Derivative - Revision history

Inna mik: (username removed) (log details removed)

2025-06-17T08:21:38Z

(username removed) (log details removed)

← Older revision

Revision as of 10:21, 17 June 2025

Inna mik at 15:08, 1 May 2025

2025-05-01T15:08:42Z

← Older revision		Revision as of 17:08, 1 May 2025
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	=== Step 2: Description of mental tasks needed to overcome the bottleneck ===		=== Step 2: Description of mental tasks needed to overcome the bottleneck ===

	# While thinking about derivative, a mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis). He clearly understands that the story is not about calculating one value. It is about how function value changes if argument value changes (see a related ~~Bottleneck~~ [[Function]])		# While thinking about derivative, a mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis). He clearly understands that the story is not about calculating one value. It is about how function value changes if argument value changes (see a related bottleneck [[Function]])
	# An expert can mask from his mind the fact, that the derivative should be calculated as a limit. He works with quotient "Change of function value" over "Change of ~~Argument~~ value" if this is useful to deduce needed information. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values. He deduces the needed values from the given graph of function by drawing a triangle in order to represent changes in values and the slope between two points.		# An expert can mask from his mind the fact, that the derivative should be calculated as a limit. He works with quotient "Change of function value" over "Change of argument value" if this is useful to deduce needed information. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values. He deduces the needed values from the given graph of function by drawing a triangle in order to represent changes in values and the slope between two points.
	# He can work with augmenting (moving to the right on the graph) and reducing (moving to the left) value of argument, and he can split the change of the value in multiple steps if he realizes that the step is too big.		# He can work with augmenting (moving to the right on the graph) and reducing (moving to the left) value of the argument, and he can split the change of the value in multiple steps if he realizes that the step is too big.
	# While doing so, he is not afraid of being unprecise, because he knows, that the approximation is locally allowed and can estimate the approximation error.		# While doing so, he is not afraid of being unprecise, because he knows, that the approximation is locally allowed and can estimate the approximation error.

Inna mik: Video Link added

2025-05-01T07:11:12Z

Video Link added

← Older revision		Revision as of 09:11, 1 May 2025
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	=== Step 3: Modelling the tasks ===		=== Step 3: Modelling the tasks ===

			==Available resources==
			[https://youtu.be/gh-cgUDM1xI Interview Video]

	===People interested in this Bottleneck===		===People interested in this Bottleneck===

Inna mik: Link to Function + less Steps

2025-04-24T08:01:40Z

Link to Function + less Steps

← Older revision		Revision as of 10:01, 24 April 2025
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	{{DISPLAYTITLE:DecodingWork:From Derivative to Proportionality}}		{{DISPLAYTITLE:DecodingWork:From Derivative to Proportionality}}

	===Description of Bottleneck===		===Step 1: Description of Bottleneck===
	Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. ~~They do not have an idea of~~ "derivative=slope" ~~or in case they have it~~ they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value"		Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. In case they know, that "derivative=slope", they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value".

	Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. ~~They~~ should be able to ~~draw for~~ a ~~given~~ graph of function ~~a triangle~~ to ~~illustrate the~~ changes in values and the slope		Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. Students should be able to construct a triangle on the graph of a function to visually represent changes in function values and the corresponding slope (rate of change).

	Concrete problems may be, for example:		Concrete problems may be, for example:
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	* C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N		* C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N

	=== Description of mental tasks needed to overcome the bottleneck ===		=== Step 2: Description of mental tasks needed to overcome the bottleneck ===

	# While thinking about derivative, mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis)		# While thinking about derivative, a mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis). He clearly understands that the story is not about calculating one value. It is about how function value changes if argument value changes (see a related Bottleneck [[Function]])
	# He clearly understands that the story is not about calculating one value~~, but~~ about ~~the process and~~ how function value ~~depends on~~ argument value		# An expert can mask from his mind the fact, that the derivative should be calculated as a limit. He works with quotient "Change of function value" over "Change of Argument value" if this is useful to deduce needed information. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values. He deduces the needed values from the given graph of function by drawing a triangle in order to represent changes in values and the slope between two points.
	# ~~He knows, that he~~ can ~~look onto small local changes if he wants to calculate~~ the ~~concrete change rate, and he knows where to place the changes into the graph of function.~~		# He can work with augmenting (moving to the right on the graph) and reducing (moving to the left) value of argument, and he can split the change of the value in multiple steps if he realizes that the step is too big.
	~~# He can forget~~, that the derivative should be calculated as a limit, if ~~the fact that derivative~~ is ~~a quotient df/dx is enough~~ to deduce needed information ~~or is the only possibility~~. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values df and dx		# While doing so, he is not afraid of being unprecise, because he knows, that the approximation is locally allowed and can estimate the approximation error.
	# He is not afraid of being unprecise, because he knows, that the approximation ~~by tangent~~ is allowed

	=== Modelling the tasks ===		=== Step 3: Modelling the tasks ===

	===People interested in this Bottleneck===		===People interested in this Bottleneck===

Riegler: Changed categories.

2025-04-22T16:34:25Z

Changed categories.

← Older revision		Revision as of 18:34, 22 April 2025
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	[[Category:Calculus]]		[[Category:Calculus]]
	[[Category:Decoding work]]		[[Category:Decoding work]]
			[[Category:Decoding Interview]]

Riegler: Changed categories.

2025-04-22T16:31:20Z

Changed categories.

← Older revision		Revision as of 18:31, 22 April 2025
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	[[Category:Mathematics]]		[[Category:Mathematics]]
	[[Category:Calculus]]		[[Category:Calculus]]
			[[Category:Decoding work]]

Inna mik: minor changes Mental Task

2025-04-16T12:00:52Z

minor changes Mental Task

← Older revision		Revision as of 14:00, 16 April 2025
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	# While thinking about derivative, mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis)		# While thinking about derivative, mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis)
	# He clearly understands that the story is not about calculating one value, but about the process and how function value depends on argument value		# He clearly understands that the story is not about calculating one value, but about the process and how function value depends on argument value
	# He knows, that he can look onto small local changes if he wants to calculate the concrete change rate and he knows ~~were~~ to place the changes into the graph of function.		# He knows, that he can look onto small local changes if he wants to calculate the concrete change rate, and he knows where to place the changes into the graph of function.
	# He can forget, that the derivative should be calculated as a limit, if the fact that derivative is a quotient df/dx is enough to deduce needed information e.g. the ~~entities~~		# He can forget, that the derivative should be calculated as a limit, if the fact that derivative is a quotient df/dx is enough to deduce needed information or is the only possibility. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values df and dx
	# He is not afraid of being unprecise, because he knows, that the approximation by tangent is allowed		# He is not afraid of being unprecise, because he knows, that the approximation by tangent is allowed

Inna mik: Describe mental tasks

2025-04-16T11:51:51Z

Describe mental tasks

← Older revision		Revision as of 13:51, 16 April 2025
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	Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. They do not have an idea of "derivative=slope" or in case they have it they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value"		Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. They do not have an idea of "derivative=slope" or in case they have it they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value"

			Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. They should be able to draw for a given graph of function a triangle to illustrate the changes in values and the slope

			Concrete problems may be, for example:

	~~Desired learning outcome: Students~~ should ~~understand that~~ the ~~change~~ of function ~~value is proportional to~~ the ~~change~~ of ~~argument value~~, ~~and~~ the derivative is ~~exactly~~ the ~~proportionality factor. They should be able to draw for a given graph~~ of ~~function~~ a ~~triangle~~ to ~~illustrate~~ the ~~changes in values and~~ the ~~slope~~		* A) Given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
			* B) Given the entities of function and entities of argument, decide on the entities of derivative. E.g. fuel price: Argument x is in Litre, function f(x) is in Euro, follows that the derivative f'(x) is in Euro/Litre
			* C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N

			=== Description of mental tasks needed to overcome the bottleneck ===

	~~Concrete~~ ~~problems may~~ be, ~~for example:~~		# While thinking about derivative, mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis)
			# He clearly understands that the story is not about calculating one value, but about the process and how function value depends on argument value
			# He knows, that he can look onto small local changes if he wants to calculate the concrete change rate and he knows were to place the changes into the graph of function.
			# He can forget, that the derivative should be calculated as a limit, if the fact that derivative is a quotient df/dx is enough to deduce needed information e.g. the entities
			# He is not afraid of being unprecise, because he knows, that the approximation by tangent is allowed

	* A) given that f(1)=~~2 and f'(1)~~=~~3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01~~=~~2.003~~		=== Modelling the tasks ===
	* B) given the ~~entities of function and entities of argument, decide on the entities of derivative~~
	* C) given the text, which describes the rate of change, deduce a differential equation.

	===People interested in this Bottleneck===		===People interested in this Bottleneck===

Inna mik: Concretization of bottleneck

2025-04-16T10:44:03Z

Concretization of bottleneck

← Older revision		Revision as of 12:44, 16 April 2025
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	===Description of Bottleneck===		===Description of Bottleneck===
	Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. ~~<math>x^2</math>~~ into ~~<math>2 \cdot x</math>~~. They do not have an idea of "derivative=slope" ~~and~~ "change of function value=slope*change of argument"		Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. They do not have an idea of "derivative=slope" or in case they have it they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value"


	Desired learning outcome: given that f(1)=2 and f'(1)=3 ~~the~~ students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
			Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. They should be able to draw for a given graph of function a triangle to illustrate the changes in values and the slope

			Concrete problems may be, for example:

			* A) given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
			* B) given the entities of function and entities of argument, decide on the entities of derivative
			* C) given the text, which describes the rate of change, deduce a differential equation.

	===People interested in this Bottleneck===		===People interested in this Bottleneck===

Riegler: + $tags$

2024-07-21T18:57:49Z

+<math> tags

← Older revision		Revision as of 20:57, 21 July 2024
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	===Description of Bottleneck===		===Description of Bottleneck===
	Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x^2 into 2x. They do not have an idea of "derivative=slope" and "change of function value=slopechange of argument"		Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. <math>x^2</math> into <math>2 \cdot x</math>. They do not have an idea of "derivative=slope" and "change of function value=slope*change of argument"

Derivative - Revision history

Inna mik: (username removed) (log details removed)

Inna mik at 15:08, 1 May 2025

Inna mik: Video Link added

Inna mik: Link to Function + less Steps

Riegler: Changed categories.

Riegler: Changed categories.

Inna mik: minor changes Mental Task

Inna mik: Describe mental tasks

Inna mik: Concretization of bottleneck

Riegler: + tags

Riegler: + $tags$