Reddit mentions of Elementary and Middle School Mathematics: Teaching Developmentally

First, you might want to start with /r/matheducation. They’re actually experts in this subject.

You can read work by hundreds of experts in child psychology/development, pedagogy, the philosophy of mathematics, the intuitive/psychological foundations of mathematics, etc. Personally I’m a fan of Piaget, Bruner, Papert, and like-minded thinkers, who advocate a child-centered “constructivist” approach to education. But there are certainly respectable educators and researchers who favor a more structured and top-down approach.

If you want to read concretely about the differences between typical US instruction and Chinese instruction in the 1990s, read Liping Ma’s book Knowing and Teaching Elementary Mathematics

Or watch this video from a few years ago discussing the TIMSS study and criticizing Khan Academy.

Or to see what a particular group of young children could learn with some expert guidance, check out Zvonkin’s book.

You might have read Lockhart’s Lament. He provides an alternative way of teaching high school mathematics in his book Measurement.

I like this concise theory of mathematical learning. YMMV. Here’s a short essay by Minksy about why mathematics is hard to learn.

If you want lesson plans and curriculum guidance, look to the American NCTM, who have been making detailed materials available for decades. Also look up math circles (both online materials and physical groups meeting in your area).

You might like this book by Van de Walle about general elementary teaching, or this book by Lenchner about problem solving.

Many people seem to like the Singapore math books. Read about Singapore’s curriculum.

If you ask homeschooling parents in your area, you can probably find strong opinions about curricula. Just searching around the web, many keywords about elementary math books etc. seem to lead to homeschooling sites. (This makes some sense: they have some free time, like to write about their experiences and form online communities, and do more personal evaluation of curricula than schoolteachers can necessarily have time/political power to do.)

There are hundreds of available books of mathematical puzzles and games, dozens of different types of physical manipulatives, and thousands of books, papers, essays, etc. about how to organize, order, and teach students of every imaginable age and background

If you have a particular age group / level of prior preparation / desired set of topics in mind, there might be some more specific materials people can point to. Are we talking about 4-year-olds? 10-year-olds? High school olympiad preparation? Are you interested in basic arithmetic? Geometry? Algebra? Do you have 1 advanced student to teach? 50 students of varying skill levels?

I'd suggest this mathematics resource book by Van de Walle and colleagues if you really want to understand mathematical concepts. It is really well-written, easy to read, based on mathematics education research, and even fun!