The mathematical background of proving processes in discrete optimization -Exemplification with Research Situations for the Classrooms

Discrete mathematics brings interesting problems to teach and learn proof with accessible objects such as integers (arithmetic), graphs (modeling, order) or polyominoes (geometry). A lot of still open problems can be explained to a large public. The objects can be manipulated by simple dynamic operations (removing, adding, "gluing", contracting, splitting, decomposing, etc.). All these operations can be seen as tools for proving. This article particularly explores the field of "discrete optimization". A theoretical background is defined by taking two main axes into account: the epistemological analysis of discrete problems studied by contemporary researchers in discrete optimization and the design of adidactical situations for classrooms in the frame of the Theory of Didactical Situations. Two problems coming from ongoing research in discrete optimization (the Pentamino Exclusion and the Eight Queens problems) are developed. They underscore the learning potentialities of discrete mathematics and epistemological obstacles about proving processes. They emphasize the understanding of a necessary condition and a sufficient condition and problematize the difference between optimal and optimum. They provide proofs involving partitioning strategies, greedy algorithms but also primal-dual methods leading to the concept of duality. The way such problems can be implemented in the classrooms is described in a collaborative work between mathematicians and mathematics education researchers (Maths à Modeler Research Federation) through the Research Situations for the Classrooms