Abstract:

The importance of intuition in mathematics, especially in geometry was always so obvious that the greatest geometrician of the nineteenth century, David Hilbert, gave the title to his famous book Anschauliche Geometrie, i.e. Intuitive Geometry (in English translation it is known as Geometry and the Imagination). He stressed that intuitive understanding played a major role in geometry.

Intuition in general means the ability to acquire knowledge without resorting to conscious reasoning. Psychologist Daniel Kahneman (2002 Nobel Prize winner in Economics) in his book “Thinking fast and slow” distinguished two systems of reaction to a problem. System 1 is the intuitive one, automatic knee-jerk reaction, and System 2 is the control, the deliberate one. According to Kahneman, sometimes we magically know things without knowing why we know them - and this is the manifestation of intuition. Mathematical intuition is understood as a series of instinctive knowledge often associated with the ability to solve mathematical challenges efficiently, validating logical arguments, and developing heuristics. In mathematics, particularly in geometry, intuition is strongly connected with imagination. In geometry, we can make intuitive discoveries (System 1) and verify them by reasoning (System 2) using mental image and visualisation.

In our research, we focused on the intuitive guessing in solution of problems related to symmetry. We analysed the process of constructive geometry problem solution given by prospective mathematics teachers. Finding some common patterns in results, we developed a series of exercises in which students can strengthen their intuitive guessing and make them more accurate in System 1 by practicing logical reasoning in System 2 in the field of symmetry-related geometric problems. In our research, we preferred problems with possible pictorial solutions providing visual evidence.

Keywords:

Intuition, imagination, geometry, problem-solving, symmetry, visual evidence.