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From action to symbols: giving meaning to the symbolic representation of the distributive law in primary school

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Abstract

The use of artifacts to introduce the distributive law of multiplication over addition in primary school is a diffused approach: it is possible to find pre-constructed learning trajectories in instructional materials. However, it is still unclear how the teacher might support his/her students in transitioning from concrete to symbolic representations of the distributive law. In the theoretical frame of the Theory of Semiotic Mediation, we report on a study where Laisant’s table, an artifact embodying the rectangular model of multiplication, is used to introduce distributive law in second grade. Taking a microanalytical approach, we show how a group of students connects the representation provided by the artifact with the symbolic representation of the arithmetic property (as equivalence of numerical sentences). Two different semiotic chains are identified and presented, showing the continuity between the activity with the artifact and the mathematical signs emerging in following activities and promoted by tasks specifically designed. The role of the teacher in triggering and scaffolding this process is highlighted.

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Notes

  1. The Italian reading of multiplications is different from the usual English one. In Italian, 4 × 9 is read as “quattro per nove,” meaning four repeated nine times.

  2. The phrase used by the researcher is “è uguale.” In Italian, it has different meanings according to the linguistic register. It means “it equals” in the formal-mathematical register, and it means “it looks the same as” in colloquial register.

  3. Likely, the association of the metaphor of putting together to the operation of adding was already known by the students. Such metaphor is common in Italian textbooks for first grade.

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Authors and Affiliations

  1. University of Pavia, Pavia, Italy

    Andrea Maffia

  2. University of Siena, Siena, Italy

    Maria Alessandra Mariotti

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  1. Andrea Maffia
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  2. Maria Alessandra Mariotti
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Correspondence to Andrea Maffia.

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Maffia, A., Mariotti, M.A. From action to symbols: giving meaning to the symbolic representation of the distributive law in primary school. Educ Stud Math 104, 25–40 (2020). https://doi.org/10.1007/s10649-020-09944-5

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