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Competence-based knowledge structuresfor personalised learning
Jürgen Heller, Christina Steiner, Cord Hockemeyer, & Dietrich Albert
Cognitive Science Section, Department of Psychology,University of Graz, Austria
ProLearn-iClass Thematic Workshop3-4 March 2005, Leuven
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Overview
Knowledge Space Theory
Competence-based Knowledge Structures
Skills and Skill Assignments
Deriving Skills from Domain Ontologies Skills as Sub-Structures of a Concept Map Component-Attribute Approach
Assigning Skills to Assessment Problems
Problem-based Skill Assessment
Assigning Skills to Learning Object
Conclusions
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Knowledge Space Theory
knowledge domain: set of assessment problems
a. ½ x 5/6 = ?
b. 378 x 605 = ?
c. 58.7 x 0.94 = ?
d. Gwendolyn is 3/4 as old as Rebecca. Rebecca is 2/5 as old
as Edwin. Edwin is 20 years old. How old is Gwendolyn?
e. What is 30% of 34?
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knowledge state of a learner: set of problems that he/she is capable of solving
mutual dependencies between problems from a correct answer to certain problems we can
surmise a correct answer to other problems
captured by surmise relation
Knowledge Space Theory
d
a
c
b
e
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not all potential knowledge states (i.e. subsets of problems) will actually be observed
knowledge structure
collection of possible knowledge states
example
K ={Ø, {a}, {b}, {a, b}, {b, c}, {a, b, c}, {b, c, e}, {a, b, c, e}, {a, b, c, d}, Q}
Knowledge Space Theory
d
a
c
b
e
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{a, b, c, d, e}
{a, b, c, e}
{b, c, e}
{b, c}
{b} {a}
{a, b}
{a, b, c}
{a, b, c, d}
knowledge structure
Knowledge Space Theory
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key features of Knowledge Space Theory
adaptive knowledge assessment
determining the knowledge state by presenting the learner with only a subset of problems
representation of individual learning paths
Knowledge Space Theory
{a, b, c, d, e}
{a, b, c, e}
{b, c, e}
{b, c}
{b} {a}
{a, b}
{a, b, c}
{a, b, c, d}
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Knowledge Space Theory in its original formalisation is purely behaviouristic
focus on solving assessment problems
Knowledge Space Theory needs to be extended to incorporate
underlying skills and competencies
learning objects
Competence-based Knowledge Structures
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relevant entities set Q of assessment problems set L of learning objects (LOs) set S of skills relevant for solving the problems,
and taught by the LOs
relevant structures knowledge structure on the set Q of assessment
problems learning structure on the set L of LOs competence structure on the set S of skills
main goal identifying the pieces of information that are needed for
establishing those structures
Competence-based Knowledge Structures
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Deriving Skills from Domain Ontologies
how to identify and structure skills? e.g. cognitive task analysis, querying experts,
systematic problem construction utilise information coming from domain ontologies
ontology specification of the concepts in a domain and
relations among them represent the structure of a knowledge domain
with respect to its conceptual organisation
concept map common way of representing ontologies network representation
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Deriving Skills from Domain Ontologies
a) skills as sub-structures of a concept map
a skill can be identified with a subset of propositions represented in a concept map
example: geometry of right triangles
skill ‚knowing the Theorem of Pythagoras‘
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Deriving Skills from Domain Ontologies
a) skills as sub-structures of a concept map a structure on the skills is induced, for example, by set-
inclusion
if skill x is subset of skill y then skill x is subordinatedto skill y
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Deriving Skills from Domain Ontologies
b) component-attribute approach
concept map represents results from curriculum and content analysis basic concepts to be taught
e.g. ‘Theorem of Pythagoras’
learning objectives related to these concepts include required activities of the learner may be captured by action verbs
e.g. ‘state’ or ‘apply’ a theorem
skill: identified with a pair consisting of a concept and an action verb e.g. ‘state Theorem of Pythagoras’
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Deriving Skills from Domain Ontologies
b) component-attribute approach
concepts with their hierachical structure
e.g. `Theorem of Pythagoras´ prerequisite for `Altitude Theorem´corresponding to curriculum
order on the action verbs
e.g.: `state´ prerequisite for `apply´
c3
c4
c1
c2
a1
a2
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Deriving Skills from Domain Ontologies
b) component-attribute approach
building the direct product of these two component orderings results in a surmise relation on the skills
e.g. skill c2a2 is a prerequisite to the skills c2a1, c1a2,
and c1a1
c2a2
c1a1
c3a2
c1a2
c2a1
c3a1
c4a2
c4a1
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Assigning Skills to Assessment Problems
relationship between assessment problems and skills is formalised by two mappings skill function s
associates to each problem a collection of subsets of skills, each of which consists of those skills sufficient for solving the problem
problem function p associates to each subset of skills the set of problems that
can be solved in it
both concepts are equivalent, i.e. given one function the other is uniquely determined
the assignment of skills puts constraints on the possible knowledge states and thus defines a knowledge structure
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Assigning Skills to Assessment Problems
example
Q = {a, b, c, d} and S = {s, t, u}
skill function:
s(a) ={{s, u}}
s(b) ={{u}}
s(c) ={{s}, {t}}
s(d) ={{t}}
p(Ø) = Ø
p({s}) = {c}
p({t}) = {c, d}
p({u}) = {b}
p({s, t}) = {c, d}
p({s, u}) = {a, b, c}
p({t, u}) = {b, c, d}
p(S) = Q
corresponding problem function:
knowledgestructure
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{a, b, c, d, e}
{a, b, c, e}
{b, c, e}
{b, c}
{b} {a}
{a, b}
{a, b, c}
{a, b, c, d}
step 1
adaptive assessment of knowledge state
problem c
solved
problem d
solved
problem e
not solved
Problem-based Skill Assessment
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Problem-based Skill Assessment
step 2
mapping of the knowledge state identified for a learner into the corresponding competence state
using the skill function
example
knowledge state {b}
knowledge state {c}
non-unique assignments have to be resolved
s(a) = {{s, u}}
s(b) = {{u}}
s(c) = {{s}, {t}}
s(d) = {{t}}
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Assigning Skills to Learning Objects
once the competence state of a learner has been determined a personalised learning path may be selected
based on assigning skills to learning objects
relationship between learning objects and skills is mediated by two mappings
mapping r associates to each LO a subset of skills (required skills), characterising the prerequisites for dealing with it
mapping t associates to each LO a subset of skills (taught skills), referring to the content actually taught by the LO
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Assigning Skills to Learning Objects
the mappings r and t induce a learning structure on the set of LOs impose constraints on the competence states that
can occur resulting competence structure characterises the
learning progress
allow deciding upon next LO, given a certain competence state
referring to learning path of the competence structure a suitable learning object is selected, featuring
required skills that the learner has already available
taught skills that correspond to next step in learning path
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Conclusions
extended Knowledge Space Theory
takes into account skills and competencies as psychological constructs underlying the observable behaviour
allows for integrating ontological information
provides a basis for efficient adaptive assessment of skills and competencies
incorporates learning objects into a set-theoretical framework
forms a basis for personalised learning
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