## Knowledge Space Theory

The original aim of knowledge space theory was to develop a framework for a non-quanitative assessment, i.e. an assessment
that delivers not only some score but more concrete items about what exactly a learner knows and what not.

### Knowledge State and Knowledge Structure

A knowledge domain is specified by a set

*Q* of problems (or items). In knowledge space theory, we structure such a
domain of knowledge by prerequisite relationships. The subset of items a learner can solve is called his

*knowledge
state*. The set of all possible knowledge state is called a

*knowledge structure*. It always contains the
empty set ∅ and the full item set

*Q*. This set of possible knowledge states is restricted by the
aforementioned prerequisite relationships.

### Knowledge Spaces

A

*knowledge space* is a specialkind of knowledge structure, namely a structure which is closed under union.
This means that for any two knowledge states

*K, L* in a knowledge space, their union

*K ∪ L* is
again contained in the knowledge space.

### Further Reading

- Falmagne, J.-C., Koppen, M., Villano, M., Doignon, J.-P., & Johannesen, L. (1990). Introduction to knowledge spaces:
How to build, test and search them.
*Psychological Review, 97,* 201-224
- Heller, J., Hockemeyer, C., & Stefanutti, L. (2017).
*Knowledge Space Theory*.
Moodle course.

## Using this App

This App illustrates the concept of knowledge spaces and their properties. It consists of five tabs:
this usage information, a very short introduction into used concepts of knowledge space theory, a list of five
example items, the interactive core of the app, and some author and context information.

If you do not know knowledge space theory, you should best start with the short introduction.

### The *"Your Turn"* Tab

On the left side, you find a group of checkboxes where you can enter a knowledge structure.
On the right side, you see in the upper part two Hasse diagrams of your knowledge structure and the
coresponding knowledge space, i.e. the closure under union of your knowledge structure. Below the
right diagram, you find some additional information about the knowledge space.

## Example Items

These items were taken from a

R
Shiny App produced by a group of students at the TquanT 2017 seminar in Deutschlandsberg, Austria.

- A bag contains 5-cent, 10-cent, and 20-cent coins. The probability of drawing a 5-cent coin is 0.20,
that of drawing a 10-cent coin is 0.45, and that of drawing a 20-cent coin is 0.35.
What is the probability that the coin randomly drawn is a 5-cent coin or a 20-cent coin?
- In a school, 40% of the pupils are boys and 80% of the pupils are right-handed. Suppose that gender and
handedness are independent. What is the probability of randomly selecting a right-handed boy?
- Given a standard deck containing 32 different cards, what is the probability of drawing a 4 in a black suit?
- A box contains marbles that are red or yellow, small or large. The probability of drawing a red marble is 0.70,
the probability of drawing a small marble is 0.40. Suppose that the color of the marbles is independent of their size.
What is the probability of randomly drawing a small marble that is not red?
- In a garage there are 50 cars. 20 are black and 10 are diesel powered. Suppose that the color of the cars is independent
of the kind of fuel. What is the probability that a randomly selected car is not black and it is diesel powered?

### Properties of the knowledge space

#### Notions

#### Basis

#### Wellgradedness

## About this App

This App was created within the

TquanT project.

TquanT was co-funded by the Erasmus+ Programme of the European Commission.

© 2017 Christoph Anzengruber & Cord Hockemeyer, University of Graz, Austria