Abstract:

The present study is part of the theme of algebraic thinking, at the level of students of the 2nd year of the Portuguese Primary School, in the component of repetition patterns. It was intended to identify the decision-making of these students when solving mathematical tasks with repetition patterns related to the theme of Science - liquid density, thus perceiving if they had the ability to perform generalizations.

We agree with Barbosa et al. (2009) when they mention that the theme Patterns, when connecting with other mathematical subjects, such as solving mathematical problems, reasoning and communication, show their potential for mathematical learning. In this sense, it was intended to identify the decision making of the students when solving tasks with patterns of repetition to see if they presented attitudes of generalization. For that, the mathematical tasks used were associated to symbols that appeared through the Science, using liquids of different densities.

In methodological terms, the study was developed in three stages: pre-test, treatment and post-test and focused on action-research, favouring a naturalistic approach. Subjects were confronted with fifteen tasks, involving repetition patterns, from ABAB to AAABBBCCC. Each task required each subject to position him / herself in the face of a near generalization - GP (identification of the next figure) and a more generalized generalization - GD (identification of the fifteenth figure).

In order to introduce the concepts and symbols used in the pre-test and in the post-test, we used the topics "Properties of materials and objects in current use - liquid density" and "Capacities with liquids". Experimental activities were carried out, with variable control: "What is density?" and “Liquid tower experiment”.

Firstly, the students were led to understand the concept of density by comparing the mass of equal volumes of water with different amounts of dissolved sugar using the dish scale. This concept was very important for the following activity, where different liquids were used: honey, olive oil, stained water, coloured alcohol, detergent and oil. Due to their different densities, the liquids did not mix, allowing the obtaining of different columns of liquids, from which the patterns were organized.

From the collected data it was verified that the resolvers obtained better results in the GP items, both in the pre-test and in the post-test, when compared with the items of DG.

The privileged strategies in GP itens were the relation with the colour of the neighbouring piece (pre-test) and the identification of the replicate module (post-test); already at the level of the items of GD, the privileged strategy, in both tests, was the counting piece by piece.
In this way, the researchers, when relating liquid density to repetition patterns, provided students with a meaningful learning in the area of science with applicability in the learning of mathematical concepts and, on the other hand, the algebraic thought was developed, using concrete situations, according to the cognitive development of children of this age.

It can be concluded, in a reflexive way, that at the level of this year of schooling students already denote the capacity to make distant generalizations, so it is advisable to involve them in activities of this pre-algebraic nature.

Keywords:

Algebraic thinking, Repetition patterns, Mathematics, Generalization, Experimental Science, Density.