Skills and Competencies Monika Pilgerstorfer 5 April 2005

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

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)

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.

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

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

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

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}

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}

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.

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

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

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

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

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

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

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

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

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

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

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 

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

a a b b c cd d e e

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

a a bb c c d d e e

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:

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

Exercise Find the possible competence states and the competence- structure for the following surmise function!

Exercise – Competence states { } {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}

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!

Exercise - Interpretation function A = {1} B 4:2+1 = {1,2,4,5} C 3*2*2 = {1,2,3} D = {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}

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

Thank you for your attention!

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,

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