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The (Homo)morphism Concept: Didactic Transposition, Meta-Discourse and Thematisation

Abstract

This article focuses on the didactic transposition of the homomorphism concept and on the elaboration and evaluation of an activity dedicated to the teaching of this fundamental concept in Abstract Algebra. It does not restrict to Group Theory but on the contrary raises the issue of the teaching and learning of algebraic structuralism, thus bridging Group and Ring Theories and highlighting the phenomenon of thematisation. Emphasis is made on epistemological analysis and its interaction with didactics. The rationale of the isomorphism and homomorphism concepts is discussed, in particular through a textbook analysis focusing on the meta-discourse that mathematicians offer to illuminate the concepts. A piece of didactic engineering, informed by the preceding analysis and using epistemological insight as a meta-lever, is presented. The empirical results of a classroom realisation are discussed in the epistemological framework, through comparison of the a priori and a posteriori analysis. This experiment shows both the potential and the difficulties in connecting the homomorphism formalism to cognitive processes of comparison and identification.

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Notes

  1. 1.

    the word “meta” being used here in a different context and with a slightly different meaning. Nevertheless, meta-theories and meta-activities meet in so far as they both introduce a reflexive point of view.

  2. 2.

    Historically, this method was initially deployed by Hilbert to remedy the imperfections of Euclid’s axioms for geometry and give rigorous descriptions of mathematical objects suitable for applying the demonstrative apparatus. In this trend, logical use of the method is about questioning consistency, mutual independence, completeness of the set of axioms, in order to give foundations to the theory. In the context of Abstract Algebra, we are concerned with a more immanent use of axiomatics: properties of concrete objects are abstracted in order to acquire a better understanding of ideas underlying mathematical constructions and proofs. This allows a unification of objects and methods, leading to the structural point of view.

  3. 3.

    as is visible, for instance, on the mathematical forum http://www.les-mathematiques.net.

  4. 4.

    Algebra 1: Group Theory and Ring Theory.

  5. 5.

    Chevallard distinguishes ostensive objects, explicitly designated by the teacher, and non-ostensive objects which remain implicit but play a role in the execution or justification of a mathematical technique; ostensives and non-ostensives emerge together and are co-activated, but Chevallard argues that a mathematical activity is sensitive to ostensives which play a semiotic role.

  6. 6.

    The word “image” in vernacular language has not the same status as the symbolic notation I which isn’t needed in van der Waerden’s case since a homomorphism is onto. The importance of kernels and images in modern expositions is related to the development of homological algebra.

  7. 7.

    \(KH/H\simeq K/(K \cap H)\), where K,H are two subgroups of G and H is normal.

  8. 8.

    G/K≃(G/H)/(K/H) where H,K are normal subgroups of G, and K contains H.

  9. 9.

    \(\forall x,x^{\prime },y,y^{\prime }\in E,\ [x\mathcal {R}x^{\prime }\text { and } y\mathcal {R}y^{\prime }]\Rightarrow [xy\mathcal {R}x^{\prime }y^{\prime }]\).

  10. 10.

    The point is that “the quotient ring \(\bar {R}=R/I\) should be viewed as the ring obtained by introducing the n relations a 1=0,…,a n =0 into R”, where I is generated by the a i ’s. According to the third-IT, introducing a relation \(\bar {b}=0\) into the ring \(\bar {R}=R/(a)\) amounts to “killing a and b at the same time”: \(\bar {R}/(\bar {b})\approx R/(a,b)\).

  11. 11.

    which wasn’t automatic unlike f(0)=0, since A was not in general a group for the multiplicative law.

  12. 12.

    I actually said : “class of zero, one, two, three”. In the sequel, students also referred to \(\bar {x}\) as “class of x” most of the time, the exceptions being obvious abbreviations in case multiple classes were mentioned consecutively.

  13. 13.

    setting up analogies leads to conceive of an abstract theory and, once an abstract theory is at hand, it is used to unearth more and deeper analogies (Sinaceur 2014, p. 98).

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Hausberger, T. The (Homo)morphism Concept: Didactic Transposition, Meta-Discourse and Thematisation. Int. J. Res. Undergrad. Math. Ed. 3, 417–443 (2017). https://doi.org/10.1007/s40753-017-0052-7

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Keywords

  • Abstract algebra
  • Homomorphism concept
  • Algebraic structuralism
  • Didactic transposition
  • Thematisation