Vestígios de uma álgebra do ensino: análise de manuais pedagógicos por futuras professoras

Abstract

This is a professional master's research project conducted within the Graduate Program in Science and Mathematics Education (PPGECM) at the Federal Institute of Education, Science and Technology of Goiás (Jataí Campus), situated in the field of the History of Mathematics Education. The guiding question was: "What traces of algebra can be interpreted as an 'algebra of teaching' based on guidelines for teaching multiplication found in 20th-century pedagogical manuals, considering a pathway designed for this purpose?" The main objective was to characterize—together with students from the Pedagogy program at the Federal University of Jataí (UFJ)—traces of an "algebra of teaching," using a pathway designed to interpret multiplication teaching guidelines found in 20th-century pedagogical manuals. The research was grounded in a theoretical-methodological framework that considers the history of knowledge and views such knowledge as constantly evolving. Thus, an educational product was developed: a pathway for collecting and organizing information, comparing teachers' knowledge (as interpreted from pedagogical manuals representing three distinct pedagogical eras), and systematizing this knowledge to characterize an algebra of teaching. This pathway, carried out by future teachers from the Pedagogy program, initially allowed them to observe visible changes in the information presented. Only after discussing differences and similarities regarding meaning, sequencing, gradation, exercises, and problems—elements constituting the "mathematics of teaching"—was it possible to establish that: I) algebra was present to represent an unknown multiplier in multiplication problems and to address the need to generalize multiplication rules; II) algebra began to appear as content to be taught, as well as implicitly within generalizations of divisibility rules and in more advanced exercises involving algebraic expressions that represent the product of a number and the difference between two others; III) Set theory served as a foundation for teaching and understanding basic algebraic concepts (comparison, inclusion, sequencing, generalization), while pattern recognition provided a basis for grasping generalizations. The pathway enabled a deeper theoretical exploration of mathematical concepts and also fostered in the participants a more attentive perspective regarding the form and rationale behind the arrangement of information in pedagogical manuals, as well as the implicit meanings therein.