Sfard (1991) - Revision history

Raymond Johnson: Removed images

2021-05-29T20:34:46Z

Removed images

← Older revision		Revision as of 20:34, 29 May 2021
Line 1:		Line 1:
	~~[[File:Sfard_(1991)_1.png\|240px\|thumb\|right\|Sfard (1991)]]~~
	{{Title\|On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin}}		{{Title\|On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin}}
	__NOTOC__		__NOTOC__

Raymond Johnson: Removed references section

2021-05-29T20:06:04Z

Removed references section

← Older revision		Revision as of 20:06, 29 May 2021
Line 60:		Line 60:
	The co-dependence of operational and structural conceptions, and the difficulty of reification, may explain why many people struggle so mightily to learn mathematics, even at the highest levels. Students may become discouraged when reification is slow to happen, and the challenge to mathematics educators is to find ways to "unravel the harmful tangle and to stimulate reification" (p. 34).		The co-dependence of operational and structural conceptions, and the difficulty of reification, may explain why many people struggle so mightily to learn mathematics, even at the highest levels. Students may become discouraged when reification is slow to happen, and the challenge to mathematics educators is to find ways to "unravel the harmful tangle and to stimulate reification" (p. 34).

	==~~References~~==		==Cite==
	~~<gallery mode=packed heights=200px>~~
	~~File:Sfard_(1991)_34.png\|Anderson-Dubinsky~~
	~~File:Sfard_(1991)_35.png\|Eisenberg-Skemp~~
	~~File:Sfard_(1991)_36.png\|Skemp-Vinner~~
	~~</gallery>~~

	~~==Corrolary==~~
	* Mendeley: http://www.mendeley.com/research/dual-nature-mathematical-conceptions-reflections-processes-objects-different-sides-same-coin/

	;APA		;APA
	: Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. ''Educational Studies in Mathematics'', 22, 1–36. ~~http~~://doi.org/10.1007/BF00302715		: Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. ''Educational Studies in Mathematics'', 22, 1–36. https://doi.org/10.1007/BF00302715
	;BibTeX		;BibTeX
	<pre>		<pre>

imported>Raymond Johnson: /* Operational and Structural Conceptions in Cognitive Processes */ fix typo

2016-06-28T05:40:53Z

Operational and Structural Conceptions in Cognitive Processes: fix typo

← Older revision		Revision as of 05:40, 28 June 2016
Line 52:		Line 52:
	Sfard now faces the challenge of answering her initial question: Why is mathematical abstraction — now understood as a process of reification — so difficult? Why must it be such a significant leap in our understanding? Sfard explains with her three-stage model, where "reification of a process occurs simultaneously with the interiorization of higher-level processes" (pp. 30-31). In the punchline of the article, Sfard describes the reification of functions:		Sfard now faces the challenge of answering her initial question: Why is mathematical abstraction — now understood as a process of reification — so difficult? Why must it be such a significant leap in our understanding? Sfard explains with her three-stage model, where "reification of a process occurs simultaneously with the interiorization of higher-level processes" (pp. 30-31). In the punchline of the article, Sfard describes the reification of functions:

	<blockquote>[In] order to see a function as an object, one must try to manipulate it as a whole: there is no reason to turn process into ~~objet~~ unless we have some higher-level processes performed on this simpler process. But here is a vicious circle: on one hand, without an attempt at the higher-level interiorization, the reification will not occur; on the other hand, existence of objects on which the higher-level processes are performed seems indispensable for the interiorization — without such objects the processes must appear quite meaningless. In other words: ''the lower-level reification and the higher-level interiorization are prerequisite for each other!'' (p. 31)</blockquote>		<blockquote>[In] order to see a function as an object, one must try to manipulate it as a whole: there is no reason to turn process into object unless we have some higher-level processes performed on this simpler process. But here is a vicious circle: on one hand, without an attempt at the higher-level interiorization, the reification will not occur; on the other hand, existence of objects on which the higher-level processes are performed seems indispensable for the interiorization — without such objects the processes must appear quite meaningless. In other words: ''the lower-level reification and the higher-level interiorization are prerequisite for each other!'' (p. 31)</blockquote>

	Sfard closes her article contemplating the implications of her theories:		Sfard closes her article contemplating the implications of her theories:

imported>Raymond Johnson at 18:59, 9 February 2016

2016-02-09T18:59:52Z

← Older revision		Revision as of 18:59, 9 February 2016
Line 90:		Line 90:
	[[Category:1991]]		[[Category:1991]]
	[[Category:Learning Sciences]]		[[Category:Learning Sciences]]
			[[Category:Summaries]]

imported>Raymond Johnson: /* Operational and Structural Conceptions in the Formation of Mathematical Concepts */

2016-02-08T23:42:58Z

Operational and Structural Conceptions in the Formation of Mathematical Concepts

← Older revision		Revision as of 23:42, 8 February 2016
Line 33:		Line 33:

	===Operational and Structural Conceptions in the Formation of Mathematical Concepts===		===Operational and Structural Conceptions in the Formation of Mathematical Concepts===
	In general, says Sfard, the operational conception comes before the structural conception, and the structural conception is viewed as more abstract and advanced. Exceptions may include figures from geometry, where the static object seems simpler than the algorithm or construction required to create it. ~~Yes~~ historically and developmentally, the operational tends to come first. Piaget ([[Piaget (1952)\|1952]]) observed that when children first learn to count, they respond to the question, "How many are there?" by recounting, not simply repeating the last number-name ~~the~~ used. Other researchers ({{Cite\|Carpenter, Corbitt, Kepner, Lindquist, & Reys\|1980}}) noticed about half of 13-year-old students could not represent a division problem like 7 divided by 4 as a fraction. It took centuries for mathematicians to treat <math>\sqrt{-1}</math> as a legitimate object, even though the operation of finding roots was well-understood. In other words, it takes time for a process on an object to become ''reified'' into a new object in and of itself. This process of ''reification'' played a prominent role in the development of functions, which began as computational processes but eventually came to be used as objects.		In general, says Sfard, the operational conception comes before the structural conception, and the structural conception is viewed as more abstract and advanced. Exceptions may include figures from geometry, where the static object seems simpler than the algorithm or construction required to create it. Yet historically and developmentally, the operational tends to come first. Piaget ([[Piaget (1952)\|1952]]) observed that when children first learn to count, they respond to the question, "How many are there?" by recounting, not simply repeating the last number-name they used. Other researchers ({{Cite\|Carpenter, Corbitt, Kepner, Lindquist, & Reys\|1980}}) noticed about half of 13-year-old students could not represent a division problem like 7 divided by 4 as a fraction. It took centuries for mathematicians to treat <math>\sqrt{-1}</math> as a legitimate object, even though the operation of finding roots was well-understood. In other words, it takes time for a process on an object to become ''reified'' into a new object in and of itself. This process of ''reification'' played a prominent role in the development of functions, which began as computational processes but eventually came to be used as objects.

	Sfard describes three stages in concept development: ''interiorization'', ''condensation'', and ''reification''. During ''interiorization'', learners become familiar with processes and operations on lower-level mathematical objects, and eventually gain skills with these operations. As examples, Sfard describes:		Sfard describes three stages in concept development: ''interiorization'', ''condensation'', and ''reification''. During ''interiorization'', learners become familiar with processes and operations on lower-level mathematical objects, and eventually gain skills with these operations. As examples, Sfard describes:

imported>Raymond Johnson: /* Operational and Structural Conceptions in Cognitive Processes */

2016-02-08T06:20:03Z

Operational and Structural Conceptions in Cognitive Processes

← Older revision		Revision as of 06:20, 8 February 2016
Line 50:		Line 50:
	Sfard suggests that mathematical conceptions only become fully formed when they are conceived both operationally and structurally, but considers what is lacking if only one or the other is realized. At some level, a solely operational approach seems necessary and perhaps sufficient, and many seem to take a view of mathematics as a collection of processes. Historically this has often appeared to be the case, but structural perspectives on mathematics in the 20th century reveal the need for structural conceptions and an understanding of the objects on which processes operate. Operational conceptions are required, but structural conceptions are equally necessary to ease the burden of endless process and organize it into more useful structures. Sfard paraphrases Henrici ([[Henrici (1974)\|1974]]) by saying, "the structural approach invites contemplation; the operational approach invites action; the structural approach generates insight; the operational approach generates result" (p. 28). Furthermore, as more processes are reified into objects, our cognitive capacity to do mathematics expands.		Sfard suggests that mathematical conceptions only become fully formed when they are conceived both operationally and structurally, but considers what is lacking if only one or the other is realized. At some level, a solely operational approach seems necessary and perhaps sufficient, and many seem to take a view of mathematics as a collection of processes. Historically this has often appeared to be the case, but structural perspectives on mathematics in the 20th century reveal the need for structural conceptions and an understanding of the objects on which processes operate. Operational conceptions are required, but structural conceptions are equally necessary to ease the burden of endless process and organize it into more useful structures. Sfard paraphrases Henrici ([[Henrici (1974)\|1974]]) by saying, "the structural approach invites contemplation; the operational approach invites action; the structural approach generates insight; the operational approach generates result" (p. 28). Furthermore, as more processes are reified into objects, our cognitive capacity to do mathematics expands.

	Sfard now faces the challenge of answering her initial question: Why is mathematical abstraction — now understood as a process of reification — so difficult? Why must it be such a significant leap in our understanding? Sfard explains with her three-stage model, where "reification of a process occurs simultaneously with the interiorization of higher-level processes" (pp. 30-31). Sfard describes the reification of functions:		Sfard now faces the challenge of answering her initial question: Why is mathematical abstraction — now understood as a process of reification — so difficult? Why must it be such a significant leap in our understanding? Sfard explains with her three-stage model, where "reification of a process occurs simultaneously with the interiorization of higher-level processes" (pp. 30-31). In the punchline of the article, Sfard describes the reification of functions:

	<blockquote>[In] order to see a function as an object, one must try to manipulate it as a whole: there is no reason to turn process into objet unless we have some higher-level processes performed on this simpler process. But here is a vicious circle: on one hand, without an attempt at the higher-level interiorization, the reification will not occur; on the other hand, existence of objects on which the higher-level processes are performed seems indispensable for the interiorization — without such objects the processes must appear quite meaningless. In other words: ''the lower-level reification and the higher-level interiorization are prerequisite for each other!'' (p. 31)</blockquote>		<blockquote>[In] order to see a function as an object, one must try to manipulate it as a whole: there is no reason to turn process into objet unless we have some higher-level processes performed on this simpler process. But here is a vicious circle: on one hand, without an attempt at the higher-level interiorization, the reification will not occur; on the other hand, existence of objects on which the higher-level processes are performed seems indispensable for the interiorization — without such objects the processes must appear quite meaningless. In other words: ''the lower-level reification and the higher-level interiorization are prerequisite for each other!'' (p. 31)</blockquote>

imported>Raymond Johnson: added references

2016-02-08T06:17:23Z

added references

← Older revision		Revision as of 06:17, 8 February 2016
Line 60:		Line 60:
	The co-dependence of operational and structural conceptions, and the difficulty of reification, may explain why many people struggle so mightily to learn mathematics, even at the highest levels. Students may become discouraged when reification is slow to happen, and the challenge to mathematics educators is to find ways to "unravel the harmful tangle and to stimulate reification" (p. 34).		The co-dependence of operational and structural conceptions, and the difficulty of reification, may explain why many people struggle so mightily to learn mathematics, even at the highest levels. Students may become discouraged when reification is slow to happen, and the challenge to mathematics educators is to find ways to "unravel the harmful tangle and to stimulate reification" (p. 34).

	== Corrolary ==		==References==
			<gallery mode=packed heights=200px>
			File:Sfard_(1991)_34.png\|Anderson-Dubinsky
			File:Sfard_(1991)_35.png\|Eisenberg-Skemp
			File:Sfard_(1991)_36.png\|Skemp-Vinner
			</gallery>

			==Corrolary==
	* Mendeley: http://www.mendeley.com/research/dual-nature-mathematical-conceptions-reflections-processes-objects-different-sides-same-coin/		* Mendeley: http://www.mendeley.com/research/dual-nature-mathematical-conceptions-reflections-processes-objects-different-sides-same-coin/

imported>Raymond Johnson: added page 1 image

2016-02-08T06:08:08Z

added page 1 image

← Older revision		Revision as of 06:08, 8 February 2016
Line 1:		Line 1:
			[[File:Sfard_(1991)_1.png\|240px\|thumb\|right\|Sfard (1991)]]
	{{Title\|On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin}}		{{Title\|On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin}}
	__NOTOC__		__NOTOC__

imported>Raymond Johnson: doi links

2016-02-08T05:52:23Z

doi links

← Older revision		Revision as of 05:52, 8 February 2016
Line 4:		Line 4:
	* Journal: [[Educational Studies in Mathematics]]		* Journal: [[Educational Studies in Mathematics]]
	* Year: 1991		* Year: 1991
	* Source: http://~~link~~.~~springer.com/article~~/10.1007/BF00302715		* Source: Springer: http://doi.org/10.1007/BF00302715

	== Abstract ==		== Abstract ==
Line 72:		Line 72:
	pages = {1--36},		pages = {1--36},
	title = {{On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin}},		title = {{On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin}},
	url = {http://~~link~~.~~springer.com/article~~/10.1007/BF00302715},		url = {http://doi.org/10.1007/BF00302715},
	volume = {22},		volume = {22},
	year = {1991}		year = {1991}

imported>Raymond Johnson: /* Summary */ finish summary

2016-02-08T05:49:20Z

Summary: finish summary

← Older revision		Revision as of 05:49, 8 February 2016
Line 51:		Line 51:
	Sfard now faces the challenge of answering her initial question: Why is mathematical abstraction — now understood as a process of reification — so difficult? Why must it be such a significant leap in our understanding? Sfard explains with her three-stage model, where "reification of a process occurs simultaneously with the interiorization of higher-level processes" (pp. 30-31). Sfard describes the reification of functions:		Sfard now faces the challenge of answering her initial question: Why is mathematical abstraction — now understood as a process of reification — so difficult? Why must it be such a significant leap in our understanding? Sfard explains with her three-stage model, where "reification of a process occurs simultaneously with the interiorization of higher-level processes" (pp. 30-31). Sfard describes the reification of functions:

	<blockquote>[In] order to see a function as an object, one must try to manipulate it as a whole: there is no reason to turn process into objet unless we have some higher-level processes performed on this simpler process. But here is a vicious circle: on one hand, without an attempt at the higher-level interiorization, the reification will not occur; on the other hand, existence of objects on which the higher-level processes are performed seems indispensable for the interiorization — without such objects the processes must appear quite meaningless. In other words: ''the lower-level reification and the higher-level interiorization are prerequisite for each other! (p. 31)</blockquote>		<blockquote>[In] order to see a function as an object, one must try to manipulate it as a whole: there is no reason to turn process into objet unless we have some higher-level processes performed on this simpler process. But here is a vicious circle: on one hand, without an attempt at the higher-level interiorization, the reification will not occur; on the other hand, existence of objects on which the higher-level processes are performed seems indispensable for the interiorization — without such objects the processes must appear quite meaningless. In other words: ''the lower-level reification and the higher-level interiorization are prerequisite for each other!'' (p. 31)</blockquote>

			Sfard closes her article contemplating the implications of her theories:

			<blockquote>The thesis of the "vicious circle" implies that one ability cannot be fully developed without the other: on one hand, a person must be quite skillful at performing algorithms in order to attain a good idea of the "objects" involved in these algorithms; on the other hand, to gain full technical mastery, one must already have these objects, since without them the processes would seem meaningless and thus difficult to perform and remember. (p. 32)</blockquote>

			The co-dependence of operational and structural conceptions, and the difficulty of reification, may explain why many people struggle so mightily to learn mathematics, even at the highest levels. Students may become discouraged when reification is slow to happen, and the challenge to mathematics educators is to find ways to "unravel the harmful tangle and to stimulate reification" (p. 34).

	== Corrolary ==		== Corrolary ==