Section (vi) The First Isomorphism Theorem as a Container of Compressed Conceptual Group-Theoretical Obstacles

Section (vi) The First Isomorphism Theorem as a Container of Compressed Conceptual Group-Theoretical Obstacles

Context: See Extract 9.6

Structure: The tutor leads the students through the proof of the First Isomorphism Theorem via very closed questioning. The overall impression of the sessions is that even when the students respond to the questions correctly they do not seem to have a global understanding of what is happening and how these questions relate to the proof.

Note to the reader: since the following is a fragmentary presentation from the four tutorials, the proof of the theorem is not fully presented in any of them. Thus, in order to follow the Factual Account, I recommend reading the proof in fig.6 first.

The Episode: A Factual Account. See Extract 9.6

An Interpretive Account: The Analysis

In the following, I comment on the sessions individually as well as examine them comparatively.

Memory Retrieval as a Necessary but not Sufficient Condition for Meaningful Understanding. Successful retrieval of the theorem varies in the fours sessions. Except Beth, the other students either remain silent or contribute very weak associations (Eleanor with normal subgroup, for instance). Patricia's association with 'division' (possibly invoked by the use of '/' in the statement of the theorem) is accompanied by her query on the meaning of ~. I note that similar evidence was given by these students with regard to Lagrange's Theorem.

Patricia's interpretation (~ means =) does not make sense because G/K contains sets of sets and Imf contains elements of the group. It nevertheless reflects the novices' muddled perception of ~ stemming from the diverse meanings that this symbol has in various mathematical contexts. Student Camille in Episode 7.4 is querying the tutor about the meaning and various uses of the ~ symbol. As elaborated upon below, Patricia's non-sensical interpretation of ~ as = conveys how problematic the perception of the newly introduced notion of isomorphism is.

In sum, retrieval of the theorem is generally problematic. The tutor, most notably in Tutorial 4, sounds alarmed and firmly reminds the students (similarly to the tutorials on Lagrange's Theorem) of the importance of 'a theorem with a name attached to it'. The tutor's firm and strict recommendation is a pragmatic and not an epistemological argument. In fact the novices have given contradictory evidence as to the power of persuasion of the different arguments: so — whilst, for instance, Connie in Extract 9.4 does not sound satisfied at all with pragmatic explanations of the type 'you need to learn this because it will be of use later' and asks for the raison-d'-être of most concepts she is introduced to — in most other occasions students seem to feel motivated to explore and understand some new concepts simply on the basis of its highly probable appearance on an exam paper. I do not see these two motivational forces as mutually exclusive but it seems to be didactically more appropriate to employ motivation of an epistemological rather than a pragmatic nature since the former is more likely to induce a more critically reflective approaching of the concepts than the former.

Incidentally I note how the students reproduce with relative ease information that has been given to them in catchy phrases:

Kg₁=Kg₂ iff g₁g₂^-1ÎK

the 'obvious map' from G/K to G is to map Kg to g

Unfortunately immediate retrieval does not precede immediate, deep or in fact any understanding of what the phrases mean. It remains at best an automatic and effective act, at worst an act void of meaning that fosters a false impression of achievement.

Similarly the students apply the rule of coset product, for instance, and calculate correctly but cannot make any decision about how to use it in order to support the completion of the proof. They comfortably provide answers to questions that constitute progressive steps of the proof as long as these steps have been predesigned by the tutor. Only once the tutor attempts a more global comment regarding the internal structure and resonance of the First Isomorphism Theorem (Tutorial 3) when she says that the 'point' of the proof lies in the strength of the homomorphic mechanism.

Another feeble but noteworthy sign of instrumental understanding of the homomorphic property is given by Abidul when she helps Frances (who is 'stuck' with f(g^-1)) by pointing out that 'it [the ^-1] doesn't matter', therefore f(g^-1) can be written as f(g)^-1. Abidul's phrasing ('doesn't matter') reflects a procedural, cause -and-effect view of f.

Finally, the tutor's comment on the mechanism of the proof as 'very standard' of proofs involving quotient groups is an attempt to generalise the approach used in the proof and hint at its potential as a methodological tool. In fact the 'standardisation' comes from the fact that the mapping Kg®g, which she has been calling the 'obvious map', is used quite often in group-theoretical proofs, when cosets need to be mapped on the elements of the group. She rarely makes similar comments and even in this case she does not elaborate further.

The Problematic ‹ Direction of (*) and the Properties of an Isomorphism. Except Frances and Beth the rest of the students have problems in interpreting the ‹ direction of (*) as the definition of 1-1 correspondence. Eleanor confuses it with onto but soon changes her mind and in Tutorial 3 a discussion is triggered that reveals a more general confusion.

First the students are interpreting (*) as providing information about homomorphism f, when in fact (*) provides information about the well-definedness and the 1-1 property of y. The students are deceived possibly because their interpretation is an interpretation by appearances. Moreover through P2-P6 Patricia and Cleo clarify, via the tutor's very leading questions (the multiple choice question they are given in T9 verges on the grotesque), their definition of an isomorphism. I note that, in this case of very directed questioning, the students' difficulties with the properties of an isomorphism (onto and 1-1 for instance) are not really explored because the teaching is solely oriented towards the elicitation of the answers that will further the progression of the proof.

Similarly in Tutorial 2 Beth appears to be severely concerned and confused as to the information contained in (*) as well as the homomorphic property of y. Like Patricia and Cleo, she does not carry out the switch from y to f flexibly and cannot understand how manipulating f can lead to an understanding of the properties of y.

Finally in tutorial 4, Abidul's A1 is another piece of evidence of the mechanical, despatched from conceptual understanding, conceptualisation of isomorphism y. Significantly in A1 the student starts dictating the necessary calculations for proving that y is a homomorphism: instead of

y(Kg₁Kg₂)=y(Kg₁)y(Kg₂)

she starts dictating

y(Kg₁g₂)=y(Kg₁)...

as if the homomorphic property has to be proved for g₁ and g₂ and not for Kg₁ and Kg₂; in other words as if y is defined on G, not G/K.

Actually in the co-ordination and understanding of the link between y and f, as well as the clarification about the definition of y lies largely the students' difficulty with the First Isomorphism Theorem and by implication with a large part of the newly introduced Group Theory. The degree of complexity in a problem which requires a well-co-ordinated manipulation of mappings between different sets is extremely high. f is defined between the elements of a group (or two groups). y is defined between the cosets of the kernel of f and the image of f. This link between y and f and the implications and importance of shifting back and forth from y to f need to be explicitly made to the learner. The shift from one level of abstraction to another is not self-evident. In the absence of a didactically illuminating decomposition of the theorem to its constituent elements, it is not surprising that the students are not capable of making these shifts to more abstract levels.

An Incident with kerf Reveals Problematic Conceptions of Mapping. In Tutorial 3, given the students' difficulty with the meaning of g₁g₂^-1ÎK, the tutor initiates a discussion of the definition of K = kerf. The tutor initially objects to Cleo's use of the term 'zero' for the identity element of G. Cleo quickly corrects (C2) — this is a common terminological mistake that the students habitually make and equally habitually correct when prompted by the tutor. Then in the explication of the definition (T3) it becomes evident that C1 was phrased in a rather problematic way not only because of the student's using the term 'zero': in Cleo's sentences the subject of the verb 'maps' in C1 and C3 is not clear at all. T4 is an interpretation that equates

Cleo's 'maps the elements to the identity'

with

'something is equal to identity'.

C4 possibly means 'element g goes to the identity element' and C5 that 'it goes via f'. The tutor does not explore any further Cleo's grammatical state of mind with regard to the definition of kerf and completes the writing on her own initiative. Therefore it remains an open question whether Cleo's perception of f as a mapping has been at all illuminated by the exchange of short verses in C1-C5. As in Extract 9.2, the student's words reflect an undecided perception of mapping as a machine. In this perception however it is not clear what is mapped where. Cleo's antonyms (it) as the subject of the verb 'maps' (C1 and C3) as well as her substituting the verbs (C4 and C5) with 'does' are lexical substitutions that possibly reflect and determine the ambiguity of her thinking.

Conclusion: In the above, evidence was given of a number of difficulties in the conceptualisation of properties associated to the notion of mapping (homomorphic property, onto, 1-1, well-definedness of a mapping); also of the varying degrees of abstraction involved in the definition of a mapping between elements of a group or the cosets of a subgroup and the elements of the group. The high degree of abstraction and the conceptual difficulties have then been linked to the students' cognitive enpuzzlement in Group Theory which culminates at the introduction and proof of the First Isomorphism Theorem for Groups. A didactical decomposition of the constituent elements of the theorem was then pointed out as a potentially helpful tool for the understanding of its content and proof.

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