I'm homeschooling my kid and have been using sample problems from both Russian and Singapore math. But I only have a single source for Russian math problems, which is a pdf of a paper by a Russian educator who taught in the US and then wrote a paper comparing the two educational systems.
It's interesting because he talks about how algebra is a big scary subject that is introduced late in the American system, while in Russia they start doing algebraic-type problems early in primary but they are solved with pictures. So when algebraic notation is introduced later it's just a next step to the types of problems that students have already been solving.
Also: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm Also: https://news.ycombinator.com/item?id=18360071
If you look at another of my response comments to my parent comment, I mention that I don't really know if the system could be replaced with something better, but I don't think standardized education is good for anyone when compared to non-standardized education.
While true, I just don't think a one-on-one education, however skilled the tutor might be, could possibly match the social development that comes as consequence of classroom education.
I admit that their ability to not embarrass themselves among their peers is greater than those who are homeschooled or from experimental private schools, and their traditionally-schooled biting, ironic sarcasm is unmatched among the more naive non-schooled population. Traditional schoolchildren are also much better at taking instructions, falling in line, acquiescing to the wishes of their superiors, and bearing assaults on decency and freedom without complaint.
I made it through 12 years of public school and am extraordinarily "well-socialized", and I am disgusted by what it took to make me that way. It's only now that I've been out of school for about a decade that I've finally started to turn into a human being.
I think we could get into an endless debate here; I would say that the classroom model is a symptom of a post-industrial society where everything must be compartmentalized, sorted, and certified by a governing authority, which thus destroys the human connection with all of human endeavor; you, and I apologize for putting words in your mouth, would likely say that we never could have gotten to the point of a post-industrial society where you and I could debate the merits of industrialized math education on the internet without the classroom. Assuming my projection of your argument is accurate, I'm not even sure you're wrong.
It's just that whatever they're teaching today is bad, and I think it's worse than not teaching it at all. People come out of school and college technically skilled but intellectually hobbled. I don't know if the system itself can fix that, or even if it should be fixed.
As a starting reference, I mention "the Seven Lesson Schoolteacher": http://www.uvm.edu/~rgriffin/GattoDumb.pdf
As a college instructor at a community college, you're teaching math to people who likely have spent their whole lives watching other kids easily ace exams that they struggled with, who have never spent any of their genuinely free time doing math, and who just want to be let out of school. They are products of the system.
1. Don't teach that math is a "ladder". Trigonometry is not "harder" than geometry, although many concepts in trigonometry are expressed as geometry. Calculus is not "harder" than algebra, although many concepts in calculus are indeed expressed as algebra. Teaching that math is a ladder allows people to believe that once they've covered the "basics" of a field of math (as if you could ever learn all of "algebra"), they shouldn't have to think about the basics again. Try telling a pianist that once they've learned a set of scales they don't need to study them any more, or try telling Richard Feynman that once he understood the standard atomic model there was no need to continue examining it in greater depth.
2. Don't teach that math was "discovered" in its current form. What most students think of when they think "math" is in fact western notation for patterns observed in reality, handed down by a bored teacher as universal law. A sheet full of squiggles isn't "math", it's just a set of the most-predictable and best-notated patterns made up by past mathematicians. "Math" is the process of discovering and refining those squiggles. The relationship between the length of the two perpendicular sides of a right triangle and the hypotenuse exists outside of human knowledge, but the Pythagorean Theorem is a only a certain method for expressing that relationship. The numbers 1 through 9 are one possible code for counting, but the quantities I through IIIIIIIIII and onward exist outside of the notation of Arabic numerals.
3. Don't immediately move on from concepts once a student has mastered them. Imagine if every time you figured out how to do your job correctly, your boss moved you to a completely new task requiring a completely new set of skills, giving you no time to enjoy applying your prior mastery. If you've just learned how to factor quadratic equations, why move on? Why not explore programs that factor equations? Why not dig up old exams and show how factoring would have solved earlier problems faster? This is always met with cries of "but they're supposed to use what they've learned to go on to even harder problems!" Sure, I agree, but how can you possibly enjoy any task if the only thing you can really expect is that even if you master your task, you'll be struggling again within the week? Would you join a rec basketball team if every time you started hitting three-pointers consistently, they moved the rim higher and farther away without giving you any time to feel what it's like to be good at basketball?
4. Don't teach that you need to know math because your grades have to be high. For nearly every profession available, grades are immediately forgotten as soon as you start receiving wages. Teach that you need to know math because the world is constructed and controlled by mathematicians and those acting on the advice of mathematicians. If you don't know math you'll be taken advantage of by those who do, whether it's in advertising, gambling, banking, medicine, insurance, politics, entertainment, engineering, programming, or any of the many other fields driven entirely by math.
Of course, not teaching those four lessons becomes difficult, because those lessons work in opposition to the central tenets of mandatory state education, which necessarily operates as a giant brainwashing factory working to justify its own existence. Modern ediucators spend most of their time trying as hard as they can to teach people that knowledge is best bestowed by authority and then proven through certificates, that various fields of inquiry exist separately from one another (for instance, that physics and biology can or should be studied independently of history, mathematics, grammar, semantics, or art), and that failure to live up to expectations must necessarily lead to shame and corrective action.
Don't send your kids to school if you can avoid it.