Learn mathematics well - Revision history

Vipul: /* Online math learning resources */

2022-12-27T22:37:22Z

Online math learning resources

← Older revision		Revision as of 22:37, 27 December 2022
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	* Wherever possible, try to acquire the conceptual intuition behind the steps. The texts or resources you use often do offer conceptual explanations, but these may not be highlighted because the authors expect that most students just want to master the algorithms and do not care about the explanations.		* Wherever possible, try to acquire the conceptual intuition behind the steps. The texts or resources you use often do offer conceptual explanations, but these may not be highlighted because the authors expect that most students just want to master the algorithms and do not care about the explanations.

	===Online ~~math~~ learning resources===		===Online mathematics learning resources===

	{{further\|[[Online ~~math~~ learning resources]]}}		{{further\|[[Online mathematics learning resources]]}}

	Note that websites often add new features and improve existing ones, so the information here may not be up-to-date.		Note that websites often add new features and improve existing ones, so the information here may not be up-to-date.
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	\| [[PatrickJMT]] \|\| [http://www.patrickjmt.com patrickjmt.com] \|\| Yes \|\| No \|\| No \|\| No		\| [[PatrickJMT]] \|\| [http://www.patrickjmt.com patrickjmt.com] \|\| Yes \|\| No \|\| No \|\| No
			\|-
			\| [[Edia]] \|\| [https://edia.app/ edia.app] \|\| Yes (as of January 2021; the videos are sourced from third parties) \|\| Yes (text explanation of each problem) \|\| Yes \|\| No (as of January 2021)
	\|}		\|}

Vipul: /* Online math learning resources */

2022-12-27T19:36:23Z

Online math learning resources

← Older revision		Revision as of 19:36, 27 December 2022
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	\| [[ALEKS]] \|\| [http://www.aleks.com www.aleks.com] \|\| No \|\| No \|\| Yes \|\| Yes, done at a considerable level of detail using knowledge spaces.		\| [[ALEKS]] \|\| [http://www.aleks.com www.aleks.com] \|\| No \|\| No \|\| Yes \|\| Yes, done at a considerable level of detail using knowledge spaces.
	\|-		\|-
	\| [[Khan Academy]] \|\| [http://www.khanacademy.org khanacademy.org]) \|\| Yes; comprehensive coverage but not high quality \|\| No \|\| Yes \|\| Yes, but this is a new feature and may not be very thorough.		\| [[Khan Academy]] \|\| [http://www.khanacademy.org khanacademy.org] \|\| Yes; comprehensive coverage but not high quality \|\| No \|\| Yes \|\| Yes, but this is a new feature and may not be very thorough.
	\|-		\|-
	\| [[Brilliant]] \|\| [http://www.brilliant.org brilliant.org] \|\| No \|\| No (with a few exceptions) \|\| Yes \|\| No (though they may be doing this soon)		\| [[Brilliant]] \|\| [http://www.brilliant.org brilliant.org] \|\| No \|\| No (with a few exceptions) \|\| Yes \|\| No (though they may be doing this soon)

Vipul: /* How can one learn math well? */

2014-03-02T22:21:22Z

How can one learn math well?

← Older revision		Revision as of 22:21, 2 March 2014
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	==How can one learn math well?==		==How can one learn math well?==

	There are some general [[good study habits]] that help with learning any subject well. There are also some general [[online learning resources]] ~~that help with all subjects~~. On this page, we describe some aspects of the strategies and resources that are specific to mathematics.		There are some general [[good study habits]] that help with learning any subject well. There are also some general [[online mathematics learning resources]]. On this page, we describe some aspects of the strategies and resources that are specific to mathematics.

	===Supplementing rote and immediate practice===		===Supplementing rote and immediate practice===

Vipul: /* Why learn math? */

2014-01-09T17:23:41Z

Why learn math?

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 ==Why learn math?==
 The following are some general reasons for learning mathematics well.
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 # '''Importance for college gateway examinations (such as SAT and ACT)''': A strong background in mathematics helps with the math (quantitative) part of the SAT, as well as with the SAT Math Subject Test. It also helps with the math part of the ACT if you choose to take that test. The situation is likely similar in many other countries.
 # '''General reasoning ability''': Mathematics, if understood well, can help with general reasoning.
 ==Psychological barriers to learning math well==

Vipul: Vipul moved page Learn math well to Learn mathematics well

2014-01-09T16:48:13Z

Vipul moved page Learn math well to Learn mathematics well

← Older revision	Revision as of 16:48, 9 January 2014
(No difference)

Vipul: /* Why learn math? */

2014-01-09T03:08:57Z

Why learn math?

← Older revision		Revision as of 03:08, 9 January 2014
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	# '''Hierarchical''': Later material in the curriculum depends heavily on earlier material (with a few important exceptions, such as many parts of geometry, plus advanced techniques within individual sections).		# '''Hierarchical''': Later material in the curriculum depends heavily on earlier material (with a few important exceptions, such as many parts of geometry, plus advanced techniques within individual sections).
	# '''Unidirectional transfer to many other domains''': Mathematics, at both the computational and conceptual level, gets used extensively in physics, chemistry, economics, and the social sciences. Mathematics, up to and including multivariable calculus and linear algebra , is particularly important for understanding statistics rigorously.		# '''Unidirectional transfer to many other domains''': Mathematics, at both the computational and conceptual level, gets used extensively in physics, chemistry, economics, and the social sciences. Mathematics, up to and including multivariable calculus and linear algebra, is particularly important for understanding statistics rigorously.
	# '''Importance for college gateway examinations (such as SAT and ACT)''': A strong background in mathematics helps with the math (quantitative) part of the SAT, as well as with the SAT Math Subject Test. It also helps with the math part of the ACT if you choose to take that test. The situation is likely similar in many other countries.		# '''Importance for college gateway examinations (such as SAT and ACT)''': A strong background in mathematics helps with the math (quantitative) part of the SAT, as well as with the SAT Math Subject Test. It also helps with the math part of the ACT if you choose to take that test. The situation is likely similar in many other countries.
	# '''General reasoning ability''': Mathematics, if understood well, can help with general reasoning.		# '''General reasoning ability''': Mathematics, if understood well, can help with general reasoning.

Vipul: /* Online math learning resources */

2013-11-25T06:46:05Z

Online math learning resources

← Older revision		Revision as of 06:46, 25 November 2013
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	===Online math learning resources===		===Online math learning resources===

	~~For more, see our~~ [[~~online~~ math learning resources]] ~~page.~~		{{further\|[[Online math learning resources]]}}

	Note that websites often add new features and improve existing ones, so the information here may not be up-to-date.		Note that websites often add new features and improve existing ones, so the information here may not be up-to-date.

Vipul: /* Hierarchical nature of the curriculum */

2013-11-01T23:42:23Z

Hierarchical nature of the curriculum

← Older revision		Revision as of 23:42, 1 November 2013
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	* Even the techniques in a part of a mathematics curriculum that do ''not'' get used later may be helpful for [[overlearning]] important techniques that are a part of the curriculum and get used a lot in later parts of the curriculum. For instance, the long division algorithm helps review key skills of subtraction, multiplication, number sense, estimation, and place value, and may be useful to master even if you never need to do long division by hand in the future.		* Even the techniques in a part of a mathematics curriculum that do ''not'' get used later may be helpful for [[overlearning]] important techniques that are a part of the curriculum and get used a lot in later parts of the curriculum. For instance, the long division algorithm helps review key skills of subtraction, multiplication, number sense, estimation, and place value, and may be useful to master even if you never need to do long division by hand in the future.

	One exception to the generally hierarchical nature of the curriculum is high school geometry. A bare minimum of geometry, including lines, circles, and angles, is essential for understanding coordinate geometry. However, many theorems about triangles and circles, are not used in the rest of mathematics. Geometry may be worthwhile as a way of improving one's general reasoning and proof discovery skills, but it is not the only way to develop such skills, and it is not ''specifically'' important to master material about triangles and circles. That said, many people may deeply enjoy learning geometry and it may ignite their passion for mathematics, so this should not be construed as a recommendation against learning geometry. The point here is simply that the general arguments for learning math well do not apply with the same force to geometry.		One exception to the generally hierarchical nature of the curriculum is high school geometry. A bare minimum of geometry, including lines, circles, and angles, is essential for understanding coordinate geometry. However, many theorems about triangles and circles are not used in the rest of mathematics. Geometry may be worthwhile as a way of improving one's general reasoning and proof discovery skills, but it is not the only way to develop such skills, and it is not ''specifically'' important to master material about triangles and circles. That said, many people may deeply enjoy learning geometry and it may ignite their passion for mathematics, so this should not be construed as a recommendation against learning geometry. The point here is simply that the general arguments for learning math well do not apply with the same force to geometry.

	Note that the arguments in favor of learning math well do ''not'' apply with the same force to other subjects such as physics, chemistry, economics, history, biology, or English literature. In most other subjects, there is a relatively small base of foundational knowledge and skills, and a large variety of other topics within the subject that build on those. The foundational base is important to understand, but dependencies between later topics are minimal.		Note that the arguments in favor of learning math well do ''not'' apply with the same force to other subjects such as physics, chemistry, economics, history, biology, or English literature. In most other subjects, there is a relatively small base of foundational knowledge and skills, and a large variety of other topics within the subject that build on those. The foundational base is important to understand, but dependencies between later topics are minimal.

Jsinick: /* Rote and immediate practice */

2013-10-21T22:09:06Z

Rote and immediate practice

← Older revision		Revision as of 22:09, 21 October 2013
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	There are some general [[good study habits]] that help with learning any subject well. There are also some general [[online learning resources]] that help with all subjects. On this page, we describe some aspects of the strategies and resources that are specific to mathematics.		There are some general [[good study habits]] that help with learning any subject well. There are also some general [[online learning resources]] that help with all subjects. On this page, we describe some aspects of the strategies and resources that are specific to mathematics.

	===~~Rote~~ and immediate practice===		===Supplementing rote and immediate practice===

	In many cases, students have access to straightforward descriptions of the procedure they need to apply to solve a particular class of mathematical problem. They read the process, look at a worked example, then try a few examples of their own. They find that their answers are correct, and conclude that they have learned the subject.		In many cases, students have access to straightforward descriptions of the procedure they need to apply to solve a particular class of mathematical problem. They read the process, look at a worked example, then try a few examples of their own. They find that their answers are correct, and conclude that they have learned the subject.

Jsinick: /* Rote and immediate practice */

2013-10-21T22:08:33Z

Rote and immediate practice

← Older revision		Revision as of 22:08, 21 October 2013
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	===Rote and immediate practice===		===Rote and immediate practice===

	In many cases, students have access to straightforward descriptions of the procedure they need to apply to solve a particular class of mathematical problem. They read the process, look at a worked example, then try a few examples of their own. They find that their answers are correct, and conclude that they have learned the subject. ~~"Boring, but straightforward."~~		In many cases, students have access to straightforward descriptions of the procedure they need to apply to solve a particular class of mathematical problem. They read the process, look at a worked example, then try a few examples of their own. They find that their answers are correct, and conclude that they have learned the subject.

	There are several problems with this type of learning:		There are several problems with this type of learning:

	* The practice is being done immediately after reading the procedure and looking at the worked examples. This means that the student may be learning the material using his/her short term memory rather than long term memory. He/she may well forget the technique a few days later ~~and blank out on equivalent problems~~.		* The practice is being done immediately after reading the procedure and looking at the worked examples. This means that the student may be learning the material using his/her short term memory rather than long term memory. He/she may well forget the technique and be unable to do problems of the same type just a few days later.
	* The textbook ~~or other source~~ from which they are obtaining the problems may have deliberately designed the worked problem, and other problems to match each other very well. ~~Students~~ may be thrown off by problems that ~~are even somewhat different numerically, even if the same algorithm works in principle~~.		* The textbook from which they are obtaining the problems may have deliberately designed the worked problem, and other problems to match each other very well. So students who learn in this way may be thrown off by similar but slightly different problems that they encounter later .
	* The student may not obtain any conceptual understanding of the material, so he/she may be completely unable to ~~modify~~ the ~~algorithm for a slightly~~ different ~~situation~~, or ~~even to fill in gaps~~ in the ~~algorithm in case~~ he/she forgets it.		* The student may not obtain any conceptual understanding of the material, so he/she may be completely unable to use the ideas behind the procedure in different contexts, or reconstruct the procedure in the event that he/she forgets it.

	The following approaches help ~~tackle~~ these ~~problems~~:		The following approaches can help you avoid these pitfalls:

	* Maintain some spacing being doing a worked problem (where you follow indicated steps) and doing practice problems. In some cases, you may prefer to do one practice problem immediately after the worked problem, and a few more after a few days. This is a special case of the more general strategy of [[spaced repetition]].		* Maintain some spacing being doing a worked problem (where you follow indicated steps) and doing practice problems. In some cases, you may prefer to do one practice problem immediately after the worked problem, and a few more after a few days. This is a special case of the more general strategy of [[spaced repetition]].
	* When you attempt practice problems a few days after doing the worked problem, pay particular attention to the parts that you are stuck at. These are probably the parts that you have conceptual trouble with. ''Do not'' just open up the worked problem again without making a serious effort.		* When you attempt practice problems a few days after doing the worked problem, pay particular attention to the parts that you are stuck at. These are probably the parts that you have conceptual trouble with. ''Do not'' just open up the worked problem again without making a serious effort.
	* Attempt practice problems that are framed somewhat differently from the original worked problem. Most books have large numbers of exercises to choose from, and good teachers generally give a varied mix of problems, but pay conscious attention to this issue if you are studying by yourself, or if your teacher isn't giving you a varied mix.		* Attempt practice problems that are framed somewhat differently from the original worked problem. Most books have large numbers of exercises to choose from, and good teachers generally give a varied mix of problems, but pay conscious attention to this issue if you are studying by yourself, or if your teacher isn't giving you a varied mix.
	* Wherever possible, try to acquire the conceptual intuition behind the steps. This does not necessarily mean obtaining a ''proof'' for why the algorithm works (though a proof would certainly be nice, it may not be worth the effort to master). There may be other ways, such as trying variations of the algorithm and understanding where they fail. The texts or resources you use often do ~~have such~~ explanations, but ~~they~~ may not be highlighted because the authors expect that most ~~people~~ just want to master the algorithms and do not care about the explanations.		* Wherever possible, try to acquire the conceptual intuition behind the steps. The texts or resources you use often do offer conceptual explanations, but these may not be highlighted because the authors expect that most students just want to master the algorithms and do not care about the explanations.

	===Online math learning resources===		===Online math learning resources===