Knowledge space

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In mathematical psychology and education theory, a knowledge space is a combinatorial structure used to formulate mathematical models describing the progression of a human learner. [1] Knowledge spaces were introduced in 1985 by Jean-Paul Doignon and Jean-Claude Falmagne, [2] and remain in extensive use in the education theory. [3] [4] Modern applications include two computerized tutoring systems, ALEKS [5] and the defunct RATH. [6]

Contents

Formally, a knowledge space assumes that a domain of knowledge is a collection of concepts or skills, each of which must be eventually mastered. Not all concepts are interchangeable; some require other concepts as prerequisites. Conversely, competency at one skill may ease the acquisition of another through similarity. A knowledge space marks out which collections of skills are feasible: they can be learned without mastering any other skills. Under reasonable assumptions, the collection of feasible competencies forms the mathematical structure known as an antimatroid.

Researchers and educators usually explore the structure of a discipline's knowledge space as a latent class model. [7]

Motivation

Knowledge Space Theory attempts to address shortcomings of standardized testing when used in educational psychometry. Common tests, such as the SAT and ACT, compress a student's knowledge into a very small range of ordinal ranks, in the process effacing the conceptual dependencies between questions. Consequently, the tests cannot distinguish between true understanding and guesses, nor can they identify a student's particular weaknesses, only the general proportion of skills mastered. The goal of knowledge space theory is to provide a language by which exams can communicate [8]

Model structure

Knowledge Space Theory-based models presume that an educational subject S can be modeled as a finite set Q of concepts, skills, or topics. Each feasible state of knowledge about S is then a subset of Q; the set of all such feasible states is K. The precise term for the information (Q, K) depends on the extent to which K satisfies certain axioms:

Prerequisite partial order

The more contentful axioms associated with quasi-ordinal and well-graded knowledge spaces each imply that the knowledge space forms a well-understood (and heavily studied) mathematical structure:

In either case, the mathematical structure implies that set inclusion defines partial order on K, interpretable as a educational prerequirement: if a(⪯)b in this partial order, then a must be learned before b.

Inner and outer fringe

The prerequisite partial order does not uniquely identify a curriculum; some concepts may lead to a variety of other possible topics. But the covering relation associated with the prerequisite partial does control curricular structure: if students know a before a lesson and b immediately after, then b must cover a in the partial order. In such a circumstance, the new topics covered between a and b constitute the outer fringe of a ("what the student was ready to learn") and the inner fringe of b ("what the student just learned").

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]

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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. ↑ "Homepage of RATH". Archived from the original on 2007-06-30.
  7. ↑ Schrepp, M. (2005), "About the connection between knowledge structures and latent class models", Methodology, 1 (3): 93–103, doi:10.1027/1614-2241.1.3.93 .
  8. ↑ Jean-Paul Doignon, Jean-Claude Falmagne (2015). "Knowledge Spaces and Learning Spaces". arXiv: 1511.06757 [math.CO].
  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
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