1.
EPFL Seminar, 2007-10-30
First Class Mathematics
The Technology Enhanced
Mathematics Rehabilitation Clinic
Ambjörn Naeve
The Knowledge Management Research Group
The Royal Institute of Technology, and
Uppsala University
Sweden
http:// kmr.nada.kth.se/wiki/Amb
http://kmr.nada.kth.se/wiki/Amb/MathematicsEducationProjects
2.
The KMR group - what do we do?
• We work with Asynchronous Public Service
in the form of infrastructures, architectures, frameworks
and tools that contribute towards the creation of a Public
Knowledge and Learning Management Environment.
• This PKLME should enable a global, (synchronous and)
asynchronous public discourse that aims to enhance the
learning of all participants.
• Our main information architecture for this PKLME
is the Knowledge Manifold (navigated in Conzilla).
• All our software is Open Source and based on Semantic
Web technology.
3.
We work to enable a fundamental shift
• From: Teacher-centric, curricular-oriented “knowledge push”
• To: Learner-centric, interest-oriented “knowledge pull”
4.
A traditional educational design pattern
(Tenured Preacher / Learner Duty)
Security Employee Conﬁnement
Prisoner
Tenure Preacher Pupil School Duties
Minimal
Life-long
learning efforts
teaching:
* *
Teacher Learner Doing time
Agent 007
in return for
with a right Course a degree
to kill interest
5.
An emerging educational design pattern
(Requested Preacher / Learner Rights)
Knowledge Seeker Interest
Resource
Pedagogical Student
Consultant School Rights
Requested Life-long
*
*
Teacher Learner
teaching: learning:
Course
Developing
You teach as long
your interests
as somebody is learning
6.
Question-Based Learning
The three fundamental questions
What am I interested in?
What is there to know about it?
What do I want to know about it?
7.
Structure of todayʼs math education system
Closed, layered architecture based on:
• curricular-oriented ”knowledge push”.
• life long teaching with:
• lack of subject understanding
within the earlier layers.
• minimization of teaching duties
within the later layers.
8.
Problems with todayʼs math education
It does not:
• stimulate interest.
• promote understanding.
• support personalization.
• support transition between the different layers.
• integrate abstractions with applications.
• integrate mathematics with human culture.
9.
Long term trend in mathematics education
Question
How Why
Algorithm Proof
Mathematical Acti
vity
10.
Solving vs Eliminating a Problem
Elimination
Problem
Solution
Symptom Cause
eliminate
dissolve
Solution Pr oblem Elimination Problem
dissolve
crystalize
Organization
Organization
11.
Applying the problem/solution pattern
to the judicial system
Legal System
dissolve
Crime Jail
crystalize Economic Dysfunctionality
Organized
12.
Applying the problem/solution pattern
to the earlier parts of math education
Pedagogical System
dissolve
Conceptual Difficulty Algorithmic Ability behave or degrade
crystalize Understanding Dysfunctionality
Anticipated
nurturing the difficulties
isA
because
Mathematics I never understood it
is difficult when I was at school
13.
Applying the problem/solution pattern
to the later parts of math education
Publicational System
dissolve
Conceptual Difﬁculty Academic Status publish or perish
crystalize Understanding Dysfunctionality
Anticipated nurturing the diffculties
i
isA
because I understand it, and
Mathematics
I am smarter than you
is difficult
14.
Applying the problem/solution pattern
to the commercial parts of math education
Extracting the Computing the
mathematical Problem algorithmic
skeleton solution
Formulation Solution
Computational Industry
dissolve
Conceptual Difficulty Algorithmic Ability solve by computation
crystalize Understanding Disfunctionality
Anticipated marketing the difficulties
isA
but
Mathematics We can help you
is difficult to solve your problems
15.
The X Calculating with X is hard.
I never understood it
when I was at school.
anxiety isA
pattern Defence Parent
Attitude Teacher
isA
Children, we will now
start to calculate
with this mysterious thing X
that you have all heard about.
Expectation Child
isA
Shit, I’m never going to
understand this stuff!
Conceptual Difficulty
isA
Slightest sign of mental resistance
Confirmation
isA
Yeah, just as I figured,
I simply can’t understand this stuff.
time
16.
Possibilities for improving math education
Promoting life-long learning based on interest by:
• using ICT to increase the ”cognitive contact” by:
• visualizing the concepts.
• interacting with the formulas.
• personalizing the presentation.
• routing the questions to live resources.
• improving the narrative by:
• showing before proving.
• proving only when the need is evident.
• focusing on the evolutional history.
17.
Mathematics as a de-semantization process
that transforms ”meaning” into ”form”
Appli-
cation
Meaning Form
Mathe-
matics
Force
Vector
Velocity
18.
Create / Apply Mathematics
Apply
Math
Form
Meaning
Create
Math
Deﬁnition
Existential Operational
What is it? How does it act?
19.
Create Mathematics
Create
Math
“Forget” meaning
Model Extract Turn into
semantics structure definition
Domain Math
model skeleton
20.
Apply
Apply Mathematics
Math
Model
Create Collect
model data
Interpret symbols
Experimental Over-determined
script script
Solvable
Solve Modify
script
script script
Minimal modiﬁcation
=> pseudo-inverse
List of
“who did it”
“CSI-Mathematics”
21.
Project Mathemagic
• 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-timequot;
who follows different trails along the evolution of mathematical ideas.
22.
Nine Mathemagic Stories
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”).
23.
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.
quot;The power of thinking is knowing what not to think about.quot;
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).
24.
Nine Mathemagic Stories
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.
25.
Ongoing mathematical ILE work at KMR
• Virtual Mathematics Explainatorium with Conzilla
• Interactive geometrical constructions with PDB
• Interacting with mathematical formulas
• using LiveGraphics3D / Graphing Calculator
• Mathemagic component archive in Confolio
• CyberMath: a shared 3D ILE for exploring math
26.
Seven different Knowledge Roles in a KM
• Knowledge Cartographer
• constructs context-maps.
• Knowledge Librarian
• ﬁlls context-maps with content-components.
• Knowledge Composer
• combines content-components into learning modules.
• Knowledge Coach
• cultivates questions.
• Knowledge Preacher
• provides live answers.
• Knowledge Plummer
• connects questions to relevant preachers.
• Knowledge Mentor
• supplies motivation and supports self reﬂection.
27.
The Conzilla “Mantra”
Content in Contexts through Concept
= Outsides of Concept
Contexts
= Inside of Concept
Content
Concept = Border between these
28.
Conzilla (www.conzilla.org)
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.
29.
Conceptual Browsing: Viewing the content
Context Content
Geometry
Projective geometry
is the study
of the incidences
Algebraic What
of points, lines
and planes
How
in space.
Differential Surf Where
It could be called
View When the geometry
Projective
of the e ye
Info
Who
30.
Conceptual Browsing: Filtering the content
Aspect Filter
Context
Elementary
Geometry
Secondary
Algebraic High
Differential
School
Surf W H W
o h
h
Projective w e
a
View Level r
t
e
Info
...
Aspect
31.
Where is mathematics done?
Depth
Contextualize
Clariﬁcation
Context Content
inspire
Mathematics Magic
invok
e
What illu
Mathematics Religion
Surf stra
te
How
ap
View ply Philosophy
Where
Info
When
Science
Who
32.
How is mathematics applied to science?
Depth
Contextualize
Clariﬁcation
Context Content
Science
inspire inspire
Mathematics Magic Mathematics Magic
assumption ∗ ∗v logical conclusion
in ok
invok
e
e A is true ⇒ B is true
illu illu
Religion Religion
stra stra
fact
te te
↓
↓
ap ap
ply p ly
Surf Philosophy Philosophy
If A were true
then
↓
View ↓
What B would be true
Science Science
Info ∗ conditional
How statement
Where
Mathematics
∗
When experiment
Falsiﬁcation of assumptions
Who
by falsiﬁcation of their logical conclusions
33.
The interplay between mathematics and science
Experimental Science Theoretical
∗ ∗
<<is an>>
Fact Assumption
A is true
∗
∗
↓
Mathematics
↓
∗ ∗
B survives If A were true,
the test
Conditional statement
then B would be true
∗
a Test
↓
Therefore B ∗ ∗
<<is a>>
must be true
∗ ∗ Logical ∗
Theory
1
∗
Experiment conclusion
↓
Falsiﬁcation of assumptions
by falsiﬁcation of their logical conclusions
40.
CyberMath: A Shared Virtual Environment
for the Interactive Exploration of Mathematics
Goals: The CyberMath system should allow:
• teaching of both elementary, intermediate
and advanced mathematics and geometry.
• the teacher to teach in a direct manner.
• teachers to present material that is hard
to visualize using standard teaching tools.
• students to work together in groups.
• global sharing of resources.
Means:
• Making use of advanced VR technology (e.g. DIVE).
49.
Mathematical ILE collaborative projects < 2002
Advanced Media Technology Laboratory (KTH)
• Mathemagic: Mathematical storytelling
Swedish Learning Lab
• Content and context of mathematics
in engineering education (with DSV, KTH/Kista)
• 3D Communication and Visualization
Environments for Learning (with DIS, Uppsala Univ)
Learning Lab Lower Saxony & Stanford LL
• Personalized Access to Distributed Learning Resorces
• MathViz: Personalized Mathematical Courselets
50.
The Greek Beginning Pythagoras
Thales
What is the basic stuff Disinterested knowledge
that the universe is made of? is the most effective
puriﬁcation of the soul.
Herakleitos
• everything changes
Ionian school Pythagorean school
• all is substance (matter) • all is form (number)
• nothing changes
Parmenides
• atoms build up shapes (bodies)
Demokritos that move around in empty space
51.
The Greek Beginning: Harmony of the Spheres
Thales Ionian School Pythagorean School Pythagoras
Substance Form Number Figure
Relation
Formal
What is the basic stuff
Proofs Number
that the universe is the essence
is made of? of all things
Geometry Astronomy Music Religion Arithmetic
Monad odd <=> unity
Philolaus
Formal #
The one that is
The earth is spherical
Deﬁnitions
everything
and rotates around the C
central ﬁre, protected
Pythagoras’ by the counter-earth b 2
Extasis
theorem 1 + 3 +5 + 7 = 4
1234
Contemplating
The quint circle
Herakleides -- quot; -- quot; -- quot; --
----
the structure 1234
5
3 of numbers
The earth also rotates
4
even <=> inﬁ nity
around its own axis.
2 2 2
3 + 4 quot;5 The central ﬁre Harmony of Armonia
is inside the earth the Spheres
The ﬁve 2:1 = octave
regular solids 3:2 = quint
The intervals between
Aristarchus
4:3 = quart 2345
the heavenly bodies -- ! -- ! -- ! --
----
The three The earth rotates daily are determined by the 1234
regular tilings around its own axis, and laws of musical harmony Katharsis
annually around the sun The holy tetrakys
together with the planets
The unspeakables Disinterested
point
knowledge
line
is the most
destroyed plane
2 effective
by Pairwise Rational Digital
1 solid
puriﬁcation
Commensurability = Mysticism = Worldview
1 of the soul 1 + 2 + 3 + 4 = 10
53.
Function
Function2
arguments and parameters
Function1
Rule Set of
Set of
A Arguments Result R
Trans-
formation
Parameters
P
Two different Function of A
perspectives: parametrized by P Function of A and P
Function2(A,P)
Function1P(A) =
Same result:
54.
Later Education
understand?
yes no
make money?
make money?
no yes no
yes
Academia Business Business
creative? creative? creative? creative?
yes no yes no no yes yes no
Researcher Teacher Consultant Employee Artist
socially conscious?
yes no
Teacher
55.
Academia
despise
Science Humaniora
Relational
Computational
stigmatize
Rational Emotional
my heart
my heart
rocks
is a rock
Artist
Non Human
Scientist
rock
heart
Sexy at heart
Nerdy
of rock
hot & cool
silicon
!
Cool Hot
Luke Warm rocks
rocks
Einstein
I am I am
Springsteen
a rock rock
Frankenstein
56.
Resource Components / Learning Modules
Learning Environment
* *
Resource Component Learning Module
connecting
separating
What to Teach
from with
What to Learn
through through
Multiple Narration Component Composition
57.
QBL: the 3 performing knowledge roles
Knowledge
Preacher Coach Plumber
´
´
´
Gardener
Master Broker
you teach you assist you ﬁnd
as long in developing someone
as somebody each indidual to discuss
is learning learning strategy the question
58.
QBL: the 3 performing knowledge roles (cont)
fascination methodology opportunity
Strategist
Source Opportunist
Knowledge
quality quality
quality
measured by measured by
measured by
raised questions
given answers lost questions
59.
Web links to some of my math projects
http://kmr.nada.kth.se/wiki/Amb/FirstClassMathematics
http://kmr.nada.kth.se/wiki/Amb/TheGardenOfKnowledge
http://kmr.nada.kth.se/wiki/Amb/VirtualMathematicsExplainatorium
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/FLIT
http://kmr.nada.kth.se/wiki/Amb/Projects-Matemagi
Complete quadrangle:
http://www.youtube.com/watch?v=63YdIUYv3qo
Polar Reciprocity:
http://www.youtube.com/watch?v=VsLnx6yAtic
Desargue's theorem:
http://www.youtube.com/watch?v=9iOxeaWrVvs
Isometry of a non-degenerated geometry:
http://www.youtube.com/watch?v=H__3Xe2tOPQ
Pappus-Pascal's theorem:
http://www.youtube.com/watch?v=VCdqJEyNSE4
Pascal's theorem:
http://www.youtube.com/watch?v=PH6_RRajHVc
Pascal-Brianchon's theorem:
http://www.youtube.com/watch?v=-vkAXFBl3bQ
The Projective Range theorem:
http://www.youtube.com/watch?v=TRf0JULnHgk
Hyperbolic circle:
http://www.youtube.com/watch?v=MU0eXvVtZrs
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