> Contrary to the point made, we do teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.
For the author's analogy, music is not being taught like what you describe. In his analogy it's being taught as several years of learning to read and transcribe music, without listening to or performing it.
Taking this analogy back to the reality of math education, the first 6 or 7 years of the standard US math curriculum is dedicated to arithmetic. Hell, it takes 4 or 5 years (3rd or 4th grade) to get to long division. The notion of variables is covered some time in middle school (6th or 7th grade) with pre-algebra (a watered down version of algebra with simple algebraic statements) being commonly taught in 7th or 8th grade, and algebra proper only showing up for 8th or 9th graders. That means we only start approaching "real math" once the students reach 13 or 14 years old. And throughout this, it's rarely hinted at how this subject can be applied. Most of the real world examples are contrived, or simple enough that the students that get it don't realize its real potential because the solution to the "problem" is practically handed to them. Showing how the sum of the angles in polygons can be determined by the number of sides and [developing a formula] via induction is a college topic in the US. Showing the sum of the first n positive integers is `n * (n + 1) / 2' and how to arrive at that is shown in a freshman or sophomore discrete math course. Bored, smart students (like I was) will recreate the tools like induction and develop these things themselves, but most won't and will get to college thinking they're "good at math" and then fail horribly because they don't have the skill set for college mathematics, they don't realize what college mathematics entailed (so many jokes about my "modern algebra" textbooks, "We took that in 9th grade!").
The thing is, I can't think of another way to teach the curriculum.
Math is abstraction on top of abstraction. You start off with counting. Once you get counting down, you abstract it with addition (You've counted 5 things, and you want to add it to a group of 7 things) and subtraction (Count 5 things and take them away from a group of 7 things). Then you abstract addition with multiplication, and then abstract that with division. Once you've done that, you abstract all of arithmetic with algebra.
And, well, it goes from there. You need some abstraction of algebra to do trig, calculus, and geometry. And to be able to abstract it, you need to understand it. This is where the disconnect happens - you get kids who have "learned" everything up to calculus, but they don't actually understand what's going on. They just know the formulae and how to plug-and-chug.
How do you get these kids to understand? My dad would relentlessly quiz me on the concepts, and he was ruthless in making sure that I understood why the formula was used just as much as how it was used. Many kids just learn the latter, and when it comes to any sort of independent thought, they're fucked.
In any case, though, I think that the current curriculum is as good as it's going to get. You can teach these concepts in a horribly boring manner, or you can teach them in an engaging, interesting manner. Either way, you aren't going to learn calculus unless you understand algebra, and you aren't going to learn algebra unless you understand arithmetic.
> 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.
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?
It ends up being on its own. You get introduced to it during geometry, but I don't think that it really gets taught until college.
I took BC calculus as a junior, so we got left with about a month between the seniors' leaving and the end of the year. My teacher said, "Okay, we're gonna learn number theory." Oh fuck, that was hard. It was completely unlike anything that we'd done before, and it required the development of completely different skills. I wasn't bad at it, but I definitely wasn't good at it either.
I'm glad that I got a taste of it, as that thought process has helped me in countless situations, but you're definitely correct in that it doesn't fit with the rest of the traditional curriculum.
For the author's analogy, music is not being taught like what you describe. In his analogy it's being taught as several years of learning to read and transcribe music, without listening to or performing it.
Taking this analogy back to the reality of math education, the first 6 or 7 years of the standard US math curriculum is dedicated to arithmetic. Hell, it takes 4 or 5 years (3rd or 4th grade) to get to long division. The notion of variables is covered some time in middle school (6th or 7th grade) with pre-algebra (a watered down version of algebra with simple algebraic statements) being commonly taught in 7th or 8th grade, and algebra proper only showing up for 8th or 9th graders. That means we only start approaching "real math" once the students reach 13 or 14 years old. And throughout this, it's rarely hinted at how this subject can be applied. Most of the real world examples are contrived, or simple enough that the students that get it don't realize its real potential because the solution to the "problem" is practically handed to them. Showing how the sum of the angles in polygons can be determined by the number of sides and [developing a formula] via induction is a college topic in the US. Showing the sum of the first n positive integers is `n * (n + 1) / 2' and how to arrive at that is shown in a freshman or sophomore discrete math course. Bored, smart students (like I was) will recreate the tools like induction and develop these things themselves, but most won't and will get to college thinking they're "good at math" and then fail horribly because they don't have the skill set for college mathematics, they don't realize what college mathematics entailed (so many jokes about my "modern algebra" textbooks, "We took that in 9th grade!").
EDIT: Grammar.