learn and play with maths daniela velichová department of mathematics institute of natural...

LEARN AND PLAY WITH MATHS

Daniela VelichováDepartment of Mathematics

Institute of Natural Sciences, Humanities and Social Sciences

Mechanical Engineering FacultySlovak University of Technology

Bratislava, Slovakia

Contents

Introduction – where do I come from General remarks – development of

mankind and impact on education Image and position of Maths today Exploring geometry Solutions and examples of

interactive maths on-line European projects and their results

Slovak University of Technology in Bratislava www.stuba.sk

Largest university in Slovakia 7 faculties – Civil, Mechanical,

Electrical eng., Informatics, Architecture, Chemical technologies, Material sciences

about 1000 university teachers and researchers

more then 20 000 students

Slovak University of Technology in Bratislava www.stuba.sk

Faculty of Mechanical Engineering more then 2 000 students 170 academic staff Institute of Natural Sciences,

Humanities and Social Sciences Department of Mathematics15 mathematicians

Slovak University of Technology in Bratislava, Mechanical Engineering Faculty

Department of Mathematics Compulsory subjects in Bachelor programmes

1st semester

Mathematics 1 4 L / 4 P

2nd semester

Mathematics 2 2 L / 2 P

3rd semester

Constructive Geometry

2 L / 2 P

3rd semester

Statistics and Probability

2 L / 2 P

4th semester

Numerical Methods 2 L / 2 P

Slovak University of Technology in Bratislava,

Mechanical Engineering Faculty

Department of Mathematics Compulsory subjects in Master programmes

7th

semesterApplied Mathematics 1 2 L / 2 P

8th

semesterLinear Algebra 2 L / 2 P

8th

semesterDifferential Equations 2 L / 2 P

8th

semesterGeometric Modelling 2 L / 2 P

9th

semesterApplied Mathematics 2 2 L / 2 P


Post-industrial development

The third wave of creating a new non-visible civilisation of the 21st century Information and post-information society RECEIVING – PROCESSING – DISSEMINATING huge amounts of information pieces in the virtual non-visible world by means of rapidly developing new information

and communication technologies using new digital media with a limited lifespan and

validity


Development with many contradictions and ambiguities

globalisation versus individualisation data manipulation versus right for true free information ,,copy and paste“ strategy versus fight for authorship

copyrights on original ideas interest in ready results not in the process of solving

problems gathering pieces of information not complex

knowledge simplification and vulgarisation of the educational

process


Information boom versus time limits daily myriads of bits appearing on the web

WWW = WORLD WIDE WAITING Need for brief and compound data Graphical representation of data structures New visual sciences, visualisation boom -

icons, graphical symbols, logos Computer graphics and virtual reality RENAISSANCE of GEOMETRY


Impact on science and education? Mathematics – highly formalised

language of science Modelling - geometric/graphical

interpretation of ideas Symbolic – illustrative presentation

of information, facts and data


Mathematics is utmost important part of (engineering) education: way of concise interpretation modelling environment and tool combination of analytic and synthetic

methods of representation


Synthetic representation geometric figure Folium of DescartesAnalytic representation Vector function

5

4,

6,

cossin

cossin2,

cossin

cossin233

2

33

2

ttt

ttal

tt

ttaktr


Synthetic representation geometric relation perspective affinityAnalytic representation transformation matrix

b

b

edac

b

000

000

0

000


Synthetic representation 3D object a surface patchAnalytic representation vector function in 2

variables

21,0),(,

4sin54cos2sin54sin6cos)2cos55(,

)14(cos54sin2sin54cos6cos)2cos55(,

,6sin)2cos55(,

,

vu

vvvvvuvuz

vvuvvuvuy

vuvux

vu

r


Synthetic representation continuous movement revolution about lineAnalytic representation matrix function (quaternion)

1,0for

1000

0100

002cos2sin

002sin2cos

uuu

uu


ICT and e-environment enable combination of both methods direct access to both representations animated illustrations on-line calculations brief and compound explanation at

hand


Mathematics consists from many branches: Algebra and Discrete Maths Differential and Integral Calculus – multivariable Differential equations Numerical Maths and Optimisation Probability and Statistics Geometry - elementary, analytic, differential,

projective Algebraic Geometry and Topology Fractal geometry Mathematical modelling Non-standard Fuzzy methods and Deterministic chaos and many others


Traditional Maths and Geometry courses at technical universities:

Basics of Linear Algebra (systems of algebraic equations, matrices and determinants, vector spaces)

Calculus + Multivariable Calculus (differentiation and behaviour of functions of one and more variables, optimisation problems, calculations of indefinite and definite integrals and applications)

Basics of Differential Equations (easy analytic solutions) Numerical mathematics (iterations, interpolations and

approximations, numerical methods for solving ODE and PDF) Basics of projection methods and 3D geometry Elementary geometric figures Geometry of curves and surfaces in technical practice Geometric modelling of free-form figures in CAD systems


New trends and needs: Mathematical and geometric modelling methods Numerical interpolation and approximation methods

for calculating solutions of problems virtualy representing and modelling real-world situations

Techniques for visualisation of abstract virtual models Knowledge about geometric spaces (also non-

Euclidean) and geometric transformations (including non-linear deformations) in virtual space

Free-form modeling - interpolation of curves, surfaces and solids and manipulations with them


AGE of VISUALISATION AND ANIMATION first and often also last information is visual graphical information boom culture of pattern recognition (logos and icons) virtual reality experience interactive manipulations with virtual models virtual modeling, shape optimisation and

continuous movement and deformations

MORE GEOMETRY IS NEEDED THAN EVER BEFORE!


Geometry is a specific European dimension attitude way of understanding recognition and differentiation key distribution pattern characteristics art and science

that is typical for European civilisationand deserves to be preserved and further cultivated.

EXPLORING GEOMETRY

Ancient Greece – from 600 B.C.Thales, Pythagoras, Euclid – Elements, 300 B.C.

EXPLORING GEOMETRY

Plato - platonic (regular) solids (philosophy)

EXPLORING GEOMETRY

Archimedes - calculating the area of a figure which was surrounded by parobola (curve) and chord (straight line)

Apollonius, 200 B.C. – conic sections

EXPLORING GEOMETRY

Hypathia from Alexandria, 370 - 415 She edited the work On the Conics of Apollonius, which divided cones into different parts by a plane. This concept developed the ideas of hyperbolas, parabolas, and ellipses. Hypatia with her graphical work on this important book, made the concepts easier to understand, thus making the work to survive through many centuries. Hypatia was the first woman to have such a profound impact on the survival of early thought in mathematics.

EXPLORING GEOMETRY

Greek art and architecture

EXPLORING GEOMETRY

Renaissance in ItalyLeonardo da Vinci, 1452-1519

Italian artist, scientist, engineer, an all-round genius whose paintings and inventions changed the world.

EXPLORING GEOMETRY Polyhedra

EXPLORING GEOMETRY Star polyhedra

EXPLORING GEOMETRY Machines

EXPLORING GEOMETRY Buildings

EXPLORING GEOMETRY Plans and maps

EXPLORING GEOMETRY Perspective, light, shadow - optics

EXPLORING GEOMETRY

Renaissance in GermanyAlbrecht Dürer, 1471-1528

German artist, scientist, mathematician, who put on scientific backgrounds in basic perspective and geometry of projection methods, interpolation and approximation.

EXPLORING GEOMETRY

Perspective

EXPLORING GEOMETRY

Melencolia IThe most famousallegory of Geometry Perspective Solid Ball Skew ladder Mill stone Magic square Special instruments

(compasses and ruler)

EXPLORING GEOMETRY

Baroque paintings and architectureEl Greco Tizian

EXPLORING GEOMETRY

Bramante, Michelangelo, Piranesi

EXPLORING GEOMETRY

Light and shadowRembrandt Velasquez

EXPLORING GEOMETRY

Non-Euclidean geometry in modern art

Cézanne Braque

EXPLORING GEOMETRY

Picasso Dali Gaudi

deformations

EXPLORING GEOMETRY

Chaos and disorderKlee Delaunay

Kandinski

EXPLORING GEOMETRY

Knots, mosaics, dense layout and space filling,hyperbolic space M. C. Escher

EXPLORING GEOMETRY

Geometry plays important role in all sciencesPhysics Einstein theory of relativity is based on idea of non-Euclidean space Riemann/Minkowski

geometry in 4 dimensional timespace

EXPLORING GEOMETRY

Biology geometric forms of

living creatures are geometric

structures, surfaces and solids with given properties

EXPLORING GEOMETRY

Chemistry (crystallography) regular polyhedra

EXPLORING GEOMETRY

Direct application in many new sciences

Robotics and Mechatronics Scientific visualisations Superstring theory and M-theory GPS and Digital mapping and

drawing with GPS tracks and many, many others

EXPLORING GEOMETRY

Technical sciences Descriptive geometry and Gaspard Monge

projection method was of utmost importance for more then 2 centuries as visualisation tool

It was called „the queen“ of technical disciplines.

Future development opened in connection to computer graphics and CAG design and

modelling virtual reality animated geometry interactive manipulations and on-line modelling


Many different solutions exist for interactive Maths on-line presentation

doc, pdf, html files versus MathML + xml files live maths formulas with semantics - example

Flash animations - example1, example2, example3 on-line Java animations – example webmathematica java server pages – example Maple – maplets – example Video presentations – example On-line drawing - example


Many different solutions exist for organisation of Maths on-line

Electronic books - web E-learning modules - in English in Slovak Information portals with hyperlinks - EVLM Educational e-learning portals with maths support

resources - MathCentre Databases of e-learning materials - Slovak EVLM

database Links to free web resources – Wolfram demonstrations

project


Many programme schemes exist at the European (and Slovak national) level supporting development of different solutions for e-learning in Maths and interactive on-line educational resources

Socrates, Minerva, Erasmus Mundus programmes Tempus IV – Higher Education Cooperation (2007 -

2013) Ceepus II - university study programmes for Central

and Eastern Europe (2005 - 2011) Lifelong Learning programmes Education and Culture (former Leonardo da Vinci) eContent UNESCO – Education for All by 2015


Xmath projectSocrates Minerva programme 5 partners: Norway - 2, Spain,

Italy, Slovakia e-learning pilot course of

mathematics MA I with on-line calculator


Aims of Xmath project Testing of development and functionality

of xml webpages with embedded MathML coding of mathematical formulas

Usage of software webMathematica for on-line calculations

Design and development of on-line calculator

Design and testing of Pilot course of Mathematics

Európa, e-learning a matematika: straty a nálezy


dMath project 9 partners: Norway - 3, Spain - 2,

Finland, Germany, Czech republic, Slovakia

European database of e-learning modules for development of electronic courses of mathematics


Aims of dMath project Testing of suitable editor for development

of webpages in uniform format as xml files

SciWriter editor, product of Soft4Science Design of e-learning modules database for

development of electronic courses of Mathematics – Orange CMS database available from the xml editor


Mathematics on-line Internet based course National project

Kega No. 3/100603

web


Aims of the project Mathematics on-line Unified format of presented learning materials xml files + MathML coding of mathematical

formulas Animations – Flash, Java Illustrated learning material with applications and

hyperlinks to the supplementary materials on web Solved problems – step-by-step solutions Collection of problems with provided results Interactive on-line calculations and live graphics –

jsp files Tests – electronic forms Interactive textbooks of mathematics and

geometry


EVLM project – 2006-2008 9 partners: Slovakia, Great Britain,

Ireland, Finland, Spain, Bulgaria, Hungary, Czech republic, Norway

European Virtual Laboratory of Mathematics – common portal in English

Net of National Centres of Mathematics database of all available source learning

materials of Mathematics


National Centres of Mathematics consultation centres providing Present consultations in national languages Learning materials in printed or electronic

form available from National portals Work on PC with all e-learning materials and

mathematical software products and CAGD packages available on the common portal


EVLM Testing possibilities how to utilise

software webMathematica for development of interactive textbooks and evaluation tests

EVLM Central portal - web

learn and play with maths daniela velichová department of mathematics institute of natural...

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