1. Skills and Competencies Monika Pilgerstorfer 5 April 2005

2. Knowledge Space Theory Knowledge: solution behaviour Knowledge state: subset of problems a person is able to solve Knowledge space: set of all possible knowledge states

3. Extensions of Knowledge Space Theory Latent cognitive structures underlying knowledge spaces Skills (Falmagne; Doignon; Düntsch & Gediga) Components and Attributes, Demand Analysis (Albert & Held) Cognitive Processes (Schrepp) Competence-Performance Approach (Korossy)

4. Basics Set S of skills that are necessary for answering certain problems. For each problem q  Q there exists a subset f(q)  S of skills that are sufficient for solving the problem.

5. Skill function assign to each problem the skills required for solving this problem Competencies = sets of skills sufficient to solve a problem

6. Example: skill function ProblemCompetencies a{1,2,4}, {3,4} b{1,2} c{3} d{3,5}

7. Problem function Set of skills (S) Set of problems (Q) assigns to each set of skills the set of problems, which can be solved in it

8. Problem function ProblemCompetencies a{1,2,4}, {3,4} b{1,2} c{3} d{3,5} {c,d}{3,5} {c}{3} {b}{1,2} {a,b}{1,2,4} Problems Competencies {a,c}{3,4}

9. Example: problem function K = { , {b}, {c}, {a, b}, {a, c}, {b, c}, {c, d}, {a, b, c}, {a, c, d}, {b, c, d}, {a, b, c, d}} b c a d v {c,d}{3,5} {c}{3} {b}{1,2} {a,b}{1,2,4} Problems Competencies {a,c}{3,4}

10. Knowledge State A subset K of problems is a knowledge state if and only if there is a subset M of skills such that K contains all those problems having at least one competency included in M and only those problems.

11. Special cases disjunctive model: only one of the skills attached to a problem q suffices to solve this problem conjunctive model: all the skills assigned to a problem q are required for mastering this problem

12. Extension: competence structure on a set of skills Competence-Performance Approach

13. Performance: observable solution behaviour Competence: underlying construct explaining performance Competence-Performance Approach

14. Performance structure (A, P) A... finite, non-empty set of problems P... family of subsets of problems A Competence-Performance Approach

15. Competence structure (E, K) E... finite, non-empty set of elementary competences K... family of subsets of elementary competences E Competence-Performance Approach

16. assigns to each problem a problem- specific set of competence states which are elements of the competence structure Interpretation function

17. assigns to each competence state the set of problems solvable in it Representation function

18. Problems given: a = 5 cm, c = 8 cm area A = ? given: b = 3 cm, c = 9 cm area A = ?

19. Elementary competences PKnowledge of the Theorem of Pythagoras KKnowledge of the Kathetensatz HKnowledge of the Höhensatz AKnowledge about calculating the area of a right- angled triangle ZKnowledge of constructing a square with the same area as a given rectangle TKnowledge of properties of tangents on circles

20. Subsets of competencies Extract subsets that are minimal concerning the subset relation Minimal: not subset of each other Surmise function

21. PKHAZTPKHAZT {P,K }, {P,H }, {P,A} {K} {H} {K,A}, {H,A} {K,Z}, {H,Z} {P,K,T,A}, {K,H,T,A} Surmise function B(K) =  K ,  H ,  P,K ,  P,H ,  P,A ,  K,A ,  H,A ,  K,Z ,  H,Z ,  P,K,T,A ,  K,H,T,A 

23. a{H}, {PK} b{HA}, {KA} c{K}, {PH} d{KZ}, {HZ) e{PKTA}, {KHTA} Interpretation function

24. a a b b c cd d e e

25. Representation function  K  cc  H  aa  K,A   b,c   H,A   a,b   H,Z   a,d   K,Z   c,d 

26. a a bb c c d d e e

27. Exercise A3+4+2 = B4:2+1 = C3*2*2 = D4+2-3 = E3+4*2 = F6:3-2 = G6:2*3 = Find the competencies that are necessary for solving following tasks:

28. Exercise - competencies A3+4+2 = B4:2+1 = C3*2*2 = D4+2-3 = E3+4*2 = F6:3-2 = G6:2*3 = + 1 - 2 * 3 : 4 * before - 5 Suggested competencies:

29. Exercise Find the possible competence states and the competence- structure for the following surmise function! 1 2 3 4 5

30. Exercise – Competence states 1 2 3 4 5 { } {1} {2} {1,2} {1,2,3} {1,2,4} {1,2,3,4} {1,2,3,5} {1,2,4,5} {1,2,3,4,5}

31. Exercise A3+4+2 = B4:2+1 = C3*2*2 = D4+2-3 = E3+4*2 = F6:3-2 = G6:2*3 = Find the Interpretation function for task A-G! 1 2 3 4 5

32. Exercise - Interpretation function A 3+4+2 = {1} B 4:2+1 = {1,2,4,5} C 3*2*2 = {1,2,3} D 4+2-3 = {1,2} E 3+4*2 = {1,2,3,5} F 6:3-2 = {1,2,4,5} G 6:2*3 = {1,2,3,4} 1 2 3 4 5

33. Exercise Find the surmise function on the problems, based on the information of the Interpretation function!

34. Thank you for your attention!

35. References Albert, D., & Held, T. (1999). Component Based Knowledge Spaces in Problem Solving and Inductive Reasoning. In D. Albert & J. Lukas (Eds.), Knowledge Spaces: Theories, Empirical Research Applications (pp. 15–40). Mahwah, NJ: Lawrence Erlbaum Associates. Düntsch, I. & Gediga, G. (1995). Skills and knowledge structures. British Journal of Mathematical and Statistical Psychology, 48,9-27. Falmagne, J.-C., Doignon, J.-P., Villano, M., Koppen, M. & Johannesen, L. (1990). Introduction to knowledge spaces: How to build, test and search them. Psychological Review, Vol.97, No.2, 201-204.

36. References Korossy, K. (1996). A qualitative-structural approach to the modelling of knowledge. Report of the Institute of Psychology, Universität Heidelberg. Korossy, K. (1997). Extending the theory of knowledge spaces: a competence-performance approach. Zeitschrift für Psychologie 205, 53-82