Starting point: a participant will master a problem
if he/she possesses a set of skills necessary for this problem.
A relation between problems and skills.
Definition:
Let
be a set of problems and
a set of skills. A mapping which
assigns to each problem
a subset
of skills is called a skill function for .
The set
is called the set of skills associated with .
To each problem
a set of latent skills
necessary to solve
is associated.
Definition:
Given a set of problems
, a set of skills ,
and a skill function ,
let .
A mapping
which assigns a subset of
to each subset
of ,
i. e., a knowledge state, is called a knowledge function.
Example:
Consider a problem set and
a skill set
.
Let
be defined by the table:
{s1,s2}
{s2,s3}
{s1}
{s1,s3}
The corresponding knowledge function
is given in that table:
{s1}
{s2}
{s3}
{s1,s2}
{s2,s3}
{s1,s3}
{s1,s2,s3}
{q1,q3,q4}
{q1,q2}
{q2,q4}
{q1,q2,q3,q4}
{q1,q2,q4}
{q1,q2,q3,q4}
{q1,q2,q3,q4}
The union of any two knowledge states is again a knowledge state.
is a knowledge space.
Generalization of a skill function: More than one subset of skills
can be assigned to a problem.
Definition:
Let
a set of problems,
a set of skills. A mapping
which assigns to each problem
a non-empty family of non-empty subsets of
is called a skill multi function. Each such subset of skills
from
is called a competence C for .
In order to master a problem, a participant has to
completely prossess at least one of the relevant competences.
Definition:
Let
a set of problems,
a set of skills
and a skill multi function. A mapping
which assigns to each subset
of a
subset of ,
i. e., a knowledge state, is called a knowledge multi function.
Doignon (1994):The family of knowledge states delineated
by ',
i. e., the image of ',
forms a knowledge structure on ,
but no longer a knowledge space.
The closure under union does no longer hold.
Example:
For a skill multi function, consider again the problem set
and the skill set
from the previous example.
Let a skill multi function be defined by
given in following Table.
{{s1,s2}}
{{s1},{s2,s3}}
{{s3}}
{{s1,s2,s3}}
The corresponding knowledge function
is given in following Table.
{s1}
{s2}
{s3}
{s1,s2}
{s2,s3}
{s1,s3}
{s1,s2,s3}
{q1,q2}
{q1}
{q3}
{q1,q2}
{q1,q2,q3}
{q1,q2,q3}
{q1,q2,q3,q4}
The union of any two knowledge states is no longer
a knowledge state, for example {q1}{q2} is not a state.
All subsets
of ,
i. e., the whole powerset of skills, have to be mapped onto subsets
of .
Approach makes no assumptions on any structural properties among the
skills or competences.
Structure among the skills (or competences) is introduced
in the work of Korossy.
A relation between problems and skills.
be a set of problems and 
a set of skills. A mapping 
which
assigns to each problem 
a subset 
of skills is called a skill function for 
is called the set of skills associated with 
.
,
and a skill function 
,
let 
.
which assigns a subset of 
of 
and
a skill set 
. 
be defined by the table:
is given in that table: 
is a knowledge space. 
which assigns to each problem 
is called a competence C for 
and a skill multi function. A mapping 
which assigns to each subset 
',
i. e., the image of 
given in following Table. 
is given in following Table. 
{q2} is not a state. 