Knowledge space

In mathematical psychology, a knowledge space is a combinatorial structure describing the possible states of knowledge of a human learner.[1] To form a knowledge space, one models a domain of knowledge as a set of concepts, and a feasible state of knowledge as a subset of that set containing the concepts known or knowable by some individual. Typically, not all subsets are feasible, due to prerequisite relations among the concepts. The knowledge space is the family of all the feasible subsets.

Knowledge spaces were introduced in 1985 by Jean-Paul Doignon and Jean-Claude Falmagne[2] and have since been studied by many other researchers.[3][4] They also form the basis for two computerized tutoring systems, RATH (defunct now) and ALEKS.[5]

It is possible to interpret a knowledge space as a special form of a restricted latent class model.[6]

Origin

Knowledge Space Theory (KST) was motivated by the shortcomings of the psychometric approach to the assessment of competence like SAT and ACT.[7] The theory was developed with an objective of designing automated procedures which -

• accurately assess the knowledge of a student, and
• efficiently provide advices for further study.

Assessments based on KST are adaptive and can account for possible slips or guesses. KST aims to give a detailed assessment of student's knowledge state as opposed to a numerical mark in traditional assessments. More specifically, the result of a KST based assessment tells two things -

• What the student can do and
• What the student is ready to learn.

Basic concepts

• Knowledge State

It is the complete set of problems that an individual is capable of solving in a particular topic (such as Algebra).

• Precedence Relation

It is the parent-child relationship between concepts. It captures the inter-dependency of concepts (prerequisite relationships).

• Knowledge Structure

It is the set of all feasible knowledge states. Because of precedence relations some of the knowledge states are infeasible.

• The outer and inner fringes

The unique items between a knowledge state and its immediate successor knowledge state is called the original knowledge state's Outer fringe. It basically tells the items that the student is ready to learn. Conversely, Inner fringe is the items that differentiate a knowledge state from its immediate predecessor. Inner fringe tells the items that the student has already learned.

Definitions

Some basic definitions used in the knowledge space approach -

• A tuple ${\displaystyle (Q,K)}$ consisting of a non-empty set ${\displaystyle Q}$ and a set ${\displaystyle K}$ of subsets from ${\displaystyle Q}$ is called a knowledge structure if ${\displaystyle K}$ contains the empty set and ${\displaystyle Q}$.
• A knowledge structure is called a knowledge space if it is closed under union, i.e. ${\displaystyle \cup F\in K}$ whenever ${\displaystyle F\subseteq K}$.[8]
• A knowledge space is called a quasi-ordinal knowledge space if it is in addition closed under intersection, i.e. if ${\displaystyle S,T\in K}$ implies ${\displaystyle S\cap T\in K}$. Closure under both unions and intersections gives (Q,∪,∩) the structure of a distributive lattice; Birkhoff's representation theorem for distributive lattices shows that there is a one-to-one correspondence between the set of all quasiorders on Q and the set of all quasi-ordinal knowledge spaces on Q. I.e., each quasi-ordinal knowledge space can be represented by a quasi-order and vice versa.

An important subclass of knowledge spaces, the well-graded knowledge spaces or learning spaces, can be defined as satisfying two additional mathematical axioms:

1. If ${\displaystyle S}$ and ${\displaystyle T}$ are both feasible subsets of concepts, then ${\displaystyle S\cup T}$ is also feasible. In educational terms: if it is possible for someone to know all the concepts in S, and someone else to know all the concepts in T, then we can posit the potential existence of a third person who combines the knowledge of both people.
2. If ${\displaystyle S}$ is a nonempty feasible subset of concepts, then there is some concept x in S such that ${\displaystyle S\setminus \{x\}}$ is also feasible. In educational terms: any feasible state of knowledge can be reached by learning one concept at a time, for a finite set of concepts to be learned.

A set family satisfying these two axioms forms a mathematical structure known as an antimatroid.

Construction of knowledge spaces

In practice, there exist several methods to construct knowledge spaces. The most frequently used method is querying experts. There exist several querying algorithms that allow one or several experts to construct a knowledge space by answering a sequence of simple questions.[9][10][11]

Another method is to construct the knowledge space by explorative data analysis (for example by Item tree analysis) from data.[12][13] A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.[14]

References

1. Doignon, J.-P.; Falmagne, J.-Cl. (1999), Knowledge Spaces, Springer-Verlag, ISBN 978-3-540-64501-6.
2. Doignon, J.-P.; Falmagne, J.-Cl. (1985), "Spaces for the assessment of knowledge", International Journal of Man-Machine Studies, 23 (2): 175–196, doi:10.1016/S0020-7373(85)80031-6.
3. Falmagne, J.-Cl.; Albert, D.; Doble, C.; Eppstein, D.; Hu, X. (2013), Knowledge Spaces. Applications in Education, Springer.
4. A bibliography on knowledge spaces maintained by Cord Hockemeyer contains over 400 publications on the subject.
5. Introduction to Knowledge Spaces: Theory and Applications, Christof Körner, Gudrun Wesiak, and Cord Hockemeyer, 1999 and 2001.
6. Schrepp, M. (2005), "About the connection between knowledge structures and latent class models", Methodology, 1 (3): 92–102, doi:10.1027/1614-2241.1.3.92.
7. Jean-Paul Doignon, Jean-Claude Falmagne (2015). "Knowledge Spaces and Learning Spaces". arXiv:1511.06757 [math.CO].
8. Falmagne, Jean-Claude; Doignon, Jean-Paul (2010-09-10). Learning Spaces: Interdisciplinary Applied Mathematics. Springer Science & Business Media. ISBN 9783642010392.
9. Koppen, M. (1993), "Extracting human expertise for constructing knowledge spaces: An algorithm", Journal of Mathematical Psychology, 37: 1–20, doi:10.1006/jmps.1993.1001.
10. Koppen, M.; Doignon, J.-P. (1990), "How to build a knowledge space by querying an expert", Journal of Mathematical Psychology, 34 (3): 311–331, doi:10.1016/0022-2496(90)90035-8.
11. Schrepp, M.; Held, T. (1995), "A simulation study concerning the effect of errors on the establishment of knowledge spaces by querying experts", Journal of Mathematical Psychology, 39 (4): 376–382, doi:10.1006/jmps.1995.1035
12. Schrepp, M. (1999), "Extracting knowledge structures from observed data", British Journal of Mathematical and Statistical Psychology, 52 (2): 213–224, doi:10.1348/000711099159071
13. Schrepp, M. (2003), "A method for the analysis of hierarchical dependencies between items of a questionnaire" (PDF), Methods of Psychological Research Online, 19: 43–79
14. Albert, D.; Lukas, J. (1999), Knowledge Spaces: Theories, Empirical Research, Applications, Lawrence Erlbaum Associates, Mahwah, NJ