Reddit reviews Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States

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'm surprised Knowing and Teaching Elementary Mathematics by Liping Ma wasn't mentioned, as she did a pretty good job at documenting this.

OK, confession time. I hated math from first grade on. I had a first grade teacher who took a dislike to me and stuck me out in the hall because I didn't understand how to add above ten. The humiliation was horrible and I suppose that set the pattern for the rest of my years in school, that I "wasn't good at math." As a sophomore in high school I was put in a remedial class, dropped out of chemistry (I was fairly good at science in general, but the math stopped me cold).

I didn't really have anything else to do with learning math until I was in my forties and took some real estate and property management classes at a technical college. It turned out that math is used in real estate all the time, from calculating mortgages, square footage of properties, figuring out taxes, and so on, and though I decided not to go into it as a profession I learned that I was actually pretty good at doing the math after all since I was motivated to learn the subjects.

Flash forward to the present. I'm just finishing the school year home schooling my eight year old daughter. My wife had been doing it but I took over in 3rd grade. We started the year using the Saxon math course for 3rd grade, but my wife had had so much push back from my daughter the previous year that we went through some horrible battles just trying to pick up from where she left off. Finally I said fine, we'll concentrate on practical, real life application kinds of things. Measuring. Using cash in a business - making change, tallying receipts, and so on. We used a chalkboard and I had her work out problems on it. She can now do addition; subtraction; multiplication; division (including long division of large numbers); adding, subtracting, multiplying and dividing using fractions and decimals, measuring weight, volume, and distance, do squares and square roots. I occasionally check our work on a calculator but mostly we check it against itself, and she never uses one. Frankly I have been learning or relearning a lot of this a few steps ahead of her, and I improvise a LOT, trying to explain things however I can do it. Sometimes I let her be the teacher and I play the dumb student who just can't seem to get it.

Interestingly, there is a book called Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States in which the author, who is Chinese, finds through data analysis that most US math teachers actually don't understand elementary math principles very well and consequently don't know how to teach the concepts. One of the observations was US teachers teach rote formulas without actually being able to explain mathematically what is being done or why you'd want to do it in the first place, which would go a long way towards explaining why US schoolchildren don't do well compared to students from other countries.

• Knowing and Teaching Elementary Mathematics, by Liping Ma. Compares how a group of math teachers from China and a group of math teachers from the US think about concepts and teaching. This book really demonstrates how deeply you can think about even the most elementary mathematical topic.
  
  
• The Teaching Gap, by Stigler and Hiebert. Compares and contrasts teaching practices and teaching philosophies in the US, Germany, and Japan.

I used Liping Ma's Knowing and Teaching Elementary Mathematics when I homeschooled my kids and still refer to it in planning remediation activities as a high school math teacher in an alternative school now.

Where did you get this? The things I've read suggest that chinese classrooms focus more on understanding than US classrooms. e.g. this.