Mathematical Reasoning in Algebraic and Geometric Contexts: an analysis with grade 9 medalist students

Abstract The purpose of this article is to understand the reasoning processes of grade 9 students employed in solving two tasks: one of an arithmetic/algebraic nature and another of flat geometry. The methodology used is qualitative, inscribed in an interpretive paradigm using document analysis of six written productions by four students: medalist students at the Brazilian Mathematics Olympiad in Public Schools (OBMEP) and at the Regional Mathematics Olympiad (ORM) in the State of Santa Catarina. The analysis of the students’ solutions was carried out according to the following categories: (i) processes and types of reasoning and (ii) representations. The results point to a great potential of the use of Algebra (with mastery of algebraic language) in the realization of the students’ justifications. Such use seems to have allowed students to diversify their semiotic representation records and to coordinate them. Furthermore, and in a different way from what is reported in the literature, the students were able to carry out their justifications without the need to be questioned by the teacher. This question seems to be associated with the fact that these students are medalists in Mathematics Olympiads, events in which the construction of a plausible justification assumes a central role.

摘要 本文旨在探究九年级学生解决两类数学问题时的推理过程：一类为算术/代数类题目，另一类为平面几何题目。本研究采用解释主义范式下的定性研究方法，以四名学生的六份书面作答作品为分析对象开展文本分析。该四名学生均为巴西公立学校数学奥林匹克（Brazilian Mathematics Olympiad in Public Schools，OBMEP）与圣卡塔琳娜州区域数学奥林匹克（Regional Mathematics Olympiad in the State of Santa Catarina，ORM）的获奖者。学生解题方案的分析依据以下两类范畴展开：(i) 推理的过程与类型，(ii) 表征形式。研究结果显示，熟练掌握代数语言并运用代数方法，能够极大助力学生构建严谨的论证。此类代数运用似乎使学生得以丰富其符号表征体系并实现不同表征间的协调。此外，与现有文献报道的结论不同，这些学生无需教师引导即可独立完成论证。这一现象似乎与他们作为数学奥赛获奖者的身份高度相关——在数学奥赛赛事中，构建合理的论证是核心考察要素。