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Groups and Group-Like Structures

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Piaget’s Genetic Epistemology for Mathematics Education Research

Part of the book series: Research in Mathematics Education ((RME))

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Abstract

Piaget defined logico-mathematical operations as reversible and composable mental actions. Because of this, formal mathematical structures, such as algebraic groups, played important roles in Piaget’s genetic epistemology. Group-like structures describe ways that mental actions can be organized and composed with one another. For example, mental actions of partitioning can be composed with one another to partition a partitioned whole, and such mental actions can be reversed by iterating one of the parts to reproduce the whole. This chapter details the ways that Piaget relied on groups and group-like structures to build models of mathematical development. Whereas research in mathematics education often refers to schemes as structures for organizing mental actions, it rarely mentions group-like structures. The chapter draws on the example of the splitting loope/group to illustrate how formal algebraic structures—as researcher constructs—can be useful in modeling children’s mathematical development. It also includes related exposition on Piaget’s INRC group, Klein’s Erlangen program, Noether’s algebraic invariants, and the mother structures identified by the Bourbaki.

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Notes

  1. 1.

    Historians credit Arthur Cayley for providing the first definition of an abstract group, in 1954. See Kleiner (1986) for a thorough and engaging review of its history, including contributions from Viète, Descartes, and (especially) Galois.

  2. 2.

    Here, Piaget is referring to Bertrand Russell’s circular definition of number, which Piaget (1971) critiques in Genetic Epistemology (see pp. 36–37).

  3. 3.

    “We are not given mathematical objects in isolation but rather in structures. That 13 is a prime number is not determined by some internal property of 13 but rather by its place in the structure of the natural numbers” (Resnick, 1981, p. 529).

  4. 4.

    Note that all finite groups satisfy the Latin squares property because they are associative, which is an even stronger condition. We can use associativity and the other properties of a group to prove that every element in a finite group appears exactly once in each row and column of the Cayley table.

  5. 5.

    Among books that have been translated to English, Psychology of Intelligence (2001/1947) and Mathematical Psychology and Epistemology (1966) stand out, so those are used as chief references on groupings.

  6. 6.

    Even within the same text, Piaget (1972a) sometimes referred to this relationship as idempotent (p. 90) and in other paces referred it to tautology (p. 97). It is not clear why.

  7. 7.

    In the case of multiplicative relations, Beth and Piaget (1966) characterized this dimension as bi-univocal (one to one, like a multiplication table) verses co-univocal (many to one, like tree branching).

  8. 8.

    Piaget (2001/1947) described the groupings of concrete operations as having a qualitative (not yet quantitative) character.

  9. 9.

    Piaget and Inhelder (1967/1948) acknowledge the influence of both Kant and Poincaré, at the start of their book, The Child’s Conception of Space.

  10. 10.

    Note that even algebraic manipulation relies on algebraic invariance: “manipulating a system of equations but maintaining its solution set” (Harel, 2008, p. 15).

  11. 11.

    Piaget notes as much in a separate account of the mother structures or “matrix structures” (Beth & Piaget, 1966, p. 164).

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Anderson Norton

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  1. Anderson Norton
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Editors and Affiliations

  1. Department of Mathematics, Texas State University, San Marcos, TX, USA

    Paul Christian Dawkins

  2. Department of Curriculum and Instruction, Indiana University, Bloomington, IN, USA

    Amy J. Hackenberg

  3. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Anderson Norton

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Norton, A. (2024). Groups and Group-Like Structures. In: Dawkins, P.C., Hackenberg, A.J., Norton, A. (eds) Piaget’s Genetic Epistemology for Mathematics Education Research. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-47386-9_7

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  • DOI: https://doi.org/10.1007/978-3-031-47386-9_7

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