Tour:Pedagogical notes one (beginners) - Revision history

Vipul: /* Tolerance thresholds for learners */

2010-02-21T01:37:01Z

Tolerance thresholds for learners

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	* Why give multiple, seemingly equivalent, definitions of group? Why are they equivalent?		* Why give multiple, seemingly equivalent, definitions of group? Why are they equivalent?

	The tour pages on verifying the group axioms and understanding the definition of a group, help resolve some of these. However, not all these concerns are adequately addressed in part one of the tour. Thus, it is best for learners ''not'' to expect a complete explanation at the end of part one.		The last two tour pages, which are on verifying the group axioms and understanding the definition of a group, help resolve some of these. However, not all these concerns are adequately addressed in part one of the tour. Thus, it is best for learners ''not'' to expect a complete explanation at the end of part one.

	To help address learners' curiosity, there is a separate entertainment menu page on the tour: [[Tour:Entertainment menu one (beginners)]]. This links to other pages dealing with the history of groups, groups as symmetry, and some fun pages about groups.		To help address learners' curiosity, there is a separate entertainment menu page on the tour: [[Tour:Entertainment menu one (beginners)]]. This links to other pages dealing with the history of groups, groups as symmetry, and some fun pages about groups.

Vipul at 16:45, 15 March 2009

2009-03-15T16:45:59Z

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	pagetype = Pedagogical notes\|		pagetype = Pedagogical notes\|
	target = beginners\|		target = beginners\|
			secnum = one\|
	next secnum = two}}		next secnum = two}}

Vipul at 16:45, 15 March 2009

2009-03-15T16:45:36Z

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			{{tour-special\|
			pagetype = Pedagogical notes\|
			target = beginners\|
			next secnum = two}}

	This page has pedagogical notes giving an explanation of the content, scope and organization of part one of the guided tour. We focus on the four concepts introduced here, and the follow-up to each of these in later parts.		This page has pedagogical notes giving an explanation of the content, scope and organization of part one of the guided tour. We focus on the four concepts introduced here, and the follow-up to each of these in later parts.

Vipul at 15:52, 8 September 2008

2008-09-08T15:52:27Z

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	The examples before the '''WHAT'S MORE''' part are not intended as a representative range, but rather, simply to give a quick idea of what a subgroup means. As in the case of group, these examples do not cover non-Abelian group. The examples after '''WHAT'S MORE''' include non-Abelian groups.		The examples before the '''WHAT'S MORE''' part are not intended as a representative range, but rather, simply to give a quick idea of what a subgroup means. As in the case of group, these examples do not cover non-Abelian group. The examples after '''WHAT'S MORE''' include non-Abelian groups.

			===Tolerance thresholds for learners===

			This particular style of definition may leave learners uncomfortable with regard to a number of things:

			* Why define subgroups in this way? What is the purpose of defining subgroups at all?
			* Why give multiple, seemingly equivalent, definitions of subgroup? Why are they equivalent?

			These concerns are not addressed in part one of the tour, except for a brief mention of this in [[Tour:Verifying the group axioms\|the tour page on verifying the group axioms]], where it is pointed out that it may be easier to prove that a subset of a group forms a subgroup under the induced operations, than prove from scratch that it is a group. That's because we only need to show closure and do not need to show associativity.

			Also, learners are promised that the concepts will be explored in more detail in part two of the tour, allowing them to defer the resolution of discomfort.

			==Abelian group==

			===Definition===

			A definition of Abelian group is presented, as a group where any two elements commute. The importance of Abelian groups is cleatr to learners as all examples of groups provided before the '''WHAT'S MORE''' section are Abelian.

			===Tolerance thresholds for learners===

			Learners may have the following natural questions:

			* How does the theory of groups and Abelian groups differ? Why create a separate notion for Abelian group?

			This is ''not'' resolved in this part of the tour.

Vipul at 13:43, 8 September 2008

2008-09-08T13:43:19Z

← Older revision		Revision as of 13:43, 8 September 2008
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	This page has pedagogical notes giving an explanation of the content, scope and organization of part one of the guided tour. We focus on the four concepts introduced here, and the follow-up to each of these in later parts.		This page has pedagogical notes giving an explanation of the content, scope and organization of part one of the guided tour. We focus on the four concepts introduced here, and the follow-up to each of these in later parts.

			{{#lst:Tour:Introduction one (beginners)\|goal}}
	==Group==		==Group==

Vipul at 18:05, 30 July 2008

2008-07-30T18:05:17Z

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	Also, learners are promised that the concepts will be explored in more detail in part two of the tour, allowing them to defer the resolution of discomfort.		Also, learners are promised that the concepts will be explored in more detail in part two of the tour, allowing them to defer the resolution of discomfort.

			==Subgroup==

			===Definition===

			Three definitions of subgroup are presented. The first definition simply talks of a subset closed under the binary operation of the group, and requires the subset to form a group under the binary operation. The second definition talks of a subset closed under all the three group operations: the binary operation, the inverse map, and the identity element. The third definition is in terms of the subgroup criterion.

			Learners are expected to go through all definitions, particularly understanding the first two. Again, it is unlikely that first-time learners will appreciate the difference between the definitions, but it is possible that seeing these two equivalent definitions of subgroup helps learners understand the difference between the two definitions of group. The third definition, involving the subgroup criterion, definitely looks different, and learners are not expected to immediately see why it is equivalent to the first two definitions. The equivalence of these definitions is presented in part two.

			===Notation===

			The notation for a subgroup is fairly straightforward, and learners are expected to grasp and remember it.

			===Examples===

			The examples before the '''WHAT'S MORE''' part are not intended as a representative range, but rather, simply to give a quick idea of what a subgroup means. As in the case of group, these examples do not cover non-Abelian group. The examples after '''WHAT'S MORE''' include non-Abelian groups.

Vipul at 17:15, 30 July 2008

2008-07-30T17:15:28Z

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	The examples after the '''WHAT'S MORE''' cover a more representative range of examples. However, these examples may require some effort and/or prerequisites to comprehend, and are best read during a second round of the tour.		The examples after the '''WHAT'S MORE''' cover a more representative range of examples. However, these examples may require some effort and/or prerequisites to comprehend, and are best read during a second round of the tour.

	===Tolerance thresholds for ~~learner~~===		===Checking that something is a group===

			This is discussed in some detail in the tour page: [[Tour:Verifying the group axioms]], along with some illustrative examples. This should give learners a reasonable introductory idea of how to determine whether something is a group, although a better intuition in the matter can only come later.

			===Understanding the definition better===

			The tour page [[Tour:Understanding the definition of a group]] looks at the various components of the definition of a group, why they are there, and why being a group is a good thing. While this page does not give a complete explanation of why groups are important, it can help resolve the initial curiosity of learners.

			===Tolerance thresholds for learners===

	This particular style of definition may leave learners uncomfortable with regard to a number of things:		This particular style of definition may leave learners uncomfortable with regard to a number of things:
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	* What are the examples of groups?		* What are the examples of groups?
	* What is the importance of groups?		* What is the importance of groups?
			* Why give multiple, seemingly equivalent, definitions of group? Why are they equivalent?

			The tour pages on verifying the group axioms and understanding the definition of a group, help resolve some of these. However, not all these concerns are adequately addressed in part one of the tour. Thus, it is best for learners ''not'' to expect a complete explanation at the end of part one.

			To help address learners' curiosity, there is a separate entertainment menu page on the tour: [[Tour:Entertainment menu one (beginners)]]. This links to other pages dealing with the history of groups, groups as symmetry, and some fun pages about groups.

	~~Some of these questions~~ are ~~addressed later~~ in the tour, ~~but for most of these, answers emerge over time, with reasonable answers emerging by~~ the ~~end~~ of ~~parts two and three~~.		Also, learners are promised that the concepts will be explored in more detail in part two of the tour, allowing them to defer the resolution of discomfort.

Vipul: New page: This page has pedagogical notes giving an explanation of the content, scope and organization of part one of the guided tour. We focus on the four concepts introduced here, and the follow-u...

2008-07-30T17:07:20Z

New page: This page has pedagogical notes giving an explanation of the content, scope and organization of part one of the guided tour. We focus on the four concepts introduced here, and the follow-u...

New page

This page has pedagogical notes giving an explanation of the content, scope and organization of part one of the guided tour. We focus on the four concepts introduced here, and the follow-up to each of these in later parts.

==Group==

===Definition===

Two definitions of group are presented. One definition starts by assuming only a binary operation, and describes the identity element and inverses as additional conditions on this binary operation. In the other definition, the binary operation, identity element and inverses are considered part of the group structure.

The difference between these definitions may not be easy to spot for first-time learners (and it may be missed by experienced mathematicians as well, since the two definitions are obviously equivalent for them). Thus, putting the two definitions is not intended to produce an immediate reaction in learners, but rather, to introduce them to the fact that there ''are'' two slightly different definitions. The difference between the definitions, and why they are equivalent, is presented in part two.

===Notation===

The typical notational conventions for a group are introduced here. Again, it is not expected that first-time learners will completely absorb all these pieces of notation. The primary purpose of introducing this notation is to give learners a first-time introduction to the notation. The notation is reinforced in the many subsequent pages.

===Examples===

The examples given in the main part of the page (before the '''WHAT'S MORE''') rely on basic concepts like integers, rational numbers and reals. These examples do ''not'' form a representative range: one common defect to them is that they are all Abelian. However, they illustrate aspects like existence of identity elements and existence of inverses (when contrasted with the accompanying non-examples).

The examples after the '''WHAT'S MORE''' cover a more representative range of examples. However, these examples may require some effort and/or prerequisites to comprehend, and are best read during a second round of the tour.

===Tolerance thresholds for learner===

This particular style of definition may leave learners uncomfortable with regard to a number of things:

* Why define groups in this way? Why define groups at all?
* Why not assume commutativity for groups?
* What are the examples of groups?
* What is the importance of groups?

Some of these questions are addressed later in the tour, but for most of these, answers emerge over time, with reasonable answers emerging by the end of parts two and three.