Skills and Competencies
Monika Pilgerstorfer5 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 underlyingknowledge 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
Problem Competencies
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
Problem Competencies
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}
ProblemsCompetencies
{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}
ProblemsCompetencies
{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
P Knowledge of the Theorem of Pythagoras
K Knowledge of the Kathetensatz
H Knowledge of the Höhensatz
A Knowledge about calculating the area of a right-angled triangle
Z Knowledge of constructing a square with thesame area as a given rectangle
T Knowledge 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
P
K
H
A
Z
T
{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 bb
c
cd d
e e
Representation function
K c
H a
K,A b,c
H,A a,b
H,Z a,d
K,Z c,d
a
a bb
c
cd d
e e
Exercise
A 3+4+2 =
B 4:2+1 =
C 3*2*2 =
D 4+2-3 =
E 3+4*2 =
F 6:3-2 =
G 6:2*3 =
Find the competencies that are necessary for solving following
tasks:
Exercise - competencies
A 3+4+2 =
B 4:2+1 =
C 3*2*2 =
D 4+2-3 =
E 3+4*2 =
F 6:3-2 =
G 6:2*3 =
+ 1
- 2
* 3
: 4
* before - 5
Suggested competencies:
Exercise
Find the possible competence states and the competence-
structure for the following surmise function!
1 2
3 4
5
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}
Exercise
A 3+4+2 =
B 4:2+1 =
C 3*2*2 =
D 4+2-3 =
E 3+4*2 =
F 6:3-2 =
G 6:2*3 =
Find the Interpretation function for task A-G!
1 2
3 4
5
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
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, 201-204.
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