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The Role of the Situation Model in Mathematical Modelling—Task Analyses, Student Competencies, and Teacher Interventions

Zur Rolle des Situationsmodells beim mathematischen Modellieren – Aufgabenanalysen, Schülerkompetenzen und Lehrerinterventionen

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Abstract

In a study within the DISUM research project, we investigated the role that the construction of situation models plays as an essential prerequisite for understanding a given mathematical modelling task, using a sample of 21 9th grade classes (N=416). Specific task characteristics, general mathematical competence, reading competence, and teacher interventions aiming at understanding the situation model were analyzed as crucial factors influencing students’ ability to solve modelling tasks. The results show that: (1) strategies for constructing an adequate situation model have a significant influence on modelling competence, (2) mathematical reading competence and intra-mathematical competence can explain almost one third of the variance of the performance on the modelling test, (3) teacher interventions may encourage students to adopt strategies facilitating the construction of situation models, but an increase of modelling competence requires separate strategy training.

Zusammenfassung

Im Rahmen einer Studie des Forschungsprojekts DISUM wurde in 21 Realschulklassen der Jahrgangsstufe 9 (N=416) untersucht, welche Rolle das Konstruieren des Situationsmodells als wesentliche Voraussetzung für das Verstehens einer gegebenen Aufgabenstellung bei der Bearbeitung von Modellierungsaufgaben spielt. Als zentrale Einflussfaktoren für die Modellierungskompetenz der Schüler wurden spezifische Aufgabeneigenschaften, die allgemeine mathematische Kompetenz und die Lesekompetenz identifiziert sowie Interventionen, mit denen Lehrpersonen auf das Verstehen einwirken. Dabei zeigen die Ergebnisse der vorliegenden Studie u.a., dass (1) Strategien zur Konstruktion eines adäquaten Situationsmodells einen signifikanten Einfluss auf die Modellierungskompetenz haben, (2) mathematische Lesekompetenz und innermathematische Kompetenzen knapp ein Drittel der Leistungsvarianz des Modellierungstests erklären können, (3) Lehrerinterventionen im Unterricht zwar die Anwendung solcher situationsmodellbezogener Strategien bei den Schülern anregen können, individuelle Leistungssteigerungen aber offenbar eines gesonderten Strategientrainings bedürfen.

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Notes

  1. Of course, one has to assume that the very mastery of those seven sub-competencies does not necessarily result in the successful treatment of a modelling task as the main complexity of such tasks consists in the necessary integration of those sub-competencies.

  2. “Didaktische Interventionsformen für einen selbständigkeitsorientierten aufgabengesteuerten Unterricht am Beispiel Mathematik“, in English: “Didactical intervention modes for mathematics teaching oriented towards self-regulation and directed by tasks”. The project has been funded by the German Research Foundation (DFG) since 2005. It is an interdisciplinary project directed by W. Blum (mathematics education), R. Messner (pedagogy both University of Kassel) and R. Pekrun (educational psychology University of Munich).

  3. The teaching methods were on the one hand a “directive”, highly teacher regulated method with students’ working individually on exercises similar to those developed earlier; and, on the other hand, an “operative-strategic” method where students mostly work independently in groups, supported by the teacher with the help of strategy oriented interventions, and where retrospective reflection takes place with the whole group.

  4. In the following, this is called modelling level 2 (modelling tasks), 1 (dressed-up tasks) and 0 (intra-mathematical tasks).

  5. Additional to the construction of a reading based situation model a partial construction of a real model is necessary for solving the task “Sloths” (cf. Fig. 8). But a small case study with 8-year-old primary school pupils (N=10), skilled enough in reading to construct the situation model but not skilled enough mathematically to construct an adequate real model, showed that it is possible to solve this mathematical reading task “correctly” without having a correct conception of how the relevant aspects are mathematically connected. This can be regarded as a hint for a sufficient validity of the introduced measurement instruments (reliability α=0.67).

  6. It can be rightly criticised that the investigation of the score was not uninfluenced by the intervention because the three items for each student had been distributed over three measuring periods (cf. Fig. 4). The students’ results show a slight increase of the solution frequency for these items within the post-test, though there are no significant differences between the two interventions.

  7. Further descriptions of tasks classifications may be found e.g. in Cohors-Fresenborg et al. (2004), Jordan et al. (2006).

  8. This procedure assumes that the problems are ones the students can—in principle—understand. In an authentic, real context, i.e. a context that was not specially prepared for the students, a completely incomprehensible term can lead to the value of the situation models having to be set at the highest level.

  9. Only the pre-test data are used for the statistical analyses in Chaps. 4.1 and 4.2.

  10. The average situation value for problems with a modelling level of 0/1/2 was 10.8/32.6/38.3.

  11. There was no empirical connection between the modelling level and the difficulty strictly in terms of the test construction which allowed a distribution as even as possible throughout the entire field of difficulty for all problem variations.

  12. There will be no more significant correlations when computing partial correlations between mathematical reading and the intra-mathematical (r=0.02/p=0.73) or the dressed-up dimension (r=0.04/p=0.42) (control variable: modelling).

  13. The term situation strategies is to be used in the following for those procedures which (are to) support the students in forming an adequate situation model.

  14. The difference to real model related interventions is that the situation model related interventions were limited to not putting the solution of the problem in the foreground by a certain form of simplification or structuring, but rather the pure understanding of the demand/ real situation.

  15. The coding mainly referred to such externalized actions that could be contextually described in the aspect (teacher intervention) depicted in Sect. 4.3.1.

  16. This increase indicates that Mrs. R. has especially broached these situation strategies within the DISUM unit.

  17. An example of a successful strategy training for problem solving are the studies of Perels et al. (2007).

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Authors and Affiliations

  1. Department of Mathematics, Leuphana University of Lueneburg, Scharnhorststraße 1, 21335, Lueneburg, Deutschland

    Dominik Leiss

  2. Department of Education, University of Kassel, Moritzstrasse 23, 34127, Kassel, Deutschland

    Stanislaw Schukajlow & Rudolf Messner

  3. Department of Mathematics, University of Kassel, Moritzstrasse 23, 34127, Kassel, Deutschland

    Werner Blum

  4. Department of Psychology, University of Munich, Leopoldstrasse 13, 80802, Muenchen, Deutschland

    Reinhard Pekrun

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  2. Stanislaw Schukajlow
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Correspondence to Dominik Leiss.

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Leiss, D., Schukajlow, S., Blum, W. et al. The Role of the Situation Model in Mathematical Modelling—Task Analyses, Student Competencies, and Teacher Interventions. J Math Didakt 31, 119–141 (2010). https://doi.org/10.1007/s13138-010-0006-y

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Mathematics Subject Classification (2000)

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