lists essential knowledge that everyone should possess who desires to advance theoretical physics, and included in that knowledge is much mathematics. There is a whole book, The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose,
that is marketed as a book about physics but includes a huge section reviewing secondary school mathematics as an essential background to physics.
The blog post submitted here has a title that is an homage to the article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by Eugene Wigner in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
People who know physics have long been delighted to find in physics applications for the mathematics they learned in mathematics courses without a hint of how useful the mathematics would be. The blog post author, however, goes beyond that perspective to urge, "Let’s think of mathematics in the abstract. Mathematics, at its most basic, is a very simple set of very well-defined rules. The rules describe the behavior and interaction of certain completely imaginary objects. Upon these rules, mathematicians have built others." And that brings to mind Paul Halmos's article (with its intentionally provocative title, an example of Halmos's spicy style in expository articles about mathematics) "Applied Mathematics Is Bad Mathematics" Halmos, P. "Applied mathematics is bad mathematics." Mathematics tomorrow (1984).
Halmos claims that mathematics is interesting and beautiful whether or not it has an apparent application.
Other replies already posted to this submission have helpfully mentioned the issue of empirical tests of what method of teaching mathematics may best help young learners appreciate (and later apply) mathematics. I have been deeply interested in cross-national comparisons of educational practice since living overseas beginning in 1982. In those days, one way in which school systems in most countries outdid the United States school system, economic level of countries being comparable, was that an American could go to many different places and expect university graduates (and perhaps high school graduates as well) to have a working knowledge of English for communication about business or research. I still surprise Chinese visitors to the United States, in 2012, if I join in on their Chinese-language conversations. No one expects Americans to learn any language other than English. Elsewhere in the world, the public school system is tasked with imparting at least one foreign language (most often English) and indeed a second language of school instruction (as in Taiwan or in Singapore) that in my generation was not spoken in most pupils' homes, as well as all the usual primary and secondary school subjects. At a minimum, that's one way in which schools in most parts of the world take on a tougher task than the educational goals of United States schools.
It was on my second stay overseas (1998-2001), that I became especially aware of differences in primary mathematics education. I began using the excellent Primary Mathematics series from Singapore
for homeschooling my own children, and I browsed Chinese-language bookstores in Taiwan for popular books about mathematics as my oldest son expressed an avid interest in mathematics. I discovered that the textbooks used in Singapore, Taiwan (and some neighboring countries) are far better designed than mathematics textbooks in the United States. (During that same stay in Taiwan, I had access to the samples United States textbooks in the storeroom of a school for expatriates, but they were never of any use to my family. I pored over those and was appalled at how poorly designed those textbooks were.) I discovered that the mathematics gap between the United States and the top countries of the world was, if anything, deeper and wider than the second-language gap.
Now I put instructional methodologies to the test by teaching supplemental mathematics courses to elementary-age pupils willing to take on a prealgebra-level course at that age. My pupils' families come from multiple countries in Asia, Europe, Africa, and the Caribbean Islands. (Oh, families from all over the United States also enroll in my classes. See my user profile for more specifics.) Simply by benefit of a better-designed set of instructional materials (formerly English translations of Russian textbooks, with reference to the Singapore textbooks, and now the Prealgebra textbook from the Art of Problem Solving),
the pupils in my classes can make big jumps in mathematics level (as verified by various standardized tests they take in their schools of regular enrollment, and by their participation in the AMC mathematics tests) and gains in confidence and delight in solving unfamiliar problems. More schools in the United States could do this, if only they would. The experience of Singapore shows that a rethinking of the entire national education system is desirable for best results,
but an immediate implementation of the best English-language textbooks, rarely used in United States schools, would be one helpful way to start improving mathematics instruction in the United States.
The blog post author begins his post with "In American schools, mathematics is taught as a dark art. Learn these sacred methods and you will become master of the ancient symbols. You must memorize the techniques to our satisfaction or your performance on the state standardized exams will be so poor that they will be forced to lower the passing grades." This implicitly mentions another difference between United States schools and schools in countries with better performance: American teachers show a method and then expect students to repeat applying the method to very similar exercises, while teachers in high-performing countries show an open-ended problem first, and have the students grapple with how to solve it and what method would be useful in related but not identical problems. From The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999): "Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x
"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77
A great video on the differences in teaching approaches can be found at "What if Khan Academy was made in Japan?"
with actual video clips from the TIMSS study of classroom practices in various countries.