Modelling by dfhdhdhdhjr



A pivotal idea for interdisciplinary
teaching in mathematics and
computer science
                           CADGME – Conference
                             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,
    appliance and advancement of theories.
       Modells provide a basis for demonstration and
       Modells advance the appreciation, because they
        reduce facts of reality and accent specific
   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

  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
   … to create a virtual reality

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


 Economy                      Mathematics                Informatics


 Reality     Abstraction    Mathematical
                                             Algorithm   Simulation
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
Combined Contents

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

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

From a modell to a differential
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
      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:
    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.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

   y '(t )   ( y (t )  20)
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
    solutions in a spreadsheet
 Data-flow diagramms and

Processes in a data-flow-diagramm can
be interpreted as mathematical
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:
 Standard software, programming only exceptionally

 Arithmetical operations on integers and floating-point
 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
Facts are admitted directly (attention
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

                  number                            multiplication



                             (2 sites)

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

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

    Dr. Hans-Stefan Siller
University of Salzburg, Austria

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