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New types of questions, e. At the same time as the development of the New Practical Arithmetic, the term modelling became better known in mathematics education see Greefrath Initially, modelling was seen as a certain aspect of applied mathematics, which, to some extent, can be seen as an independent process within applications or as a perception of applications Fischer and Malle In the s and s, Blum and Kaiser gradually introduced the term modelling into the German debate. Classical applied mathematics classical branches of analysis, parts of analysis that apply to physics.

Mathematics with significant practical applications statistics, linear algebra, computer science, analysis. Perspectives on applied mathematics by Pollak , p. Modelling then was considered to be a cycle between reality and mathematics, which is repeated several times Greefrath Two volumes of documentation of selected literature on application-oriented mathematics instruction Kaiser et al.

They provided an excellent overview of the national and international debate on applied mathematics education and also took into account selected publications on modelling that were written up to the beginning of the 20th century. The classification of works presented there incorporated ideas regarding goals, types of application, relation to reality, and embedding of the curriculum and analysed selected publications on applied mathematics teaching in more depth than ever before.

The classification system was presented at the First International Conference on the Teaching of Mathematical Modelling in in Exeter and had a significant impact on a closer integration of German researchers, especially Werner Blum and Gabriele Kaiser, into the international debate on modelling Blum and Kaiser Henn gave an example of using mathematical modelling at school.

He proposed the study of the theory of the rainbow as a piece of mathematics fraught with relations. This contribution was a revised version of his lecture delivered in in Freiburg at the German mathematics education conference. Many aspects of the rainbow were examined here and a mathematical model was presented. The model used an incident light beam and rays of first to fourth order. In addition, a detailed analytical model of a rainbow was developed.

Thus, the occurring intensities could be described in detail. Furthermore, a model illustrating the reflected first-order ray was presented using dynamic geometry software.

Thus it became obvious that explanations written in schoolbooks often contain mistakes. Modelling cycle by Blum Blum and Kirsch , p. For the first time the visualisation shown in Fig.

Blum not only distinguished between applications and tasks, where the problem is wrapped into the context of another discipline or of everyday life, he furthermore delivered arguments and aims regarding applications in mathematics teaching i. In addition, he summarised arguments against applications such as time problems or less suitable examples. For details see Kaiser This caused an intensified debate on modelling in Germany. Students should learn to understand environmental and real-life situations by means of mathematics and develop general mathematical skills e.

They should thereby establish an appropriate comprehension of mathematics including the actual use of mathematics. Learning mathematics should be supported by using relation to real life Blum A new series established in and published by Springer since enables the ISTRON Group, having already produced 20 volumes, to be present and visible in mathematics teaching as well as in the academic community.

These contributions are intended to support teachers in dealing with real-life problems in school. Teachers are considered to be experts in teaching; therefore, teaching proposals should be modifiable so that teachers can adapt them to a specific situation.

They should suggest uncommon ways of teaching mathematics and support preparing lessons e. They looked for contributions referring to teaching and learning mathematics that were combined with real-life applications, e. The worksheet presented there is even today often used in mathematics education. The accompanying ICMI Study volume fully presents the state of the discussion on modelling and applications at a high level. It became a standard reference work for the teaching and learning of applications and modelling.

Over the following years mathematical modelling was incorporated into the curriculum and into the standards for mathematics education see Sect. The debate about the term mathematical model plays an important role in the research on mathematical modelling in Germany. The term modelling describes the process of developing a model based on an application problem and using it to solve the problem Griesel Therefore, mathematical modelling always originates from a real-life problem, which is then described by a mathematical model and solved using this model.

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The entire process is then called modelling. Different virtual images are possible and they can even differentiate from various directions. Images not compatible with our commonly accepted rules of thinking should not be accepted. Therefore, all virtual images should be logically compliant or at least acceptable in the short term.

Virtual images are false if their internal interdependencies are contradictory to the interdependencies of the external objects: they should be true. However, even two images both true and acceptable could differentiate in terms of expedience. Normally an image would be preferable that reflects more interdependencies than another, i. If both images are equally compliant and concrete, the image of choice would be the least complex one.

Hertz , p.

Hertz mentions logical admissibility , accuracy , and expediency as criteria. A mathematical model is admissible if it does not contradict the principles of logical thinking. In this context, it is accurate if the relevant relations of a real-world problem are shown in the model. Finally, a model is expedient if it describes the matter by appropriate as well as relevant information.

If a model proves to be expedient, it can only be judged in comparison with the real-life problem. It can be expressed by an economical model or in a different situation by the richness of relations Neunzert and Rosenberger A new problem might require a new model, even if the object is the same. Furthermore, Hertz emphasises as conditio sine qua non that the mathematical model has to match the real-life items Hertz The term mathematical model has been described in the German literature in many ways.

Models are simplified representations of the reality, i. For this purpose, the observed part of reality is isolated and its relations are controlled. Mathematical models are a special representation of the real world enabling the application of mathematical methods. If mathematical methods are used, mathematical models that just represent the real world can even deliver a mathematical result Zais and Grund Thus, a mathematical model is a representation of the real world, which—although simplified—matches the original and allows the application of mathematics.

However, the processing of a real problem with mathematical methods is limited, as the complexity of reality cannot be transferred completely into a mathematical model. This is usually not even desired. Another reason for generating models is the possibility of processing real data in a manageable way. Thus, only a selected part of reality will be transferred into mathematics through modelling Henn Descriptive models aim to simulate and represent real life. This can happen in a descriptive or even explanatory way Winter , Therefore, one kind of descriptive model does not intend only to describe a selected part of reality but to help understanding the inner coherence.

Furthermore, it is possible to distinguish between models aiming for understanding and models predicting a future development Burscheid These predictions might be completely determined as well as to some extent probable. To summarise, there are descriptive models that are just descriptive in character, others that have additional explanations for something explicative descriptive models , and, finally, those that even predict a development deterministic and probabilistic models.

Tasks on descriptive and normative mathematical models can be quite different. Whereas descriptive models are used to describe and finally solve real-life problems, normative models aim to create mathematical rules as help in decision making in certain situations. For example, to distribute the cost of heating in a house with several apartments, a normative model is needed. Actually, this is a real problem that students at the junior secondary level are able to understand and solve.

In this example, the reality was only created after deciding on a certain mathematical model, e.

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As modelling is characterised as a procedure for processing a problem, it can be seen as a difference between a conscious and an unconscious process. Reflection of the proceeding not being considered as a criterion for implementing mathematical modelling is called general perception. According to this general perception, a modelling process even occurs if it happens unconsciously Fischer and Malle In the framework of this perception of modelling, students working on real-life problems without consciously simplifying the situation on a higher mathematical level are performing modelling.

The problem involving the volume of the sand in the container is a real-world problem. The first simplifications on a factual level lead to what is called a real-world model. Afterwards this is transferred to a mathematical model, which is used to calculate a mathematical solution. The result is then applied to the real-life problem.

It is also possible to idealise the solution process in other ways. For example, collecting the data could be shown separately or steps in developing the mathematical model could be omitted. We present different descriptions of modelling processes in the following ordered by the complexity of steps in developing a mathematical model. Modelling cycle by Schupp , p.

The modelling cycle need not always be fully completed or be repeated several times. After each run, experience with regard to solving the problem is gained.

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However, specifying the problem is separated as an individual step between reality and model. There are also particular modelling cycles that include a simple mathematizing step.

Bardy, P. Two groups of ninth graders in six classes were compared. Differenzielle Bildungsprozesse und Probleme der Verteilungsgerechtigkeit. I assume by handwriting you mean cursive. Brown Eds. Leistungsbewertung Im Sportunterricht in Der Grundschule

Modelling cycle of Kaiser and Stender , p. The model by Fischer and Malle described how to transfer a real-life situation to a mathematical model in detail. Interestingly enough, the process of collecting data was added to this model, which was specifically helpful in specifying the simplification step. This description of the modelling process is especially suitable for Fermi problems, because most of the data have to be estimated.

Depending on target group, research topic, and research interest, the described models focus on different aspects. Often they also have a different purpose. Normative and descriptive models should especially be distinguished. For example, a certain model could be used to describe student activities within an empirical study. For this purpose, even very complex models are suitable see Fig.

In a normative way, modelling cycles such as those shown in Fig. Different goals at various levels are pursued while using applications and modelling in mathematics teaching.

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