Abstract:

This study is about algebraic thinking in textbooks adopted by an educational institution in Brazil. The importance of this analysis lies in the use that teachers make of textbooks, which make them mediators between the curriculum and the classroom. Since, in many cases, the textbook does this mediation, it is important to study textbooks based on a grid analysis with theoretical support. The current National Curricular Common Base(BNCC, Portuguese acronym) is the official education program proposing consolidation, expansion, and learning involving algebraic thinking. Therefore, the 1st book adopted in 2021 by the Institute of Bahia(IFBA, Portuguese acronym) is analyzed in this work. Since it was used to teach the Integrated Middle-Level Technical Course in Buildings(CTNMIE, Portuguese acronym) mathematics course from the IFBA, we also compared the textbook analyses with the course syllabus concerning algebraic thinking. In this 1st work, the theoretical framework was Kaput’s 4 elements of algebra content in elementary and high school to analyze algebraic thinking skills: Generalized Arithmetic, which includes generalizing arithmetic operations and their properties, and the structure of arithmetic relations. Functional thinking refers to situations in which ways are sought to express a systematic variation of instances and involves the ideas of causality, growth, and continuous variation. The modeling language and how to apply it, the student is expected to start by interpreting the context, identifying the quantities involved, observing their relationships, representing this relationship through an appropriate mathematical model, solving the problem, and justifying their answer concerning the original context. Finally, abstract algebra and algebraic proof can take three forms:
i) using generalizations to build other generalizations;
ii) generalization of processes or mathematical formulas;
iii) testing conjectures, justifying, and proving.

Thus, arguments and justifications of an abstract nature for symbolic algebra are expected to be constructed in the algebraic proof aspect. The study by Pitta-Pantazi et al. (2020) empirically validates the existence of these four strands of algebraic thinking. In addition, students can solve functional thinking tasks first. They can solve generalized arithmetic tasks only when they have appropriated the modeling language and abstract algebra and algebraic proof tasks.

The study methodology is qualitative because it is mainly descriptive and based on content analysis of 2 of the 1st textbook chapters and the school’s current CTNMIE mathematics course syllabus: geometric and arithmetic progressions. The Epistemic Analysis Tool (EAT) from the Ontosemiotic Approach (OSA), in parallel with the algebraic thinking components, was used in the analysis. The EAT is based on components and indicators proposed by Godino et al..

The textbook analysis revealed that all elements of the EAT were presented and are in line with the OSA framework and the four aspects of algebraic thinking. However, the same cannot be said for the CTNMIE course syllabus. Some elements of the EAT were not found, specifically in the categories of ‘propositions’ and ‘procedures.’ The IFBA’s CTNMIE syllabus for mathematics was based on the pre-BNCC document of Brazilian education. A critical reevaluation of this plan is imperative to adapt it to the current demands of the BNCC, thereby ensuring alignment with the current educational program.

Keywords:

Algebraic thinking, Epistemic analysis tool, Textbook, Syllabus.