How the handmade patterns can develop a real intelligence in the 3D geometric thinking of the learners?

Abstract

This paper investigates the teaching and learning of third-degree remarkable identities at the middle school level, with particular attention to students’ transition from second-degree to third-degree generalizations. The study aims to analyze the errors and difficulties encountered by students when extending the identity (a+b)²  to  (a+b)³ focusing on the coordination between geometric and algebraic reasoning.The research is based on a didactic intervention integrating symbolic expansion and volumetric modeling through the manipulation of cardboard cubes representing the decomposition of a cube of edge length . The analysis of students’ productions reveals conceptual misunderstandings, insufficient combinatorial reasoning, and significant difficulties in coordinating semiotic representation registers.

The results indicate that many observed errors stem from the overgeneralization of two-dimensional formulas and from disruptions in the articulation between geometric and symbolic registers. These difficulties reflect deeper epistemological and didactic obstacles that hinder the construction of structural understanding of remarkable identities in three dimensions. The findings highlight the importance of embodied and spatial mediation in supporting the development of algebraic thinking.