Fringe & Neighbourhood

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. In knowledge space theory, a domain of knowledge is described by a set Q of test problems or items. A learner is desribed by his/her knowledge state, i.e. the subset K ⊆ Q 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 prerequisite relationships between the items. Such a knowledge structure is called a knowledge space.

Fringe and Neighbourhood

For the learning process, the fringe of the learner's current knowledge state K is very interesting. For that, we look at the 1-neighbourhood of K, i.e. the set of knowledge states K' which differ from K by exactly one item. The fringe K^F is then the set of those items by which the neighboring knowledge states K' differ from K. It is divided into tge inner and the outer fringe. The inner fringe K^I = {q ∈ K: K\{q} is a knowledge state} contains those items which the learner has probably recently learned and, therefore, should train a bit more. The outer fringe K^O = {q ∈ Q\K: K∪{q} is a knowledge state} contains those items for which the learner has all the prerequisites and which s/he is ready to learn.

Example Spaces

As example data, knowledge spaces provided by the R package pks (Heller & Wickelmaier, 2013; Wickelmaier et al., 2016) are used. Concretely, the following spaces are used:

Chess: Held, Schrepp and Fries (1995) derive several knowledge structures for the representation of 92 responses to 16 chess problems. See Schrepp, Held and Albert (1999) for a detailed description of these problems. This app uses the projection of their DST1-structure reduced to the fiurst five items.
Density: Taagepera et al. (1997) applied knowledge space theory to specific science problems. The density test was administered to 2060 students, a subtest of five items each is included here.
Matter: Taagepera et al. (1997) applied knowledge space theory to specific science problems. The conservation of matter test was administered to 1620 students, a subtest of five items each is included here.
Doignon & Falmagne: Fictitious data set from Doignon and Falmagne (1999, chap. 7).

References

Doignon, J.-P., & Falmagne, J.-C. (1999). Knowledge spaces. Berlin: Springer.

Held, T., Schrepp, M., & Fries, S. (1995). Methoden zur Bestimmung von Wissensstrukturen — Eine Vergleichsstudie [Methods for determining knowledge structures — a comparing study]. Zeitschrift für Experimentelle Psychologie, XLII (2) ,205–236.

Heller, J. & Wickelmaier, F. (2013). Minimum discrepancy estimation in probabilistic knowledge structures. Electronic Notes in Discrete Mathematics, 42, 49-56.

Schrepp, M., Held, T., & Albert, D. (1999). Component-based construction of surmise relations for chess problems. In D. Albert & J. Lukas (Eds.), Knowledge spaces: Theories, empirical research, and applications (pp. 41--66). Mahwah, NJ: Erlbaum.

Taagepera, M., Potter, F., Miller, G.E., & Lakshminarayan, K. (1997). Mapping students' thinking patterns by the use of knowledge space theory. International Journal of Science Education, 19, 283--302.

Wickelmaier, F., Heller, J., & Anselmi, P. (2016). pks: Probabilistic Knowledge Structures. R package version 0.4-0. https://CRAN.R-project.org/package=kst

Using this App

This App illustrates the neighbourhood and fringe of knowledge states. It consists of five tabs: a very short introduction into used concepts of knowledge space theory, information about the example spaces used in this app, this usage information, 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 see two dropdown selections where you can select a knowledge space and a knowledge state within this space.

On the right side, you see a Hasse diagram of the knowledge space where the selected state and its 1-neighbourhood are marked. Below the diagram, the fringe is printed.

The above Hasse diagram shows the knowledge space with the selected state marked in green and its 1-Neighbourhood marked in yellow

Fringe

About this App

This App was created within the TquanT project.
TquanT was co-funded by the Erasmus+ Programme of the European Commission.