helping students to learn matehmatics beyond lms

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

2Martin Homik 5th Sakai Conference 2006, Vancouver

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


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


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


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


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


by author/teacherby author/teacher

created by learner

Hands-On ActiveMath: Main PageHands-On ActiveMath: Main Page


Book PageBook Page


Chinese StudentChinese Student


German StudentGerman Student


Spanish StudentSpanish Student


““Internationalized” PicturesInternationalized” Pictures

• English• German• Spanish


““Internationalized”Internationalized” Applets Applets

• English• German• Spanish


Exercise with different feedbackExercise with different feedback

3. Correct answer

2. Correct, but not simplified.

1. False, but error identified


Exercise with Input EditorExercise with Input Editor


Exercise with hintsExercise with hints

Problem statement


Exercise with hintsExercise with hints

Hint!

Problem statement


ExerciseExercise

Student asks foranother hint


ExerciseExercise

Chooses wronganswer

Repeat problemstatement


ExerciseExercise

Student asks foranother hint


ExerciseExercise

Answer is correct

Continue withsub problem


ExerciseExercise

Rest is correct


Drag-and-drop

Interactive Concept MappingInteractive Concept Mapping

Drag-and-dropWorkspaceWorkspace

PalettePalette


derivation

SearchSearch


SearchSearch


Search (continued)Search (continued)


Function PlotterFunction Plotter


1. Select area of interest1. Select area of interest

Course GenerationCourse Generation


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



5. Select book topics5. Select book topics




ApproveApprove


Course Generation: Final BookCourse Generation: Final Book


More ToolsMore Tools


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


OverviewOverview






Why and how?

4. Conclusion


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


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


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


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>


<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


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


Math: XHTML+MathML, PDFMath: XHTML+MathML, PDF



User ModelUser Model

KnowledgeBase

KnowledgeBase

PedagogicalRules

PedagogicalRules

CourseGenerator

CourseGenerator


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)



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)



Goal concept

3. Linearize graph • according to mathematical dependency• according to pedagogical dependency


iCMap VerificationiCMap Verification

Match directly against knowledge base and predefines exercise

Some reasoning: transitivity, equivalence

Fault-tolerance

Suggestions

<exercise ..> …</exercise>

KnowledgeBase


MVC ArchitectureMVC Architecture


OverviewOverview






Why and how?

4. Conclusion


ActiveMath and Sakai: Why?ActiveMath and Sakai: Why?


ActiveMath and Sakai: Why ?ActiveMath and Sakai: Why ?


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


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?

helping students to learn matehmatics beyond lms

Technology