Abstract:

Nowadays, technological development has changed the way people relate and organize themselves in society. As today's students differ from those of a decade ago, it's crucial for schools and educators to adapt their teaching methods and integrate innovative technologies.

An effective mathematics learning assumes that students are able to not only interpret and relate various syllabus contents but also apply them to real-life situations of varying complexity, depending on their grade level. Additionally, they should be able to clearly communicate their reasoning for their problem-solving strategies. To achieve these learning outcomes, it is important that students have the possibility to experiment, investigate and conjecture ideas, which is crucial in consolidating knowledge in an effective and lasting way.

In this sense, it is essential to use active learning methodologies integrating new technologies, that allow autonomous learning, that capture the attention and interest of students, taking into account their different learning paces, and leading to a long-term knowledge.

In this paper we present two experiments carried out using the GeoGebra platform, applied in mathematics classes of Portuguese public education, to a total of 82 students from the 9th, 11th and 12th grades. In this context, two dynamic books, Discovering Trigonometry with GeoGebra and The Exponential function, were created, compiling several activities on these topics. They were explored in two different scenarios: within a normal classroom and in the perspective of a flipped classroom.

Within the scope of trigonometry, a didactic sequence of dynamic activities was developed for the 9th and 11th grades, using GeoGebra and a set of training sheets, with different levels of complexity, implemented through the interactive application MILAGE Learn+, with immediate feedback. This set of activities was carried out in the classroom in small heterogeneous groups, but also explored autonomously by each student. After the completion of the tasks proposed, there was a collective systematization of the learning outcomes.

The dynamic book The Exponential Function was created to teach this 12th grade subject, using the inverted classroom methodology, and consists of a sequence of activities to introduce this content. The flipped classroom is characterized by two moments, one before the face-to-face class in which students previously study the contents provided by the teacher and the face-to-face class where practical activities are carried out. The material produced was used in the pre-class phase and was made available to students through the GeoGebra Classroom, allowing the teacher to remotely monitor the development of students' work.

The students' feedback regarding the application of these activities was quite positive, with the majority considering that these resources facilitated learning and made the classes more captivating and dynamic. The teachers found that their students were committed and motivated, substantially more attentive than in a usual class, developed their autonomy and confidence, worked as a team to achieve the intended results and, in some cases, went beyond teachers’ expectations.

Later, when it was necessary to recover concepts from these topics to be used in other curricular contents, most students did not present any difficulties. This fact allowed us to conclude that the acquired knowledge was well consolidated, a keystone for a long-term knowledge.

Keywords:

Mathematics, GeoGebra, Trigonometry, Exponential function, Active learning.