Skip to main content

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


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.

This is a preview of subscription content, access via your institution.


  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

  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).


  1. Artaud, M. (1998). Introduction à l’approche écologique du didactique - L’écologie des organisations mathématiques et didactiques. In M. Bailleul, C. Comiti, J.-L. Dorier, J.-B. Lagrange, B. Parzysz, & M.-H. Salin (Eds.), Actes de la neuvième Ecole d’ete de didactique des mathématiques (pp. 101–139). Caen: ARDM & IUFM.

  2. Artigue, M. (1991). Epistemologie et didactique. Recherches en Didactique des Mathématiques, 10(2.3).

  3. Artigue, M. (2009). Didactical design in mathematics education. In C. Winsløw (Ed.), Proceedings of NORMA08 Nordic Research in Mathematics Education (pp. 7–16). Rotterdam: Sense Publishers.

  4. Artin, E. (1962). Zur Problemlage der Mathematik In Lang, S., & Tate, J. (Eds.), The Collected Papers: Addison-Wesley.

  5. Artin, M. (1991). Algebra. Englewood Cliffs: Prentice-Hall.

    Google Scholar 

  6. Bourbaki (1948, reed. 1998). L’architecture des mathématiques. In F. Le Lionnais (Ed.), Les grands courants de la pensée mathématique. Paris: Hermann.

  7. Brousseau, G. (1986). Fondements et méthodes de la didactique des mathématiques. Recherches en Didactique des Mathématiques, 7(2), 33–55.

    Google Scholar 

  8. Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer.

    Google Scholar 

  9. Cavaillès, J. (2008). Oeuvres complètes de Philosophie des Sciences. Paris: Hermann.

    Google Scholar 

  10. Chevallard, Y. (1985). La transposition didactique - Du savoir savant au savoir enseigné. Grenoble: La Pensée sauvage.

    Google Scholar 

  11. Chevallard, Y., & Bosch, M. (1999). La sensibilité de l’activité mathématique aux ostensifs. Recherches en didactique des mathématiques, 19(1), 77–124.

    Google Scholar 

  12. Corry, L. (2004). Modern Algebra and the Rise of Mathematical Structures, 2nd edn. Basel: Birkhäuser.

    Book  Google Scholar 

  13. Dorier, J. -L., Robert, A., Robinet, J., & Rogalski, M. (2000). The meta lever. In Dorier (Ed.), On the teaching of linear algebra (p. 151–176). Dordrecht: Kluwer Academic Publisher.

  14. Douady, R. (1986). Jeux de cadres et dialectique outil-objet. Recherches en didactique des mathématiques, 7(2), 5–31.

    Google Scholar 

  15. Dubinsky, E. (1991). Advanced Mathematical Thinking In Tall, D.O. (Ed.), Reflective abstraction in mathematical thinking, (pp. 95–123). Dordrecht: Kluwer Academic Publisher.

    Google Scholar 

  16. Durand-Guerrier, V., Hausberger, T., & Spitalas, C. (2015). Définitions et exemples : prérequis pour l’apprentissage de l’algèbre moderne. Annales de Didactique et de Sciences Cognitives, 20, 101–148.

    Google Scholar 

  17. Dyck, v. W. (1882). Gruppentheoretische studien. Mathematische Annalen, 20 (1), 1–44.

    Article  Google Scholar 

  18. Gueudet, G. (2008). Investigating the secondary-tertiary transition. Educational Studies in Mathematics, 67(3), 237–254.

    Article  Google Scholar 

  19. Guin, D. (1997). Algèbre Tome 1 : groupes et anneaux. Paris: Belin.

    Google Scholar 

  20. Hausberger, T. (2012). Enseignement des mathématiques et contrat social, Enjeux et défis pour le 21 e siècle, Actes du colloque EMF2012 In Dorier, J.-L., & Coutat, S. (Eds.), Le challenge de la pensée structuraliste dans l’apprentissage de llalgèbre abstraite : une approche épistémologique, (pp. 425–434). Genève: Université de Genève.

    Google Scholar 

  21. Hausberger, T. (2016a). Abstract algebra, mathematical structuralism and semiotics. In Krainer, K., & Vondrová, N. (Eds.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (pp. 2145–2151). Prague: Faculty of Education, Charles University.

  22. Hausberger, T. (2016b). A propos des praxéologies structuralistes en algèbre abstraite. In Nardi, E., Winsløw, C., & Hausberger, T. (Eds.), Proceedings of the 1st Congress of the International Network for Didactic Research in University Mathematics (pp. 296–305). Montpellier: University of Montpellier and INDRUM.

  23. Hazzan, O. (1999). Reducing abstraction level when learning Abstract Algebra concepts. Educational Studies in Mathematics, 40, 71–90.

    Article  Google Scholar 

  24. Lautmann, A. (2006). Les mathématiques, les idées et le réel physique. Paris: Vrin.

    Google Scholar 

  25. Leron, U., & Dubinsky, E. (1995). An Abstract Algebra story. American Mathematical Monthly, 102(3), 227–242.

    Article  Google Scholar 

  26. Leron, U., Hazzan, O., & Zazkis, R. (1995). Learning group isomorphism: a crossroads of many concepts. Educational Studies in Mathematics, 29, 153–174.

    Article  Google Scholar 

  27. Mac Lane, S. (1996). Structure in mathematics. Philosophia Mathematica, 4 (2), 174–183.

    Article  Google Scholar 

  28. McLarty, C. (2006). The architecture of Modern Mathematics: Essays in History and Philosophy In Ferreiros, J., & Gray, J.J. (Eds.), Emmy Noether’s “Set Theoretic” Topology: From Dedekind to the Rise of Functors, (pp. 187–208). New York: Oxford University Press.

    Google Scholar 

  29. Nardi, E. (2000). Mathematics Undergrates’ Responses to Semantic Abbreviations, Geometric Images and Multi-level Abstractions in Group Theory. Educational Studies in Mathematics, 34, 169–189.

    Article  Google Scholar 

  30. Piaget, J. (1972). The principles of genetic epistemology. London: Routledge and Kegan Paul.

    Google Scholar 

  31. Piaget, J. (1985). The Equilibration of Cognitive Structures, Harvard University Press.

  32. Robert, A. (1987). Cahier de didactique des mathématiques. De quelques spécificités de l’enseignement des mathématiques dans l’enseignement post-obligatoire Vol. 47. Paris: IREM de Paris 7.

  33. Rogalski, M. (1995). Seminaire DidaTech. Que faire quand on veut enseigner un type de connaissances tel que la dialectique outil-objet ne semble pas marcher et qu’il n’y a apparemment pas de situation fondamentale? L’exemple de l’algèbre linéaire (Vol. 169, pp. 127–162). Grenoble: Université Joseph Fourier.

  34. Russell, B. (1903). The Principles of Mathematics, Cambridge University Press.

  35. Scheiner, T., & Pinto, M. (2014). Cognitive processes underlying mathematical concept instruction: the missing process of structural abstraction. In Nicol, C., & et al. (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA, (Vol. 36 pp. 105–112). Vancouver: PME.

  36. Simpson, A., & Stehlíková, N. (2006). Apprehending mathematical structures: a case study of coming to understand a commutative ring. Educational Studies in Mathematics, 61, 347–371.

    Article  Google Scholar 

  37. Sinaceur, H. (2014). Facets and levels of mathematical abstraction. Philosophia Scientiæ, 18(1), 81–112.

    Article  Google Scholar 

  38. Waerden, B. L. v. (1930). Moderne Algebra Vol. 2. Berlin: Springer.

  39. Winsløw, C. (2008). Actes de la XIIIème Ecole d’Eté de Didactique des Mathématiques In Rouchier, R., & et al. (Eds.), Transformer la théorie en tâches : la transition du concret à l’abstrait en analyse r’eelle, (pp. 1–12 Cédérom). Grenoble: La Pensée Sauvage.

    Google Scholar 

  40. Wussing, H. (2007). The Genesis of the Abstract Group Concept, Dover Publications.

Download references

Author information



Corresponding author

Correspondence to Thomas Hausberger.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hausberger, T. The (Homo)morphism Concept: Didactic Transposition, Meta-Discourse and Thematisation. Int. J. Res. Undergrad. Math. Ed. 3, 417–443 (2017).

Download citation


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