Using this App
This App demonstrates the connection between surmise relations and knowledge spaces. 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
In the top left corner, you find a group of checkboxes where you can enter a prerequisite relation.
The checkboxes are preset with the reflexive pairs of the relation. Riht next to it, you see your relation as a
binary matrix. Next to it, there is the binary matrix of the coresponding surmise relation, i.e. the reflexive and
transitive closure of your prerequsite relation.
In the lower part of the tab, there are some diagrams. On the left side, you find a "raw" diagram of your relation.
In the middle, there is a Hasse diagram of the surmise relation, and on the right side a Hasse diagram of the
corresponding knowledge space.
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.
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 a prerequisite relation. A pair (a,b)
of items is in the prerequisite relation if a
prerequisite of b
, i.e. if somebody solves b
we can surmise tha s/he is also able to solve a
In knowöe3dge space theory, we speak of a surmise relation
as technical term for prerequisite relation.
A surmise relation is, from a mathematical point of view, a quasi-order, i.e. a reflexive and transitive relation.
- For all items a, (a,a) is in the surmise relation (if somebody solves a we can surmise
that s/he can also solve a).
- For all items a, b, c, if (a, b) and (b, c) are in the surmise relation, then also (a, c)
is in the surmise relation.
A learner is desribed by his/her knowledge state, i.e. the subset of items s/he is able to solve. A set of knowledge states is
called a knowledge structure
if it contains the empty set and the full set of items. The set of possible knowledge states
can be delimited by a surmise relation. Such a knowledge structure is called a quasi-ordinal knowledge space. The larger a surmise relation is (i.e. the more prerequisite relationships hold), the smaller the corresponding quasi-ordinal knowledge space becomes.
- 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.
These items were taken from a R
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?
Enter your Prerequisite Relation!
Corresponding Surmise Relation
About this App
This App was created within the TquanT
TquanT was co-funded by the Erasmus+ Programme of the European Commission.
© 2017 Cord Hockemeyer, University of Graz, Austria