Japan has been well-known for its high achievement in Mathematics, particularly in the Trends in International Mathematics and Science Study (TIMSS).
I had been in Japan for the past one year and had been studying their curriculum, observing classes, as well as reading elementary school textbooks (yes, I can read and understand Japanese a bit.) Since some of the books have English translations, I was able to have a glimpse at how they develop mathematical concepts. I can say, that their textbooks are the best among all that I have read so far.
For teachers who want to take a look at how Japan develop mathematical concepts, I strongly recommend that you read the following Teaching Guides for the Japanese Course of Study (the term used for curriculum). You may not agree with me, but I believe that can learn a lot from these documents.
- Elementary School Teaching Guide for Japanese Course of Study
- Junior High School Teaching Guide for Japanese Course of Study
Another supplement to these readings is the book Mathematical Thinking: How to Develop it in the Classroom by Masami Isoda and Shigeo Katagiri. The book describes three perspectives of mathematical thinking: Mathematical Attitude (mindset), Mathematical Methods in general, and Mathematical Ideas with content. It also explains how to develop them in the classroom with illuminating examples. This book is a result of Lesson Studies of the past 50 years in Japan.
I am currently reading the book (almost finished) and I also strongly recommend it to all elementary school and high school mathematics teachers.