Helping Students to Learn Mathematics Beyond LMSHelping Students to Learn Mathematics Beyond LMS
Martin Homik
Sakai Conference 2006, Vancouver
Saarland University, Saarbrücken
German Research Center for Artificial Intelligence, DFKI GmbH
www. .org
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OverviewOverview
1. What is ActiveMath?
Motivation, tools and content
2. Technological basis
Knowledge representation and architecture
3. Integration into Sakai
Why and how?
4. Conclusion
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What/Who is ActiveMath?What/Who is ActiveMath?
User-adaptive, personalization
Web-based, open architecture
Interactive exercises
Feedback and suggestions
Semantic knowledge representation
Integrated learning tools and repositories
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Artificial Intelligence in ActiveMathArtificial Intelligence in ActiveMath
User modeling
Content planning
Adaptive user interfaces
Problem solving systems, deduction systems
Knowledge representation
Error diagnosis
Agent-based feedback
Natural language processing
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Pedagogical GoalsPedagogical Goals
Personalization
Contents, exercises
Presentation, language
Feedback, support
Interaction and self-regulation
For content
In interactive exercises
Learning tools
Reflection
Diagnosis and feedback
Motivational support
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Pedagogical ApproachesPedagogical Approaches
OECD PISA framework: competencies problem solving,
the use of mathematical language
mathematical modeling
Problem-based learning
Formalization of pedagogical strategies
Meta-cognitive abilities
Thoughtful processing of information, tutorial dialogue
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by author/teacherby author/teacher
created by learner
Hands-On ActiveMath: Main PageHands-On ActiveMath: Main Page
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Book PageBook Page
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Chinese StudentChinese Student
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German StudentGerman Student
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Spanish StudentSpanish Student
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““Internationalized” PicturesInternationalized” Pictures
• English• German• Spanish
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““Internationalized”Internationalized” Applets Applets
• English• German• Spanish
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Exercise with different feedbackExercise with different feedback
3. Correct answer
2. Correct, but not simplified.
1. False, but error identified
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Exercise with Input EditorExercise with Input Editor
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Exercise with Input EditorExercise with Input Editor
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Exercise with hintsExercise with hints
Problem statement
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Exercise with hintsExercise with hints
Hint!
Problem statement
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ExerciseExercise
Student asks foranother hint
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ExerciseExercise
Chooses wronganswer
Repeat problemstatement
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ExerciseExercise
Student asks foranother hint
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ExerciseExercise
Answer is correct
Continue withsub problem
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ExerciseExercise
Rest is correct
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Drag-and-drop
Interactive Concept MappingInteractive Concept Mapping
Drag-and-dropWorkspaceWorkspace
PalettePalette
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derivation
SearchSearch
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SearchSearch
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Search (continued)Search (continued)
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Search (continued)Search (continued)
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Function PlotterFunction Plotter
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1. Select area of interest1. Select area of interest
Course GenerationCourse Generation
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2. Give your book a title2. Give your book a title
3. Describe your book3. Describe your book
4. Select book type4. Select book type
Course GenerationCourse Generation
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5. Select book topics5. Select book topics
Course GenerationCourse Generation
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Course GenerationCourse Generation
ApproveApprove
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Course Generation: Final BookCourse Generation: Final Book
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More ToolsMore Tools
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Content ProjectsContent Projects
LeActiveMath CalculusUniversität Augsburg University of Glasgow
150 students…
450 pagesde, en, es
Statistics HTW Saarland50 students,
200 pages de
Optimization,
Operations ResearchMary State University
100 pages )
3x50 studentsru, en
1st year Calculus U. Westminster, London50 pages (exc)
250 studentsen
Algebra Interactive! TU/Eindhoven, DFKI 30 pages en
Analysis IndividuellUni Koblenz
Uni Saarland
20 students
300 pages de
MatheführerscheinFH Dortmund
3 schools
50 pages
100 exercisesde
Fractions Gesamtschule Bellevue100 pages
70 studentsde
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OverviewOverview
1. What is ActiveMath?
Motivation, tools and content
2. Technological basis
Knowledge representation and architecture
3. Integration into Sakai
Why and how?
4. Conclusion
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Technological PrinciplesTechnological Principles
Reuse of contents through standards (OpenMath, OMDoc, partly LOM) Reuse and interoperability of tools and components
Semantics for formulae Various output formats and appearances
Open web-architecture, modular design, configurability Standard XML-communication Platform-independence
Open source
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Knowledge RepresentationKnowledge 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
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MetadataMetadata
XLearning contextschool, university, ..
Difficultyeasy, medium, difficult
Abstractnessabstract, neutral, concrete
Typical learning time
Fieldmathematics, biology, physics, ..
Representationspeech, images, numbers, …
Competencythink, argue, model, solve, ..
Competency levelknowledge, multistep, complex
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Example OMDoc<definition id="def_diff" for="deriv_symbols/diff">
<metadata>
<Title xml:lang="en">Definition of the derivative, resp., differential quotient</Title>
<Title xml:lang="de">Definition der Ableitung bzw. des Differentialquotienten</Title>
<extradata>…</extradata>
</metadata>
<CMP xml:lang="en">
A <textref xref="functions_symbols/function">function</textref> $f$ is called
<highlight type="important">differentiable at $x_0$</highlight> …
</CMP>
<CMP xml:lang="de">
Eine <textref xref="functions_symbols/function">Funktion</textref> $f$ heißt
<highlight type="important">differenzierbar an der Stelle $x_0$</highlight> …
</CMP>
<CMP xml:lang="x-all">
$ap(diff(f),x_0)=lim(x_0,both_sides,lambda(x,(ap(f,x)-ap(f,x_0))/(x-x_0)))$.
</CMP>
</definition>
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<metadata>
…
<extradata>
<relation type="domain_prerequisite">
<ref xref="diffquot_symbols/diff_quot"/>
<ref xref="maplimits_symbols/maplimit"/>
</relation>
<learningcontext value="secondary_education"/>
<learningcontext value="higher_education"/>
<learningcontext value="university_first_year"/>
<field value="all"/>
<typicallearningtime value="00:01:00"/>
<representation value="verbal"/>
<representation value="symbolic"/>
<abstractness value="abstract"/>
</extradata>
</metadata>
Example OMDoc MetadataExample OMDoc Metadata
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Content PresentationContent Presentation
KnowledgeBase
KnowledgeBase
Fetching
Pre-processing
Transformation
Assembly
Personalisation
Compilation
Presentation
1. Retrieve single learning items from knowledge base
2. Inserts server-specific information into the XML content
3. Transformation into the output format by XSLT
4. Assemble the fragments to form the requested page
5. Add personalized data to the document
6. Compile in case of LaTeX
7. Present
ViewTemplates
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Math: XHTML+MathML, PDFMath: XHTML+MathML, PDF
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Course GenerationCourse Generation
User ModelUser Model
KnowledgeBase
KnowledgeBase
PedagogicalRules
PedagogicalRules
CourseGenerator
CourseGenerator
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Course Generation (1)Course Generation (1)
Goal concept
1. Retrieve content from knowledge base• Start with goal concept• Collect recursively concepts the goal depends on• … plus additional information (e.g. examples)
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Course Generation (2)Course Generation (2)
Goal concept
2. Filter concepts• According to pedagogical rules• According to the user model
Defrule PatternExamPrepAllow (definition, exercise)Order (definition, exercise)
Defrule ReqAppExAllow (definition, exercise)Test (user-kb(definition)<0.3)Add ex-for(definition 0.3) ex-for(definition 0.5)
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Course Generation (3)Course Generation (3)
Goal concept
3. Linearize graph • according to mathematical dependency• according to pedagogical dependency
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iCMap VerificationiCMap Verification
Match directly against knowledge base and predefines exercise
Some reasoning: transitivity, equivalence
Fault-tolerance
Suggestions
<exercise ..> …</exercise>
KnowledgeBase
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MVC ArchitectureMVC Architecture
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OverviewOverview
1. What is ActiveMath?
Motivation, tools and content
2. Technological basis
Knowledge representation and architecture
3. Integration into Sakai
Why and how?
4. Conclusion
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ActiveMath and Sakai: Why?ActiveMath and Sakai: Why?
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ActiveMath and Sakai: Why ?ActiveMath and Sakai: Why ?
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ActiveMath integration into Sakai: how ?
Many common technologies
web-application, logging, velocity, hibernate, ....
Make ActiveMath a web-application tool
drop-in-Sakai installation
single sign-on
push all user-manipulations to Sakai good for privacy good for administration
provide tools for links to ActiveMath facilities list my books, link to this book, ....
Thus far:
single-sign-on working
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ActiveMath is a complex learning environment
Integrates a variety of tools
Offers diverse open content
Sakai provides missing user-centered management
Sakai + ActiveMath = Powerful Symbiosis
ConclusionConclusion
Thank you!Thank you!Questions?