What do students do in other mathematical areas if they don't do proofs?
The book I already mentioned, Knowing and Teaching Elementary Mathematics, will give much of the sad story. A really sad description of how mathematics textbooks are written in the United States[1] and a contrasting description of what kinds of problems are found in Russian textbooks[2] fill in more details.
It was on my second stay overseas (1998-2001), that I became especially aware of differences in primary mathematics education. 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),[3] 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 book I already mentioned, Knowing and Teaching Elementary Mathematics, will give much of the sad story. A really sad description of how mathematics textbooks are written in the United States[1] and a contrasting description of what kinds of problems are found in Russian textbooks[2] fill in more details.
It was on my second stay overseas (1998-2001), that I became especially aware of differences in primary mathematics education. 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),[3] 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.
[1] http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_o...
[2] http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd...
[3] http://www.artofproblemsolving.com/Store/viewitem.php?item=p...