As I did the last time I posted a reply about mathematics education, I read the original post and the comments here before posting this reply.

First things first. Mathematical notation is now universal. A student in China learns in middle school (junior high) most of the mathematical notation that an American is expected to learn by graduation from high school. I know a large number of Chinese people who have taken the GRE test for admission to United States graduate schools. All of them, even those pursuing graduate studies in humanities, deride the GRE math section as "junior high math," which it literally is in terms of the standard school curriculum in China. Not all people in China have access to schooling beyond junior high, but through junior high the instruction in mathematics is generally excellent, and the United States could learn from the methods of mathematics teaching used in schools in China.

http://stuff.mit.edu:8001/afs/athena/course/6/6.969/OldFiles...

http://www.ams.org/notices/199908/rev-howe.pdf

Second things second. There is indeed a distinction between doing the kinds of calculations in arithmetic that may be tested in elementary school, are often done by adults with electronic calculators, and may or may not be a practiced skill of professional mathematicians and the kind of mathematical reasoning that makes up university-level study of mathematics and quantitative sciences. But that is not to say that learning arithmetic is not important. W. Stephen Wilson, a professor of mathematics at Johns Hopkins University, surveyed mathematics researchers about a year after the webpage submitted here was written, and asked them to agree or disagree with the statement

"In order to succeed at freshmen mathematics at my college/university, it is important to have knowledge of and facility with basic arithmetic algorithms, e.g. multiplication, division, fractions, decimals, and algebra, (without having to rely on a calculator."

http://www.math.jhu.edu/~wsw/ED/list

His colleagues around the world unanimously agreed, answered him in terms such as

"I am shocked that there is any issue here. I absolutely agree with your statement."

"That it is even slightly in doubt is strong evidence of very distorted curriculum decisions. I do not know even one university-level teacher of mathematics who would disagree with it. I would be truly astonished to meet a person who disagrees."

The charming story about Kummer is one I tell my own students, but I also tell them that the mathematics I teach them (prealgebra mathematics, in a class for self-selected elementary-age pupils looking for a challenging mathematics course) is based on their doing their own calculations with their minds alone, or with pencil and paper, never with a calculator.

Other mathematicians point out that learning the long division algorithm is itself a basis for the development of mathematical understanding.

http://www.csun.edu/~vcmth00m/longdivision.pdf

The late mathematician W. W. Sawyer spent decades thinking about how to teach mathematics effectively.

http://www.marco-learningsystems.com/pages/sawyer/sawyer.htm

Back in 2004, when I joined the Art of Problem Solving forums, I chose the screen name "tokenadult" (which so annoys some people here, chosen there because many participants on the forums are much younger than I am), and also chose a tagline quotation from Sawyer:

"The proper thing for a parent to say is, 'I did badly at mathematics, but I had a very bad teacher. I wish I had had a good one.'" W. W. Sawyer, Vision in Elementary Mathematics (1964), page 5.

Sawyer didn't want parents to give their children an excuse for thinking "I don't have a head for mathematics." Instead, a learner can keep searching for an effective teacher, and learn more than at first seems possible. The curriculum expectations in much of the English-speaking world are meager. In both Singapore and Taiwan (and in some other countries), every seventh grader is expected to learn algebra, and a fair amount of geometry--even all of the below-average students. That is possible. Not everyone in those countries is brilliant in mathematics, but many, many people in those countries have a day-by-day correct understanding of mathematics that helps in their daily life activities. My wife received that kind of mathematical education back when Taiwan is wretchedly poor. Taiwan is no longer poor, in part because it has developed rapidly through its educated workforce.