Learner Modelling and Adaptation in Math-Bridge

1. Learner Modelling and
Adaptation in Math-Bridge Sergey Sosnovsky!
Saarland University

2. Learning Content Presentation: Dashboard

3. Learning Content Presentation: Main View

4. Personalised Course Generation
1 2
3
4

5. Adaptive Link Annotation

6. Micro-course generation

7. Intelligent Adaptive e-Learning System:
Main Components
Instructional
Content
Interaction
0..1..1
..0..1..
1..
Domain
Model
!
!
!
Learner
Model
Pedagogical
Model
Adaptation
Me t a d a t a

8. Learning Events of
Math-Bridge
0..1..1..0
..1..1..

9. Rich continuos stream of learning data
❖ Any interaction of the student with Math-Bridge causes
an event in the system logs;!
❖ More than 30 types of events (e.g., system login/logout,
course started/finished, exercise started/finished, etc.);!
❖ More than 50 attributes (e.g., for the exerciseStep event:
time, user, session, courseId, successRate, metadataText,
userInputDelay, userInputText,…);

10. Content and
knowledge modelling
in Math-Bridge
Instructional
Content
Domain
Model
!
!
!
Me t a d a t a

11. Knowledge Items
Abstract Concepts
Concepts with content
Content

12. Metadata
❖ Descriptive!
❖ author!
❖ date…!
❖ Pedagogical!
❖ difficulty!
❖ competency!
❖ educational level…!
❖ Semantic!
❖ is prerequisite for!
❖ is exercise for!
❖ is introduction for…

13. Ontology
❖ 536 symbols!
❖ will, probably, need to be
extended

14. Martin Homik 5th Sakai Conference 2006, Vancouver
!14
Knowledge Representation
D
S
EX
P
T
S S
S
isA
D
D T
XE
Definition
E
Symbol
Example
Theorem
ProofExercise
X
forfor
forforfor
D D
for counter
P
for
S S
for depends on
depends on
Abstract Layer
Content Layer
Satellite Layer

15. OMDoc
❖ All content and its metadata, are
represented in OMDoc!
❖ OMDoc is an XML dialect developed for
math documents !
❖ Formulas are written in OpenMath!
❖ OpenMath is an extensible standard for
representing the semantics of mathematical
objects
<definition id="c6s1p4_Th2_def_monoid" for="c6s1p4_monoid„
<metadata>
<depends-on>
<ref theory="cp1_Th3" name="structure" />
</depends-on>
<Title xml:lang="en">Definition of a monoid</Title>
</metadata>
<CMP xml:lang="en" format="omtext">
A monoid is a <ref xref="cp1_Th3_def_structure"> structure </ref>
<OMOBJ>
<OMS cd="elementary" name="ordered-triple"/>
<OMV name="M"/> <OMS cd="cp4_Th2" name="times"/> <OMS cd="cp4_Th2" name="unit"/>
</OMOBJ>
in which
<OMOBJ>
<OMS cd="elementary" name="ordered-pair"/>
<OMV name="M"/> <OMS cd="cp4_Th2" name="times"/>
</OMOBJ>
is a semi-group
with <ref xref="c6s1p3_Th2_def_unit">e</ref>
<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMS cd="cp4_Th2" name="unit"/>
</OMOBJ>.
</CMP>
<FMP><OMOBJ> ... </OMOBJ></FMP>
</definition>
Definition of
a Monoid

16. Learner Modelling in
Math-Bridge

17. Background
❖ Dynamic Overlay Model with forgetting!
❖ Specific challenges !
❖ Dynamic Domain Model!
❖ Dynamic Content Base and Metadata annotations!
❖ SLM (Eric, Arndt, Salim)

18. Evidences
❖ (s,e,c,p,l,a)!
❖ Student!
❖ Exercise!
❖ Concept!
❖ Competency!
❖ Achievement

19. Updates
❖ Direct evidence - individual events for 1 concept, 1
process!
❖ Indirect evidence - propagation!
❖ Intra-Concept: across competencies!
❖ Inter-Concept: prerequisite, for

20. Amplitude of the update
❖ IRT:
psychometric
theory for
testing!
❖ Used
successfully
since 20+
years

21. IRT Usage
❖ Pool of calibrated items with known ICC!
❖ Logistic function (difficulty, discrimination, guess)!
❖ Idea: Measure latent trait 𝞱!
❖ Administer sequence of test items!
❖ 𝞱 uncovered by responses to items

22. IRT vs. MthBridge—IRT
Propoer IRT MathBridge—IRT
ICC Empirical Theoretical
Input
Item Response
Sequence
Sparse
Evidences
Answers Dichotomous Continuous
Difficulty Single factor
Difficulty/
Competency
Independence
Items are
independent of each
other
Exercises are often
related
Learning
No learning between
or during
assessment
Learning is essential
for Math-Bridge

23. Belief Masses
❖ Round achievement to {1,0}!
❖ if r=1:
m(H(b)) = P(correct | 𝞱 =b)!
❖ if r=0:
m(H(b)) = 1-P(correct | 𝞱 =b)!
!
❖ restrict updated hypotheses
to Information Radius
interval: [irtdiff ±δ]
𝞱
p(correct)

24. Mastery inference
Learning Event
(Raw evidence)
𝞱
p(correct)
Dempster-Shafer
Best belief about
mastery

25. Competency Models
❖ Bloom (subset: K / C / A)!
❖ PISA (☈ operationalization)!
❖ Math-Bridge !
❖ TU-D cognitive operators!
❖ Commonality: Multi-dimensional overlays
Exercise
Concept
is For
Competency

26. Mastery Aggregate
❖ Single “mastery” value!
❖ Necessary for Course Generation!
❖ Implemented as (weighted) average of competencies

27. Adaptation in Math-
Bridge
Pedagogical
Model
Adaptation

28. Scenario LearnNew
Introduce
Develop
Practice
Connect
Reflect

29. Scenario LearnNew
Introduce
Develop
Practice
Connect
Reflect
Motivate
Context
Illustrate
Prerequisites

30. Scenario LearnNew
Introduce
Develop
Practice
Connect
Reflect
Motivate
Context
Illustrate
Prerequisites
(:method (motivate! ?c)
((learnerProperty hasEducationalLevel ?el)
(learnerProperty hasAnxiety ?c ?an)
(?an <= 2)
(GetElement
((class Exercise)
(class Introduction)
(relation isFor ?c)
(property hasLearningContext ?el)
(property hasDifficulty very_easy))))
((insert! ?element)))
IF
THEN

31. Scenario LearnNew
Introduce
Develop
Practice
Connect
Reflect
Motivate
Context
Illustrate
Prerequisites
(:method (motivate! ?c)
((learnerProperty hasEducationalLevel ?el)
!
!
(GetElement
((class Example)
(class Introduction)
(relation isFor ?c)
(property hasLearningContext ?el)
(property hasDifficulty very_easy))))
!
((insert! ?element)))
IF
THEN
(:method (motivate! ?c)
((learnerProperty hasEducationalLevel ?el)
(learnerProperty hasAnxiety ?c ?an)
(?an <= 2)
(GetElement
((class Exercise)
(class Introduction)
(relation isFor ?c)
(property hasLearningContext ?el)
(property hasDifficulty very_easy))))
((insert! ?element)))