Developing Mathematical Competences, Learning Linear Equations, Functions and the relation among these Concepts

In this article we report part of the findings of a research project focused on the developing of mathematical competences and knowledge about concepts as linear equations, functions, variables, unknown quantities and solution. The research topic is algebraic thinking. We discuss about the research questions: What are the cycles of understanding exhibited by students when they solve an algebraic problem? What kind of competences do the students exhibit in each cycle? How can we develop competences? The participants were college students from University of Quintana Roo, Mexico. They were 18 and 19 years old and they were starting degree courses in social science. They solved problems in a Problem Solving environment encouraged in the classroom. The students could extend, revise and refine their own conceptual systems. The Problem formed part of a series of problems implemented in the classroom. We used qualitative methodology to analyze the results obtained. We used Models and Modelling Perspective to analyze student's procedures and learning processes. The results show us cycles of understanding exhibited by the students when they solved the Problem. In each cycle we observed how college students were developing understanding of the mathematical concepts while they were solving problems. We concluded that the meanings can evolve to a more structured and organized way while students solve problems, and a Problem Solving environment is useful to develop Mathematical competence