teacher sent us out around the school to create and solve a problem using calculus. I did a problem of the tip of a persons shadow as they run towards, under, and past a streetlight. Using some calculus and trig an assuming a constant pace for the runner, I was able to find an equation to represent the tip of the shadow throughout that process. I loved that class. I'm the opposite to most people (in my social groups) in math though, it didn't make full sense to me until I took my first geometry course with proofs. Most people I talk to started having a hard time when the curriculum got to geometry, but I found it began clicking with me when I got to that point.

[@chayanika8155]

Same here!

[@IranOutofUserNames11]

Taking differential equations and calc 3 at the same time was when things started to click for me. Especially diff eq when it came to recognizing patterns in relationships as they change. However calc 3 did wonders for my spatial reasoning especially when we got to spherical coordinates.

[@laulaja-7186]

Graphical thinking versus linguistic thinking.

[@JanWnogu]

@@laulaja-7186 I remember solving an exercise (which was about some physical problem) during a mathematical analysis exam using conservation of energy and symmetries found in the problem in a few lines, after which I proceeded to solve it using mathematical analysis methods which took solid two pages 🤣🤣🤣 Both result coincided 🤣🤣🤣

[@FirstNameLastName-wt5to]

You lose the average student at the rectangle because you didn’t explain why a rectangle. That’s pulled out of thin air. The average student will not be able to catch up after this because they’ll get stuck at “where the heck did she get the rectangle from?” That’s when they check out. This is why students start to bow out of math in high school. Not because it’s boring or hard. But because the why is not being satisfied by teachers who don’t see the steps they are skipping.

[@JohnM...]

The reason teachers can’t effectively teach maths is because they don’t understand it deeply enough themselves, and so are incapable of distilling it into easily understandable chunks - the basis of the Feynman technique (if you can’t explain it well enough, you’re shi* at it).

[@acrane3496]

That’s why the best math books are simply the books that take their time explaining as much as possible about the math being shown. OR Books with tons of problems that come with detailed solutions I swear these books are sometimes needle in a haystack

[@grillinnchillin4009]

Yeah, I don't think the average student would find the approach in this video much more welcoming to math just because a variable is named flower, and everything is drawn a e s t h e t i c a l l y. I do like the general approach though, as she says after the demonstration, but it seems like just basic math at the end of the day, but with more of an emphasis on self-discovery. Perhaps because I've already begun studying proof-level maths, I have the bias that this is "just doing math," forgetting that most earlier schooling focuses primarily on stale, rote memorization. But this approach feels like it would take forever for the average student who doesn't even pay attention to the 'softer sciences' of history or language arts to even begin to appreciate the artistic element of pure maths. Overall though, I think a building block of personal discovery of math works well, but it's probably best done when giving someone a sort of 'real-world' problem, and having them play around with ideas while being guided by previous work and a teacher/tutor/group-lead that understands the actual answer

[@donotRussiamyMerica]

I recall learning all the tricks in algebra, calc, trig but asking myself, “What an I supposed to do with this information?.”

[@janthomas468]

​@@acrane3496 Hi I couldn't agree more. Could you please name some books which actually follows such a structure? Thanks

[@jorgetorresramos902]

Beautiful video. The shorter side of the rectangle is the one that should be labeled flower minus one, and the longer one flower.

[@CliffSedge-nu5fv]

_Figure not drawn to scale._

[@gcewing]

First lesson of geometry exams -- don't trust the diagram!

[@grillinnchillin4009]

I was so confused by how the longer one gained size, and the shorter one lost it after the average. I thought this was just more complex than it seemed on the surface.

[@jonathanodude6660]

@@grillinnchillin4009 the great thing about maths is that the drawing doesnt actually matter if the principles are correct.

[@grillinnchillin4009]

@jonathanodude6660 yeah, fair enough, but the logic of the drawing still took me for a spin. Part of the reason I came down to the comments 😅

[@vivekpujaravp]

Phenomenal presentation and book recommendation. Please do more like this.

[@gebruikerarjan]

As a math teacher i myself learnd it by reading the solution to problems, but it was a struggle to get enough solutions because a lot of teachers think you give the principal and the proof and then you can solve the problems....but not everyone has time or patience to think about a problem for days etc...learning math is like walking through an unknown city and learn your way by walking a lot of different paths and getting to know what you see around, but its a like a language to. Give students a lot of paths in the mind or story's in the new language (a lot of solutions to mathproblems). Every branch of math is like a different language to or a different landscape (from the city to the mountains)

[@gaius_enceladus]

I just never really got to grips with mathematical proofs. I think that one reason for this is that I only really started doing them when starting calculus, and I think you need to start learning about how to do proofs much earlier than that. I think proofs would be a lot easier to understand if (say) you were given the set of rules that govern triangles. Those would be your "tools". You'd learn those "off by heart" and then you'd make a gentle start on creating a proof using those "tools". The same with circles and the rules that govern them (chords, tangents etc). I think simple geometry would be a good place to start learning about proofs. I think shapes would be a lot less intimidating to people than numbers can be.

[@JanWnogu]

7:45 I think the sides of the left-hand rectangle are wrongly labelled. The vertical one should be Flower (F) and the horizontal one F - 1. Then you take away 1/2 (dashed line) from F (vertical) and add it to F - 1 (horizontal) thus getting (F - 1/2) times (F - 1/2). Interesting technique, though.

[@protocol6]

I've felt for long time it might be better to bring back compass and straightedge construction and proofs quite early and then introduce abstract algebras in that context.

[@absolutelyanisa]

Learning math is like learning to be a therapist for “math” who has a lot of “issues” and solving them in their own “brain patterns”. I swear, when you said flower instead of A or B, Immediately felt relaxed and went into novel reading mode 🌼. This is one of the best motivation I’ve ever received for starting to do math

[@pingupenguin2474]

My Ah ha moment was when I did physics, the day we rolled balls down a shute and plotted time/ distance. Then plotted the results. The teacher pointed out the maths pattern on a cartesian graphs and told us what equation matched it. The maths teacher reminded us how to recognise their associated equations for other graph shapes ( inc. Sine waves). Seeing the physics scatterplot and recognising the matching equation for that expirement ( and those of subsequent physics and chemistry expirements) I began to understand the point of higher level maths. Seeing the teachers use these recognised graph patterns to identify equations, and employ these with calculus to predict actual physical results beyond our test range was a revelation !

[@LifeLikeSage]

The problem is that those who are smart enough don't need to go through the negative experience of making mistakes in mathematics; They can experience the creative flow in maths, because they already understand the rules. In order to get to this state, it takes most people a lot of training, and a lot of failure (being wrong). This negative experience, naturally pushes people away into things they are good at (things they don't get as much negative feedback from).

[@sprout8426]

nailed it.

[@bluehair0476]

I completely disagree with your starting premise that people smart enough don't make mistakes. The difference lies in how people look at the mistakes. A picture I like to use is the exploration of a labyrinth. Each dead end you reach (each mistake you make) gives you more information about the layout of the labyrinth. The goal (and the joy that lies in math) is not to get out as quickly as possibe on a way without mistakes. The real goal is to understand the layout of all the labyrinth which would be impossible without the knowledge about the "mistakes". Figuring out the layout, figuring out the rules, solving unknown problems from mistake to mistake to mistake until you not only reach the solution but a deep understanding of the problem - there lies the joy and the creative flow of math.

[@user-rm2qj2jh4l]

I agree with @bluehair0476. And I also think that the negative experience isn't actually from making mistakes, it's from being told you're stupid because you've made mistakes. If everyone thought of mistakes as the important learning tools they really are and treated people with enthusiasm and respect instead of bad grades and humiliation when they made them, math would be quite fun indeed. I really like drawing and singing even though I'm not that good and am off key a lot.

[@user-rm2qj2jh4l]

I agree with @bluehair0476. And I also think that the negative experience isn't actually from making mistakes, it's from being told you're stupid because you've made mistakes. If everyone thought of mistakes as the important learning tools they really are and treated people with enthusiasm and respect instead of bad grades and humiliation when they made them, math would be quite fun indeed. I really like drawing and singing even though I'm not that good and am off key a lot.

[@user-rm2qj2jh4l]

@@bluehair0476 I agree with @bluehair0476. And I also think that the negative experience isn't actually from making mistakes, it's from being told you're stupid because you've made mistakes. If everyone thought of mistakes as the important learning tools they really are and treated people with enthusiasm and respect instead of bad grades and humiliation when they made them, math would be quite fun indeed. I really like drawing and singing even though I'm not that good and am off key a lot.

[@aromview]

Math can be rewarding, but most people end up getting lost because instructors tend assume everyone knows and just go n ahead with the lectures

[@veqv]

I got most of the way through an undergrad in math before life got in the way. It's been a lifetime since then - I'm now a developer, and there are still problems that I return to from time to time. I've always referred to this as "back burnering", it has to simmer for a while before it's ready. And this is true in every field I've experienced - there will always be problems that one is unsure how to approach, that one can throw the entire accumulation of their learning at and still come back puzzled. That just means you need to return to cooking other parts of the meal. Luckily nothing here spoils, it just gets more tender with time.

[@HartleySan]

Bravo! Wonderful video. I've been trying for years to help my daughters see the joy in math, and this video (and the book you talked about) may provide some other great ideas. Thank you.

[@mnmlst1]

I literally had the exact same conversation about that yesterday. This is so true. Too unfortunate the majority of global educational system is developed to be a bore.

[@1johnnygunn]

And not by accident

[@martm216]

The best maths book I have come across for non-specialist students who need to strengthen their maths is Maths - A Student's Survival Guide by Jenny Olive, published by Cambridge. It's a substantial work, very friendly for folk like me whose maths is shaky. Olive really speaks to you, like a teacher at your side. And there are masses of exercises all with solutions, including worked solutions where necessary.

[@brianlhughes]

one of my favorites is how to draw a perfect equilateral triangle inside a circle. I was a DJ and we did an AA (alcoholics anonymous) dance and I wanted to put a huge AA symbol, a circle with a triangle inside it, on the wall with twinkle lights. I put the circle up no problem using a length of string as a radius but I couldn't figure out how to do the triangle! I happened to post about my old dilemma on YT and someone said, just tape the string to the top of the circle and mark the spot where it intersects, tape the string to that spot and do it again to mark the bottom of the triangle, repeat it for the other side and there you are! I tested it out with a compass and found it quite interesting that the radius of the circle can be used to divide a circle into 6 parts. Are there any other fun secrets shared between a circle and it's radius?

[@sharonjuniorchess]

Yes. I find it more interesting to talk about the person who discovered the fact (as you are teaching history & Geography at the same time) than just a fact in a text book that is striped of all the interesting details of time & place. Take Thales of Miletus for example; who discovered the 90 degree angle in a semicircle, known as Thales's theorem. Its even more impressive to try it out with a piece of string pinned to opposite ends of a line that bisects the circle in half (i.e. the diameter line which is two radii in a straight line).

[@kevinpaulus4483]

Is that right ? Isn't it more like the radius divides the circle in 6.283... parts (2*pi). And that just happens to be close enough for your application ? You need a triangle with 3 x 60 degree angles well with that method you'll probably get 2 angles of 1 radian (~57 degrees) and one with the rest added ~ 66 or so.

[@sharonjuniorchess]

@@kevinpaulus4483 No I am following the curve on the semi-circle with string fixed at either end of the diameter. You get a 90 degree angle wherever you are on the circumference.

[@user-rm2qj2jh4l]

"People can do hard things if they want to, and if they feel supported". So good!!

[@kloassie]

7:16 one question though ... Why are you drawing (❀-1) to be *_longer_* than ❀?? Shouldn't ❀ be the longest side? As it is exactly 1 longer than (❀-1)?

[@mathculture7615]

Yes, it feels like this could have led to taking 1/2 off the long side of the rectangle to adding it to the short side, making it even more intuitive.

[@BryanLu0]

Not drawn to scale, as is the case in many geometry problems. Usually this doesn't cause problems, rarely you'll have problems with where lines intersect, such as the every triangle is equilateral proof

[@0Smile0]

(❀-1) is clearly longer than ❀ by about 0.68 centimeters on my screen, I got out my ruler and checked! Art doesn't always need to make sense 😁

[@tiborbogi7457]

Tibees clearly knows that even with this mistake she get her desired outcome, so she don't bother assign it correctly. Such detail drew me crazy.

[@richkroberts]

Yes, that’s why I had trouble following her idea. I liked her overall approach though.

[@scanurasiri1]

Good to hear back from you as always with great contents.....

[@duncanhill4434]

My kids are probably a bit young for this, but the way I'm teaching my kids at the moment is through exploration. Giving them a nudge to find things like Kaprekar's constant by themselves, all the time practising the basic skills (who wants to do a massive sheet of practising subtraction, when you can find if some other 4 digit number has the same result, or if the first one they picked is somehow special). This definitely looks like a book I need to invest in for them though.

[@scottguitar8168]

It's been a while since I have been in a math class but if I remember correctly it seemed that in the very basic math, most students could see the value of learning this. It was only when we got away from numbers and into variables that the number one question was "what am I going to use this for?". The teachers often did not have any good answers, nor anything to spark interest in learning higher math. Many kids had problems with the question of why, when it came to the rules of higher math where the typical response from teachers is just accept those rules and move on. I had a natural interest in math and could just accept the rules and move on. It was my pursuit of an engineering degree that I finally reached the level in math that answered all those questions of why and gave me that huge a-ha moment of understanding math to be a beautiful language. I don't think most math teachers have that comprehension, which is why they never had good answers for their students as to why. Math is very apparent with things like measurements and keeping track of finances, but it is found in art and music, which is less obvious.

[@mujtabaalam5907]

What did you learn in your engineering degree that gave you an a-ha moment?

[@scottguitar8168]

@@mujtabaalam5907 I was studying to be an electrical engineer, but a lot of professions like this required the same prerequisite courses, including a lot of higher math. If my memory serves me correctly, it was one of the calculus courses where we had to do the proofs for many of the rules we were told just to accept where now we knew how they came to be. I think it was shortly after that I realized math was actually a language and a bit of a conversion language where you could describe reality conceptually and turn conceptual ideas into reality. I can't remember which level of Calculus. I remember Calculus 1 was Calculus done on algebraic equations, Calculus 2 on trigonometric equations and Calculus 3 on 3 dimensional equations. There were more math classes after all of the Calculus classes that were more of a deeper dive into previous classes. Such as differential equations or Linear Algebra that used things like matrices to solve simultaneous equations which was faster and more efficient than solving them in early algebra classes the traditional way. I think sometimes people have a misconception that higher math is harder, but it is not, it is just different and adds tools to your tool belt to get the job done. I suppose I had other aha moments, but math was the biggest one because it was a mystery about the rules of math and being told to just accept them compared to the moment of understanding how and why the rules came to be. It's the difference between accepting your car gets you from A to B and understanding the entire process of how that is possible. As an electrical engineer it is interesting that we cannot see electrons but have equations to describe their movement and reactions and how much trigonometry about triangles is involved in circles.

[@AusFastLife]

Yes that would be interesting if maths was taught in that way. In maths classes you seem to be taught maths rules and you get maths problems which you are then supposed to solve and you take a maths exam which consists of yet more maths equations which you then are supposed to solve. But the part that is missing is the application of maths. How is it used to do whatever it is that you are meant to do with it? Maths teachers, certainly at the school level are not mathematicians, they are education students who have chosen to teach maths, and so they have learnt how to create lessons that cover a predesigned maths curriculum. But as far as I am aware, no one seems to even know what the different sections of maths are for, so of course how to apply maths in the situations it is meant to be used for is absent.

[@scottguitar8168]

@@AusFastLife I guess my thought is that if real world applications where taught, math may have more appeal to students not as interested in maths once they see what it can do. I sort of liked physics class better because the math did have real world implications.

[@neutronenstern.]

and then there are actual teachers that deny 63 on the question: " Write down a two digit number, where the tens place is double the one place". Why you ask? Because the sample solution suggested 21.

[@MuthuKumaran-hb6ku]

Very interesting perspective....so happy to have known u & this channel...i hope my 6 year old will soon get to watch some of your stuff..

[@ChristopheStrobbe]

Thank you for adding proper captions to the video. KZbinrs don't do that enough.

[@teaformulamaths]

I thought it was automatic now?

[@CarlJackson92]

Great video, thanks so much for sharing! Definitely going to check out Lockhart's book now :)

[@Lance54689]

This explains a lot. I've always been the odd "Math Guy". People seem to think I'm solving math problems from a text book, and I explain "No, no, math isn't about numbers, it is something I play around with in my head." Naturally I was good in math class, but that was because I turned it into something like this. Somehow I intuitively understood what "math" was about. Maybe I'm not all that bad at art after all, its just that my art is math!

[@supergirl1892]

Your Voice is so Soothing, and comforting ...maths anxiety took the back bench...😂😂😂

[@paulfrindle7144]

Oh dear - I so love this video! This is basically how I taught myself and one sibling why and how we solve things from first principles, just my looking at a visualisation of the problem that we can actually 'play with' in our heads. I have had no 'real' formal education myself - but this is the way I've done everything in my career in several disciplines with great results - and even developing new ideas. It's like playing with virtual 'friends' in my head :-)

[@malcolmanderson6735]

My Brain locked up when I saw that flower minus one was longer than flower on the square. That confused me a lot

[@rorrzoo]

As a professional mathematician, I can tell you that confusion happens a lot, and we learn to accept it and to take the time to overcome it. Confusion can actually be very useful as a means to identifying the part of a problem that you have not yet understood properly.

[@mikescarborough9196]

@@rorrzoo confusion = I need to keep thinking - understanding

[@williammorton8555]

Tibees. You finally sound so confident and happy. Joy to you. My Cat in the Klein Bottle tee is still worn to this day. 🥰

[@PlasmaOscillations]

Looking forward to reading your book someday

[@brentsoper345]

So much fun this way - I loved the flower to demystify algebra symbols! A nice way to find what the flower is!

[@vishalkumar040393]

After reading A Mathematician Lament, I wanted to make a video on how mathematics should be taught. I am so inspired that in future I want to start a school in Bihar, where students can be taught following his ideas.

[@stevenboelke6661]

In school I often only half paid attention in math classes. Solving math problems were not only a matter of remembering a formula and applying it, but often forgetting formulas and finding my own way to the answer. I never had the sense of math being stale because I had found that there were so many different ways to arrive at answers, that there was room for creativity and expression.

[@elysianfields1671]

When I was at school, I hated it. I really couldn't feel interested in anything else but History, Literature and sports. At the University, where I studied Philosophy and Sociology, I was lucky to have teachers that were into Philosophy of Science. And people who studied the philosophy of Mathematics, Geometry and Physics. Revisiting all those school topics, from a philosophical perspective, made me fall in love with mathematics and physics.

[@gastube22]

Wonderful, as always. Thank you and keep on mathing so beautifully calmly!

[@modolief]

K-12 math teacher here. 15 months ago I was interviewing with the principle of one of San Francisco's best elementary schools. We were talking about the efficacy of math and I told him that I basically saw math as an art. Yes it has lots of applications, but I like it for the "cool theorems" and interesting concepts that we encounter. Square root of negative one. Functions that are nowhere continuous. Proving that e is transcendental. There's so much depth to it - Newton's famous quote: "I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

[@stevencarr4002]

In California, in the school Kamala attended, Thousand Oaks school, currently white students score 84% and Black students get 17% , according to reports I saw. Fix that first.....

[@TheYurubutugralb]

Principal

[@modolief]

@@TheYurubutugralb omg, how did I miss that ....

[@arbiforumnow]

I have to tell you as an artist who struggled with mathematics even in college I love your channel! You make me feel like I could gain from studying it again!

[@aniksamiurrahman6365]

No matter what you do, learning math will always take a lot of elbow grease.

[@HartleySan]

True. But like any other pursuit, if you enjoy it, the elbow grease don't feel so greasy... or elbow-y?!

[@railspony]

@@HartleySan Most people don't enjoy it, though, so what is the part that makes them want to put more grease in?

[@HartleySan]

@@railspony Like most things in life, I think sometimes there has to be a little bit of a "I don't like this, but I'll force myself to do it a little bit and see how it goes" mentality.

[@tcjpn]

Great perspective on being creative and promoting creativity in any endeavor. (Don't forget the yin and yang, the joy and pain, in the struggle that can accompany being creative. For me surfing serves as a good metaphor.)

[@JohnDanielBryant]

Paul Lockhart is GOATed. Guy works independently for years, submits his research to Columbia University, and they give him a PhD in math. He says math is taught all wrong in schools, and he's completely right.

[@dhickey5919]

Thank you, Tibees. Looking forward to the book as well. I must admit I'd never considered the difference between a math solution in the back of the textbook and not having the 'solution' on the back of painting. Excellent point.

[@kulbhushansingh]

.

[@unclerat2131]

This is fun. I have retired and have decided to play with the subjects I have yet to grasp, like mathematics. Your channel is really helpful. Thank you.

[@Terry8TheBat]

Finally, she’s backkkkk❤❤🎉

[@leagarner3675]

Yay!!

[@JesusPlsSaveMe]

​@@leagarner3675 Revelation 3:20 Behold, I stand at the door, and knock: if any man hear my voice, and open the door, I will come in to him, and will sup with him, and he with me. HEY THERE 🤗 JESUS IS CALLING YOU TODAY. Turn away from your sins, confess, forsake them and live the victorious life. God bless. Revelation 22:12-14 And, behold, I come quickly; and my reward is with me, to give every man according as his work shall be. I am Alpha and Omega, the beginning and the end, the first and the last. Blessed are they that do his commandments, that they may have right to the tree of life, and may enter in through the gates into the city.

[@VinTheDirector]

wait was Tibees gone??

[@Gigusx]

@@VinTheDirector No...

[@alangable9555]

Awesome!! I just ordered Measurement!! Thanx So Much!!!

[@The_Gallowglass]

I remember the math classes I excelled in were taught by teachers who actually gave a sh-t. The ones who didn't just teach theory and abstraction, but made us apply it to something useful and/or creative.

[@sohailakandil2046]

I am a senior computer engineering student and i have a passion towards physics and astrophysics. I would like to shift to be an astrophysics computer engineer. to work on building astronomy simulations for stars births, blackholes and supernovas. but i don't know where to start and what to learn first. I am specialized in machine learning and data analysis where can i apply my knowledge for astronomy applications.

[@TheGuruNetOn]

Try having a look at Kip Thorne's book on the physics behind the movie "The Science Of Interstellar". For more math/physics related stuff like into the book and lecture series by Leonard Susskind called The Theoretical Minimum. Kurzgesagt videos are more basic though really well researched. Neil deGrasse Tyson's podcast, Star Talk and his books.

[@nentendomofo]

Love your junk journal ❤

[@airsicklowlander7756]

I struggled with memorizing multiplication tables as an eight-year-old. My teacher had a poster in the back of the classroom with all of the kids’ names on it. Whenever someone got 100% on our weekly multiplication tests, they’d get a sticker next to their name. By the end of the year everyone had like 10-15ish stickers next to their name except for me who maybe had 1 or 2. It was humiliating. It was an always-present public shaming. I distinctly remember one day when some older kids came into our classroom and started laughing at how that one kid with no stickers must be stupid. It gave me life-long self-esteem issues that I didn’t identify and start to address until my late 20’s. Math permanently traumatized me.

[@phillipphil1615]

I feel you having been through this humiliation pre 1968 school in France was brutal.

[@Blastgun1]

It wasn't math that traumatized you friend, I mean that in the most respectful way.

[@user-sx9hq7qwert]

Yep, I agree. Stickers r stupid. They r a failed idea that needs to be left behind on the trash heap of human history, like so much of "education", which is really just factory training.

[@heronimousbrapson863]

I had a similar experience. I still can't understand why some teachers in the past felt that humiliation was motivational.

[@user-sx9hq7qwert]

Deleted comment. How many times will YT delete this, I wonder? Yep, I agree. Stickers r ------. They r a failed idea that needs to be left behind on the trash heap of human history, like so much of "education", which is really just factory training.

[@only_trades]

My most genuine thanks to you. 🙏🏾

[@MewinMoose]

Tibees 🗣️🗣️🗣️, glad to see a new vid

[@Imassin]

Good to see you Tibees , I really missing you 💌

[@me0101001000]

"There are no wrong answers, but there are bad answers" -a my former Math professor of mine. His attitude to math was, "You can tell me 2+2=5. You just have to prove it." I really liked his approach.

[@penultimateh766]

Somebody who obviously needs to repeat kindergarten.

[@goldowolabi7377]

I would really like to meet that type of teacher one day

[@protocol6]

2+2=5 for large values of 2 and small values of 5. All you have to do is accept the specified precision as a rule, as is common in engineering and empirical science. Barring any statement to the contrary, you can assume all values are +/- half of the specified precision.

[@nsh1772]

My Physics teacher is like that lol

[@BooBaddyBig]

2+2=5 modulo 1, and that's not even a joke.

[@peterbonnema8913]

Please make more video's of this kind. These are my favorites :D

[@t1sk1jukka]

My brain completely shuts down when there is anything math related and didn’t understand a thing from this video lol. I’d want to try relearn from the very basics with my own pace but sadly there isn’t much material on the internet in my native language and definity not going to use a plagiarism machine to be my tutor. I’m an artist who can draw very well but i didn’t go to an art school. My goal was to learn to draw realistic animals and that’s what i did learn on my own

[@till8413]

what is your native language? I would recommend wikipedia - even if it is not the most reliably source for the humanities, it is very good for STEM topics especially maths

[@pingupenguin2474]

Try Khan Academy, either on youtube or the app. Lessons are in English but he takes it slowly and gently, explaining every step very simply. If you understand this message you will understand him. Turn on transcript if watching on youtube, on your phone, to help with translation of spoken words if necesary.

[@TheGuruNetOn]

Try the books by Morris Kline "Mathematics for Non Mathematicians" and if you like to proceed from there, "Mathematics and the Physical World".

[@the-trusteeship]

Thank you! This is spot on!

[@ianoliverbailey6545]

I was fine until the square with the bit missing, then I was lost...

[@gcewing]

She's solving a quadratic equation by completing the square, but doing it geometrically.

[@shortlessonshardquestions8105]

That was a very charming and creative way to show how the golden ratio comes from the geometry of the pentagon. Thank you for putting that together so nicely and simply.

[@jonathanndawula8584]

Math teachers were always mean and violent in Africa, always telling me how i can't do math, attitude sunk to 0%, i started loving math in college, when i met a teacher who was kind and made me see the beauty of the subject, now am a software engineer and math is my best friend

[@portreemathstutor]

First Steps for Problem Solvers is the best maths book. I don't like badly defined questions. I like questions to be unambiguous but puzzling and with solutions so that you know if you have the right answer. It is great when children say "That was a good one." and show the question to their friends.

[@AlbanAwan]

Aaaaaand it’s out of print

[@AntonKochkov]

Another great book that is inspiring is the Concepts of Modern Mathematics (though it's quite old by now, not a "modern" anymore - it's from 1975) by Ian Stewart. I wish there were more books like this.

[@gwang3103]

Bach's music actually seemed to instill in me a certain liking for math when I was young. The electronic performances of his Brandenburg Concertos by Carlos and Folkman just evoked in my mind fantasies of equations dancing around. :)

[@vortexlegend101]

3:58 I would have just used (5-2)180 /5 to find the internal angles to be 108 degrees, so the top triangle has a top angle 108 degrees so the other 2 are 36 degrees. splitting the top triangle in half gives a right triangle with a hypotenuse of 1 (the side length), and a base of half the diagonal, so the diagonal = 2 x 1 x cos(36 degrees) = Phi

[@BryanLu0]

How do you get cos(36) though?

[@thejimmymeister]

How do you get to your starting point of (5-2)180/5? Is it a formula you've been handed down or something you've discovered yourself and understand the proof of?

[@kidsteach938]

I taught my children from "Singapore Math" up to grade eight, and from then on, from "Elementary Algebra" by Wade Ellis and Denny Burzynski. Two are already in med school.

[@MadaraUchihaSecondRikudo]

I always looked at math as an extension of my logic and common sense. It gives me tools to formulate an argument about reality, and test if that argument holds. All of the techniques you learn at school are tools to that effect, and knowing how to use them correctly is important, but knowing when to spot when to use which tool when encountering a problem is equally as important and not as well taught.

[@TheGuruNetOn]

What, How are taught in school. Not Why, When and Where.

[@JayakrishnanNairOmana]

You have the voice of an angel. You should do ASMR videos narrating scientific audiobooks.

[@sampan04]

I recently decided to swap from Aerospace to Maths as my degree, hope its fun

[@marshalmcdonald7476]

It will be.

[@sonypony69420]

I thought about switching to a math major too. Fascinating job & good pay, but I have a prosthetic & my dream is to be a Certified Prosthetist-Orthotist (CPO)

[@boogie.devera]

It’s heaps of fun

[@52flyingbicycles]

If math was your favorite part of engineering, it probably will be. It’s going to be a lot of very abstract ideas and proofs.

[@JesusPlsSaveMe]

​Where are you going after you die? What happens next? Have you ever thought about that? Repent today and give your life to Jesus Christ to obtain eternal salvation. Tomorrow may be too late my brethen😢. Hebrews 9:27 says "And as it is appointed unto man once to die, but after that the judgement

[@superball2452]

To me, the essential tool to understand a certain mathematical concept became to find out what question it answers: to which question is this definition the right answer? How could one come up with these definitions? Compactness for example answers the question of generalizing minima and maxima results of continuous functions. these definitions don't fall from clear sky, but are the results of many smart people trying to pin down certain phaenomena they observed while practicing mathematics. If one follows their footsteps, and finds the question that is answered by this mathematical concept, they become natural.

[@rhodaborrocks-dy3fb]

In the UK we call it maths…because we do it more than once😎

[@Ken-pi7qk]

She sounds Australian and Australians say “maths” so I’m surprised to hear her say “math”. Perhaps she’s living in the US or directing her video the US market?

[@astrocslopes863]

Toby is from New Zealand

[@brianvernon7754]

just try to get it right the first time. no need to keep swinging the hammer if you hit the nail.

[@robert-wr6md]

@@brianvernon7754 Might not stay in place with one hit.

[@waterhead001]

I was a poor student, and I struggled with math. i had a math teacher in the 5th grade who was cruel to me. She would call on me knowing I did not know the answer, I think she did this to embarrass me. she would yell at me and when I would ask a question, she would talk to me like I was stupid. I had a Geometry teacher in High School, most of her students were struggling, she used her class as an instrument to throttle us. There is no such thing as a bad student only bad teachers.

[@johannesvanderhorst9778]

Was it intended that the chosen name of that variable, "flower", turns out to have a value that appears often in flowers?

[@JanWnogu]

Yes, I think it is intentional. I used a similar technique: denoting an operation in abstract algebra by some funny symbol instead of + or * in order not to make assumptions about it's properties (e.g. commutativity), because seeing a "+" you may intuitively rearrange the terms which is obviously wrong is the operation is not commutative. Drawing e.g. a flower makes you much more aware of the caveats.

[@siddhubhai2508]

I failed every year in mathematics in school from 6th to 9th, thank you to the school I was in which didn't remove me due to the principal was my father's close friend, in the 10th I was about to fail, but only passed with 1 extra number, one number was left to fail in boards and waste my one year, after that I'm in 11th right know, even this year studied nothing in maths due to interest in coding, I have forgotten almost everything what I learned, but now its time to start myself from beginning and to reach graduate level maths till this year ends, I know my goal is ambitious but nothing is there which humans can't achieve, what matters is your determination and dessiplean, I'm starting and going to change my maths forever. Thank you for reading my comment, support me if you can, and if you don't want to, then also fine I won't stop now.

'The Pleasures of Counting' is the best math book.

[@BlueGiant69202]

No, Anno's Three Little Pigs is the best math book. 😊

[@Jancel705]

@@BlueGiant69202 nah there was this one section in a nursery rhyme book called "10 little ducks", that was the best as it shows that addition is the opposite of subtraction.

[@Ph4n_t0m]

7:52 onwards : *warning!* misleading wording. It is not the case that 'if you take a bit off the top and stick it on the side [...] its length will actually be the average of the first two sides..." . It is only if you _make_ it so --- i.e. if you deliberately choose to cut off 'enough' of the rectangle such that the new length is the average of both, then the new square's sides will be the average of the two rectangle's sides. By making it seem as if any amount arbitrarily chosen to be cut off somehow 'magically' turns out to give you (i.e. 'will actually be') a square (ok, even with the 'bite cut out of it') whose sides are now the average of the two initial sides, you risk setting your audience's heads spinning as mine did for a good 15 minutes... By deliberately choosing to make a square whose sides are the average of the two rectangle's sides, you are choosing a very convenient relationship that does indeed help in determining the missing side's length (flower).

[@zmkdco8956]

Yes, and we also assumed that the sides of two different triangles were the same

[@PurpleYoshiEgg]

they aren't making it seem like any arbitrary amount cut off will be a square, they are making it such that the cut off amount will produce a square minus a smaller square. i think you're reading too much into it to find any nit to pick.

[@luchesartomov]

The role of the teacher is to inspire and lead and this is irreplaceable

[@howeworth]

9:43 The golden ratio is not the ratio between two succesive Fibonacci numbers, but the limit of the ratio between two succesive Fibonacci numbers.

[@MT.2012]

There are two inherent problems with mathematics: 1. It requires high mental effort, which not everyone is willing to do. 2. Largely, extensive math is not required in everyday life. Most people can very well get along with the four simple operations and possibly a knowledge of fractions.

[@railspony]

And they may only need to understand them well enough to enter them into a calculator. Or, as the video foreshadows, speak the problem as a question to their digital assistant.

[@elijaprice]

Tbh this video was worth it just for the way the lady says the word 'measurement' in her accent.

[@WritingOnCDs]

The reason I stay away from math is because I don’t even understand this lol

[@fran6b]

🤣😅

[@leagarner3675]

Honestly, math is fun if you like things that make sense and show beautiful things. The problem is, we never get to get to that..... It's always about timed scores...bleh..

[@live4gears40]

You should check out the book on audible. it’s really nice plus it’s free.

[@redlight3932]

its easy your teachers didnt care if you actually know something or not

[@witokija]

nothings stopping you from learning

[@kokomanation]

I was really excellent at math I always had problem with solving very difficult classical mechanics based physics problems I wish there was a book for that

[@CashueTM]

Great video as always Tibees.

[@Greymantle77]

Wonderful video! Absolutely positively wonderful- thank you!!❤😊

[@TeknoMage13]

This may be a side effect of my autism, but I find this is far less intuitive than normal math.

[@no.6377]

Calling it flower just makes me keep thinking of flowers, instead of calculation or logic.

[@InvaderZed]

Thats kind of the point...art isn't intuitive, life isn't intuitive. The whole idea is that math is only a set of tools to apply to your own imagination of the abstract ideas of reality. Being lost in the sea of meaning and finding your way is the journey you're meant to be on. The computerized way we teach the parts of math now doesn't actually let you think for yourself.

[@vsm1456]

what was "not normal" in the video? the flower instead of a letter?

[@ephyrian]

I got interested in math because Vi Hart, she talks about math in such a similar way, telling engaging stories by exploring mathematic principles. Still not great with math but I'm so much more willing to engage in math because of learning how to approach it from this perspective, really takes the sting out of making mistakes haha.

[@vibingshroom]

ChatGPT has a history of getting basic addition wrong, I don't think asking it for maths tutoring is a good idea.

[@gloriakalengelayi8294]

Why don’t you use the tutor me side of it instead. So far it’s been good to me. I also use the universal primer.

[@3than0lvideos22]

u dont use it for answers you just ask it to make study plan and schedules

[@starplatinum727]

It is improving now

[@FireyDeath4]

You should use it along with WolframAlpha or something. It's kind of good at explaining and manipulating concepts, but it can't handle numbers since it can't learn and apply formal systems and there are multiple groups of digits that have been tokenised without having examples of every possible manipulation in the training data

[@bhargavsai8014]

it handles things much better now imo

[@michaelcornish2299]

I love the connection you make between doing maths and reading fantasy. Wonderfully true but then again I like what maths can tell us about the world and the same can also be true for fantasy.

[@bobedwards8896]

i have been looking for something like this forever. I appreciate u talking about it. I actually like math and was pursue engineering (which I really wanted to do) but my failure at math (possibly because i was never taught it well) made me change my major.

[@drpkmath12345]

Very nice video Toby! Glad you made this video😊

[@EricLeePiano]

as a math/science educator and self-taught musician, that essay in the beginning spoke to me on a personal level lol

[@jetboyracing]

Thank you for this well thought out video. As an aerospace engineer turned math teacher, I can appreciate this approach. This question going g forward is how can we implement this affectively in a classroom.

[@tamirlyn]

I don't think the problem is the mathematics, nor how it is taught. If you wrote 'a' instead of 'flower', nobody would get any more or less confused, it would just be simpler to write. From my experience, the problem with school children and mathematics is twofold: 1. missing classes (even if they're present, they might not be paying attention) - which will make them unable to follow future classes, so they're out right there, and 2. lack of intelligence - I learned the hard way that people genuinely find difficult some things you and I might take for granted. For these people to catch up, you'd need to bring the level of all other children down with them, or make a separate course. How do we tell which are the 'stupid' ones? We can't do this easily. But we can do what we're doing. The dumb ones are among those that failed. What's the percentage of dumb compared to those that dropped off the tracks and couldn't follow anymore? Hard to say. I had a high-school student supposed to solve quadratic equations and the like, solve 11*7 by writing seven times "11" one under the other, to sum it up. That's the procedure she learned. I didn't get the impression she lacked intelligence. She just never learned how to multiply, and good luck learning the rest... On the matter of asking for proofs over asking them to follow procedure - following procedure requires less intelligence (and is suitable for many jobs) - more of them will manage this, than proving things. When asked to prove something, it's these tasks kids hated the most. Can proofs be beautiful? Definitely, for us. But it just might not be for everyone.

[@sprout8426]

nailed it

[@jimf2525]

Things you’ve said in this video are important. Great job.