Graph theory and combinatorial calculus: an early approach to enhance robust understanding

The objective of this work is to show an educational path for combinatorics and graph theory that has the aim, on one hand, of helping students understand some discrete mathematics properties, and on the other, of developing modelling skills through a robust understanding. In particular, for the path proposed to middle-school students, we used a connection between k-permutations and colourings of graphs: we indicated a way to solve problems related to counting all the possible arrangements of given objects in a k-tuple under given constraints. We solve this kind of problem by associating a graph with the constraints related to the k-tuple and by using graphs' colourings, in which every colour is associated with one of the objects. The number of arrangements is given by finding the number of colourings through an algorithm called the Connection-Contraction Algorithm. The educational path is set within the Teaching for Robust Understanding framework and the goal, from the mathematical skills perspective, is to enhance modelling, passing from real situations (the fish problem in our experiment) to mathematical problems (the graph's colouring in our experiment) and vice versa through the use of technology (the Connection-Contraction Algorithm with yEd editor, in our experiment), by using an extended modelling cycle. The meetings with students were videotaped and some results of the experimentation are given.