Educational Design Patternsin Mathematics

Ambjörn Naeve

The Knowledge Management Research groupCentre for user-oriented IT Design (CID)

Numerical Analysis and Computer Science (NADA)Royal Institute of Technology (KTH)

Stockholm / Sweden

amb@nada.kth.se

http://kmr.nada.kth.se

Structure of today’s math education system

Closed layered architecture based on:

• lack of subject understanding in the early layers.

• minimization of teaching duties in the final layers.

• life long teaching with:

• curricular-oriented ”knowledge pushing”.

Problems with today’s 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.

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.

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

* *

A Knowledge Manifold

• is designed in a way that reflects a strong effort to comply with emerging international IT standards.

• can be regarded as a Knowledge Patchwork, with a number of linked Knowledge Patches, each with its own Knowledge Gardener.

• gives the users the opportunity to ask questions and search for live certified Knowledge Sources.

• is a learner-centric educational architecture that supports question-based learning.

A Knowledge Manifold (cont.)

• allows teachers to compose components and construct customized learning environments.

• makes use of conceptual modeling to support the separation of content from context.

• contains a conceptual exploration tool (concept browser) that supports these principles.

• has access to distributed archives of resource components.

The seven different Knowledge Roles of a KM

• The knowledge cartographer

• The knowledge composer

• The knowledge librarian

• The knowledge coach

• The knowledge preacher

• The knowledge plumber

• The knowledge mentor

• constructs context-maps.

• fills the context-maps with content.

• composes content components into learning modules.

• cultivated questions.

• provides live answers.

• connects questions with appropriate preachers.

• provides motivation and supports self-reflection.

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

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

Problem / Solution versus Problem / Elimination

dissolve

Crime Jail

crystalize

Organized

Economic Dysfunctionality

Legal System

Problem / Solution applied to Crime

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

Problem/Solution applied to early math education

dissolve

Conceptual Difficulty Academic Status

crystalize

Anticipated

Understanding Dysfunctionality

Publicational System

isA

Mathematicsis difficult

nurturing the difficulties

I understand it, andI am smarter than you

because

publish or perish

Problem/Solution applied to late math education

dissolve

Conceptual Difficulty Algorithmic Ability

crystalize

Anticipated

Understanding Disfunctionality

Computational Industry

Mathematicsis difficult

isA

marketing the difficulties

solve by computation

Formulation Solution

ProblemExtractingmathematicalskeleton

Computing thealgorithmicsolution

the

We can help youto solve your problems

but

Problem/Solution applied to calculator industry

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

Linear

Total Digital

Fourier

HermitianReal

Matrix

Operator

Eigenvalues

Symmetric

Newtonian

Harmony(1, 4/3, 3/2)

Mechanics

Quantum Mechanics

Harmony of the Spheres

Analysis

CommensurabilityWorldview

Rational Music

Heisenberg’s Uncertainty Principle

Pythagoras’ Philosophy

The digital view - from Pythagoras to Heisenberg

Harmony

AtomicCrystal Arithmetic Musical

SubstancePattern Sub

Bohr’s electron model:

orbit1234

speed (km/s)21601080720540

Lavoisier:

HydrogenOxygenNitrogenH2O

Dalton:

Every chemical combinationtakes place only in its particularand precise weight proportions

Haüy (Law of Rational Indices):

The lattice spacings of atomspermit certain peculiarcrystal symmetriesand no other

Mendelejev:

Prediction of3 new elements(eka-aluminium)

octavequintquartters

21--- 3

2--- 4

3--- 5

4---≠≠≠

1786

1807

1869

1780

1920

-600

:Pythagoras

P eriodical System of the Elements

law ofoctaves

:Harmonic Mean

= + /a b b h= + /b c c h

=>eliminate h( - )/( - ) = /a b b c a c

Fedoro :vTher e ar e exactly17 wallpaper symmetry groups230 crystal symmetry groups

1891

Information Knowledge Understandingreflectingweeding

Concept

conceive

Percept

perceive

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

Attityd

Förväntan

Försvar Förälder

Lärare

Bekräftelse

Barn

tid

Att räkna med x är svårt.Jag förstod det aldrignär jag gick i skolan.

Barn, vi ska nu börja räkna med det där mystiska talet x som ni alla har hört talas om.

Jag kommer aldrigatt begripa det här!

Just det, precis som jag trodde, jag kan helt enkelt inte förstå det här!

Begreppsmässig svårighet

Minsta tecken på mentalt motstånd

Låd-Multiplikation

Låda<<ärEn>>

inlärningsordning

Låd-Relation

Låd-Kvadrering

Grafisk Låd-Relation

Funktion

Pil

x

x • y

x2

y = x2

x

y

x2

1

1

( )2a a2

x - fixering

<<leder till>>

x

y = x2

R+R+

<<ärEn>>

<<ärEn>>

<<ärEn>>

<<ärEn>>

<<ärEn>>

<<ärEn>>

References

• Naeve, A., IT-baserade matematikverktyg på KTH, CID-49,

NADA/KTH, 2001.

These reports are available in pdf at http://kmr.nada.kth.se

• Naeve, A., The Garden of Knowledge as a Knowledge Manifold - a conceptual framework for computer supported subjective education, CID-17, NADA/KTH, 1997.

• Naeve, A., Conceptual Navigation and Multiple Scale Narration in a Knowledge Manifold, CID-52, NADA/KTH,1999.

• Naeve, A., The Work of Ambjörn Naeve in the Field of Math. Educational Reform,CID-110, NADA/KTH, 2001.

• Naeve, A., Begreppsmodellering och matematik, CID-109, NADA/KTH, 2001.