My kids are homeschooled, and I'm in charge of teaching Math, because my wife is one of those "I suck at math" people. Her parents focused exclusively on memorization when they helped her with homework, and then she got a few bad teachers, so her mental models are completely off. She's getting better, but that's not really my point.

The math program I use with my kids is this one: http://www.defimath.ca/ecole-maison/ (in French)

It was developed by two mathematicians who actually studied in classrooms what worked on kids when teaching math. They've come up with a method where kids learn negative numbers and multiplication _before_ learning positional numeration (numbers greater than 10, with units, tens, hundreds, thousands, etc.). I've seen first-hand how these concepts just click in my daughter's mind, and how she often comes up with the new concept herself when you introduce the prerequisites in an order that makes sense to her.

They learn about negative numbers and multiplication before they learn to count past ten? Or they just learn counting without learning how to write the numbers?

The latter. My daughter actually learned it because we mixed different learning methods early on (we don't anymore, it confused her), but you have to admit that writing numbers greater than 10 kinda involves multiplication (by powers of ten, but still), so it makes sense to learn multiplication first. Negative numbers were seen while learning addition / subtraction (the concept was basically adding a negative number), which made a lot of sense too.

That sounds wonderful. My kids are about the right age to do this, and I'm curious, are there any English-language versions of this text?

Cheaper? I also don't know if that's necessarily true. If it's ineffective for many, is it a cheaper solution or just simply cheaper?

Personally, I think Khan Academy is one of the few forward-thinking organizations doing something right in education. It tries to skew learning towards a 'mastery-based' model which keeps kids from falling behind by not advancing them too early. Of course, teachers and parents still need to implement the usage of Khan Academy in this way for it to be effective.

>Further, socially-speaking, being with your actual peers is extremely valuable, obviously.

I would disagree... I think that there is huge value to being around a variety of people, and a strong negative value to only being around people in the same phase of life as yourself, especially when you are in the nasty phase. I think I made it through that part of my life because I had an after-school job where I fixed computers for a local office. I even enrolled in the vocational program at my highschool that let me out of school early to go to work (except every second Monday, when we learned how to spot shoplifters, short-change scams, and other hazards of the retail life.)

I think the number of people who felt good about their social lives, the things they did socially and the things, socially that were done to them in high school is... small. As far as I can tell, for most people, college is important because it is a kind of recovery from this, and prepares you for a workplace where conflict is muted, where yes, if you are good enough, you can still be an asshole, but where being an asshole has a pretty heavy cost that must be made up in other ways.

For me? Being expected to behave like an adult while being treated like an adult around a bunch of adults was amazing. It gave me a reason to keep going in high school, and when I came of age, I was all set to get a really nice job.

Not being argumentative -- is the current system actually cheaper? I'm not enormously familiar with Montessori, but is the student/teacher ratio vastly different, or are the various learning materials any more expensive over time than the books, computers, and reams of paper that traditional schools use?

I went to a public elementary school which had only mixed-grade classes (2–3 grades at a time), and it was just fine, and didn’t cost any more than having one grade per class with kids all precisely the same age.