Learning Mathematics with

(1)

Learning

Mathematics

with

Deutsches Forschungszentrum f

(2)

e

-

Learning: Systems and Platforms

Systems

:

- Pact-Tutors,

-

Geometry Tutor, ActiveMath,

-

ELM-Art . . . . etc

Platforms:

-

Clix

-

SAKAI

-

BLACKBOARD. . . etc

Portals:

-

for example: SaarLearn Net

-

regional nets, European e-Learning Nets, Commercial

Nets,...

(3)

W

hat is an

“

Intelligent

”

Tutor System

?

• Cassical Tutor Systems:

-

precanned solutions

• “Intelligent“

Features:

–

Adaptive sequencing

–

Interactive

problem solving

–

Error diagnosis

–

User Model

• Personalization of:

ITS:

(4)

(5)

• Personalization

–

Contents, exercises

–

Presentation

• Interaction

–

for individual content

–

in interactive exercises

–

learning tools, reflection tools

–

Natural language dialogs and input editor

• Diagnosis and feedback

(6)

Technological Goals

• Separation of knowledge and functionality

• Reuse of contents (via standards)

• Reuse and interoperability of tools and components

• Semantics for formulae, searchable mathematics content

• Various output formats and appearances

• Open web-architecture, modular design, configurability

(7)

Interdisciplinary Work

AI

Learning

Psychology

Education

Web Technology Computational Linguistics

Domain

ITS

AI

(8)

Artificial Intelligence Techniques

• user modeling

• presentation planning

• adaptive user interfaces

• problem solving systems

• deduction systems

• knowledge representation

• error diagnosis

• agent-based feedback

(9)

(10)

by author/teacher

created by learner

ActiveMath: Start Page

(11)

Choice of

(12)

Wahl des

(13)

Choice of

(14)

Choice of

(15)

Choice of

„

Book

“

: Fran

ç

ais

(and Spanish and . . . (and Spanish and . . . .)

(16)

Adaptivity:

The Student

´

s

Background

Mathematics

Biology

(17)

Adaptivity:

The Student

´

_{s Competency}

Anton

• Physics

• University, Master

• Prep for Examination

Eva

• Mathematics

• High School, A-level

• Learning/Understanding

Same Content, different Learning Goals

Introduction to Calculus

(18)

Scenario for Anton: Definitions and Exercises

(19)

Scenario for Eva: Learning in Depth

(20)

The different

„

Books

“

(21)

A „Book“: How good are you?

Good mastery

Medium mastery

Weak mastery

(22)

(23)

(24)

How can this be implemented?

MBase

OMDoc

MBase

OMDoc Pedagogical_Rules

Pedagogical Rules Course Generator Course Generator

Course Generation:

(25)

Course Generation

• Course Generator: assembles learning objects according to learner’s

– Goals

– Capability

– Context/scenario

• Using an Operator like:

ProvideAdequateExerciseFor(C):

– if mastery(C)<0.3 then Exercise(easy)

– if 0.3≤mastery(C)<0.7 then Exercise(medium)

– if 0.7≤mastery(C)<1 then Exercise(hard)

• With instructional ontology: vocabulary at the adequate level of

(26)

Course Generation (1)

Goal concept

1. Retrieve content from knowledge base

• Start with goal concept

(27)

Course Generation (2)

Goal concept

2. Filter concepts

Defrule PatternExamPrep

Allow (definition, exercise) Order (definition, exercise) Defrule ReqAppEx

Allow (definition, exercise) Test (user-kb(definition)<0.3) Add ex-for(definition 0.3)

(28)

Course Generation (3)

Goal concept

3. Linearize graph

(29)

(30)

Knowledge Representation

• Metadata

–

Mathematical dependencies

for, theory

–

Pedagogical dependencies

prerequisites, against…

–

Pedagogical characteristics

difficulty, competencies. . .

• Domain Data: OMDoc / OpenMath

–

Structures:

theory, definition, axiom, example…

(31)

Knowledge Representation

• OpenMath/OMDoc

Representation of the semantics of mathematical knowledge

-

Domain ontology

-

Structures

-

Semantics of mathematical objects

• Metadata

characterization and relationships between instructional

objects

(32)

→ Definition “ε-neighborhood” → Theorem ”neighborhood-equality” → Proof for ”neighborhood-equality” → Example for “ε-neighborhood” Exercise

(33)

OMDoc Knowledge Representation

<definition id="c6s1p4_Th2_def_monoid" for="c6s1p4_monoid„

<Title xml:lang="en">Definition of a monoid</Title> </metadata>

A monoid is a <ref xref="cp1_Th3_def_structure"> structure </ref>

<OMOBJ>

in which

<OMOBJ>

is a semi-group

(34)

(35)

Feedback in Exercises

(36)

Feedback in Exercises

(37)

Feedback in Exercises

(38)

(39)

Interactive Mathematics

Interactive

(40)

Interactive Math

(41)

Interactive Math

(42)

Interactive Math

(43)

Interactive Math

(44)

My Profile

(45)

My Profile

(46)

(47)

Information Flow in

browser

Course

generator Pedagogicalrules

MBase Se ss io n M an ag er W eb Se rv er User model http request CASMath systems XSLT html xml

(48)

SE: Event Framework

Components can

publish

and

subscribe

-

Event object: Type, timestamp, source, …

-

Loose coupling between components

Used for:

-

Generic component integration

-

User logging

- User modeling

(49)

Example: Mastery Update

Event:

UserMastery-

Change

Event:

Exercise-

(50)

Presentation Process for

Mathematics

(51)

Presentation Process

• Naive approach:

–

Start with table of contents

–

Bring XML-fragments together

–

Apply XSLT transformation

• Issues

–

Low performance

–

Adaptivity logic defined in XSLT

(52)

ActiveMath Presentation Component

• Idea: 2-stage approach for presentation

• First stage deals with individual content fragments

• Second stage combines fragments to user-specific

(53)

Presentation Pipeline: Fetching

• Collects content from knowledge base

• The output of this step are XML fragments

<definition id="def1">

Definition 1 with a reference to

<ref xref="def2">Definition 2.</ref>

</definition>

(54)

Presentation Pipeline: Pre-Processing

• Inserts server-specific information into the XML

content

<definition id="

kb1://

def1">

Definition 1 with a reference to

<ref xref="

kb1://

def2">Definition 2.</ref>

</definition>

(55)

Presentation Pipeline: Transformation

• Conversion into the output format by XSLT

• Output: content fragments

<div class="definition" id="

kb1://def1

">

Definition 1 with a reference to

$link.dict("Definition 2", "

kb1://def2

")

.

(56)

Presentation Pipeline: Assembly

• Joins the fragments to form the requested page

<html> <head/><body>

This page is generated for $user.Name. 

<div class="definition" id="kb1://def1"> Definition 1 with a reference to

$link.dict("Definition 2", "kb1://def2"). </div>

 </body></html>

(57)

Presentation Pipeline: Personalization

• Adds personalized data to the document

This page is generated for $user.Name.

<div class="definition" id="kb1://def1"> Definition 1 with a reference to

<a onClick="openInDictionary(’kb1://def2’)"> <img src="green.jpg"/>

Definition 2 </a>.

</div>

(58)

Advantages of this Approach

• Multi-formats

–

HTML, XHTML+MathML, PDF, SVG, slides

• Caching

–

real performance much higher

• Incremental rendering

–

perceived performance much higher

• High flexibility

–

Separation of concerns (MVC)

(59)

More specifically: Presentation of Mathematics

• Mathematics on the Web is a problem:

–

Mostly just as an image

–

No semantics

• ActiveMath:

–

HTML, MathML, (SVG)

–

Cross-browser: Internet Explorer, Mozilla

–

Usage of semantics to add invisible information

(60)

(61)

(62)

Math: PDF

(63)

(64)

Applications:

BRÜCKENKURSE

(65)

Student Numbers in Engineering

• Total Number of Students

A possible Remedy:

Brückenkurse with e-Learning

• Final No: in 1996 there were 52.278 graduated

engineers

(66)

Another Application:

E-Chalk

(67)

E-Chalk: Raul Rojas in Berlin

Cooperation

with

Freie Universität

Berlin

(68)

(69)

(70)

Cooperation between Maths Department

Saarbrücken and DFKI

Prof. F. Schreyer:

Computeralgebra

and

(71)

Cooperation between Maths Department and DFKI

Oliver Labs:

3-D Object

Definition

and

Rendering

(72)

(73)