Overcoming the intuitive rule of three: improving the false direct and inverse proportion

Abstract

This article analyzes the teaching of the rule of three and its application in solving mathematical problems involving magnitudes. It is also called direct and inverse proportion. It questions traditional didactics, which rely on intuition, arguing that they omit mathematical justification and lead to conceptual errors. It is proposed an alternative method based on the "First Algebra of Magnitudes", introducing a more rigorous and logical approach. The study highlights how current teaching methods fail to adequately operate physical magnitudes by deleting the magnitudes of operations, leading to the "arithmetization of physics". Traditional resolution methods (unit reduction, proportions, and practical methods) are criticized for their lack of logical foundation. The proposed new method structures reasoning through the algebraic formalization of magnitudes and demonstrates its effectiveness with an empirical study conducted on 97 students. A compound rule of three problem is presented, and different approaches are compared, concluding that the method based on the First Algebra of Magnitudes is more precise and logical. The article emphasizes the need to rethink the teaching of proportionality in mathematics to avoid mechanical and intuitive procedures and promote learning based on deduction and a deep understanding of the relationships between magnitudes.