Tour:Pedagogical notes two (beginners)
Group
The main purpose of this part of the guided tour, with respect to the definition of group, is to give learners a clear understanding of how the different parts of the definition of group fit together and why each component is important.
For this purpose, some variations of the notion of group are introduced, including: maga (a set with a binary operation), semigroup (A set with an associative binary operation), monoid (a set with an associative binary operation having a two-sided identity element). Similarly, notions of neutral element/identity element, invertible and cancellative element are introduced.
Identity elements and the way they behave
The first fact page of the tour shows learners the statement that if a magma (set with binary operation) has a left neutral element and a right neutral element, they are both equal. While this is an extremely simple statement, the purpose of this page is to set the tone for learners, and make learners pause and think oabout the interplay between the binary operation and neutral elements. Learners are also encouraged to see for themselves how this implies that a binary operation can have at most one two-sided neutral element.
Some pedagogical comments:
- There is an extraordinary amount of parsing and reduction here. Rather than simply prove that the identity element of a group is unique, this gives a statement in a much greater generality. The purpose of this is to illustrate to learners that some statements can be viewed in substantially greater generality than we ordinarily perceive them. Of course, it is unlikely that learners will recognize this consciously.
- An entire page is devoted to a fact that would otherwise seem trivial. The idea is to impress upon learners that some facts are important, not because of their degree of difficulty, but because they control the language and framework of the subject. Again, it is unlikely that learners will recognize this consciously.
Inverses and the way they behave
Following closely on the heels of equality of left and right neutral elements is the statement of equality of left and right inverses with respect to a neutral element. Learners are encouraged to make careful note of the use of associativity, and are provided a justification for the manipulatino behind the proof. Learners are also encouraged to see how this implies that an element can have at most one two-sided inverse.
Some pedagogical comments:
- The similar pattern between this fact and the previous one, as well as the similar way in which both facts give rise to correspond facts about "two-sided" constructions, emphasizes the similarity and common pattern across mathematical results. It is unlikely that learners will recognize this consciosuly.
- The key difference between the facts, namely, the use of associativity in one case, also serves to highlight the importance of the axiomatic assumptions made in proving basic things. This might deepen learners' appreciation of associativity.
Definition equivalences for group and subgroup
The equivalence of definitions of group simply pieces together the newly learned facts with the definitions learned in part one. This serves as a double-revision as well as an exercise in piecing together ideas.
The equivalence of definitions of subgroup follows the establishment of the property that invertible elements can be cancelled -- another application of associativity. This equivalence of definitions further cements learners' understanding of the basic manipulations in groups.
Some pedagogical comments:
- It is likely that the similarity as well as difference between multiple definitions of group, as well as between multiple definitions of subgroup, will start to click by this stage in the learner's minds.
- Learners might also have a greater appreciation of the interplay between associativity, identity element and inverses by this stage.