I just read the first 6 pages of that pdf and had to stop. I'm just blown away that you think that pdf has any valuable insights.
All of the premises so far are incorrect. The idea that math is supposed to be this playful fun activity that people just kind of discover on their own, perhaps with little nudges, would lead to a society where almost nobody learns math because most people aren't that smart or creative. The premise that math is not useful to society is one of the dumbest things I've ever heard. The premise that math is not a tool for science and technology is maybe tied for that.
Those are obviously wrong so I won't elaborate, but I want to talk more about the music and painting allegory in the intro which seems to inform the author's view on how math "should" be. I don't know much about music (the same as the author, I'm guessing), but my wife spent a few years at a painting atelier learning classical realist painting, and let me tell you, it's absolutely not about "expressing yourself and your feelings and things like that—really way-out-there abstract stuff." Nope. You know why? Because that produces trash. It's what got us the modern art movement, which is a laughingstock to most people in this world. (If you genuinely find yourself swooning over the "expression" and "feelings" in a Mondrian painting of squares or the swooping colors and "eroticism" of a Georgia O'Keeffe flower painting, you need your head examined.)
So yeah, in your first 6 months to a year of learning painting it's pretty common to not pick up a paintbrush unless you're a prodigy. Learning to paint starts with learning to see and learning to measure. Then you go on to copying simple figures. Not drawing from life! No don't even think of that, it's a waste of your time and you'll learn bad habits that you need to unlearn later. You copy classical studies like the Bargue drawings. The first Bargue that you copy is probably an eye or a forearm made with just a few lines that the master threw down as a basic exercise for his students. It took him seconds to produce a general eye-like shape. But the student is going to spend hours copying that. You master the ability to have a drawing sitting next to your paper, and copying the lengths and angles and weights exactly, just by looking. Then later, you move the drawing you're copying a few feet away so that you have to adjust the size in your head. Notice we're not even close to starting to learn how to shade. And boy, actually painting with paints is a lonnnng way off still.
Does this kill the retarded exuberance of just throwing paint on paper, literally splashing it, and then looking at it and saying "OMG I totally expressed myself I'm so awesome I'm a special snowflake" -- yes, absolutely. That's the point. It's a craft that has to be mastered, and it's not easy. The fun part begins after that.
I don't think it's possible for a person today to reach the same artistic heights as the classical masters on their own. It's exceedingly rare to do so. All the classical masters spent years, usually from their early childhoods to adulthood, studying in someone else's studio, before making their own style.
It would be similar to finding someone who independently reinvented modern math. You may be thinking of Srinivasa Ramanujan here, but would he have reached the heights he did if he hadn't been given that initial formal training and been exposed to those advanced trig books? Who knows. But in any case, how many Srinivasas are there in the world at a given time?
Most people need to be trained to master something. The author disagrees derisively when he says:
"SIMPLICIO: I don’t think that’s very fair. Surely teaching methods have improved since then.
SALVIATI: You mean training methods. Teaching is a messy human relationship; it does not require a method. Or rather I should say, if you need a method you’re probably not a very good teacher"
A) What a jackass
B) Totally wrong
C) Sheer arrogance... I mean even Socrates had a method, who does this guy think he is?
But anyway... most people need training, and most people can be trained. It's a great combination. In painting, academic art proved that a larger number of people could paint incredibly well. Today many academic artists are unknown, as a result of the modern art movement. Do an image search for Bouguereau (who??) sometime. Nobody's heard of this guy in school. His career happened to peak at the end of the classical realism period and the academic art movement. The newer artists like Monet and Degas (who you probably have heard of) who were moving away from classical realism though, much like this author, that the old way was dumb and boring and took the joy out of painting and viewing paintings.
And also like this author... these artists were trained in the classical method at the time. And their paintings are infinitely better than the drivel that was produced 60 or 70 years after their "reforms" when entire generations of artists had been taught (not trained!) under the new methods.
I think it's a pretty common thing... people reach some great height with all the benefits of a system that they end up despising for some reason. Then they think they have the answer... do away with that system! Go for stupid crap like expression and feelings right away, get right to the fun part! Why bother with boring old stuff like memorization or learning to draw a straight line free-hand? That's "old" and "useless"!
And then it turns out, without that rigorous training, you end up with a bunch of idiots who don't know how to paint (or do math).
Luckily, the drudgery part of math is much shorter and much easier to learn than painting. Pretty much everybody can do it, even at a young age, and it's enormously fruitful to society, which is why we teach it in schools. But it IS drudgery, and it's required. It's hilarious that you think that paper is connected with identifying the problem in math education, or trying to fix it. That paper doesn't even... oh man I don't even have the words to describe what a fail that paper is.