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