One thing I'm often asked about, when it comes to teaching math & science, is why the Chinese do so much better at math than the US. There are a few things that they do differently than we do here but for the most part it is cultural. They truly believe it is no ok to say 'I'm just not good at math'. Perhaps this is why in 2011 40% of Chinese college graduates were in degrees of Math, Sciences & Engineering, compared to the 5-9% in the US.
*Math is taught only by specialists in China- even in elementary school. Whereas in the US elementary teachers teach everything. IN fact a poll a few years ago determined that many teachers felt uncomfortable teaching math. Especially since most teachers with a bachelors degree can have as little as 6-12 hours of mathematics education themselves.
*Environment -in China school rooms are clean and empty. Students stay in one room and teachers travel from room to room. This means no one teacher can decorate the room with pretty posters or eye catching things. In fact, until the last couple of years there was no area in a Chinese classroom that even promoted group discussion. THe focus is on what is being taught.
*Emotion -One major difference is teachers (and even HS mom's can be guilty) in the US pus for comfort and happiness in the Present. Whereas Chinese teachers feel that focusing on academics - developing a solid understanding of math and science, will allow for easy entry to college, and therefore translate into a comfortable job - thus happiness and comfort in the Future.
*Testing - Testing in China is done twice, and they are entrance exams - to get into Middleschool and to get into High School. You qualify, at the High School level to either go to a general education high school or to a trade school. They feel that focusing on entrance exams only will also help a student better prepare for entrance exams for college. Where as testing in the US can determine everything from a teachers pay, to the amount of funding given to schools. I think everyone (at least in all the research I've done) agrees standardized testing doesn't really give an accurate representation of the Knowledge a child has, or their true progress.
So, how do I as a child-led learner, reconcile my push for greater math & science understanding? Well, one thing is in the practical application of math & science. Kids are like little curious sponges. My kids want remote controlled helicopters or cars? Then they have to build them. They want a 3DS? Then I bought them a broken on and they had to learn to fix it. They wanted to build a time traveling machine in the back yard? Well they watched Dr. Who as well as a documentary on how the science of Dr. Who is being worked on by scientists today.
I've taught algebra (and even trig & calculus) to 10 y/o's by just putting the terminology into vocabulary they can understand. How many times in high school did one say or hear "When will I ever use this, really?" I know I did it a lot. Even though we don't use worksheets, and very few textbooks. There are may instances where we more resemble Chinese style of methodology in mathematics and sciences than we do US.
For more on Chinese mathematics check out the following: