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Disclosure: I am a mathematics teacher. (I teach prealgebra to elementary-age pupils.) I am NOT a programmer. But I think the correct answer to the question is

Disclosure: I am a mathematics teacher. (I teach prealgebra to elementary-age pupils.) I am NOT a programmer. But I think the correct answer to the question is "it depends." In other words, even though you have specified a level of programming, "bog standard, journeymen, CRUD programming in Objective-C," and even though you have laid out the usual progression through the usual school mathematics topics in the English-speaking world, you haven't given enough information to match the levels. Some people program insightfully, or so I am told by people who program, and other people program nothing but kludges. Among mathematics learners, there are great variations in the levels of "profound understanding of fundamental mathematics" (PUFM) among people who are studying the "same" level of mathematics. It interested me that you mentioned neither number theory nor combinatorics among your list of secondary mathematics courses, because in some other countries, those courses would come well before calculus. In general, any reader of Hacker News could learn a lot about what is missing in the United States (or, more generally, in the English-speaking world) in elementary mathematics instruction by reading Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma.[1] There is a lot to basic mathematics that is far from easy, if deep understanding is what you are looking for. In the countries that excel over the United States in mathematics instruction, proof is part of every mathematics course, not just part of geometry, for example. [1] http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843 http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

thaumasiotes 13y ago
> In the countries that excel over the United States in mathematics instruction, proof is part of every mathematics course, not just part of geometry, for example. There's room to wonder about how much of a causative role this has, and how much it reflects consumer demand. When my sister started taking algebra, my parents asked me to help her, and I was careful to provide proofs of all the algebraic properties of exponents (e.g. x^m * x^n = x^{m+n}; x^0 = 1; like that). That's not usual in the US, and didn't feature in her textbooks or supplements, but I did it basically just because it was important to me.
cocoflunchy 13y ago
> In the countries that excel over the United States in mathematics instruction, proof is part of every mathematics course, not just part of geometry, for example. Can you expand on that? What do students do in other mathematical areas if they don't do proofs? I did all of my studies in France and I don't how much of a difference there is (though all my friends who have studied both in the US and in France found the maths classes a lot easier in the US).
- tokenadult 13y ago
  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. [1] http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_of_your_childs_math_textbook_you_should_be http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_o... [2] http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pdf http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd... [3] http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra http://www.artofproblemsolving.com/Store/viewitem.php?item=p...
lotharbot 13y ago
Under the new "Common Core" standards for mathematics in the US, there's a lot more emphasis on proof in elementary-school mathematics than there was before. The terminology is usually more along the lines of "explain your reasoning", but the process is definitely proof-oriented. (Whether teachers will be able to incorporate those standards into their teaching, and whether students can live up to them, remains to be seen.) Disclosure: I'm a full-time tutor for third and fourth grade mathematics in the inner city through the Denver Math Fellows program ( http://www.teachindenver.com/teachindenver/denver-fellows-program.html http://www.teachindenver.com/teachindenver/denver-fellows-pr... ).

thaumasiotes 13y ago

> In the countries that excel over the United States in mathematics instruction, proof is part of every mathematics course, not just part of geometry, for example. There's room to wonder about how much of a causative role this has, and how much it reflects consumer demand. When my sister started taking algebra, my parents asked me to help her, and I was careful to provide proofs of all the algebraic properties of exponents (e.g. x^m * x^n = x^{m+n}; x^0 = 1; like that). That's not usual in the US, and didn't feature in her textbooks or supplements, but I did it basically just because it was important to me.

cocoflunchy 13y ago

> In the countries that excel over the United States in mathematics instruction, proof is part of every mathematics course, not just part of geometry, for example. Can you expand on that? What do students do in other mathematical areas if they don't do proofs? I did all of my studies in France and I don't how much of a difference there is (though all my friends who have studied both in the US and in France found the maths classes a lot easier in the US).

tokenadult 13y ago
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. [1] http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_of_your_childs_math_textbook_you_should_be http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_o... [2] http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pdf http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd... [3] http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra http://www.artofproblemsolving.com/Store/viewitem.php?item=p...

lotharbot 13y ago

Under the new "Common Core" standards for mathematics in the US, there's a lot more emphasis on proof in elementary-school mathematics than there was before. The terminology is usually more along the lines of "explain your reasoning", but the process is definitely proof-oriented. (Whether teachers will be able to incorporate those standards into their teaching, and whether students can live up to them, remains to be seen.) Disclosure: I'm a full-time tutor for third and fourth grade mathematics in the inner city through the Denver Math Fellows program ( http://www.teachindenver.com/teachindenver/denver-fellows-program.html http://www.teachindenver.com/teachindenver/denver-fellows-pr... ).