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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
Mathematics is utmost important part of (engineering) education: way of concise interpretation modelling environment and tool combination of analytic and synthetic
methods of representation
LEARN AND PLAY WITH MATHS
Synthetic representation geometric figure Folium of DescartesAnalytic representation Vector function
5
4,
6,
cossin
cossin2,
cossin
cossin233
2
33
2
ttt
ttal
tt
ttaktr
LEARN AND PLAY WITH MATHS
Synthetic representation geometric relation perspective affinityAnalytic representation transformation matrix
b
b
edac
b
000
000
0
000
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
Synthetic representation continuous movement revolution about lineAnalytic representation matrix function (quaternion)
1,0for
1000
0100
002cos2sin
002sin2cos
uuu
uu
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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!
LEARN AND PLAY WITH MATHS
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 Polyhedra
EXPLORING GEOMETRY Star polyhedra
EXPLORING GEOMETRY Machines
EXPLORING GEOMETRY Machines
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
Xmath projectSocrates Minerva programme 5 partners: Norway - 2, Spain,
Italy, Slovakia e-learning pilot course of
mathematics MA I with on-line calculator
LEARN AND PLAY WITH MATHS
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
web
LEARN AND PLAY WITH MATHS
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
Európa, e-learning a matematika: straty a nálezy
LEARN AND PLAY WITH MATHS
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
Európa, e-learning a matematika: straty a nálezy
LEARN AND PLAY WITH MATHS
Mathematics on-line Internet based course National project
Kega No. 3/100603
web
Európa, e-learning a matematika: straty a nálezy
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
LEARN AND PLAY WITH MATHS
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
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