technology enhanced math rehab

First Class Mathematics The Technology Enhanced

Mathematics Rehabilitation Clinic

Ambjörn Naeve

The Knowledge Management Research GroupThe Royal Institute of Technology, and

Uppsala UniversitySweden

http:// kmr.nada.kth.se/wiki/Ambhttp://kmr.nada.kth.se/wiki/Amb/MathematicsEducationProjects

EPFL Seminar, 2007-10-30

The KMR group - what do we do?• We work with Asynchronous Public Service

in the form of infrastructures, architectures, frameworksand tools that contribute towards the creation of a PublicKnowledge and Learning Management Environment.

• This PKLME should enable a global, (synchronous and)asynchronous public discourse that aims to enhance thelearning of all participants.

• Our main information architecture for this PKLMEis the Knowledge Manifold (navigated in Conzilla).

• All our software is Open Source and based on SemanticWeb technology.

We work to enable a fundamental shift

• From: Teacher-centric, curricular-oriented “knowledge push”

• To: Learner-centric, interest-oriented “knowledge pull”

A traditional educational design pattern

(Tenured Preacher / Learner Duty)

School Duties

Course

Preacher

PrisonerEmployee

Teacher Learner

Life-longteaching:

Minimallearning efforts

Security

Doing time in return fora degree

Pupil

Confinement

Tenure

Agent 007with a rightto kill interest

* *

An emerging educational design pattern

(Requested Preacher / Learner Rights)

School Rights

Resource Seeker

Learner

Student

Knowledge

Life-longTeacher

Consultant

learning:

Pedagogical

Interest

Developingyour interests

Requestedteaching:

You teach as longas somebody is learning

Course

* *

What am I interested in?

What is there to know about it?

What do I want to know about it?

Question-Based Learning The three fundamental questions

Structure of todayʼs math education system

Closed, layered architecture based on:

• lack of subject understanding within the earlier layers.• minimization of teaching duties within the later layers.

• life long teaching with:

• curricular-oriented ”knowledge push”.

Problems with todayʼs math education

It does not:

• promote understanding.

• support personalization.

• integrate mathematics with human culture.

• stimulate interest.

• integrate abstractions with applications.

• support transition between the different layers.

Question

How Why

Mathematical Activity

Algorithm Proof

Long term trend in mathematics education

Solution Elimination Problem

eliminate

ProblemElimination

Cause

Organization

edissolv

Pr oblemSolution

crystalize

Symptom

Organization

edissolv

Solving vs Eliminating a Problem

dissolve

Crime Jail

crystalize

Organized

Economic Dysfunctionality

Legal System

Applying the problem/solution patternto the judicial system

dissolve

Conceptual Difficulty Algorithmic Ability

crystalize

Anticipated

Understanding Dysfunctionality

Pedagogical System

isA

Mathematicsis difficult

nurturing the difficulties

I never understood itwhen I was at school

because

behave or degrade

Applying the problem/solution patternto the earlier parts of math education

dissolve

Conceptual Difficulty Academic Status

crystalize

Anticipated

Understanding Dysfunctionality

Publicational System

isA


nurturing the difficulties

I understand it, andI am smarter than you

because

publish or perish

Applying the problem/solution patternto the later parts of math education

dissolve

Conceptual Difficulty Algorithmic Ability

crystalize

Anticipated

Understanding Disfunctionality

Computational Industry


isA

marketing the difficulties

solve by computation

Formulation Solution

ProblemExtractingmathematicalskeleton

Computing thealgorithmicsolution

the

We can help youto solve your problems

but

Applying the problem/solution patternto the commercial parts of math education

Attitude

Expectation

Defence Parent

Teacher

Confirmation

Calculating with X is hard.I never understood itwhen I was at school.

Child

Children, we will now start to calculatewith this mysterious thing Xthat you have all heard about.

Shit, I’m never going tounderstand this stuff!

Yeah, just as I figured, I simply can’t understand this stuff.

isA

isA

isA

isA

time

Conceptual Difficulty

Slightest sign of mental resistance

isA

The Xanxietypattern

Possibilities for improving math education

• visualizing the concepts.• interacting with the formulas.

• using ICT to increase the ”cognitive contact” by:

• personalizing the presentation.

Promoting life-long learning based on interest by:

• improving the narrative by:• showing before proving.

• focusing on the evolutional history.

• routing the questions to live resources.

• proving only when the need is evident.

Mathematics as a de-semantization processthat transforms ”meaning” into ”form”

Meaning

Force

Velocity

Form

Vector

Mathe- matics

Appli- cation

Meaning Form

Create / Apply Mathematics

Create Math

Apply Math

Definition Operational

What is it? How does it act?

Existential

Create Math

Model semantics

Domain model

Extract structure

Math skeleton

Turn into definition

Create Mathematics

“Forget” meaning

Apply Math

Create model

Collect data

Experimental script

Over-determined script

Model

Modify script

List of “who did it”

Solvable scriptSolve

script

“CSI-Mathematics”

Apply Mathematics

Interpret symbols

Minimal modification=> pseudo-inverse

• Ideology: Within the mathemagic project we want to emphasize the speculative and creative aspects of mathematics.• Aim: To stimulate interest in mathematics among young and old by emphasizing “week-end mathematics”.• Basic idea: Problematize and dramatize the major mathematical concepts by anchoring them in the history of ideas.• Metdod: Improving the narrative - showing without necessarily proving.• Form: The news of yesterday: Proust: “In Search of Lost Mathematics.” Knowledge components (featuring Pythagoras, Archimedes, Newton, …) are “tied together” by a ”news anchor in space-time" who follows different trails along the evolution of mathematical ideas.

Project Mathemagic

1. The story of the people who thought the world was understandable. From Thales and Pythagoras to Demokritos and Aristarkos.

2. The story of the people that wanted to escape the realm of the senses. From Plato via Augustinus and Aquino to the “scolastic age”.

3. The mathematics of the eye: The development of true perspective. From Pappos via de la Franchesca and da Vinci to Desargues, Pascal, Poncelét, Plücker, Grassmann and Klein.

4. Einstein for Flatlanders: Two-dimensional relativity theory. The story about the flatlanders that lived on a sphere and the flatlanders that lived on a torus (“dough-nut”).

Nine Mathemagic Stories

5. The story of the people that disregarded almost everything. The evolution of abstract thinking: From induction and abduction to abstraction and deduction. "The power of thinking is knowing what not to think about."

6. About the difficulties in overcoming psychological complexes. The story about the development of the concept of number: From “positive” to “negative”, from “rational” to “irrational”, from “real” to “imaginary” and “complex.”

7. What is there between the atoms? Does the world consist of particles or waves - or maybe something else? The historical debate from Thales versus Pythagoras via Newton versus Huygens to Einstein versus Bohr and Heisenberg and the break-up of the particle concept (super-string theory).


8. The mysterious law about the degradation of work: The principles of energy and entropy. The development of the energy concept from Leibniz via Rumford and Carnot to Maier, Joule and Bolzmann.

9. The story of the long-lived demon that was unable to forget. Maxwellʼs demon and the deep connections between information theory and thermodynamics.


Ongoing mathematical ILE work at KMR

• Mathemagic component archive in Confolio

• Interactive geometrical constructions with PDB

• Interacting with mathematical formulas

• CyberMath: a shared 3D ILE for exploring math

• using LiveGraphics3D / Graphing Calculator

• Virtual Mathematics Explainatorium with Conzilla

Seven different Knowledge Roles in a KM• Knowledge Cartographer

• Knowledge Composer

• Knowledge Librarian

• Knowledge Coach

• Knowledge Preacher

• Knowledge Plummer

• Knowledge Mentor

• constructs context-maps.

• fills context-maps with content-components.

• combines content-components into learning modules.

• cultivates questions.

• provides live answers.

• connects questions to relevant preachers.

• supplies motivation and supports self reflection.

Content

Contexts

Concept

The Conzilla “Mantra”

Content in Contexts through Concept

= Outsides of Concept

= Inside of Concept= Border between these

Right-clicking on a concept or concept-relation brings up a menu with three choices: Contexts, Content, and Information.

• Selecting Contexts opens a sub-menu, which lists all the other contexts where this concept or concept-relation appears.

• Selecting Content opens a window (to the right) where the content-components of the concept or concept-relation are listed.

• Pointing to a content-component brings up information about it, and double-clicking on a content-component opens another window where the content is shown.

Conzilla (www.conzilla.org)

Context Content

Conceptual Browsing: Viewing the content

Projective

Geometry

Algebraic

Differential Surf

View

Info

What

How

Where

When

Who

Projective geometry is the studyof the incidencesof points, lines

in space.

It could be calledthe geometryof the eye

and planes

Surf

View

Info

Geometry

Projective

Algebraic

Differential

Context

Aspect

Level

School

Elementary

Secondary

High

W H W

...

hat

ow h

ere

Aspect Filter

Conceptual Browsing: Filtering the content

Surf

View

Info

WhatHow

WhereWhen

Who

Mathematics

Where is mathematics done?

Content

Clarification

Depth

Context

Science

Magic

Religion

Philosophy

Mathematicsinvoke

illustrateapply

inspire

Contextualize

How is mathematics applied to science?

Content

Surf

View

InfoWhat

How

Where

When

Who

Magic

Philosophy

Religion

Science

Mathematicsinvoke

illustrateapply

inspire

Clarification

DepthContextualize

Context

A is true

Science

∗assumption

∗ conditional statement

logical conclusion∗⇒

↓

B is true

If A were truethen

B would be true

Mathematics

↓

Falsification of assumptionsby falsification of their logical conclusions

experiment∗

↓

↓

fact

Science

Magic

Religion

Philosophy

Mathematicsinvoke

illustrateapply

inspire

Mathematics

Therefore Bmust be true

∗

ScienceExperimental Theoretical

↓

↓

B survivesthe test

Assumption∗

Fact∗

Conditional statementIf A were true,then B would be true

A is true

Experiment∗ Logical

conclusion

<<is an>>

<<is a>>

∗

∗

a Test

∗

∗ ∗ Theory∗∗

∗

∗ 1

↓

↓

The interplay between mathematics and science

Falsification of assumptions by falsification of their logical conclusions

Virtual Mathematics Exploratorium - Entrance

Virtual Mathematics Explainatorium - Filtering

Virtual Mathematics Explainatorium - Viewing

Dynamic Geometry with PDB

Taylor Expansion with the Graphing Calculator

www.nada.kth.se/~osu/math/Geometry/Quadrics/one_sheeted_hyp_tang1.html.

Mathematical Component Archive

CyberMath: A Shared Virtual Environmentfor the Interactive Exploration of Mathematics

• teaching of both elementary, intermediate and advanced mathematics and geometry.

Goals: The CyberMath system should allow:

Means:• Making use of advanced VR technology (e.g. DIVE).

• global sharing of resources.

• the teacher to teach in a direct manner.

• students to work together in groups.

• teachers to present material that is hard to visualize using standard teaching tools.

CyberMath: an avatar using a laser pointer

CyberMath: finding the kernel of a linear map

=

=

=

CyberMath: importing a Mathematica object

CyberMath: the cylindrical exhibition hall

CyberMath: Avatars visiting the Virtual Museum

CyberMath: The Solar Energy Exhibit

CyberMath: The Cooperative Learning Mode

CyberMath: The WASA Platform

• Mathemagic: Mathematical storytelling

Mathematical ILE collaborative projects < 2002

Advanced Media Technology Laboratory (KTH)

Swedish Learning Lab • Content and context of mathematics in engineering education (with DSV, KTH/Kista)

Learning Lab Lower Saxony & Stanford LL • Personalized Access to Distributed Learning Resorces

• MathViz: Personalized Mathematical Courselets

• 3D Communication and Visualization Environments for Learning (with DIS, Uppsala Univ)

Thales Pythagoras

Herakleitos

Parmenides

What is the basic stuffthat the universe is made of?

Disinterested knowledgeis the most effectivepurification of the soul.

The Greek Beginning

• everything changes

• nothing changes

• all is substance (matter) • all is form (number)

Ionian school Pythagorean school

• atoms build up shapes (bodies) that move around in empty spaceDemokritos

Pythagorean School

Geometry

Form RelationFormal

ArithmeticReligion

Ionian School

Formal

Figure

Armonia

Extasis

Music

Substance Number

Astronomy

Katharsis

Thales Pythagoras

Philolaus

Herakleides

Aristarchus

Definitions

Proofs

Harmony ofthe Spheres

RationalMysticism

DigitalWorldview

Contemplatingthe structureof numbers

Disinterestedknowledgeis the mosteffectivepurificationof the soul

1 3 5 7+ + + 42

=

2

1---

3

2---

4

3---

5

4---! ! !

1

1---

2

2---

3

3---

4

4---" " "

1 2 3 4+ + + 10=

32

42

+ 52

"

34

5

Pythagoras’ theorem

The five regular solids

The three regular tilings

The intervals betweenthe heavenly bodiesare determined by thelaws of musical harmony

2:1 = octave3:2 = quint4:3 = quart

The earth also rotatesaround its own axis.The central fire is inside the earth

C

#

b

The quint circle

The earth is sphericaland rotates around thecentral fire, protectedby the counter-earth

The earth rotates dailyaround its own axis, andannually around the suntogether with the planets

odd <=> unity

even <=> infi nity

pointline

plane

The holy tetrakys

solid

Monad

The one that iseverything

The unspeakables

2

1

1

Numberis the essenceof all things

What is the basic stuffthat the universeis made of?

PairwiseCommensurability

destroyed

by==

The Greek Beginning: Harmony of the Spheres

Energy - an overview

Trans- formation

Functionarguments and parameters

Arguments

Parameters

ResultSet of Set ofRule

Function1

A

P

R

Function2

Two different perspectives:

Function of Aparametrized by P Function of A and P

Same result: Function1P(A) Function2(A,P) =

Researcher ArtistConsultantTeacher

Academia Business

Later Education

Business

Employee

nono no no

no

nono

yes no

understand?

make money?

socially conscious?

creative? creative? creative? creative?

make money?

Teacher

yes yes

yesyes yesyes

yes

despise

Science Humaniora

stigmatize

Rational

Non Human

Emotional

Academia

Nerdy

Luke Warm

Scientist Artist

Sexy

Cool Hot!

I ama rock

siliconrocks

Computational

my heartis a rock

heartof rock

Relational

my heartrocks

rockat heart

hot & coolrocks

I amrock

Einstein

FrankensteinSpringsteen

Resource Components / Learning Modules

Learning Environment

Learning Module

What to Teach

Resource Component* *

What to Learn

separating connecting

from with

through

Multiple Narration Component Composition

through

QBL: the 3 performing knowledge roles

Preacher

Knowledge

Coach Plumber

Master Gardener Broker

you teachas longas somebodyis learning

you assistin developingeach indiduallearning strategy

you findsomeoneto discussthe question

´ ´ ´

QBL: the 3 performing knowledge roles (cont)

Source Strategist Opportunist

Knowledge

fascination methodology opportunity

qualitymeasured bygiven answers

qualitymeasured bylost questions

qualitymeasured byraised questions

http://kmr.nada.kth.se/wiki/Amb/TheGardenOfKnowledge

http://kmr.nada.kth.se/wiki/Amb/VirtualMathematicsExplainatorium

http://kmr.nada.kth.se/wiki/Amb/FirstClassMathematics

http://kmr.nada.kth.se/wiki/Amb/MathPRAO

http://kmr.nada.kth.se/wiki/Amb/CyberMath

http://kmr.nada.kth.se/wiki/Amb/ProjectiveDrawingBoard

http://kmr.nada.kth.se/wiki/Amb/MatriksProjektet

http://kmr.nada.kth.se/wiki/Amb/FLIThttp://kmr.nada.kth.se/wiki/Amb/Projects-Matemagi

Web links to some of my math projects