Mathematical Foundations¶
Basic definitions¶
Definition — Knowledge Structure
A knowledge structure on a finite domain \(Q\) is a pair \((Q, \mathcal{K})\) where \(\mathcal{K} \subseteq 2^Q\) is a family of subsets (called knowledge states) such that:
\(\emptyset \in \mathcal{K}\) (the empty state)
\(Q \in \mathcal{K}\) (the full domain)
The elements of \(Q\) are called items — they represent problems, questions, or competencies that can be mastered.
Special structures¶
Definition — Knowledge Space
A knowledge structure \((Q, \mathcal{K})\) is a knowledge space if \(\mathcal{K}\) is closed under set union:
Definition — Learning Space
A knowledge space \((Q, \mathcal{K})\) is a learning space if it is well-graded: for every non-empty state \(K \in \mathcal{K}\), there exists an item \(q \in K\) such that \(K \setminus \{q\} \in \mathcal{K}\).
A learning space guarantees that every skill combination is reachable from \(\emptyset\) by adding items one at a time — there is always a learning path.
Surmise relations¶
A surmise relation (or prerequisite relation) is a partial order \(\preceq\) on \(Q\). If \(a \preceq b\), then mastering \(a\) is a prerequisite for mastering \(b\).
The ordinal space generated by a surmise relation is the family of all downsets (downward-closed subsets):
This is always closed under both union and intersection (a distributive lattice).
Competence-Based KST¶
The CB-KST framework separates skills from items:
\(S\): a set of skills
\(\mu: Q \to 2^S\): for each item, the skills required to solve it
A surmise relation on \(S\) (skill prerequisites)
The problem function maps a competence state \(C \subseteq S\) to the set of solvable items:
The knowledge structure is:
where \(\mathcal{C}\) is the competence structure (downsets of the skill order).
References¶
Doignon, J.-P., & Falmagne, J.-C. (1999). Knowledge Spaces. Springer-Verlag.
Falmagne, J.-C., & Doignon, J.-P. (2011). Learning Spaces. Springer-Verlag.