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> 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 examp

> 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...
Jtsummers 13y ago
US math courses for high schoolers often goes along the following path: Algebra, Geometry, Algebra II (basic linear algebra, but often not discussed as such, and other concepts), Trigonometry, Precalculus (1/2 semester each of the last two) and then Calculus. Of course, that adds up to 5 years so unless a student took Algebra I in middle school (year 8) they won't complete the full sequence, leaving calculus for college (depending on major). The structure of formal proofs is taught in geometry. That's pretty much the only course where students are expected to prove theorems about the subject via proofs. In most of the other courses students are taught the formulas and techniques, axioms and theorems, but then apply them, rather than attempt to develop or prove them themselves. They're also, typically, not shown a proof of the theorems they learn, they're just told to accept them.
- weland 13y ago
  > The structure of formal proofs is taught in geometry. That's pretty much the only course where students are expected to prove theorems about the subject via proofs. In most of the other courses students are taught the formulas and techniques, axioms and theorems, but then apply them, rather than attempt to develop or prove them themselves. They're also, typically, not shown a proof of the theorems they learn, they're just told to accept them. Wow! I can't even imagine how that works. When I went to school (and that's in Europe, not Asia), the notions of theorem and proof were taught in the fifth year of school (out of 12 pre-college years -- 8 in elementary, 4 in high school). That was, indeed, as part of geometry. Starting from high school, however, pretty much no formula is given without prior demonstration. Subjects where they can't do that (like mechanics -- that's taught in Physics prior to pre-calculus and calculus) are a royal fuckup. You aren't usually expected to be able to reproduce the proofs for those formulae, but a) you're strongly encouraged to study them, because they help you with the exercises and b) it's usually implied that, if you can't remember the formula, but you can prove it, it's OK. For example, if you can't finish a problem in a test because you don't know the formula, but you obviously try to find it, the professor will sometimes give you full credit, depending on circumstances and following a short talk. I didn't even like math back then, save for geometry, but I reckon this was the most important thing I learned there. Unless you're dumber than a toaster, you're typically able to apply what you learned to everyday phenomenae after ten years of school; the bonus you get is also being able to reason rigorously and correctly, which is an important takeaway no matter what path you choose to follow.
6d0debc071 13y ago
When I took a maths GCSE, (UK qual,) most of the work was memorising equations and using them. You didn't prove them or understand how they worked or anything. Like teaching someone to program by memorising the functions in the standard library.

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...

Jtsummers 13y ago

US math courses for high schoolers often goes along the following path: Algebra, Geometry, Algebra II (basic linear algebra, but often not discussed as such, and other concepts), Trigonometry, Precalculus (1/2 semester each of the last two) and then Calculus. Of course, that adds up to 5 years so unless a student took Algebra I in middle school (year 8) they won't complete the full sequence, leaving calculus for college (depending on major). The structure of formal proofs is taught in geometry. That's pretty much the only course where students are expected to prove theorems about the subject via proofs. In most of the other courses students are taught the formulas and techniques, axioms and theorems, but then apply them, rather than attempt to develop or prove them themselves. They're also, typically, not shown a proof of the theorems they learn, they're just told to accept them.

weland 13y ago
> The structure of formal proofs is taught in geometry. That's pretty much the only course where students are expected to prove theorems about the subject via proofs. In most of the other courses students are taught the formulas and techniques, axioms and theorems, but then apply them, rather than attempt to develop or prove them themselves. They're also, typically, not shown a proof of the theorems they learn, they're just told to accept them. Wow! I can't even imagine how that works. When I went to school (and that's in Europe, not Asia), the notions of theorem and proof were taught in the fifth year of school (out of 12 pre-college years -- 8 in elementary, 4 in high school). That was, indeed, as part of geometry. Starting from high school, however, pretty much no formula is given without prior demonstration. Subjects where they can't do that (like mechanics -- that's taught in Physics prior to pre-calculus and calculus) are a royal fuckup. You aren't usually expected to be able to reproduce the proofs for those formulae, but a) you're strongly encouraged to study them, because they help you with the exercises and b) it's usually implied that, if you can't remember the formula, but you can prove it, it's OK. For example, if you can't finish a problem in a test because you don't know the formula, but you obviously try to find it, the professor will sometimes give you full credit, depending on circumstances and following a short talk. I didn't even like math back then, save for geometry, but I reckon this was the most important thing I learned there. Unless you're dumber than a toaster, you're typically able to apply what you learned to everyday phenomenae after ten years of school; the bonus you get is also being able to reason rigorously and correctly, which is an important takeaway no matter what path you choose to follow.

6d0debc071 13y ago

When I took a maths GCSE, (UK qual,) most of the work was memorising equations and using them. You didn't prove them or understand how they worked or anything. Like teaching someone to program by memorising the functions in the standard library.