Function - Revision history

Riegler: Changed categories.

2025-04-22T16:36:00Z

Changed categories.

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	[[Category:Bottleneck]]		[[Category:Bottleneck]]
	[[Category:Mathematics]]		[[Category:Mathematics]]
			[[Category:Decoding Interview]]

Riegler at 16:30, 22 April 2025

2025-04-22T16:30:50Z

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	In terms of APOS-theory the identified bottleneck is related to the conceptual transition from the action to the process stage.		In terms of APOS-theory the identified bottleneck is related to the conceptual transition from the action to the process stage.

	===Description of bottleneck===		===Step 1: Description of bottleneck===
	Students don't grasp the concept of functions in a calculus class.		Students don't grasp the concept of functions in a calculus class.

	===Description of mental moves needed to overcome the bottleneck===		===Step 2: Description of mental moves needed to overcome the bottleneck===
	The two interview excerpts revealed the following mental moves:		The two interview excerpts revealed the following mental moves:

Riegler: -typos

2025-04-22T13:27:18Z

-typos

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	==Core findings from mathematics education research==		==Core findings from mathematics education research==
	Students conceptual difficulties with respect to functions have been investigated at least since the 1980s. These findings have led to the development of APOS Theory<ref name=":0" /> which is a general constructivist theory explaining how mathematical understanding works and develops. It models what might happen in a person's mind when one tries to learn a mathematical concept. It postulates that a person's understanding of a mathematical topic develops as ~~one~~ constructs or reconstructs certain mental structures and organizes them into schemas. The mental structures proposed by APOS theory are actions, processes, objects and schemata; hence the acronym APOS.		Students conceptual difficulties with respect to functions have been investigated at least since the 1980s. These findings have led to the development of APOS Theory<ref name=":0" /> which is a general constructivist theory explaining how mathematical understanding works and develops. It models what might happen in a person's mind when one tries to learn a mathematical concept. It postulates that a person's understanding of a mathematical topic develops as the person constructs or reconstructs certain mental structures and organizes them into schemas. The mental structures proposed by APOS theory are actions, processes, objects and schemata; hence the acronym APOS.

	In case of functions in ~~mathemtics~~, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).		In case of functions in mathematics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).

	Note that there are two core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[Treshold concept\|treshold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.		Note that there are two core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[Treshold concept\|treshold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.

Riegler: typo

2025-03-28T08:20:45Z

typo

← Older revision		Revision as of 10:20, 28 March 2025
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	In case of functions in mathemtics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).		In case of functions in mathemtics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).

	Note that there are two core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[~~Treshhold~~ concept\|~~treshhold~~ concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.		Note that there are two core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[Treshold concept\|treshold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.

	==Decoding work done==		==Decoding work done==

Riegler: -typos

2025-03-27T17:52:08Z

-typos

← Older revision		Revision as of 19:52, 27 March 2025
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	In case of functions in mathemtics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).		In case of functions in mathemtics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).

	Note that there are ~~twoe~~ core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[Treshhold concept\|treshhold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.		Note that there are two core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[Treshhold concept\|treshhold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.

	==Decoding work done==		==Decoding work done==
	In their book [[Overcoming Student Learning Bottlenecks]] [[:Category:Middendorf, J\|Joan Middendorf]] and [[:Category:Shopkow, L\|Leah Shopkow]] give excerpts of two [[Step 2 - Decoding Interview\|Decoding Interviews]] dealing with the concept of function.<ref>Middendorf, J., & Shopkow, L. (2017). [[Overcoming Student Learning Bottlenecks\|''Overcoming student learning bottlenecks: Decode the critical thinking of your discipline''.]] Stylus Publishing, LLC, pp. 55-57.</ref> There, however, the focus is on posing suitable [[interview questions]].		In their book [[Overcoming Student Learning Bottlenecks]] [[:Category:Middendorf, J\|Joan Middendorf]] and [[:Category:Shopkow, L\|Leah Shopkow]] give excerpts of two [[Step 2 - Decoding Interview\|Decoding Interviews]] dealing with the concept of function.<ref>Middendorf, J., & Shopkow, L. (2017). [[Overcoming Student Learning Bottlenecks\|''Overcoming student learning bottlenecks: Decode the critical thinking of your discipline''.]] Stylus Publishing, LLC, pp. 55-57.</ref> There, however, the focus is on posing suitable [[interview questions]].

	In terms of APOS-theory the identified bottleneck is related to the conceptual transition from action to process stage.		In terms of APOS-theory the identified bottleneck is related to the conceptual transition from the action to the process stage.

	===Description of bottleneck===		===Description of bottleneck===
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	The two interview excerpts revealed the following mental moves:		The two interview excerpts revealed the following mental moves:

	*Trying not to think of a function in terms of an algorithmic relationship between input and output, for instance via visualing functions as graphs		*Trying not to think of a function in terms of an algorithmic relationship between input and output. This can be achieved, for instance, via visualing functions as graphs.
	*Having available examples of input-output relationsships which can be described as functions but not algorithmically (e.g. random processes in the stock market)		*Having available for oneself examples of input-output relationsships which can be described as functions but not algorithmically (e.g. random processes in the stock market).

	==Available resources==		==Available resources==

Riegler: +typo

2025-03-26T14:57:32Z

+typo

← Older revision		Revision as of 16:57, 26 March 2025
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	In case of functions in mathemtics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).		In case of functions in mathemtics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).

	Note that there are twoe core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[treshhold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.		Note that there are twoe core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[Treshhold concept\|treshhold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.

	==Decoding work done==		==Decoding work done==

Riegler: proof reading

2025-03-26T14:51:18Z

proof reading

Riegler: Changed categories.

2025-03-26T14:09:23Z

Changed categories.

← Older revision		Revision as of 16:09, 26 March 2025
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	<references />		<references />
			[[Category:Bottleneck]]
			[[Category:Mathematics]]

Riegler: init

2025-03-26T14:09:05Z

init

New page

'''Functions''' are a core concept of mathematics and disciplines using mathematics. Difficulties related to the concept of function are well documented and analyzed in mathematics education research<ref>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), 1-36.</ref><ref>Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. ''Educational studies in mathematics'', ''23''(3), 247-285.</ref> <ref name=":0">lana Arnon, Jim Cottrill, Ed Dubinsky, Asuman Oktaç, Solange Roa Fuentes, Maria Trigueros, Kirk Weller: APOS theory: a framework for research and curriculum development in mathematics education. Springer, New York Heidelberg 2014, [https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9781461479659 ISBN 978-1-4614-7965-9].</ref> and other disciplines (e.g. computer science<ref>Philipp Marwan, Peter Riegler (2011): Entwicklung des Funktionenkonzepts bei Studierenden der Informatik, Wismarer Frege–Reihe, vol. 02/2011.</ref>, physics<ref>Combining mathematics and physics beyond the introductory level: the case of partial differential equations, https://research.kuleuven.be/portal/en/project/3E211102</ref>).

==Core findings from mathematics education research==
Students conceptual difficulties with respect to functions have been investigated at least since the 1980s. These findings have led to the development of APOS Theory<ref name=":0" /> which is a general constructivist theory explaining how mathematical understanding works and develops. It models what might happen in a person's mind when they try to learn a mathematical concept. It postulates that a person's understanding of a mathematical topic develops as they construct or reconstruct certain mental structures and organize them into schemas. The mental structures proposed by APOS theory are actions, processes, objects and schemata; hence the acronym APOS.

In case of functions in mathemtics a different mental structure is in the foreground depending on the developmental stage of understanding. At the action stage, the function is seen merely as an instruction that specifies how the function value is to be calculated. In the process stage, the function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule. At the object stage, the function becomes an object, which in turn can itself be affected by processes (e.g. differentiation). Going even further, functions can be organised as objects with other mathematical processes or objects to form schemes (e.g. function spaces).

Core transitions in a person's conceptualization of function are from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[treshhold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.

==Decoding work done==
In their book [[Overcoming Student Learning Bottlenecks]] [[:Category:Shopkow, L|Leah Shopkow]] and [[:Category:Middendorf, J|Joan Middendorf]] give excerpts of two [[Step 2 - Decoding Interview|Decoding Interviews]] dealing with the concept of function.<ref>Middendorf, J., & Shopkow, L. (2017). [[Overcoming Student Learning Bottlenecks|''Overcoming student learning bottlenecks: Decode the critical thinking of your discipline''.]] Stylus Publishing, LLC, pp. 55-57.</ref> There, however, the focus is on posing suitable [[interview questions]].

In terms of APOS-theory the identified bottleneck is related to the conceptual transition from action to process stage.

===Identification of bottleneck===
Students don't grasp the concept of functions in a calculus class.

===Description of mental tasks needed to overcome the bottleneck===
The two interview excerpts revealed the following mental moves:

*Trying not to think of a function in terms of an algorithmic relationship between input and output, e.g. by visualing functions as graphs
*Having available examples of input-output relationsships which can be described as functions but not algorithmically (e.g. random processes in the stock market)

==Available resources==
Researchers have developed teaching materials to help students overcome conceptual difficulties with functions, see in particular the text books by Arnon et. al.<ref name=":0" /> and Dubinsky et al.<ref>Dubinsky, E., Schwingendorf, K. E., & Mathews, D. M.: ''Calculus concepts and computers'' (2nd ed.). McGraw-Hill College 1994, [https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9780314011299 ISBN 978-0-314-01129-9]</ref>

==References==

[[Category:Decoding work]]

<references />

← Older revision		Revision as of 16:51, 26 March 2025
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	'''Functions''' are a core concept of mathematics ~~and~~ disciplines using mathematics. Difficulties related to the concept of function are well documented and analyzed in mathematics education research<ref>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), 1-36.</ref><ref>Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. ''Educational studies in mathematics'', ''23''(3), 247-285.</ref> <ref name=":0">lana Arnon, Jim Cottrill, Ed Dubinsky, Asuman Oktaç, Solange Roa Fuentes, Maria Trigueros, Kirk Weller: APOS theory: a framework for research and curriculum development in mathematics education. Springer, New York Heidelberg 2014, [https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9781461479659 ISBN 978-1-4614-7965-9].</ref> and other disciplines (e.g. computer science<ref>Philipp Marwan, Peter Riegler (2011): Entwicklung des Funktionenkonzepts bei Studierenden der Informatik, Wismarer Frege–Reihe, vol. 02/2011.</ref>, physics<ref>Combining mathematics and physics beyond the introductory level: the case of partial differential equations, https://research.kuleuven.be/portal/en/project/3E211102</ref>).		'''Functions''' are a core concept of mathematics as well as of disciplines using mathematics. Difficulties related to the concept of function are well documented and analyzed in mathematics education research<ref>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), 1-36.</ref><ref>Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. ''Educational studies in mathematics'', ''23''(3), 247-285.</ref> <ref name=":0">lana Arnon, Jim Cottrill, Ed Dubinsky, Asuman Oktaç, Solange Roa Fuentes, Maria Trigueros, Kirk Weller: APOS theory: a framework for research and curriculum development in mathematics education. Springer, New York Heidelberg 2014, [https://de.wikipedia.org/wiki/Spezial:ISBN-Suche/9781461479659 ISBN 978-1-4614-7965-9].</ref> and other disciplines (e.g. computer science<ref>Philipp Marwan, Peter Riegler (2011): Entwicklung des Funktionenkonzepts bei Studierenden der Informatik, Wismarer Frege–Reihe, vol. 02/2011.</ref>, physics<ref>Combining mathematics and physics beyond the introductory level: the case of partial differential equations, https://research.kuleuven.be/portal/en/project/3E211102</ref>).

	==Core findings from mathematics education research==		==Core findings from mathematics education research==
	Students conceptual difficulties with respect to functions have been investigated at least since the 1980s. These findings have led to the development of APOS Theory<ref name=":0" /> which is a general constructivist theory explaining how mathematical understanding works and develops. It models what might happen in a person's mind when ~~they try~~ to learn a mathematical concept. It postulates that a person's understanding of a mathematical topic develops as ~~they construct~~ or ~~reconstruct~~ certain mental structures and ~~organize~~ them into schemas. The mental structures proposed by APOS theory are actions, processes, objects and schemata; hence the acronym APOS.		Students conceptual difficulties with respect to functions have been investigated at least since the 1980s. These findings have led to the development of APOS Theory<ref name=":0" /> which is a general constructivist theory explaining how mathematical understanding works and develops. It models what might happen in a person's mind when one tries to learn a mathematical concept. It postulates that a person's understanding of a mathematical topic develops as one constructs or reconstructs certain mental structures and organizes them into schemas. The mental structures proposed by APOS theory are actions, processes, objects and schemata; hence the acronym APOS.

	In case of functions in mathemtics a different mental ~~structure is in the foreground~~ depending on the developmental stage of understanding. At the action stage, ~~the~~ function is seen merely as an instruction that specifies how the function value is to be calculated. In the process stage, ~~the~~ function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule. At the object stage, the function becomes an object, which in turn can itself be affected by processes (e.g. differentiation). Going even further, functions can be ~~organised~~ as objects with other mathematical processes or objects to form ~~schemes~~ (e.g. function spaces).		In case of functions in mathemtics, different mental structures are required depending on the developmental stage of understanding. At the action stage, a function is seen merely as an instruction that specifies how the function value (output) is to be calculated. In the process stage, a function is seen as a process that assigns an output to every possible input. This process can be thought of without the need to execute or know the instructions specified in the function's calculation rule/algorithm. At the object stage, the function becomes an object, which in turn can itself be affected or manipulated by processes (e.g. differentiation). Going even further, functions can be organized as objects with other mathematical processes or objects to form larger schematas (e.g. function spaces).

	~~Core~~ transitions in a person's conceptualization of function ~~are~~ from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[treshhold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.		Note that there are twoe core transitions in a person's conceptualization of function: from action to process stage (referred to as interiorization in APOS) and from process to object stage (referred to as encapsulation). In terms of [[treshhold concepts]] both transitions can be considered as thresholds. This indicates the cognitive complexity of the function concept since acquiring it requires multiple (re-)conceptualizations.

	==Decoding work done==		==Decoding work done==
	In their book [[Overcoming Student Learning Bottlenecks]] [[:Category:~~Shopkow~~, L\|~~Leah Shopkow~~]] and [[:Category:~~Middendorf~~, J\|~~Joan Middendorf~~]] give excerpts of two [[Step 2 - Decoding Interview\|Decoding Interviews]] dealing with the concept of function.<ref>Middendorf, J., & Shopkow, L. (2017). [[Overcoming Student Learning Bottlenecks\|''Overcoming student learning bottlenecks: Decode the critical thinking of your discipline''.]] Stylus Publishing, LLC, pp. 55-57.</ref> There, however, the focus is on posing suitable [[interview questions]].		In their book [[Overcoming Student Learning Bottlenecks]] [[:Category:Middendorf, J\|Joan Middendorf]] and [[:Category:Shopkow, L\|Leah Shopkow]] give excerpts of two [[Step 2 - Decoding Interview\|Decoding Interviews]] dealing with the concept of function.<ref>Middendorf, J., & Shopkow, L. (2017). [[Overcoming Student Learning Bottlenecks\|''Overcoming student learning bottlenecks: Decode the critical thinking of your discipline''.]] Stylus Publishing, LLC, pp. 55-57.</ref> There, however, the focus is on posing suitable [[interview questions]].

	In terms of APOS-theory the identified bottleneck is related to the conceptual transition from action to process stage.		In terms of APOS-theory the identified bottleneck is related to the conceptual transition from action to process stage.

	===~~Identification~~ of bottleneck===		===Description of bottleneck===
	Students don't grasp the concept of functions in a calculus class.		Students don't grasp the concept of functions in a calculus class.

	===Description of mental ~~tasks~~ needed to overcome the bottleneck===		===Description of mental moves needed to overcome the bottleneck===
	The two interview excerpts revealed the following mental moves:		The two interview excerpts revealed the following mental moves:

	*Trying not to think of a function in terms of an algorithmic relationship between input and output, ~~e.g. by~~ visualing functions as graphs		*Trying not to think of a function in terms of an algorithmic relationship between input and output, for instance via visualing functions as graphs
	*Having available examples of input-output relationsships which can be described as functions but not algorithmically (e.g. random processes in the stock market)		*Having available examples of input-output relationsships which can be described as functions but not algorithmically (e.g. random processes in the stock market)