An Application of Knowledge Engineering to Mathematics Curricula Organization and Formal Verification

Abstract

The authors present a theoretical proposal for the organization of mathematical contents, more precisely to curricula development formalization and formal verification, inspired by knowledge engineering techniques. The situation addressed is the following: the starting point is a mathematical “official curriculum” (or part of it), not necessarily completely detailed. In our proposal, a group of experts would have to first build a detailed formulation of this curriculum (including the “prerequisite” relation between contents), which we will denominate “preprocessed official curriculum.” We detail how any “official curriculum development” could then be rigorously formalized and formally verified in a way inspired by rule-based expert system formal verification. We have defined the following terms: “contents soundness,” “contents completeness,” “relation soundness,” “relation completeness,” and “absence of cycles.” We believe that this is a completely new formalization within mathematics teaching theory that, once computer is implemented, would be very helpful. That would be the case, for instance, in countries where government sets the “official curricula” for Primary and Secondary Education and textbook contents have to be manually checked and approved by academic authorities: evaluators would “only” have to extract the textbook contents and set the “prerequisite” relation among them and let the computer do the rest.

Towards a Notion of Basis for Knowledge-Based Systems—Applications

“…The modularity enjoyed by logic-based representation can facilitate not only the specialization of the knowledge. It also aids verifying the derived methods, such as partitions [11], contextualizations [7,12], or customizations [12] of the knowledge, as well as providing efficient representations for reasoning [13]. The aforementioned methods can be designed using Computer Algebra Systems (CAS) when the logic is interpretable in Algebra.…”