What types of mathematical modeling are there

Model

This module provides an overview of the modeling cycle.

  1. definition
  2. Objectives of models
  3. Classification of models
  4. What is modeling?
  5. Example of the modeling cycle
  6. swell

1. Definition

Models are images of a real object. The model can be an imitation of the original or a theory. Every modeling contains an abstraction. With this abstraction, certain properties of the original are lost, i.e. not all properties of the object can be transferred to the model.

The model has at least one property in common with the original. Which properties these are depends on the problem and the aim of the modeling. Different models can arise for the same object; depending on the context, these models have different properties in common with the object.

2. Objectives of models

You create and use models to achieve a specific goal.

Such goals can be:

  • Functionality: Models are made to perform certain functions. They are often preferred to their original because they are simpler and easier to use.
  • simulation: Operations are to be carried out and tested on the model that cannot or can only be carried out with great difficulty on the original object itself.
  • Explanation: The model is supposed to explain certain phenomena or the behavior of objects.
  • Prediction: Models must be able to make predictions about the future behavior of the objects.

3. Classification of models

A distinction is made between two types of models on the basis of their goals named above, descriptive and normative models (according to Henn, 2000).

To the descriptive models include predictive, explanatory or descriptive models, such as weather forecasts or city maps. Under normative models one understands models that prescribe something, for example building plans or construction drawings.

4. What is modeling?

To the individual steps of the modeling cycle

  1. Construct / understand
    • Reading and understanding the task
    • Obtaining the necessary information
    • Identify where the problem is and what information is still missing
  2. Simplify / structure
    • Simplify and structure
    • Separate important from unimportant information
    • Obtain missing data
    • Name math related to the problem
    • Look for analogies and relationships
    • jot down creative ideas
    • make estimates if necessary
  3. Mathematisation
    • "Translate" relevant quantities and relationships into mathematical terms
    • Look for adequate mathematical notation
    • Graph the situation if necessary
    • use known mathematical tools
  4. Work mathematically
    • Calculate task
    • Apply heuristic strategies
    • use new technologies if necessary
  5. Interpret
    • "Retranslate" the mathematical result in everyday language
    • Relate results to the real model
    • Generalize solutions developed for special situations
  6. Validate
    • Critically review and reflect on the solution found
    • Consider whether other solutions are possible
    • Fundamentally question the model
    • compare with measurement data if possible
    • if the solution is not appropriate, repeat parts of the modeling cycle
  7. Expose
    • Document results
    • Present results if necessary

5. Example of the modeling cycle

Task:

The elephant Elsa from the Frankfurt Zoo wants to go swimming, but the zookeeper fears that so much water will run out of the pool that he will have to laboriously fill it up with buckets. How much water flows out of the pool when the elephant is completely submerged?

  1. Construct / understand
    • Calculate the volume of the elephant, then you know how much water it is displacing
    • the size of the elephant can be estimated with the help of the height of the keeper
  2. Simplify / structure
    • Simplify and structure
  3. Mathematisation
  4. Work mathematically
  5. Interpret
    • The elephant displaces 3,854m3 Water from the pool, as this is its own volume.
  6. Validate
    • Can the result be correct? Comparison with human volume: 0.073m3
    • What scale did we expect?
  7. Expose
    • Document results
    • Present results if necessary

6. Sources

  • Definition and goals of models: http://www.usf.uni-kassel.de/ftp/lehre/schaldach/vl_modellbildung_simulationI/MoSim3.pdf
  • Classification of models:
    • http://www.mathematik.hu-berlin.de/~filler/lv_ph/sem-begabfoerd/Vortrag-Modellierung-Baumann-Pfeil.pdf
    • https://www.ph-freiburg.de/fileadmin/daten/fakultaet3/mathe/Mathe_fuer_alle/Modellieren_in_der_Sekundarstufe_1.pdf
  • Regarding the individual steps of the modeling cycle:
    • http://www.mathematik.hu-berlin.de/~filler/lv_ph/sem-begabfoerd/
    • http://kola.opus.hbz-nrw.de/volltexte/2010/503/pdf/Publikation.pd
  • Figure 1: http://www.mathematik.uni-kassel.de/didaktik/HomePersonal/lilitakis/home/uebungen/uebungenWS0607/material/Modellierungskreislauf.pdf
  • Figure 2: http://www.morgenpost.de/multimedia/archive/00289/b_elefantin_BM_Berl_289494b.jpg