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

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 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.

Esperienze didattiche di modellizzazione emergente: uno sguardo ai processi di matematizzazione

“…L'obiettivo di apprendimento per questa attività di modellizzazione emergente è stato identificato nella proprietà distributiva (nel seguito P. D.) della moltiplicazione rispetto all'addizione: a × (b + c) = a × b + a × c, con a, b, c numeri naturali. Si ricorda brevemente come la P. D. sia fondamentale in algebra, per lo studio dei polinomi e spazi vettoriali, e caratterizzi le operazioni tra numeri interi, e per tale motivo sia importante introdurla sin dai primi anni della scuola primaria (Maffia & Mariotti, 2020).…”

A Literature Review Of Multimodal Semiotic Reasoning In Mathematics

“…In mathematics class, teachers involve more concrete objects, pictures, signs, or symbols to help students do mathematical reasoning (Fiantika et al, 2018;Maffia & Mariotti, 2020). The reasoning that students do by involving signs and symbols is used to get the best explanation based on the ability to remember facts that have been seen and observed by students (Reisberg, 2017;C.…”

Iconic Mathematics: Math Designed to Suit the Mind

“…Hands-on mathematics has a long history and features prominently in Montessori education ( Laski et al, 2015 ). Earlier in the 20th century, Laisant’s table ( Laisant, 1915 ; Maffia and Mariotti, 2020 ) allowed students to physically explore—by counting squares—the meaning and validity of the distributive law of multiplication ( Figure 1 ). And physically rearranging simple geometrical shapes has been used as far back as during the Han dynasty, 206 B.C.−220 A.D. ( Wang, 2009 ), to prove the Pythagorean theorem and resolve problems like finding the side of an unknown square that just fits a known right triangle ( Figure 2 ; for video demonstrations, see footnote 1 ).…”

“… Laisant’s table [adapted from Maffia and Mariotti (2020) ]. Each black-framed rectangle represents a multiplication—that of its width in little squares by its height in little squares.…”

“…One major problem remains, however, a meta-analysis found that whereas hands-on interaction considerably improves retention of abstract mathematical facts or operations, it does not have as big a positive effect on students’ ability to solve new abstract mathematical problems ( Carbonneau et al, 2013 ). Combining embodied and symbolic mathematics helps establish a link between the two and facilitates the desired transfer of learned skills from the one to the other ( Laski et al, 2015 ; Coles and Sinclair, 2019 ; Maffia and Mariotti, 2020 ). Still, there obviously remains a gap between embodied cognition and embodied education on the one hand and un-embodied, formal mathematics on the other.…”