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Modelling A pivotal idea for interdisciplinary teaching in mathematics and computer science CADGME – Conference 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, 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 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- 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 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 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-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 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 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 admittance! Dr. Hans-Stefan Siller University of Salzburg, Austria stefan.siller@gmx.at

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posted: | 8/13/2012 |

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