What is mathematics and how can we change the teaching of it for the better?

I read Jo Boaler's book, based on the recommendations in this post. I was right with her until I read the chapter on Ability Grouping. Her arguments against AG assume that high level and low level students are eager to work together towards the same goal. She assumes all children and parents are motivated. Its just not so. I like her ideas of group problem solving instead of just rote learning, but there really needs to be a mix of both.

I have noticed that schools broadly cover many math concepts in a school year, without mastery of any. The math consultant idea and year round schedule are possible solutions to this problem.

Well said

I partly agree with energizer comments. It may not be fair to compare US to Asian country. But teacher in Asian country also faces students with broad capability. In my experience(I'm Asian), about 3rd grade some of students clearly starting to fall behind in math. Over US, teachers have advantages because they can just iignore these group. Education policy does not really interested in this group (they are not bright, who cares attitude), and parents of these students felt shameful, do not even think about raise their voice to teachers. I do not want to generalize this to all Asian country and this is more than 20 years ago. This is very sad situation and I'm glad my kids are not growing up under that circumstances.

However, in US I feel like things reversed. Need of bright kids(or sometime regular kids) may be ignored and they are not challenged enough, especially in math. If they are not challenged enough, math skill will not be improved and the worst part of it is they even do not know their math skill is not enough for some carriers (How do they know, if they got A in math in all of their K-12 years and meet the all the milestones).

And another problems is in reality, it does not matter whether comparison is fair or not, there is huge gap between in math performance in international students in US students. Trust me on this, I'm teaching Engineering in University. Whatever reason is after K-12 math education, math skills of our kids are not developed enough and this is not because they are less capable of, but because how US school teaches and VALUES math..

I was reading the information about mathematics being taught in other countries and the need of extending school hours and the school days. This is being discussed heavily right now in my district. As a teacher, I feel frustrated because we are comparing students in the U.S. to students in other countries. It is true that students in other countries have a more well-developed foundation in math due to the extensive time on task of learning math, however; we must remember that not all students are allowed to attend school in other countries. Only the students who are very bright with usually high IQ's attend school. Compared to U.S. schools where everyone attends school, thank goodness, because the laws demand it, we are asking teachers to do almost the impossible. Because not all students learn the same way, or at the same pace, U.S. teachers find teaching very difficult. In many cases we are continually dealing with issues such as single parent families, drug babies, ADD, ADHD, obsessive compulsive disorders, emotionally disturbed, handicapped, down sydrome, and the list goes on and on. Not only do we have to make an educational plan for every child within the classroom, but we also have to make sure they pass the yearly test because our jobs are on the line. It is no wonder students in other countries do better in testing areas because they have completely cut out most of the kids which most teachers find difficulty teaching in the first place. People may say whatever they want about our educational system in the U.S., but as far as I am concerned their is great teaching going on everyday across the country! Sadly, it is true we are lagging behind when it comes to outsourcing jobs in the U.S. to overseas workers. But let's face it, we are not comparing apples to apples and oranges to oranges.