Umiejętność matematycznego uogólniania wśród nauczycieli i studentów matematyki specjalności nauczycielskiej (na przykładzie serii zadań "schodki")

Abstract

Generalizations are very common in mathematics. A generalization may be a process (mathematical activity) and a product in the form of mathematical concepts, tasks, theorems, hypotheses, methods of reasoning and argumentation. The article points out that the basic skills in the process of generalization of concepts and theorems consist of observation focused on distinguishing mathematical relations and the ability to represent the results by means of letters, algebraic expressions, equations and other symbols. It is particularly significant in mathematics education at the later stages of elementary school and in middle school (12-16 years). The article analyzes the generalization abilities and difficulties that were exhibited by university students of mathematics education and mathematics teachers from the middle school level while solving the "step pattern" task. A concrete situation, presented on a schematic graph together with explanatory information and questions, was a starting point for the formulation of general relations and recurrence and induction generalizations within the set of natural numbers. As a result of these generalizations, the condition defining the triangular numbers was formulated. Almost 65% of the students (out of 106) and 50% of the teachers (out of 130) successfully represented the generalizations. Almost 50% of the students managed to represent the reasoning that led to the generalization using the method of mathematical induction or the theorem of the sum to n terms of an arithmetic progression; while about 20% of the teachers represented the justification leading to the generalization by using the theorem of the sum of the first n integers or the induction generalization of the reasoning based on the example concerning the area of the figure created by the "step pattern."