Pedagogy:Product rule for differentiation - Revision history

Vipul: /* Conceptual mastery */

2023-12-05T18:12:47Z

Conceptual mastery

← Older revision		Revision as of 18:12, 5 December 2023
Line 47:		Line 47:
	===Conceptual mastery===		===Conceptual mastery===

	A good test of conceptual mastery is a free recall (specifically, a [[learning:recursive recall\|recursive recall]]) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.		A good test of conceptual mastery is a free recall (specifically, a [[learning:recursive recall\|recursive recall]]) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page. A variant may be to use the [[learning:Feynman technique\|Feynman technique]] to try to explain and progressively simplify the explanation of the product rule.

Vipul: /* Conceptual mastery */

2023-12-05T18:07:35Z

Conceptual mastery

← Older revision		Revision as of 18:07, 5 December 2023
Line 47:		Line 47:
	===Conceptual mastery===		===Conceptual mastery===

	A good test of conceptual mastery is a free recall (specifically, a [[~~market~~:recursive recall\|recursive recall]]) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.		A good test of conceptual mastery is a free recall (specifically, a [[learning:recursive recall\|recursive recall]]) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.

Vipul: /* Conceptual mastery */

2023-12-05T18:07:26Z

Conceptual mastery

← Older revision		Revision as of 18:07, 5 December 2023
Line 47:		Line 47:
	===Conceptual mastery===		===Conceptual mastery===

	A good test of conceptual mastery is a free recall (specifically, a [[~~learning~~:recursive recall\|recursive recall]]) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.		A good test of conceptual mastery is a free recall (specifically, a [[market:recursive recall\|recursive recall]]) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.

Vipul: /* Conceptual mastery */

2023-12-05T18:06:38Z

Conceptual mastery

← Older revision		Revision as of 18:06, 5 December 2023
Line 47:		Line 47:
	===Conceptual mastery===		===Conceptual mastery===

	A good test of conceptual mastery is a free recall (specifically, a recursive recall) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.		A good test of conceptual mastery is a free recall (specifically, a [[learning:recursive recall\|recursive recall]]) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.

Vipul: /* Conceptual mastery */

2023-12-05T18:06:27Z

Conceptual mastery

← Older revision		Revision as of 18:06, 5 December 2023
Line 47:		Line 47:
	===Conceptual mastery===		===Conceptual mastery===

	A good test of conceptual mastery is a free recall (specifically, a ~~[[learning:~~recursive recall~~\|recursive recall]]~~) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.		A good test of conceptual mastery is a free recall (specifically, a recursive recall) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.

Vipul: /* Error types */

2023-12-05T17:56:53Z

Error types

← Older revision		Revision as of 17:56, 5 December 2023
Line 38:		Line 38:
	\| ''differentiation by parts'', trying to do integration by parts for differentiating a product\|\| This is a rare error, and is likely to occur only if the person has been doing too much integration in the recent past. More attentiveness is the best bet against this.		\| ''differentiation by parts'', trying to do integration by parts for differentiating a product\|\| This is a rare error, and is likely to occur only if the person has been doing too much integration in the recent past. More attentiveness is the best bet against this.
	\|}		\|}

			==Mastery==

			===Computational mastery===

			{{#lst:Practical:Product rule for differentiation\|mastery}}

			===Conceptual mastery===

			A good test of conceptual mastery is a free recall (specifically, a [[learning:recursive recall\|recursive recall]]) of as much as possible of the [[product rule for differentiation]] page or equivalent material. The goal isn't to regurgitate the exact structure and sequence of the page, but to get as much coverage as possible of the ideas covered in the page.

Vipul: /* Pedagogical contexts */

2012-03-22T16:56:58Z

Pedagogical contexts

← Older revision		Revision as of 16:56, 22 March 2012
Line 8:		Line 8:
	! Course level !! Goal !! Coverage suggestions		! Course level !! Goal !! Coverage suggestions
	\|-		\|-
	\| high school calculus (e.g., AP Calculus, IB Calculus, or A levels) \|\| Computational fluency with using product rule numerically and algebraically \|\| Focus on the formula aspect for product of two functions. Relevant sections: [[product rule for differentiation#Statement with symbols]], [[product rule for differentiation#Computational feasibility significance]] (provide examples and don't overemphasize the theoretical significance), [[product rule for differentiation#Examples]] (exclude sanity checks in first treatment except as asides).<br>Do more worked examples or assign worked examples as homework. The examples should cover all the function types that the students are familiar with, and as they see new functions, they should see at least one application of the product rule to each new type of function.		\| high school calculus (e.g., AP Calculus, IB Calculus, or A levels) \|\| Computational fluency with using product rule numerically and algebraically \|\| Use [[School:Product rule for differentiation]]<br>Long version: Focus on the formula aspect for product of two functions. Relevant sections: [[product rule for differentiation#Statement with symbols]], [[product rule for differentiation#Computational feasibility significance]] (provide examples and don't overemphasize the theoretical significance), [[product rule for differentiation#Examples]] (exclude sanity checks in first treatment except as asides).<br>Do more worked examples or assign worked examples as homework. The examples should cover all the function types that the students are familiar with, and as they see new functions, they should see at least one application of the product rule to each new type of function.
	\|-		\|-
	\| single variable calculus class (college class or gifted/talented class) with partial focus on ''understanding'' theory (but not coming up with proofs of theorems by oneself) \|\| Build on existing computational fluency and provide insight into subtleties and existential questions, i.e., into ''what the results tell us qualitatively''. \|\| Review formula aspects quickly, then cover a selection of these based on time constraints: [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]], [[product rule for differentiation#Sanity checks\|sanity checks]]. Exclude the compatibility checks as they may be too tricky to motivate, at least in the initial treatment.<br>Exclude proof from the initial treatment. If you wish to include some proof, consider a proof of [[differentiation is linear]], which is easier to understand and covers the same ideas.		\| single variable calculus class (college class or gifted/talented class) with partial focus on ''understanding'' theory (but not coming up with proofs of theorems by oneself) \|\| Build on existing computational fluency and provide insight into subtleties and existential questions, i.e., into ''what the results tell us qualitatively''. \|\| Use [[College:Product rule for differentiation]]<br>Long version: Review formula aspects quickly, then cover a selection of these based on time constraints: [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]], [[product rule for differentiation#Sanity checks\|sanity checks]]. Exclude the compatibility checks as they may be too tricky to motivate, at least in the initial treatment.<br>Exclude proof from the initial treatment. If you wish to include some proof, consider a proof of [[differentiation is linear]], which is easier to understand and covers the same ideas.
	\|-		\|-
	\| single variable calculus class, review time \|\| provide more insight into subtleties and existential questions \|\| Review the rule while doing [[integration by parts]] (see [[product rule for differentiation#Reversal for integration]])<br>Cover more subtleties/aspects of the rule at opportune moments, e.g., when covering [[vertical tangent]]s and [[vertical cusp]]s, when discussing one-sided differentiability, when discussing piecewise functions.<br>Cover the product rule when comparing with the [[product rule for differentiation of formal power series]].		\| single variable calculus class, review time \|\| provide more insight into subtleties and existential questions \|\| Review the rule while doing [[integration by parts]] (see [[product rule for differentiation#Reversal for integration]])<br>Cover more subtleties/aspects of the rule at opportune moments, e.g., when covering [[vertical tangent]]s and [[vertical cusp]]s, when discussing one-sided differentiability, when discussing piecewise functions.<br>Cover the product rule when comparing with the [[product rule for differentiation of formal power series]].
Line 16:		Line 16:
	\| multivariable calculus class \|\| cover [[product rule for partial differentiation]], [[product rule for differentation of dot product]], etc. \|\| Emphasize key features of the product rule and which of them do and don't generalize to the multivariable calculus situation.<br>At a later stage in the class, consider covering the [[proof of product rule for differentiation using chain rule for partial differentiation]].		\| multivariable calculus class \|\| cover [[product rule for partial differentiation]], [[product rule for differentation of dot product]], etc. \|\| Emphasize key features of the product rule and which of them do and don't generalize to the multivariable calculus situation.<br>At a later stage in the class, consider covering the [[proof of product rule for differentiation using chain rule for partial differentiation]].
	\|-		\|-
	\| single variable calculus class for math majors, with strong focus on proofs (including coming up with proofs oneself) \|\| Teach how one would prove it, how one would think of it, and how it fits in with the other rules. \|\| cover one or more proof ideas. Start off with [[proof of product rule for differentiation using difference quotients]].<br>Cover one or more of the topics: [[product rule for differentiation#Compatibility checks\|compatibility checks]], [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]] <br>After introducing the [[natural logarithm]], consider the [[proof of product rule for differentiation using logarithmic differentiation]].		\| single variable calculus class for math majors, with strong focus on proofs (including coming up with proofs oneself) \|\| Teach how one would prove it, how one would think of it, and how it fits in with the other rules. \|\| Use [[Mathmajor:Product rule for differentiation]]<br>Long version: cover one or more proof ideas. Start off with [[proof of product rule for differentiation using difference quotients]].<br>Cover one or more of the topics: [[product rule for differentiation#Compatibility checks\|compatibility checks]], [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]] <br>After introducing the [[natural logarithm]], consider the [[proof of product rule for differentiation using logarithmic differentiation]].
	\|}		\|}

Vipul at 03:06, 28 November 2011

2011-11-28T03:06:58Z

← Older revision		Revision as of 03:06, 28 November 2011
Line 2:		Line 2:

	==Pedagogical contexts==		==Pedagogical contexts==

			===Course contexts===

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
Line 16:		Line 18:
	\| single variable calculus class for math majors, with strong focus on proofs (including coming up with proofs oneself) \|\| Teach how one would prove it, how one would think of it, and how it fits in with the other rules. \|\| cover one or more proof ideas. Start off with [[proof of product rule for differentiation using difference quotients]].<br>Cover one or more of the topics: [[product rule for differentiation#Compatibility checks\|compatibility checks]], [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]] <br>After introducing the [[natural logarithm]], consider the [[proof of product rule for differentiation using logarithmic differentiation]].		\| single variable calculus class for math majors, with strong focus on proofs (including coming up with proofs oneself) \|\| Teach how one would prove it, how one would think of it, and how it fits in with the other rules. \|\| cover one or more proof ideas. Start off with [[proof of product rule for differentiation using difference quotients]].<br>Cover one or more of the topics: [[product rule for differentiation#Compatibility checks\|compatibility checks]], [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]] <br>After introducing the [[natural logarithm]], consider the [[proof of product rule for differentiation using logarithmic differentiation]].
	\|}		\|}

			===Independent learners===

			For self-directed learning, consider the following:

			* First follow the coverage suggestions for the "high school calculus" version.
			* Then, if you are highly interested in proof-based mathematics, skip directly to the "single variable calculus for math majors, with strong focus on proofs" version.
			* Otherwise, continue to the more basic "single variable calculus (college class)" version, and if interested further, fill in the proof-related gaps later.

	==Error types==		==Error types==

Vipul: /* Pedagogical contexts */

2011-11-28T02:53:33Z

Pedagogical contexts

← Older revision		Revision as of 02:53, 28 November 2011
Line 14:		Line 14:
	\| multivariable calculus class \|\| cover [[product rule for partial differentiation]], [[product rule for differentation of dot product]], etc. \|\| Emphasize key features of the product rule and which of them do and don't generalize to the multivariable calculus situation.<br>At a later stage in the class, consider covering the [[proof of product rule for differentiation using chain rule for partial differentiation]].		\| multivariable calculus class \|\| cover [[product rule for partial differentiation]], [[product rule for differentation of dot product]], etc. \|\| Emphasize key features of the product rule and which of them do and don't generalize to the multivariable calculus situation.<br>At a later stage in the class, consider covering the [[proof of product rule for differentiation using chain rule for partial differentiation]].
	\|-		\|-
	\| single variable calculus class for math majors, with strong focus on proofs (including coming up with proofs oneself) \|\| Teach how one would prove it, how one would think of it, and how it fits in with the other rules. \|\| cover one or more proof ideas. Start off with [[proof of product rule for differentiation using difference quotients]].<br>Cover one or more of the topics: [[product rule for differentiation#Compatibility checks]], [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]] <br>After introducing the [[natural logarithm]], consider the [[proof of product rule for differentiation using logarithmic differentiation]].		\| single variable calculus class for math majors, with strong focus on proofs (including coming up with proofs oneself) \|\| Teach how one would prove it, how one would think of it, and how it fits in with the other rules. \|\| cover one or more proof ideas. Start off with [[proof of product rule for differentiation using difference quotients]].<br>Cover one or more of the topics: [[product rule for differentiation#Compatibility checks\|compatibility checks]], [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]] <br>After introducing the [[natural logarithm]], consider the [[proof of product rule for differentiation using logarithmic differentiation]].
	\|}		\|}

Vipul: /* Pedagogical contexts */

2011-11-28T02:52:46Z

Pedagogical contexts

← Older revision		Revision as of 02:52, 28 November 2011
Line 6:		Line 6:
	! Course level !! Goal !! Coverage suggestions		! Course level !! Goal !! Coverage suggestions
	\|-		\|-
	\| high school calculus (e.g., AP Calculus, IB Calculus, or A levels) \|\| Computational fluency with using product rule numerically and algebraically \|\| Focus on the formula aspect for product of two functions. Relevant sections: [[product rule for differentiation#Statement with symbols]], [[product rule for differentiation#Computational feasibility significance]] (provide examples and don't overemphasize the theoretical significance), [[product rule for differentiation#Examples]] (exclude sanity checks in first treatment except as asides).		\| high school calculus (e.g., AP Calculus, IB Calculus, or A levels) \|\| Computational fluency with using product rule numerically and algebraically \|\| Focus on the formula aspect for product of two functions. Relevant sections: [[product rule for differentiation#Statement with symbols]], [[product rule for differentiation#Computational feasibility significance]] (provide examples and don't overemphasize the theoretical significance), [[product rule for differentiation#Examples]] (exclude sanity checks in first treatment except as asides).<br>Do more worked examples or assign worked examples as homework. The examples should cover all the function types that the students are familiar with, and as they see new functions, they should see at least one application of the product rule to each new type of function.
	\|-		\|-
	\| single variable calculus class (college class or gifted/talented class) with partial focus on ''understanding'' theory (but not coming up with proofs of theorems by oneself) \|\| Build on existing computational fluency and provide insight into subtleties and existential questions, i.e., into ''what the results tell us qualitatively''. \|\| Review formula aspects quickly, then cover a selection of these based on time constraints: [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]], [[product rule for differentiation#Sanity checks]]. Exclude the compatibility checks as they may be too tricky to motivate, at least in the initial treatment.<br>Exclude proof from the initial treatment. If you wish to include some proof, consider a proof of [[differentiation is linear]], which is easier to understand and covers the same ideas.		\| single variable calculus class (college class or gifted/talented class) with partial focus on ''understanding'' theory (but not coming up with proofs of theorems by oneself) \|\| Build on existing computational fluency and provide insight into subtleties and existential questions, i.e., into ''what the results tell us qualitatively''. \|\| Review formula aspects quickly, then cover a selection of these based on time constraints: [[product rule for differentiation#Statement for multiple functions\|multiple functions]], [[product rule for differentiation#One-sided versions\|one-sided versions]], [[product rule for differentiation#Significance\|significance (qualitative/existential, computational feasibility, computational results)]], [[product rule for differentiation#Sanity checks\|sanity checks]]. Exclude the compatibility checks as they may be too tricky to motivate, at least in the initial treatment.<br>Exclude proof from the initial treatment. If you wish to include some proof, consider a proof of [[differentiation is linear]], which is easier to understand and covers the same ideas.
	\|-		\|-
	\| single variable calculus class, review time \|\| provide more insight into subtleties and existential questions \|\| Review the rule while doing [[integration by parts]] (see [[product rule for differentiation#Reversal for integration]])<br>Cover more subtleties/aspects of the rule at opportune moments, e.g., when covering [[vertical tangent]]s and [[vertical cusp]]s, when discussing one-sided differentiability, when discussing piecewise functions.<br>Cover the product rule when comparing with the [[product rule for differentiation of formal power series]].		\| single variable calculus class, review time \|\| provide more insight into subtleties and existential questions \|\| Review the rule while doing [[integration by parts]] (see [[product rule for differentiation#Reversal for integration]])<br>Cover more subtleties/aspects of the rule at opportune moments, e.g., when covering [[vertical tangent]]s and [[vertical cusp]]s, when discussing one-sided differentiability, when discussing piecewise functions.<br>Cover the product rule when comparing with the [[product rule for differentiation of formal power series]].