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share as a common feature is that they start from a real situation and lead to a math-ematical model. Mller and Wittmann (1984) present a cyclic model of modelling,which only uses one step to move from the situation to the model. An especiallyclear description of modelling showing one step from reality to the model is pre-sented by Schupp (1988). This model separates mathematics and reality within onedimension, something which is customary for modelling models. Additionally, anequal distinction is made between problem and solution in a second dimension.

One well-known cycle of model building is described by Blum (1985, p. 200).This cycle represents in some respect a standard modelling model ( Fig. 23.1 ). Amore recent modelling model by Borromeo Ferri (2006, p. 92) has been designedfrom a cognitive point of view. Compared to the Blum model it has been extended byincluding the mental representation of the situation (situation model). The model byFischer and Malle (1985, p. 101) also describes in detail the step from the situation

to the mathematical model ( Fig. 23.2 ). Especially the inclusion of the data collectionprocess is of interest for the problems used in our study. With reference to the modelsmentioned above, it is not always possible to follow the entire cycle or to repeatit several times. Depending on the target group, the question to be researched orthe special interest, the models mentioned earlier focus on different aspects of themodelling process.

Fig. 23.1 Modelling cycle, Blum (1985, p. 200)

Problem-solving processes are also often described using a model. In his book How to Solve It , Polya (1971) developed a catalogue of heuristic questions designedto help in solving problems. For this purpose the problem-solving process is dividedinto four phases: Understanding the problem, devising a plan, carrying out the planand looking back.

Within the literature there are many similar examples for the structuring of problem-solving processes. Schoenfeld (1985), for instance, describes the rst phaseas reading, followed by analysing, exploring and planning. Garofalo and Lester(1985) combine these rst steps as orientation and organisation. In both modelsthe next step is execution.

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Fig. 23.2 Modelling cycle, Fischer and Malle (1985, p. 101)

The characterisations of problem solving thus refer to the description of the solu-tion process. This is also the case with modelling models. Here, however, an addi-

tional content-related aspect (i.e. the distinction between reality and mathematics)is relevant.

23.3 Open, Reality-Related Problems

Open reality-related problems were used to analyse model-building and problem-solving processes. The following problems ( Figs. 23.3 and 23.4 ) are examples of theopen and fuzzy problems with reality references used in the analysis.

In order to characterize the house-plastering problem, I use the description of aproblem as initial state, target state and transformation, borrowed from the psychol-ogy of problem solving (Bruder, 2000, p. 70). The problems initial state is unclearbecause the relevant information is missing. Also unclear is the transformation frominitial state to target state which students can employ. However, the nal state isclearly dened, for instance, by asking for a price.

23.4 Analysis Design and Evaluation Methods

Two pupils at a time were monitored while they worked on their problems. Thestudents were asked to undertake the task in pairs without any further help. Thestudents work was recorded using a video camera.

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How much will it cost to plaster this house?

Fig. 23.3 The house-plastering problem

How many peopleare caught up in atraffic jam 180 km

long?

Fig. 23.4 The trafc jam problem

For evaluation the entire video data were transcribed. Within the framework of open coding with three raters, the individual expressions of the pupils were allo-cated conceptual terms, which were discussed and modied during several runsthrough the data. These terms for individual text passages were then assigned tothe following categories: planning, data acquisition, data processing and checking.

The process category planning describes text passages in which the pupils dis-cuss the path to complete the task or which in the broadest sense relate to the path

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of completing the task. The process category data acquisition describes text pas-sages in which pupils procure data for their further work on the problem. This caninvolve guessing, counting, estimating, measuring or recalling intermediate resultsthat had been achieved earlier. The process category data processing describesthe calculation with concrete values. This can be done either with or without a cal-culator. For all problems the pupils were provided with a (conventional scientic)calculator. The process category checking includes text passages in which the dataprocessing, data acquisition or planning is questioned or controlled.

The choice of categories was done in such a way that the categories could be allo-cated in a consistent manner, independently from the problem. During this phase thepreliminary categories were therefore combined and modied. Following this allo-cation, the entire transcripts were coded using the now nalized categories. Laterwe found that one rater was able to code all categories with adequate condence

on his/her own. The degree of agreement was checked by performing a samplecorrelation analysis (see Bortz et al., 1990, p. 460f), which showed statistically sig-nicant concordance at the 0.05 level. As a result, I then coded all of the remainingtranscripts on the basis of the developed categories.

23.5 Results

Because we are interested in those planning processes that have special impor-tance for the completion of modelling tasks and that have frequently appeared thecompletion of problem-solving tasks, we chose to correlate that descriptions of model-building and problem-solving processes. Consequently, these elements of theplanning process are described and allocated to the phases of the modelling process.

23.5.1 Orientation Phases

Typically the orientation phases form the start of the work on the problem. Theyshould relate directly to the material that is handed out and thus intrinsically belongto the domain of reality. If orientation phases appear again in the later stages of thecompletion process, it is a sign that some of the comprehension issues were not dealtwith at the beginning of the work. These orientation phases can then be related tothe situation model in the modelling cycle.

23.5.2 Transition of Planning in Reality and Mathematics

In some sections of the transcripts, planning processes in reality and in mathematicssucceed each other closely. More precisely, there are three kinds of transition.

One type of transition is the one from reality to mathematics. Starting from thestandard modelling model, this is the type that we would expect to appear most

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frequently, as the mathematical model is only supposed to be developed after thereal model.

The second type is the transition from mathematics to reality. According to thestandard modelling model this transition within the planning is rather unusual,since the creation of a mathematical model would be followed by carrying outthe calculations and not by planning in reality. Such a step was found more fre-quently within the completion processes, especially for those completions, in whichplanning processes were discussed very precisely.

The third type is a multiple transition between reality and mathematics. Thisis a combination of the two types of transition mentioned above. It could be anindication of an intensive examination of the real and mathematical contents of theproblem. In the standard modelling model this is the section between the real andthe mathematical model, which cannot be separated any longer with this type of

transition.

23.5.3 Partial Models

We speak of partial models in reality or mathematics, if simplications and assump-tions are made during the planning phases that lie in the area of reality ormathematics respectively. These planning steps can be identied with the creation

of the real model or mathematical model, respectively, during the model-buildingcycle.According to initial observations, the elements mentioned above are suitable for

the characterization of the following planning types.

Type I: Following a target-led orientation phase pupils take time for planningand discuss simplications of reality in depth. They often use mathemati-cal terms and correctly apply them to reality. They also correctly associate

objects from reality with the relevant mathematical actions and simpli-cations. Planning processes are discussed in depth and are generallysuccessful.

Type II: The pupils orientation phase is quiet and very short. In their dis-cussions they refer mainly to the real situation. The mathematical modelsare used but not discussed. Reality is not consciously simplied or thesesimplications are not expressed. Terms from reality are integrated intomathematical process descriptions. Here, no clear indications can be maderegarding the eventual success of the students modelling efforts.

Type III: Pupils frequently need orientation phases in order to see the problemin context. The discussion is very much reality-oriented. No mathematicalterms are used and mathematical process description only takes place on alow level. Simplications are rarely discussed. On the whole, the planning isnot very abstract, but rather supercial and generally not successful.

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As a consequence of our ndings, we recommend that these different plan-ning types provide indications for education. For example, pupils can discuss aboutmathematical and real simplications of open problems. The Type I is a prototypefor such a discussion. In our view the Type II is of special interest in diagnosingmodelling competencies, because no clear indications can be made regarding thesuccess. Whether the number of planning types is to be increased will be subject tofuture investigation.

References

Blum, W. (1985). Anwendungsorientierter Mathematikunterricht in der didaktischen Diskussion. Mathematische Semesterberichte , 32(2), 195232.

Blum, W., and Lei, D. (2005). Modellieren im Unterricht mit der Tanken-Aufgabe. Mathematik lehren , 128, 1821.

Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modellingprocess. Zentralblatt fr Didaktik der Mathematik , 38(2), 8695.

Bortz, J., Lienert, G. A., and Boehnke, K. (1990). Verteilungsfreie Methoden in der Biostatistik .Berlin, Heidelberg: Springer.

Bruder, R. (2000). Akzentuierte Aufgaben und heuristische Erfahrungen. Wege zu einemanspruchsvollen Mathematikunterricht fr alle. In: L. Flade, and W. Herget, (Eds.), Mathematik lehren und lernen nach TIMSS: Anregungen fr die Sekundarstufen (pp. 6978). Berlin: Volk und Wissen.

Fischer, R., and Malle, G. (1985). Mensch und Mathematik . Mannheim Wien, Zrich:Bibliographisches Institut.

Garofalo, J., and Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematicalperformance. Journal for Research in Mathematics Education , 16(3), 163176.

Greefrath, G. (2006). Modellieren lernen mit offenen realittsnahen Aufgaben . Kln: Aulis VerlagDeubner.

Mller, G., and Wittmann, E. (1984). Der Mathematikunterricht in der Primarstufe. Braunschweig,Wiesbaden: Vieweg.

Polya, G. (1971). How to Solve It . Princeton, NJ: University Press.Schoenfeld, A. H. (1985). Mathematical Problem Solving . Orlando, FL: Academic Press.Schupp, H. (1988). Anwendungsorientierter Mathematikunterricht in der Sekundarstufe I zwischen

Tradition und neuen Impulsen. Der Mathematikunterricht , 34(6), 516.Strauss, A., and Corbin, J. (1996). Grounded Theory. Grundlagen Qualitativer Sozialforschung .

Weinheim: Beltz Psychologie Verlags Union.Wiegand, P., and Blum, W. (1999). Offene Probleme fr den Mathematikunterricht Kann man

Schulbcher dafr nutzen? Beitrge zum Mathematikunterricht (pp. 590593). Hildesheim:Franzbecker.