*In my experience, it's because high school math is taught equally terribly. No... more terribly, because the subject matter is more complex. Useless busywork and rote memorization abound.*

See, on this point we actually agree. I was appalled at Physics 11 and 12 for example; once I hit first year calculus and all the stupid formulas we were applying and memorizing v=1/2at^2 for velocity of an accelerating object etc.. just fell out of simple calculus cases. But Organic Chemistry and balancing reactions, that needed to be exactly what it was.

The paper you linked had a musical example... and berated the fixation on music theory, and was a good read. But at the same time, theory is good too, and the history of music too. It is not bad to teach and test those, its bad to ONLY teach those.

But elementary school math? I'm just NOT seeing the issue you have. They are drawing things, and piling them up, and working with sequences, nearly everything they do at the beginning is based around patterns and symmetry. All the times tables are introduced gradually, and as sequences, and visually. The relationships established between numbers, grids of squares, piles of beads. It doesn't seem bad to me at all.

Yes, memorization of basic arithmetic facts kicks in grade 3 and 4 but I just can't get upset by that. Its a small but important piece. And even if they "fixed" the latter years education, I'm hard pressed to imagine a curriculum that wouldn't be facilitated by having single digit arithmetic as a basis skill to draw from. Just as I can't imagine a written language course that didn't require you to have at some early point memorized the alphabet and their canonical sounds. (Or in the case of a language like Mandarin, the basic set and the rules that govern the alphabet..)

Just as your document mocked painting in terms of theory, and rightly so, there is a need to be able to name colours taught alongside the freeform expression of fingerpainting. Does a child need to know that colour they smeared from here to there in a pleasing squiggle is blue to make that blue squiggle? No all they need is paint and imagination. But they still DO need to be taught that the color is blue to be able to communicate. And that has to be memorized. There is no deeper understanding of the names of colours -- you just have to remember which are called blue and which are called green, etc.

Your linked paper went into detail talking about the joy of discovering analytic geometry by drawing a rectangle around a triangle, but how would you teach this if your students hadn't previously memorized what a rectangle and triangle actually were? And how would you teach the names of shapes? They are occasionally descriptive... quadrilateral, triangle, parallelogram... but why is it canonically called a triangle and rarely a trilateral? And what the fuck is a rhombus or a trapezoid or a hexagon? And usually what is meant by a hexagon is a regular hexagon, god help the kid who tries to bisect an irregular hexagon into 6 equilateral triangles...

"A similar problem occurs when teachers or textbooks succumb to âoecutesyness.â This is

where, in an attempt to combat so-called âoemath anxietyâ (one of the panoply of diseases which

are actually caused by school), math is made to seem âoefriendly.â To help your students

memorize formulas for the area and circumference of a circle, for example, you might invent this

whole story about âoeMr. C,â who drives around âoeMrs. Aâ and tells her how nice his âoetwo pies

areâ (C = 2Ïr) and how her âoepies are squareâ (A = Ïr2) or some such nonsense"

Yikes. I've never seen something so banal in my own or my kids education. We can agree that's terrible. But I can also stipulate that my kids weren't exposed to it either... has anybody actually been taught that? Was it ever more than a failed experiment? Tried for a few years, found wanting, and then abandoned?

The upshot, in my opinion is that something like the area of a circle, just like my physics 11/12 formulas really SHOULDN'T be taught until after the kids have learned trigonometry, periodic functions, and calculus... because those are necessary to really understand the answer.

There's no reason to memorize the forumula though. Ever. And I'm not sure they are expected to now.

Your linked article also writes:

"Mathematics is the purest of the arts,"

I'd argue that philosophy (logic) is purer still. Mathematics itself is a construct of logic. (And for truly fun mind games, take meta-logic.)

-cheers