ABSTRACT

Published for the first time in Spanish by the Araluce publisher in 1934, Maria Montessori’s Psycho Arithmetic and Psychogeometry presents an intensification of the Italian author’s educational research on school teaching, in a season characterized by numerous attempts at renewal of children’s mathematics education. The author herself, in the introduction to Psycho Arithmetic, writes that it is a “book that departs from the traditional conventions of teaching”. The book proposes an early approach to numbers, following the approach already presented in The Method of Scientific Pedagogy (1909), and introduces some aspects of algebra and divisibility not usually considered in popular primary education.

 

In this presentation, we propose an analysis of the concept of ratio in Psycho Arithmetic. This concept – that Charles Ange Laisant described as the basis of all elementary mathematics in Initiation mathématique – hinges the integration between arithmetic and geometry conceived for children’s education.

 

Geometrical ratio is present since the first pages of the essay, hidden in the physical materials devised by Montessori for introducing counting and natural numbers, notably in Montessori’s red and blue rods inspired by Édouard Séguin’s rods (Gil Clemente & Millán Gasca, 2021). Moreover, it is also used in a naïve form in the penultimate chapter entitled “Metric Decimal System”, where an appeal to geometric intuition regarding the ratio for segments, plane figures and solids is subjacent, albeit supported by drawings. The final chapter is entitled “Ratio and Proportion”. Apparently indicating the intention to offer a general mathematical approach that goes beyond mere practical needs, although fundamentally focused on explaining the rule of three with numerical proportions (all examples relate to numerical counting). The chapter is characterized by numerous simple numerical examples belonging to different application contexts and by a stylistic effort to establish a colloquial, challenging yet friendly tone, while also providing some basic definitions.

 

After some considerations on the circumstances of the publication of this groundbreaking essay, we analyze its content, highlighting the conceptual and expressive obstacles faced by the author.

 

SELECTED BIBLIOGRAPHY

Bjarnadóttir, Kristin (2014). History of Teaching Arithmetic. In A. Karp & G. Schubring (Eds.), Handbook of the history of mathematics education (pp. 431–457). New York: Springer.

De Giorgi, Fulvio (2012). Montessori, Maria. In Dizionario Biografico degli Italiani, vol. 76. https://www.treccani.it/enciclopedia/maria-montessori_%28Dizionario-Biografico%29/

Gil Clemente, Elena & Millán Gasca, Ana (2021). Geometry as “forceps of intelligence”: lines, figures and the plane in Édouard Séguin’s pedagogical thought. Bollettino di storia delle scienze matematiche, 41(2), pp. 315–339.

Millán Gasca, Ana (2015). Mathematics and children’s minds: The role of geometry in the European tradition from Pestalozzi to Laisant. Archives Internationales d'Histoire des Sciences, 65(2), pp. 261–277.

Montessori, Maria (1934). Psicoaritmética. Casa editorial Araluce, Barcelona. (Italian edition 1971, English critical edition 2000).

Parenti, Laura (2016). La proporzionalità come introduzione al pensiero scientifico in età infantile. Unpublished Master Thesis, Roma Tre University.