Problems with more than one solution - its contribution to the metacognitive development of students and their capacity to communicate matematically

Abstract

The subject of mathematical communication has been an important aspect to be promoted in Mathematics classes in Portugal of this long time until the present date. Placing students on the condition that they can and should use valid communicative arguments to justify their decision making in solving mathematical tasks is something that has been promoted at the level of Portuguese educational policies, namely through the latest Basic Education Programs. Thus, this article aims precisely to contribute to an important reflection on this theme, proposing as something innovative that teachers seek to provide their students with problem solving that allow more than a valid solution, to the detriment of the problem solving with only one response. With this pedagogical intention there will certainly be conditions for students to effectively propose resolutions different from those of their colleagues, as well as to propose each of them more than a solution, having to always present the arguments that validate their decision-making. From the methodological point of view, nine problems will be presented that enable more than one resolution, problems that were authored by the authors of this article. The selection of these problems, which are now the subject of our analysis, also took into account the idea of ​​addressing more than one thematic area of ​​Mathematics. Thus, the first three problems are related to the theme of Geometry and Measurement, the next three to the theme of Numbers and Calculus and the last three to the theme of Mathematical Logic. We will present, for each case, several possible resolutions, of our authorship, to illustrate the pedagogical wealth that these problems raise, both at the level of Mathematical Communication and at the level of Metacognition. As a consequence of this methodological suggestion, it will also contribute to the metacognitive development of each student, as they become aware of their cognitive processes and use them properly so they can promote their problem solving condition and math communicators.