> The thing is, I can't think of another way to teach the curriculum.

But there's already a rich literature of real-world results for better ways to teach mathematics. See for example Seymour Papert's "Mindstorms" as a starting point (and much has been done since it was written ~30 years ago).

All children already learn quite a lot of fairly deep mathematical intuitions. We just take them for granted because everybody learns them.

For example: conservation of volume, the concept of "integer", order independence of cardinality, projecting orientation onto other reference frames, the equivalence between ordinal and cardinal numbers.

Everybody learns these things because they're embedded in our environments, and we can learn them playfully as children. When we create environments that embed even richer concepts, children learn those concepts just as easily. This is the explicit design goal of LOGO, and the whole family of descendants it has inspired.

Teaching in this way requires a degree of freedom and play that normal schools generally don't tolerate, which is why these proven, powerful tools still haven't taken over the world.

I don't disagree with the order of teaching, in general, but with its pacing. Arithmetic shouldn't be given 6 (K-5) years. Algebra shouldn't be 3-4 years (6-8 or 6-9). We bore students with the same material slightly stepped up each year for years at a time until they get to high school. Then we try and hit the accelerator and make them jump from the most rudimentary concepts in arithmetic and algebra and get them through trigonemetry or calculus in 3 more years. The concepts of trig, calculus, probability and stats, linear algebra can all be taught earlier. Students are capable of this, but the curricula aren't designed around it.

I didn't even realize until college that Algebra II was linear algebra. The notations used in college linear algebra would've been difficult for me to grasp fully at the time, but they make solving those systems of equations so much easier. And learning the notation [in high school], getting to the courses in college they'd be far less intimidating. We have 13 years with students before college, plenty of time to introduce notation and higher order concepts slowly rather than dumping it on them freshman year of college.

EDIT: Clarification on time of something

A few weeks ago there was an article here on HN that suggested kindergarten students might do better learning an intuitive form of calculus and algebra before arithmetic. Yes, math is built out of layered abstractions, but we can rotate the entire conceptual space to use a different foundation and still get a complete picture in the end.

Kids deal with abstraction every day. The very idea of "color" and "number" are abstractions of concrete experience.

Teaching kids basic group theory is very possible. You can play games with shapes in the plane to learn about dihedral groups (without ever using those words). Graph theory, as the author says, is another avenue.

The problem is that what students are practicing isn't math, any more than running after the ball when you miss a swing in tennis is practicing tennis. And you improve at what you practice.

Where does, say, proof by induction fit into that linear set of progressing abstraction? Surely you don't need to understand calculus to understand inductive reasoning?