Profound Understanding of Fundamental Mathematics (PUFM) is a phrase coined by Liping Ma in her landmark book, Knowing and Teaching Elementary Mathematics, to describe the deep, broad, and thorough understanding exhibited by several of the Chinese teachers she interviewed.
The Chinese teachers with PUFM didn’t get it automatically. It grew over many years of teaching several levels of elementary math and of studying their textbooks and teaching materials. They met weekly in teaching research groups to learn from each other’s experience, to find multiple ways to solve problems, and to broaden their mathematical understanding.
More than eight years ago, a group of homeschooling friends started a Yahoo “teaching research group” to discuss math in hope of deepening our own understanding and learning to better help our students. We had a good time, but the busy-ness of everyday life eventually won out. The group has mostly disbanded, though the archives remain. Now I’d like to bring that study to my blog, bit by bit, updated with things I’ve learned in the years since.
Denise Gaskins' Let's Play Math 