Comparison of types of generalizations and problem-solving schemas used to solve a mathematical problem

A cognitive schema is a mechanism which allows an individual to organize her/his experiences in such a way that a new similar experience can easily be recognised and dealt with successfully. Well-structured schemas provide for the knowledge base for subsequent mathematical activities. A new experience can be assimilated into a previously existing schema or, if that is not possible, an existing schema must be accommodated to a new experience. These processes are applied in all the different areas of mathematics, including mathematical problem solving. The paper presents a study conducted with primary teacher students, the aim of which was to gain a deeper insight into the way of reasoning of individuals when presented with a mathematical problem. The types of generalization used by individuals and their problem solving schemas were analysed. The research results lead to some interesting conclusions. The type of generalization used for solving a problem is related to the type of the problem: if the relationship between the problem and the mathematical concept can be established, the reasoning type of generalization prevails. The results also confirm the importance of a well-structured schema for the successful solving of a complex mathematical problem. The students applying unstructured or partially formed schémas had problems when addressing a complex problem, whereas the students who were able to solve it mostly accommodated or assimilated their knowledge, which again proves these processes to be a necessary prerequisite for successful problem solving.