# Modelling by dfhdhdhdhjr

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Modelling

A pivotal idea for interdisciplinary
teaching in mathematics and
computer science
Pecs
20. – 22. June 2007

Hans-Stefan Siller
   Modelling as a fundamental idea
   Modelling in DYNASIS
   Functional Modelling
Why modelling?
   A modell is a pictorial, symbolic or
conceptual representation of an real object
or a procedure.
   A modell is neither true or false, it is only
appropriate or inexpedient.
   Modells are additives for the array,
   Modells provide a basis for demonstration and
description.
   Modells advance the appreciation, because they
reduce facts of reality and accent specific
aspects.
   Modelling is a fundamental idea in science!!
What is the fundemental idea of
modelling in mathematical education?

In math modells are taken for …
 … a formal and compact description of
the reality
 … the exploration of the attributes

Aims of modelling in maths are …
 … to explain connections between
operations and results
 … to allow prognoses

   Results are provided interdisciplinary
What is the fundemental idea of
modelling in informatical education?
In computer science modells …
 … support the subject to control and to use
processes and structures high efficiently
 … can be modells which have its origin in
mathematics

Aims of moddeling in informatics are …
 … the replacement of reality, e.g. human cognition,
through a marginal abstracted new design
 … that the modells are designable by and at
computers
 … to create a virtual reality

   Results of modelling in informatics are pivotal and
fundamental for all other subjects
Connections
Simplified
problem                     (computer
Language                                      aided)
problem
solving

Sociality

Given
problem
Economy                      Mathematics                Informatics

Science

Reality     Abstraction    Mathematical
Algorithm   Simulation
Modell
Mathematics and computer science in
interdisciplinary education

   development and preparation of a
computer-assisted instrument, so that an
given mathematical modell can be
simulated and simulation outcomes can
be visualized
   development and preparation of a
system, which allows a defined user
group, e.g. pupils, without known
mathematical structures or knowledge in
mathematics to solve a problem on their
own.
Combined Contents

   problem solving
   heuristical acting and thinking
   compose concepts
   mathematize
   algorithmical acting and thinking
   creative acting and thinking
   verifying, proving
Curriculum
Modelling …
 … is a general mathematical competence

 … can be found in several curriculums for
schools
   Austrian curriculum
   German federal states, e.g. Schleswig-
Würtenberg, …
   … allows to use GTR, like Casio ClassPad
300 or CAS
What schould learners see?
Learners schould be able to …
 … apply math in several different
blowers
 … learn critical thinking
 … learn self dependent
 … accept math as an own language
An example in DYNASIS

From a modell to a differential
equation
Graphical Modelling

 Developing a modell on a at first
mathematic free level till a flowchart
 Easy possibilities for changing the
connections in the modell and for
changing the parameters in the
simulation cycle
 Result in graphical and schedular
description
Process of cooling down

   Hot coffee has the temperature of 80°C. The
cooling down should be done in a way so
that the coffee is loosing 2K temperature in
the first time-unit.
   Modell of a beginner:
Time-Temp.-diagram
Review on the modell

 Brickbat
   The temperature cannot drop down arbitrarily
   The temperature of a coffee is limited through the
environmental temperature
   The cooling down factor is not constant, it
decelerates in the course of time

 A linear falling of the temperature of a coffee
is because of these reasons not realistic
Process of cooling down

   The graph of operation
Process of cooling down

      New modell

Easiest case: factors of
cooling down and
temperature difference
are proportional
Process of cooling down

   Re-designed model
Cooling down- differential equation

Constitutive equation
Temperature.new <-- Temperature.old + dt*(Rate of Cooling down)
Start Value Temperature = 80

y(t+t)  y(t)+ t*(Rate of Cooling down)

At the limiting value
y´(t) = Rate of Cooling down
Cooling down- differential equation

y´(t) = Rate of Cooling down
Constitutional change
Rate of cooling down = PPF*Temperature difference

Absolute terms
Environmental temperature = 20
PPF = -2/60

Interim Values
Temperature difference = Temperature - Environmental temperature

2
y '(t )   ( y (t )  20)
60
Functional modelling and
Computer science

A function can be seen as a data process with input
and output. In school we are able to solve examples
in application of this perception. We are able to
combine several functions and to show the data-
flows between those functions in a diagram. The
solution can be realized through a spreadsheet. With
this functional mode of operation, pupils can
combine mathematics and computer science easily.
Motivation for pupils
   Elementary knowledge in computer
literacy produce impressive effects
   Curiosity in solution of complex
duties and responsibilities
   Enjoyment in implementation of
Data-flow diagramms and
functions

Processes in a data-flow-diagramm can
be interpreted as mathematical
functions.
We know:
A function is an image, which associates
every item of a set A, clearly an item of
a set B.
Example: square of a number
Basis            Square

number            number
Realising a data-flow diagram
For the efficient implementation of a data-flow
diagram on a computer, we need several systems,
which supply several functions:
Requirements:
 Standard software, programming only exceptionally

 Arithmetical operations on integers and floating-point
numbers
 Statistical operations

 Elementary data types in everyday life (currency, date,
time, etc.)
 Converting of textes

 Possibilities to knot the data-flow on conditions

Translation of data-flow-diagrams

In a first step the geometrical structure
can be transmitted directly to a
Facts and functions are identified
through the cells of sheet in a
to the format!), functions are identified
through the „=„-sign; after it you can
find the algebraic function.
An easy example: Interests of a credit
Date of the        Date of          Interest        100      Capital
beginning          ending             loan

date                    date      number               number     currency

Fraction of                        division
years

number                            multiplication

currency
multiplication

currency

Approximation
(2 sites)

currency
An easy example: Interests of a credit (Implementation)
Starches in the creation of
concepts

   The character of mathematical
functions is directly obvious
   Through this approach mathematics
and computer science get a little bit
closer
Thank you for your