Introduction to Knowledge Space Theory

https://www.flickr.com/photos/mikemacmarketing/30212411048

The Knowledge Space Theory (KST) is a set concepts and structures that aims to assess and represent one’s knowledge state. Assess people’s knowledge is an important task in order to check if someone is capable to perform some activity or to provide guidance towards the mastery of some subject. We face these assessments regularly in our daily lives, in the form of school tests or job interviews. KST intends to generate a higher detailed report of someone’s knowledge as opposed to single metrics used in a lot of tests. These detailed reports can then be further used for better analysis of someone’s capacity and provide more eficient guidance to the learner to master a subject.

In KST, the knowledge states comes in the form of a collection of problems that the individual is qualified to solve. KST heavily relies on Set Theory in order to provide its axioms and definitions.

In this post we will go through the main aspects of what composes the KST.

Items and Instances

In KST the knowledge states are represented in a very straightforward manner: as a set of types of problems that the individual is qualified to solve. These types of problems are statements about one specific task to be performed within a domain, for example:

• Invert a matrix using cofactors (concept from linear algebra)
• Calculate the resulting forces over a particle (concept from physics)
• Invert a chord (concept from music)

These types of problems are called items. So in KST each knowledge state is a set of items. An item is just a general concept description. An specific problem related to some item is called an instance. So for example of item and intance would be:

• Item: Calculate the roots of a quadratic polynomial
• Instance: Find the roots of the equation x² +5x + 6 = 0

Knowledge Structures

A knowledge atructure is defined as a pair (Q, K) where Q is a set of items and K is a collection of subsets of Q. As you may have noticed, the elements of K are the possible knowledge states, that is, the combination of items that one individual can belong to. In a knowledge structure the collection K must include the empty set and Q. For example:

• Q = {a,b,c,d} (Each item represented by one letter)
• K = {{}, {a}, {d}, {a,b}, {a,d}, {a,b,c}, {a,b,d}, Q} (Each subset of Q is the knowledge states, for example the state {a,b,d} means the maestry of the items a, b and d.

Knowledge Spaces

A knowledge space is a more restrict knowledge structure. A knowledge space is a knowledge structure that is closed under union. Closed under union means that every possible outcome from a union of any two states that belongs to K, must also belong to K. This property allow us to define the entire knowledge structure by a combination of a small set of elements.

Learning Spaces

Then we come to learning spaces. A learning space aims to, apart from also represent knowledge states, represent the expected learning dynamics. A learning space is an even more restrict knowledge structure than knowledge space: In addition than also being closed under union, a learning space must include also two additional properties, learning smoothness and learning consistency.

Learning Smoothness

This property estabilishes the existence of enough knowledge states such that every state can be mastered learning one concept at a time. This way it prohibits the existence of a state that can only be achieve if we learn more than one thing a time. And this make sense. We can be able to learn a lot of things within a small period of time, but still, we learn one thing at time. So it is reasonable the existence of states that represent all these intermediary steps.

Here, whereas the K1 respects the learning smoothness, K4 does not since from the state {d} to state {a,b,d} we had to master 2 items (a and b) at the same time. Example taken from https://arxiv.org/abs/1511.06757.

Learning Consistency

This property ensures the existence of enough states to represent any feasible order of item mastery. This is, the fact that we know something cannot prevent us from learning something new.

Here, whereas K2 respects the learning consistency, K3 does not since once in the state {a} the only possibility of mastering state {d} is by mastering all the items, since there is no state {a,d}. Example taken from https://arxiv.org/abs/1511.06757.

These graph like representations are called “covering diagrams”.

Atoms

In KST, atoms are the minimum states in which a given item q is present. In other words, among all the states where an item q exists, the states with the smallest number of items are called atoms (more specifically, atoms at q).

The definition of atoms is important because it has a direct pedagogical meaning since the items within an atom at q are the set of items that are required to be mastered before mastering q. For example, suppose we have the following items:

• Item A: sum two integer values
• Item B: multiply two integer values
• Item C: square an integer value

Intuitively we can say that before mastering item C, we need to master item B and before mastering item B, we need to master item A. So by representing this with KST, the minimum state containing A must be the state {A} (since we don’t need to master anything before mastering it) and the minimum statue containing B is {A, B} and not {B} (since before mastering B we need to have mastered A). As a consequence the atom for C would have to be {A, B, C}.

Fringe Theorem

When we are within a state there is two things we can ask:

1. From all the possible predecessors state from the current one, what are the items that once mastered would put me on this state?
2. Once in this state, what are them possible next items that I can master to proceed to another state?

The set of items that answers the first question is called the inner fringe, whereas the set of items that answers the second question is called outer fringe. The outer fringe has an important pedagogical meaning also. The outer fringe of a given state K is the set of items that someone in state K is ready to master. So, if after an assessment we conclude that a student is in a state G, the set of items that we should recommend the student to master next will be the outer fringe of G.

Learning Sequences

When we are in a learning journey, we cannot learn things in parallel (unfortunately). To master a given domain, we need to go through a sequence of learning materials, in a serial fashion, and master one item after the other. Since before master some items, other items must be mastered first, this sequence of mastering cannot be made in any order. A valid order of item mastering in a learning space is called a learning word. If the learning word contains all the items within the knowledge structure, then it is called a learning string.

Once we have a very large learning space, keep representing its form as covering diagram can be very problematic since for each state node, the set of items that compose the state may be very large. Learning sequences makes possible to represent the learning space in a very reduced form called learning diagram:

In a learning diagram, instead of labeling the nodes, we label the edges. The state that represents each node is given by grouping the edges that compose a path from the empty state towards the target state. Example taken from https://arxiv.org/abs/1511.06757.