Adding to the Numerator and Denominator (Or ode to a Mathematician's Lament)

Paul Lockhart’s A Mathematician’s Lament:

Joey learns to play guitar:

[BrianSpurrier]

This is actually a god way to introduce limits. You show your example of a fraction going up to one, but then you show what happens when you start with a fraction greater than one, then show how it is perfectly stable starting with one

[TypoKnig]

I love how your video is supportive of students. You aren't saying "you're wrong!", you are saying "let's explore that direction and see whether it helps." You point out flaws with standard teaching: "Do the same thing to top and bottom" is not as useful guidance as "multiply by a form of 1 that helps you." Great job! I hope to see more videos like this one, about other common misconceptions in math learning.

[pamdemonia]

That was lovely! As someone who was "good at math" in high school, I was very very lucky to have a teacher for pre-calc and calculus who also got the beauty part. So much so that she both actually asked the people who always had their hands up first the magical question "why?" and allowed me and a friend of mine to chat quietly in the back, knowing we were actually talking about math (fractals, mainly. James Gleick's book "Chaos" had just come out and we were entranced.) Her name was Sue Farmer. She was amazing. I even went off to college as a math major, only to be disappointed by Calc 3 ( too many engineers in that class) and swept away by the prospect of film school. But I still love math, and still totally agree with your thesis! Thanks!

[lavan]

this was so nice. i'm a 12th grade student with only social science subjects (in my country we choose specific subjects in 11th grade and drop the rest). i hated maths till 10th grade (it's mandatory to study maths till 10th) but just as if i would no longer be forced to study the subject, i developed a newfound appreciation for its beauty. now i want to do maths just for my own enjoyment; reading Lockhart's Lament only made this conviction grow stronger. your video was a lovely starting point. please make more such videos!!! i really need guidance on how to do maths the way Lockhart explained; if you could make some videos/exercises to try on one's own, maybe some problems to try to solve/prove on one's own, that would be so so so helpful. thank you for this video :))

[laiceps]

This is a very nice video. The visuals are pleasant. The explanation is succinct but still meticulous on why things happen they way they do. It felt very professional and I wanted to see what else you have done but it seems you quite recently started. Because of that, I just wanted to say good work! If you decide to continue, I hope it does well.

[eFiddle]

I enjoyed the Joey Learn guitar clip, I can see the connection to what you said about math being taught. very beautiful ribbon in the end, subbed.

[MathEnthusiast]

cool video.
Seems interesting, I shall look into reading "A Mathematician's Lament"

[moskthinks]

Amazing video👏. This actually builds up to several important concepts.

First, calculus: if you have a fraction a/b and you add p on the top and q on the bottom n times, the fraction (a+np)/(b+nq) approaches p/q as n approaches infinity, so in a sense, adding like that is like a way of turning a/b into p/q.

Second, this type of addition actually has a name: Farey Addition, which comes up a lot in some important advanced math topics. Really sad how an idea avoided by school could have actually been so important, not just about only exploring an idea for the sake of exploring (although that's also very amazing).

Really amazing video. As a top math student in my school, I really despise how my classmates learn math, but it's just that there is not enough time, effort, or interest for exploring, just like what was discussed for art and music.

[mikecaetano]

Tell the kids that taking successive mediants of two fractions is one way to enumerate the positive rational numbers.

[MathsUnpluggedIndia]

Damn! Amazing I just read Lockhart's Mathematician's Lament and was searching something on the web about it and found u. Well I have myself thought about this fraction problem and 100s of such problems when introduced the very first time by my teachers in elementary and high school(of-course, i was scolded, "You Idiot! Stop wasting time on this BS, else this capitalistic world would eat you up")
In this one its like x/y=(x+a)/(y+a) (the one case where it might be true)
hmm
cross multiplying gives,
x(y+a)=y(x+a)
xy+ax=xy+ay
which gives x=y that snatches the ability of x/y of being a true fraction so thats why this wont be true for any fraction whatsoever.

Now this deduction is even more interesting bcuz I dont need to check this addition thing in trillions of fractions bcuz maths armed us with something called PROOFS. Proof is like this FINITE thing containing an argument which satisfied our souls for INFINITE. UUr thoughts?

Anyways loved ur video...keep going...

[Reddles]

I don't really like the comparison to music or art here. In art there it is hard to say that something is objectively wrong, so you can let people just mess around and if they come up with something they like then their opinion is just as valid as anyone critiquing them. So then art education becomes more about showing people new possibilities and/or teaching them better ways to get the final product they way they intended it.

But math is different, most possible equations you could write down are just flat out wrong. It can be interesting and educational to examine why exactly a particular thing is wrong, as you show in the video, but that only works because you already know the correct answer. There's a big difference between someone who already knows the math playing around with the consequences of doing something wrong, versus a student doing the thing wrong without realizing it. Of course we should try to encourage people to think outside the box, but you still need to spend most of your time inside the box proving that whatever new idea you have actually works.

I think it would be more accurate to compare math to language. Obviously it takes a lot of creativity to do something like tell a story or write poetry, but It usually isn't creative or interesting to misspell words or use bad grammar. And, in the rare case that that is done creatively, it is because the author knows the rules and is deliberately disobeying them for some reason.

[Tabu]

I recommend you denoise your recording and remove mouth noise and breathing.