Discovery in Mathematics Lessons II

Abstract

The article follows on from the previous section on "discovery" in mathematics lessons and shows why graph theory is a suitable environment for constructivist teaching. The author works with the topic of drawing with a single stroke and offers tasks based on real situations in which students naturally experiment, look for rules, and then refine them into mathematical language. In the first task, students investigate whether it is possible to walk through an apartment building passing through each door exactly once, which leads to the formulation of conditions for the existence of an Eulerian path in an undirected graph. The second task transfers the same idea to the environment of one-way streets and deliberately creates space to reveal the difference between an undirected and a directed graph (the balance of "entrances" and "exits"). The third, more formal part is an addition: the discovery and didactically guided proof of the relationship between the number of vertices, edges, and walls in a planar graph, which naturally turns into an inductive proof in class discussion. At the same time, the text does not idealise practice: it emphasises the need for time, appropriate task selection, and the mentoring role of the teacher, who must respond flexibly to students' strategies and mistakes.