Contest mathematics - Revision history

Vipul at 02:16, 6 March 2014

2014-03-06T02:16:04Z

@@ Line 3: / Line 3: @@
 Contest math differs from [[high school math]] both in terms of the nature of topics covered and in terms of the methods of thinking used. It also differs from [[higher math]] in these respects.
-Specific contests discussed on this page are specific to the United States, but the considerations outlined are fairly general.
+See:
-For recommendations on ''how'' to prepare for contest mathematics, see our [[contest mathematics learning recommendations]].
+* [[Contest mathematics learning benefits]]
-==Learning mathematics better==
+* [[Contest mathematics learning recommendations]]

Vipul at 23:31, 22 February 2014

2014-02-22T23:31:28Z

← Older revision		Revision as of 23:31, 22 February 2014
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	Specific contests discussed on this page are specific to the United States, but the considerations outlined are fairly general.		Specific contests discussed on this page are specific to the United States, but the considerations outlined are fairly general.

			For recommendations on ''how'' to prepare for contest mathematics, see our [[contest mathematics learning recommendations]].
	==Learning mathematics better==		==Learning mathematics better==

Vipul: /* Developing problem-solving skills */

2014-02-22T23:31:02Z

Developing problem-solving skills

← Older revision		Revision as of 23:31, 22 February 2014
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	Some people have claimed that the lower levels of contest math preparation help increase one's ''potential'' to score highly on the SAT (quantitative section), even though realizing the potential requires preparation specific to the SAT. However, the statement has not been independently verified. Please also see the problem-solving skills section below.		Some people have claimed that the lower levels of contest math preparation help increase one's ''potential'' to score highly on the SAT (quantitative section), even though realizing the potential requires preparation specific to the SAT. However, the statement has not been independently verified. Please also see the problem-solving skills section below.

	===Developing problem-solving skills===		==Developing problem-solving skills==

	Contest math requires people to develop the skill of approaching a problem using a general collection of heuristics and a base of factual knowledge rather than by applying a pre-specified procedure mechanically. This enables the development of the general skills as well as the intellectual temperament to approach problems using a trial-and-error, heuristic-based approach.		Contest math requires people to develop the skill of approaching a problem using a general collection of heuristics and a base of factual knowledge rather than by applying a pre-specified procedure mechanically. This enables the development of the general skills as well as the intellectual temperament to approach problems using a trial-and-error, heuristic-based approach.

	The extent to which there exist ''general'' problem-solving skills (that transfer across domains) as well as the extent to which these skills can be ''trained'' through practice in one area is much debated. It is likely, however, that because school education focuses ''so little'' on the development of these skills, a decent effort at contest math preparation can significantly increase one's exposure to problem-solving skills. That being said, people who have already been exposed to situations that have called upon extensive use of their problem-solving skills may have less to gain from contest math preparation.		The extent to which there exist ''general'' problem-solving skills (that transfer across domains) as well as the extent to which these skills can be ''trained'' through practice in one area is much debated. It is likely, however, that because school education focuses ''so little'' on the development of these skills, a decent effort at contest math preparation can significantly increase one's exposure to problem-solving skills. That being said, people who have already been exposed to situations that have called upon extensive use of their problem-solving skills may have less to gain from contest math preparation.

Vipul: /* Developing problem-solving skills */

2014-02-22T23:30:47Z

Developing problem-solving skills

← Older revision		Revision as of 23:30, 22 February 2014
Line 40:		Line 40:
	Some people have claimed that the lower levels of contest math preparation help increase one's ''potential'' to score highly on the SAT (quantitative section), even though realizing the potential requires preparation specific to the SAT. However, the statement has not been independently verified. Please also see the problem-solving skills section below.		Some people have claimed that the lower levels of contest math preparation help increase one's ''potential'' to score highly on the SAT (quantitative section), even though realizing the potential requires preparation specific to the SAT. However, the statement has not been independently verified. Please also see the problem-solving skills section below.

	==Developing problem-solving skills==		===Developing problem-solving skills===

	Contest math requires people to develop the skill of approaching a problem using a general collection of heuristics and a base of factual knowledge rather than by applying a pre-specified procedure mechanically. This enables the development of the general skills as well as the intellectual temperament to approach problems using a trial-and-error, heuristic-based approach.		Contest math requires people to develop the skill of approaching a problem using a general collection of heuristics and a base of factual knowledge rather than by applying a pre-specified procedure mechanically. This enables the development of the general skills as well as the intellectual temperament to approach problems using a trial-and-error, heuristic-based approach.

	The extent to which there exist ''general'' problem-solving skills (that transfer across domains) as well as the extent to which these skills can be ''trained'' through practice in one area is much debated. It is likely, however, that because school education focuses ''so little'' on the development of these skills, a decent effort at contest math preparation can significantly increase one's exposure to problem-solving skills. That being said, people who have already been exposed to situations that have called upon extensive use of their problem-solving skills may have less to gain from contest math preparation.		The extent to which there exist ''general'' problem-solving skills (that transfer across domains) as well as the extent to which these skills can be ''trained'' through practice in one area is much debated. It is likely, however, that because school education focuses ''so little'' on the development of these skills, a decent effort at contest math preparation can significantly increase one's exposure to problem-solving skills. That being said, people who have already been exposed to situations that have called upon extensive use of their problem-solving skills may have less to gain from contest math preparation.

Vipul: Vipul moved page Contest math to Contest mathematics

2014-02-14T02:33:27Z

Vipul moved page Contest math to Contest mathematics

← Older revision	Revision as of 02:33, 14 February 2014
(No difference)

Vipul: /* Preparing for a mathematical or mathematics-based career */

2013-12-30T02:33:35Z

Preparing for a mathematical or mathematics-based career

← Older revision		Revision as of 02:33, 30 December 2013
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	===Preparing for a mathematical or mathematics-based career===		===Preparing for a mathematical or mathematics-based career===

			See also the Quora question [http://www.quora.com/Mathematics/How-relevant-have-you-found-your-contest-math-experience-to-math-study-and-research How relevant have you found your contest math experience to math study and research?]. The information below is quite similar to Vipul Naik's answer.

	Some parts of contest mathematics, despite their lack of direct overlap with school math and higher math, are very helpful in preparing for mathematics-based careers. A review of the relevance of different parts of contest math is below:		Some parts of contest mathematics, despite their lack of direct overlap with school math and higher math, are very helpful in preparing for mathematics-based careers. A review of the relevance of different parts of contest math is below:

Vipul: /* Preparing for a mathematical or mathematics-based career */

2013-12-28T18:24:48Z

Preparing for a mathematical or mathematics-based career

← Older revision		Revision as of 18:24, 28 December 2013
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	* Study thoroughly the "parts that are helpful" from the above table. In particular, study most of the underlying theory for number theory and combinatorics, and study relevant parts of algebra. Geometry is somewhat optional, and you should study it in depth if it fascinates you. Consider studying more about geometric transformations and their invariants instead of diving deep into geometric facts about triangles and circles, unless the latter interests you.		* Study thoroughly the "parts that are helpful" from the above table. In particular, study most of the underlying theory for number theory and combinatorics, and study relevant parts of algebra. Geometry is somewhat optional, and you should study it in depth if it fascinates you. Consider studying more about geometric transformations and their invariants instead of diving deep into geometric facts about triangles and circles, unless the latter interests you.
	* To hone your problem-solving skills, consider taking an Olympiad-focused mathematics class with the Art of Problem Solving. Based on how well you do and how interested you continue to be, you can decide whether to try for something bigger.		* To hone your problem-solving skills, consider taking an Olympiad-focused mathematics class with the Art of Problem Solving. Based on how well you do and how interested you continue to be, you can decide whether to try for something bigger.

			See also the blog post [http://lesswrong.com/lw/2v1/great_mathematicians_on_math_competitions_and/ Great Mathematicians on Math Competitions and "Genius"] that includes quotations of statements by many mathematicians regarding the value of contest math.

	===Preparation for the SAT===		===Preparation for the SAT===

Vipul: /* Learning mathematics better */

2013-12-28T05:53:28Z

Learning mathematics better

← Older revision		Revision as of 05:53, 28 December 2013
Line 31:		Line 31:
	* Study thoroughly the "parts that are helpful" from the above table. In particular, study most of the underlying theory for number theory and combinatorics, and study relevant parts of algebra. Geometry is somewhat optional, and you should study it in depth if it fascinates you. Consider studying more about geometric transformations and their invariants instead of diving deep into geometric facts about triangles and circles, unless the latter interests you.		* Study thoroughly the "parts that are helpful" from the above table. In particular, study most of the underlying theory for number theory and combinatorics, and study relevant parts of algebra. Geometry is somewhat optional, and you should study it in depth if it fascinates you. Consider studying more about geometric transformations and their invariants instead of diving deep into geometric facts about triangles and circles, unless the latter interests you.
	* To hone your problem-solving skills, consider taking an Olympiad-focused mathematics class with the Art of Problem Solving. Based on how well you do and how interested you continue to be, you can decide whether to try for something bigger.		* To hone your problem-solving skills, consider taking an Olympiad-focused mathematics class with the Art of Problem Solving. Based on how well you do and how interested you continue to be, you can decide whether to try for something bigger.

			===Preparation for the SAT===

			Some people have claimed that the lower levels of contest math preparation help increase one's ''potential'' to score highly on the SAT (quantitative section), even though realizing the potential requires preparation specific to the SAT. However, the statement has not been independently verified. Please also see the problem-solving skills section below.

			==Developing problem-solving skills==

			Contest math requires people to develop the skill of approaching a problem using a general collection of heuristics and a base of factual knowledge rather than by applying a pre-specified procedure mechanically. This enables the development of the general skills as well as the intellectual temperament to approach problems using a trial-and-error, heuristic-based approach.

			The extent to which there exist ''general'' problem-solving skills (that transfer across domains) as well as the extent to which these skills can be ''trained'' through practice in one area is much debated. It is likely, however, that because school education focuses ''so little'' on the development of these skills, a decent effort at contest math preparation can significantly increase one's exposure to problem-solving skills. That being said, people who have already been exposed to situations that have called upon extensive use of their problem-solving skills may have less to gain from contest math preparation.

Vipul: /* Preparing for a mathematical or mathematics-based career */

2013-12-28T05:27:57Z

Preparing for a mathematical or mathematics-based career

← Older revision		Revision as of 05:27, 28 December 2013
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	\| Algebra \|\| Polynomials, equation-solving, inequalities, functional equations \|\| Basic theory of polynomials, basic algebraic manipulation techniques, rough idea (not proficiency) of how to approach inequalities and functional equations \|\| Proficiency in equation-solving tricks<br>Proficiency in inequalities and functional equations		\| Algebra \|\| Polynomials, equation-solving, inequalities, functional equations \|\| Basic theory of polynomials, basic algebraic manipulation techniques, rough idea (not proficiency) of how to approach inequalities and functional equations \|\| Proficiency in equation-solving tricks<br>Proficiency in inequalities and functional equations
	\|-		\|-
	\| Geometry \|\| Planar Euclidean geometry, including many facts about triangles and circles<br>For some contests (not IMO) basics of coordinate geometry \|\| Understanding geometric transformations and geometric invariants can be helpful for geometry as seen in higher math. This is not a central topic of contest math, but is a supplementary topic that many students learn because it often provides shorter, more elegant solutions to contest math problems.<br>The intricacies of triangle geometry can build ~~the~~ appreciation of ~~structure~~ and ~~factual knowledge~~ combined with ~~technique~~. \|\| Most facts about triangle geometry and circles.		\| Geometry \|\| Planar Euclidean geometry, including many facts about triangles and circles<br>For some contests (not IMO) basics of coordinate geometry \|\| Understanding geometric transformations and geometric invariants can be helpful for geometry as seen in higher math. This is not a central topic of contest math, but is a supplementary topic that many students learn because it often provides shorter, more elegant solutions to contest math problems.<br>The intricacies of triangle geometry can build ''appreciation'' of how a large factual base can be structured in the mind and combined with techniques to solve problems, even though the actual facts of triangle geometry are not useful. \|\| Most facts about triangle geometry and circles.
	\|-		\|-
	\| Number theory \|\| Elementary number theory (not using any abstract algebra ideas explicitly), including congruences and Diophantine equations \|\| The basic theorems of number theory, as well as facility with manipulating congruences, are very important in abstract algebra and thereby in much of mathematics that grows out of abstract algebra. The basic idea of how to approaching Diophantine problems is worthwhile. \|\| Some of the techniques to solve Diophantine equations are artificial.		\| Number theory \|\| Elementary number theory (not using any abstract algebra ideas explicitly), including congruences and Diophantine equations \|\| The basic theorems of number theory, as well as facility with manipulating congruences, are very important in abstract algebra and thereby in much of mathematics that grows out of abstract algebra. The basic idea of how to approaching Diophantine problems is worthwhile. \|\| Some of the techniques to solve Diophantine equations are artificial.

Vipul at 05:23, 28 December 2013

2013-12-28T05:23:14Z

← Older revision		Revision as of 05:23, 28 December 2013
Line 4:		Line 4:

	Specific contests discussed on this page are specific to the United States, but the considerations outlined are fairly general.		Specific contests discussed on this page are specific to the United States, but the considerations outlined are fairly general.

			==Learning mathematics better==

			===Learning math for pleasure===

			Contest math can be a fun way of learning mathematical topics and techniques not covered in the school syllabus. However, it is not the only possible or even the optimal way of getting pleasure from learning mathematics. See our [[math reading recommendations]] for other ways people can expose themselves to interesting and challenging mathematics for fun and learning.

			===Preparing for a mathematical or mathematics-based career===

			Some parts of contest mathematics, despite their lack of direct overlap with school math and higher math, are very helpful in preparing for mathematics-based careers. A review of the relevance of different parts of contest math is below:

			{\| class="sortable" border="1"
			! Area !! Main subtopics !! Parts that are helpful in mathematics-based careers !! Parts that are not helpful
			\|-
			\| Algebra \|\| Polynomials, equation-solving, inequalities, functional equations \|\| Basic theory of polynomials, basic algebraic manipulation techniques, rough idea (not proficiency) of how to approach inequalities and functional equations \|\| Proficiency in equation-solving tricks<br>Proficiency in inequalities and functional equations
			\|-
			\| Geometry \|\| Planar Euclidean geometry, including many facts about triangles and circles<br>For some contests (not IMO) basics of coordinate geometry \|\| Understanding geometric transformations and geometric invariants can be helpful for geometry as seen in higher math. This is not a central topic of contest math, but is a supplementary topic that many students learn because it often provides shorter, more elegant solutions to contest math problems.<br>The intricacies of triangle geometry can build the appreciation of structure and factual knowledge combined with technique. \|\| Most facts about triangle geometry and circles.
			\|-
			\| Number theory \|\| Elementary number theory (not using any abstract algebra ideas explicitly), including congruences and Diophantine equations \|\| The basic theorems of number theory, as well as facility with manipulating congruences, are very important in abstract algebra and thereby in much of mathematics that grows out of abstract algebra. The basic idea of how to approaching Diophantine problems is worthwhile. \|\| Some of the techniques to solve Diophantine equations are artificial.
			\|-
			\| Combinatorics \|\| Counting rules (enumerative combinatorics) and existential combinatorics (e.g., Ramsey theory) \|\| Important because dealing with abstract mathematical structures often requires using abstract counting procedures even to get a sense of what's going on. \|\| Attaining proficiency in some of the clever combinatorial constructions may not be worthwhile.
			\|}

			We would generally recommend the following for people interested in potentially pursuing a mathematics-based career:

			* Study thoroughly the "parts that are helpful" from the above table. In particular, study most of the underlying theory for number theory and combinatorics, and study relevant parts of algebra. Geometry is somewhat optional, and you should study it in depth if it fascinates you. Consider studying more about geometric transformations and their invariants instead of diving deep into geometric facts about triangles and circles, unless the latter interests you.
			* To hone your problem-solving skills, consider taking an Olympiad-focused mathematics class with the Art of Problem Solving. Based on how well you do and how interested you continue to be, you can decide whether to try for something bigger.