Isomorphism: Abstract and Concrete Representations

  • Omid Khatin-ZadehEmail author
  • Hassan Banaruee
  • Zahra Eskandari
  • Fernando Marmolejo-Ramos
Ideas and Opinion


Looking at concrete representations of mathematical problems from an isomorphic perspective, this article suggests that every concrete representation of a mathematical concept is understood by reference to an underlying abstract representation in the mind of the comprehender. The complex form of every abstract representation of a problem is created by the gradual development of its elementary form. Throughout the process of cognitive development, new features are added to the elementary form of abstract representation, which leads to gradual formation of a fully developed abstract representation in the mind. Every developed abstract representation of a problem is the underlying source for understanding an infinite number of concrete isomorphic representations. Deep or abstract representations of a problem are shared by the concrete realizations or concrete forms of that problem. In other words, concrete representations of a problem are the realizations of a single abstract representation. This discussion is extended to mind-brain relationship and the possible isomorphism that could exist between mind and brain.


Cognitive development Isomorphism Representation Abstract algebra 


Compliance with Ethical Standards

Conflict of Interest

The authors declare that they have no conflict of interest.


  1. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241.CrossRefGoogle Scholar
  2. Benovsky, J. (2016). Dual aspect monism. Philosophical Investigations, 39(4), 335–352.CrossRefGoogle Scholar
  3. Bob, P. (2011). Brain, mind and consciousness: Advances in neuroscience research. Springer Science & Business Media.Google Scholar
  4. Booth, R. L., & Thomas, M. J. (1999). Visualization in mathematics learning: Arithmetic problem-solving and student difficulties. Journal of Mathematical Behavior, 18(2), 169–190. Scholar
  5. Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Durán, R., Reed, B., & Webb, D. (1997). Learning by understanding: the role of multiple representations in learning algebra. American Educational Research Journal, 34(4), 663–689. Scholar
  6. Cobb, P. (2002). Reasoning with tools and inscriptions. Journal of the Learning Sciences, 11(2), 187–215.,2-3n_3.CrossRefGoogle Scholar
  7. Disessa, A. A. (2000). Meta-representation: an introduction. Journal of Mathematical Behavior, 19(4), 385–398. Scholar
  8. Disessa, A. A. (2004). Metarepresentation: native competence and targets for instruction. Cognition and Instruction, 22(3), 293–331. Scholar
  9. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.CrossRefGoogle Scholar
  10. Epstein, W., & Hatfield, G. (1994). Gestalt psychology and the philosophy of mind. Philosophical Psychology, 7(2), 163–181.CrossRefGoogle Scholar
  11. Fanselow, M. S., Zelikowsky, M., Perusini, J., Barrera, V. R., & Hersman, S. (2013). Isomorphisms between psychological processes and neural mechanisms: From stimulus elements to genetic markers of activity. Neurobiology of Learning and Memory, 108, 5–13.CrossRefGoogle Scholar
  12. Fraleigh, J. B. (2003). A first course in abstract algebra (7th ed.). Saddle River, NJ: Pearson.Google Scholar
  13. Gray, E., Pitta, D., & Tall, D. O. (2000). Objects, actions, and images: A perspective on early number development. Journal of Mathematical Behavior, 18(4), 401–413. Scholar
  14. Judson, T. W. (2012). Abstract algebra: theory and applications. Orthogonal Publishing L3C.Google Scholar
  15. Kandel, E. R., & Squire, L. R. (2000). Neuroscience: Breaking down scientific barriers to the study of brain and mind. Science, 290(5494), 1113–1120.CrossRefGoogle Scholar
  16. Khatin-Zadeh, O., & Vahdat, S. (2015). Abstract and concrete representations in structure-mapping and class-inclusion. Cognitive Linguistic Studies, 2(2), 349–360.CrossRefGoogle Scholar
  17. Khatin-Zadeh, O., Vahdat, S., & Yazdani-Fazlabadi, B. (2016). An algebraic perspective on implicit and explicit knowledge. Cognitive Linguistic Studies, 3(1), 151–162. Scholar
  18. Khatin-Zadeh, O., Yarahmadzehi, N., & Banaruee, H. (2018). A neuropsychological perspective on deep or abstract homogeneity among concretely different systems. Activitas Nervosa Superior, 60(2), 68–74.CrossRefGoogle Scholar
  19. Marmolejo-Ramos, F., Khatin-Zadeh., O, Yazdani-Fazlabadi, B., Tirado, C., & Sagi, E. (2017). Embodied concept mapping: blending structure-mapping and embodiment theories. Pragmatics & Cognition, 24(2), 164–185.Google Scholar
  20. Moschkovich, J. N. (2008). “I went by twos, he went by one”: Multiple interpretations of inscriptions as resources for mathematical discussions. Journal of the Learning Sciences, 17(4), 551–587. Scholar
  21. Mulligan, J. T. (2011). Towards understanding of the origins of children’s difficulties in mathematics learning. Australian Journal of Learning Difficulties., 16(1), 19–39. Scholar
  22. Mulligan, J. T., Mitchelmore, M. C., English, L. D., & Robertson, G. (2010). Implementing a Pattern and Structure Awareness Program (PASMAP) in kindergarten. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education: Proceedings of the 29th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 797–804). Fremantle: MERGA.Google Scholar
  23. Pitta-Pantazi, D., Gray, E., & Christou, C. (2004). Elementary school students’ mental representations of fractions. In M. J. Høines & A. Fuglestad (Eds.), Proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 41–48). Bergen: Bergen University College.Google Scholar
  24. RAND Mathematics Study Panel. (2003). Mathematical proficiency for all students. Santa Monica: RAND.Google Scholar
  25. Rolls, E. T. (2012). Neuroculture: on the implications of brain science. Oxford: Oxford University Press.Google Scholar
  26. Rolls, E. T., & Deco, G. (2010). The Noisy brain: stochastic dynamics as a principle of brain function. Oxford: Oxford University Press.CrossRefGoogle Scholar
  27. Selling, S. K. (2016). Learning to represent, representing to learn. Journal of Mathematical Behavior, 41, 191–209. Scholar
  28. Thomas, N., & Mulligan, J. T. (1995). Dynamic imagery in children's representations of number. Mathematics Education Research Journal, 7(1), 5–25. Scholar
  29. Warren, E. (2005). Young children’s ability to generalize the pattern rule for growing patterns. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of mathematics education (Vol. 4, pp. 305–312). Melbourne: Program Committee.Google Scholar
  30. White, T., & Pea, R. (2011). Distributed by design: on the promises and pitfalls of collaborative learning with multiple representations. Journal of the Learning Sciences, 20(3), 489–547. Scholar

Copyright information

© Neuroscientia 2019

Authors and Affiliations

  • Omid Khatin-Zadeh
    • 1
    Email author
  • Hassan Banaruee
    • 1
  • Zahra Eskandari
    • 1
  • Fernando Marmolejo-Ramos
    • 2
  1. 1.Chabahar Maritime UniversityChabaharIran
  2. 2.University of AdelaideAdelaideAustralia

Personalised recommendations