Development of the schema for the differential of a function of several variables

Abstract

The Purpose of this study aimed to analyze the understanding of the concept of the differential of a function of several variables among university students, adopting a constructivist and cognitive approach focused on the development of mental schemas.  The research was conducted sequentially: 1) A theoretical analysis of the concept; 2) The design and implementation of specific instruction; 3) The collection and analysis of empirical data from questionnaires and interviews. The development of the differential scheme was characterized by the mental structures students used to relate its constituent elements. The schema's evolution through intra-, inter-, and trans-levels was evidenced by a process of synthesis and increased cognitive complexity. It is concluded that understanding the concept advances through the progressive evolution of the mental schema. The articulation of the intra, inter, and trans levels is fundamental for a complete grasp of the differential. The study supports designing didactic sequences based on the concept's genetic decomposition. This approach, aided by materials that facilitate transitions between representations, enhances the teaching of multivariable calculus.