modelling a pivotal idea for interdisciplinary teaching in mathematics and computer science cadgme...
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Modelling
A pivotal idea for interdisciplinary teaching in mathematics and computer science
CADGME – ConferencePecs
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
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
Mathematics Informatics
SimulationAlgorithmMathematical Modell
Abstraction
Science
Economy
Reality
Sociality
Language
Given problem
Simplified problem (computer
aided) problem solving
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-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
blowers … 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 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
At the limiting value
y(t+t) y(t)+ t*(Rate of Cooling down)
y´(t) = Rate of Cooling down
Cooling down- differential equation
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'( ) ( ( ) 20)
60y t y t
y´(t) = Rate of Cooling down
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 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-flowdiagram 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
SPREADSHEETS
Translation of data-flow-diagrams
In a first step the geometrical structure can be transmitted directly to a spreadsheet.Facts and functions are identified through the cells of sheet in a spreadsheet.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 beginning
Fraction of years
date
Interest loan
Date of ending
100 Capital
division
multiplication
multiplication
Approximation(2 sites)
date number number currency
currency
number
currency
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 admittance!
Dr. Hans-Stefan SillerUniversity of Salzburg, Austria