Ensino-aprendizagem-avaliação de razões trigonométricas através da proposição e resolução de problemas

Abstract

Among the approaches focused on mathematics education, the Methodology of Teaching Learning-Evaluation through Problem Solving has stood out in research and pedagogical practice. In line with recent Brazilian curriculum guidelines, Problem Posing has also been gaining ground, emerging as a promising strategy, although it remains little known and under explored. Within this context, the present study is guided by the following research question: “How can problem posing, as explored in stages 1 and 10 of the MEAAMaRP, contribute to the construction of knowledge regarding trigonometric ratios?” The study adopted a qualitative approach, and the participants were 30 11th-grade students, selected because they were students of the researcher-teacher. The main data collection instruments were problem-posing activities associated with MEAAMaRP, aimed at analyzing students’ problem-posing processes during MEAAMaRP implementation, to identify specific aspects of its development in the first and tenth stages and its contributions to the construction of trigonometric ratio content. Data were collected through written records and transcriptions of audio recordings of dialogues between the researcher-teacher and the participants. In the field study, students were initially invited to construct a sundial and take measurements of the projected shadow. Based on these elements and an initial prompt provided by the teacher, the students proposed two problems, which were subsequently solved. One of the problems was selected and adapted by the teacher to serve as a “generator,” since it could introduce trigonometric ratios - sine, cosine, and tangent. The students solved and discussed the problem before the teacher formally presented the content. Next, in the tenth stage of MEAAMaRP, two new prompts were provided to the groups, and new problems were proposed. Data analysis consisted of the students’ written records and the transcripts of the audio recordings of the dialogues that took place during the class. The results indicate that the problems proposed and solved in the first stage enabled the students to identify missing or contradictory data in their own creations. Selecting and reformulating a problem proposed by the students into the generative problem gave the students a leading role. Analysis of the stages revealed an evolution in the quality and complexity of the proposed problems: in the initial stage, the problems were simpler and, at times, incomplete or contained excessive data. As the students progressed to the final stage, greater consistency was observed in the data provided in the problem statements. In Stage 1, the prompt was open-ended, allowing students to freely explore the data collected using the sundial and encouraging creativity and initial investigation. However, because the scope was broad, not all groups initially addressed trigonometric ratios, focusing instead on concepts such as angles, shadow lengths, and time. By stage 10, the prompts had become targeted and closed-ended, which allowed for the application of formalized content (sine, cosine, and tangent). Thus, the process of posing and solving their own problems promotes an understanding of trigonometric ratios through problem-posing. In connection with this research, an e-book on Problem Posing and Problem Solving was developed for teachers.