Abstract
This paper is adapted from a book and many scholarly articles. It reviews the main ideas of a theory for the assessment of a student’s knowledge in a topic and gives details on a practical implementation in the form of a software. The basic concept of the theory is the ‘knowledge state,’ which is the complete set of problems that an individual is capable of solving in a particular topic, such as Arithmetic or Elementary Algebra. The task of the assessor—which is always a computer—consists in uncovering the particular state of the student being assessed, among all the feasible states. Even though the number of knowledge states for a topic may exceed several hundred thousand, these large numbers are well within the capacity of current home or school computers. The result of an assessment consists in two short lists of problems which may be labelled: ‘What the student can do’ and ‘What the student is ready to learn.’ In the most important applications of the theory, these two lists specify the exact knowledge state of the individual being assessed. Moreover, the family of feasible states is specified by two combinatorial axioms which are pedagogically sound from the standpoint of learning. The resulting mathematical structure is related to closure spaces and thus also to concept lattices. This work is presented against the contrasting background of common methods of assessing human competence through standardized tests providing numerical scores. The philosophy of these methods, and their scientific origin in nineteenth century physics, are briefly examined.
We wish to thank Chris Doble, Dina Falmagne, and Lin Nutile for their reactions to earlier drafts of this article.
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References
Albert, D. (ed.): Knowledge Structures. Springer, New York (1994)
Albert, D., Lukas, J. (eds.): Knowledge Spaces: Theories, Empirical Research, Applications. Lawrence Erlbaum Associates, Mahwah (1999)
Birkhoff, G.: Rings of sets. Duke Mathematical Journal 3, 443–454 (1937)
Cosyn, E., Uzun, H.B.: Axioms for Learning Spaces. Journal of Mathematical Psychology (2005) (to be submitted)
Doignon, J.-P., Falmagne, J.-C.: Spaces for the assessment of knowledge. International Journal of Man-Machine Studies 23, 175–196 (1985)
Doignon, J.-P., Falmagne, J.-C.: Knowledge Spaces. Springer, Berlin (1999)
Dowling, C.E.: Applying the basis of a knowledge space for controlling the questioning of an expert. Journal of Mathematical Psychology 37, 21–48 (1993a)
Dowling, C.E., Hockemeyer, C.: Computing the intersection of knowledge spaces using only their basis. In: Dowling, C.E., Roberts, F.S., Theuns, P. (eds.) Recent Progress in Mathematical Psychology, pp. 133–141. Lawrence Erlbaum Associates Ltd., Mahwah (1998)
Dowling, C.E.: On the irredundant construction of knowledge spaces. Journal of Mathematical Psychology 37, 49–62 (1993b)
Falmagne, J.-C., Doignon, J.-P.: A class of stochastic procedures for the assessment of knowledge. British Journal of Mathematical and Statistical Psychology 41, 1–23 (1988a)
Falmagne, J.-C., Doignon, J.-P.: A Markovian procedure for assessing the state of a system. Journal of Mathematical Psychology 32, 232–258 (1988b)
Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Berlin (1999); Mathematical foundations, Translated from the 1996 German original by C. Franzke
Heller, J.: A formal framework for characterizing querying algorithms. Journal of Mathematical Psychology 48, 1–8 (2004)
Kelvin, W.T.: Popular Lectures and Addresses, vol. 1-3. MacMillan, London (1889); Electrical Units of Measurement. In: Constitution of Matter, vol. 1
Koppen, M.: Extracting human expertise for constructing knowledge spaces: An algorithm. Journal of Mathematical Psychology 37, 1–20 (1993)
Koppen, M.: The construction of knowledge spaces by querying experts. In: Fischer, G.H., Laming, D. (eds.) Contributions to Mathematical Psychology, Psychometrics, and Methodology, pp. 137–147. Springer, New York (1994)
Pearson, K.: The Life, Letters and Labours of Francis Galton, vol. 2. Cambridge University Press, London (1924) Researches of Middle Life
Roberts, F.S.: Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences. Addison-Wesley, Reading (1979)
Rusch, A., Wille, R.: Knowledge spaces and formal concept analysis. In: Bock, H.-H., Polasek, W. (eds.) Data analysis and information systems, pp. 427–436. Springer, Heidelberg (1996)
Suck, R.: The basis of a knowledge space and a generalized interval order. In: Abstract of a Talk presented at the OSDA 1998, Amherst, MA, September 1998. Electronic Notes in Discrete Mathematics, vol. 2 (1999a)
Suck, R.: A dimension–related metric on the lattice of knowledge spaces. Journal of Mathematical Psychology 43, 394–409 (1999b)
Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore (1992)
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Falmagne, JC., Cosyn, E., Doignon, JP., Thiéry, N. (2006). The Assessment of Knowledge, in Theory and in Practice. In: Missaoui, R., Schmidt, J. (eds) Formal Concept Analysis. Lecture Notes in Computer Science(), vol 3874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11671404_4
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