Strategies used by 11th grade Portuguese students in solving mathematical problems

Abstract

Problem-solving, organised as individual or group activity, should be highlighted in the practice of Mathematics. This is emphasized by the Portuguese Mathematics Working Group (GTM) in the report entitled "Recomendações para a melhoria das aprendizagens dos alunos em Matemática" [Recommendations for improving student learning in Mathematics] (2019). In addition to problemsolving, the authors of this document also mention the importance of communication and of resorting to different representations, among other mathematical skills. It was to highlight these capacities that the research presented in this paper was carried out, as part of a PhD project. The project's main objective is to understand the how problem-solving and written communication skill mutually reinforce each other. Within this broad context, one of the research questions enquires "What are the strategies used by students in problem-solving?". Problem-solving strategies is precisely the focus of this paper, where following strategies are considered: trial and error, search for a pattern, generalization, deduction, endto-start resolution, construction of diagrams or figures, construction of tables, construction of a model, resolution by parts, application of formulas, exhaustion, and particularization. Twenty-nine 11th grade students from two classes of the same school participated in this study. Throughout the academic year, students were invited to participate in 16 problem-solving sessions, lasting about 90 minutes each, conducted virtually over an online platform. These students were divided into six working groups which remained unchanged until the end of the project. In each session, a mathematical problem was proposed to be addressed and solved by the groups. At the end of each session, all groups delivered a single resolution to the proposed problem, which resulted in 92 resolutions at the end of the project. This data collection was carried out by the researcher and first author of this communication, who was present at all times of data collection, using a qualitative methodology within an interpretative paradigm. After collecting all the resolutions over the 16 sessions, the same author analysed the resolutions and identified which strategies the students used. After analysing the resolutions of all groups in all sessions, we concluded that all strategies listed were used by at least one group throughout the 16 sessions. Furthermore, it was found that all groups resorted to almost all the strategies listed, which means that they felt the need to address the problems with different processes. It was also possible to detect that the most used strategy by the students was the "construction of diagrams or figures" – 35 resolutions resort to it – while the least used were "search for a pattern", "generalization", "end-to-start resolution" and "particularization" – all with 4 resolutions only.