The Knowledge Vault is a personal, unique notebook full of facts that you find useful.
Children can start with their own knowledge vault. As they grow older, they add to it and improve the things stored in there.
For instance, they can store the fact that "Carbon is Element 6" when they first learn the Periodic Table.
In addition to basic facts, they can learn and use algorithms. For instance, they could write an algorithm that would decline a Latin noun, and improve upon that algorithm over time.
The Knowledge Vault depends on the Facts project. Instead of universal facts, it contains personal facts. Some of the universal facts may be imported, which implies a sort of fact sharing algorithm that the Facts project would need to use anyway.