In Germany, the focus on mathematical modelling has intensified considerably since the 1980s. Earlier, in 1976, Pollak gave a talk at the ICME 3 in Karlsruhe, where he contributed to defining the term “modelling” (Pollak 1977). Different modelling cycles (for example Schupp 1989) were developed and discussed, in order to describe modelling processes and goals, as well as arguments for using applications and modelling in mathematics teaching. After subject-matter didactics (Stoffdidaktik) had affected mathematics education with pragmatic and specific approaches in Germany, there was a shift in the last quarter of the 20th century towards a competence orientation, focusing on empirical studies and international cooperation.
Background of the German Modelling Debate
In fact, the discussion of applications and modelling in education has played an important role in Germany for more than 100 years. The background to the German modelling debate at the beginning of the 20th century can be divided into a practical arithmetic approach (Sachrechnen) at the public schools (Volksschule, primary school and lower secondary school) and an approach of applications supported by Klein and Lietzmann in the higher secondary school (Gymnasium).
At the beginning of the 20th century, mathematics education was influenced by the reform pedagogy movement. Johannes Kühnel (1869–1928) was one of the key figures in this movement. Kühnel criticised teaching problems that were basically irrelevant and called for problems that were truly interesting and relevant for students. During this period, applications were considered to be more important for the learning process. They were used to help visualise issues and motivate the students, rather than prepare them for real life (Winter 1981).
In contrast to the practical arithmetic approach at the Volksschule, the formal character of mathematics was the centre of attention at the Gymnasium. Mathematical applications were mostly neglected. Whereas Kühnel and other educators (representing the reform pedagogy movement) had a greater influence on the Volksschule, Klein started a reform process in the Gymnasium. At the beginning of the 20th century, a more appropriate balance between formal and material education was requested, due to the impact of the so-called “reform of Merano”. The main focus was on “functional thinking”. In the context of Merano’s reform, a utilitarian principle was propagated “which was supposed to enhance our capability for dealing with real life through a mathematical way of thinking” (Klein 1907, p. 209, translated). Because of the industrial revolution, more scientists and engineers were needed in the economy and society. This is why applied mathematics gained in importance and real-life problems were used more often. Lietzmann (1924) made some important proposals for the implementation of Merano curricula and constituted an implementation of applications in the classroom. Finally, the contents of the Merano reform in 1925 were included in the curricula of Prussian secondary schools.
This trend continued until the 1950s. In the late 1950s, Lietzmann stressed the need for stronger inner-mathematical objectives (Kaiser-Messmer 1986). After World War II, some ideas that had evolved from the progressive education movement and the reform of Merano were picked up again, but with applications losing importance. More emphasis was again placed on subject classification rather than on applications (Kaiser-Messmer 1986).
In 1976, Pollak gave a talk at ICME 3 in Karlsruhe, where he defined the term modelling. He pointed out that at that time, people were less familiar with how applications were used in mathematics teaching. To clarify the term, he distinguished between four definitions of applied mathematics (Pollak 1977, s. Fig. 4.1):
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)
Single modelling (the modelling cycle is only conducted once)
Ongoing modelling (the modelling cycle is repeated several times).
In the 1980s, the so-called New Practical Arithmetic (Neues Sachrechnen) evolved at all types of schools in Germany (Franke and Ruwisch 2010). The principles of the reform pedagogy movement were emphasized again and schools started to use applications in mathematics education more frequently. The New Practical Arithmetic aimed at finding authentic topics for students and to conduct long-term projects that were separated from the current mathematical topic and offered a variety of solutions. New types of question, such as Fermi problems, which are formulated as one question only and can be solved by estimating different physical quantities, were used accordingly (Herget and Scholz 1998). At the same time as the development of the New Practical Arithmetic, the term modelling became better known in mathematics education, and the two have partially complemented each one another. However, the main focus of New Practical Arithmetic and modelling was on different types of schools (Greefrath 2018).
The development in German-speaking countries was diverse. For example, in the field of stochastics, there were modelling approaches emphazising stochastic aspects (e.g. Eichler and Vogel 2016).
In 1991, the German ISTRON Group was founded by Werner Blum and Gabriele Kaiser. This caused an intensified debate on modelling in Germany. The idea behind ISTRON was that—for various reasons—mathematics education should focus more on practical applications. Students should learn to understand environmental and real-life situations by means of mathematics and develop general mathematical skills (e.g., transfer between reality and mathematics) and become open-minded regarding new situations. They should thereby establish an appropriate comprehension of mathematics including the actual use of the concepts. Learning mathematics should be supported relating it to real life (Blum 1993). A new series established in 1993 and published by Springer since 2014 has enabled the ISTRON Group, having already produced 20 volumes, to be present and visible in mathematics teaching, as well as in the academic community. Their contributions are intended to support teachers in dealing with real-life problems in school. Teachers are considered to be experts in teaching, so that teaching proposals should be modifiable, enabling teachers to adapt them to a specific situation. They should suggest innovative ways of teaching mathematics and support lesson preparation (e.g., Bardy et al. 1996).
Modelling as a Competency and German Educational Standards
Based on results from the Danish KOM project (Niss 2003) and accompanied by international comparative studies, mandatory educational standards for mathematics were introduced in Germany in 2003 (first at the non-university entrance level). Mathematical modelling is now one of the six general mathematical competencies that the education standards for mathematics regard as obligatory for intermediate school graduation. This approach can also be found in the educational standards for primary school, as well as for upper secondary school.
By means of varied mathematical content, students are to acquire the ability to translate between reality and mathematics in both directions. In the work of Blum (Blum et al. 2007), modelling skills are described in a more detailed way as the ability to adequately perform the necessary steps in the process of changing back and forth between reality and mathematics, as well as comparatively analysing and evaluating models.
It is possible to consciously divide modelling into partial processes for reducing complexity for teachers and students, and for creating suitable exercises (see Table 4.1). This view of modelling especially enables training individual partial competencies and establishing a comprehensive modelling competency in the long term. For more information on modelling competencies, we refer to the comprehensive overview by Kaiser and Brand (2015).
The German educational standards for mathematics at the secondary level for 2003—as well as those at the primary level for 2004 and for the higher education entrance qualification of 2012—describe mathematical modelling as a competency. The educational standards for the general higher education entrance qualification, for example, specify the requirements regarding the modelling competency in the three following areas:
Requirement areas of study I: Students can:
Apply familiar and directly immediately recognisable models
Translate real situations directly into mathematical models
Interpret mathematical results in the context to the real situation.
Requirement areas of study II: Students can:
Conduct modelling processes consisting of several steps and with a few and clearly formulated limitations
Interpret the results of such modelling processes
Adopt mathematical models to changing situations.
Requirement areas of study III: Students can:
Model complex real situations for which variables and conditions need to be specified
Check, compare, and evaluate mathematical models considering the real situation (KMK 2012, p. 17, translated).
Since 2006, an overall strategy for educational monitoring in Germany has been pursued by the Standing Conference of the Ministers of Education and Cultural Affairs. The aim is to strengthen the competence orientation within the educational system. The general modelling competency plays an important role in mathematics. In addition to international school achievement studies (PISA, TIMSS), there are national achievement studies as well as comparative studies (VERA). These tests are carried out in class in Grades 3 and 8 in all general education schools, in order to investigate which competencies students have achieved at a particular point in time. The comparative studies aim to give teachers individual feedback on the educational standards requirements that students can handle.
Beginning in 2017, a pool with audit tasks for the Abitur examination has been be provided for Germany, from which all states can obtain audit tasks for the Abitur. This was an important step in improving the quality of audit tasks and gradually adjusting the level of requirements in all states. Tasks are developed based on the educational standards. Thus, by default, some of the tasks for the Abitur include modelling as a competency. The use of modelling in examination problems, however, is not unreservedly viewed positively. The fact that in many cases, the relevance of the factual context used is not the focus of examination problems, has given rise to criticism on the part of some expert representatives with regard to modelling in examinations: On the one hand, there is criticism of the fact that “modelling competence” is not examined at all through the used audit tasks. On the other hand, other authors point to the categorical refusal of modelling problems also in examinations. Strong criticism is also directed against the fact that examination problems tend to contain to much text (s. Greefrath et al. 2018).
Implementation of Modelling in Everyday Lessons
Fostering students’ modelling competence is compulsory for all mathematics teachers in all grades. But classroom observations regularly reveal only a low proportion of modelling of working on a holistic modelling task in everyday-lessons and class exams in Germany (Blum 2011). Several reasons may apply:
Modelling has been part of the national standards for only about fifteen years. Therefore, many teachers are not trained to teach modelling. Although there are many in-service teacher trainings in modelling, German teachers are not obliged to attend them. Accordingly many simply do not know how to implement modelling in everyday-lessons, how to behave during students’ work on modelling problems, and generally how to support their students best. Modelling is a competence that is difficult not only for students, but for teachers as well. Because students are encouraged to develop their own models, teachers can only anticipate what students will do. They therefore have to be able to diagnose and intervene spontaneously, but often, do not feel confident in doing so (Tropper et al. 2015).
In addition, concurrently to the implementation of modelling in the national standards, state-wide comparison tests have been established in Germany. As modelling is one of the six competencies of the national standards, it is included in these tests. Thus, teachers who want to prepare their students for these tests must implement modelling in their classes to a certain extent. But, as Henn and Müller (2013) stated, most of the so-called modelling problems at school and particular in exams, are not modelling at all, according to the description of modelling problems given above. Mostly, not a complete complex modelling task is tested, but only sub-competencies of modelling. Therefore, teachers do not have to tackle entire modelling problems in their mathematics classes to prepare for the central exams.
Furthermore, teachers claim that there is insufficient knowledge about how to foster students’ modelling competence best and most effectively. At first glance, this is surprising, as many studies have researched single aspects (for an overview of research results, see Greefrath and Vorhölter 2016). But clearly, until now, these findings have not been integrated in such a manner as to be useful for teachers. However, Böhm (2013) developed a theoretical approach for improving students’ modelling competencies systematically and permanently. Furthermore, Blum (2015) presented—based on empirical findings—ten important aspects of a teaching methodology for modelling. In addition, there are various task collections (for example, the ISTRON series and the collection of tasks by MUED, see Greefrath and Vorhölter 2016) that can be used as a teaching resource for modelling problems.
Summing up, there has been much research on different aspects of fostering students modelling competencies. But until now, this knowledge has not been applied in practice (at least not as much as one would wish). One indication that teachers want to implement modelling, but do not know how to do so is the fact that at least in Hamburg, it is not difficult for researchers to convince teachers to take part in research projects on modelling. Furthermore, teachers in Hamburg often wish to participate in modelling days or even weeks. These projects are introduced in the next section.
Implementation of Modelling via Modelling Projects
Modelling cannot only be conducted during regular mathematics instruction. In Germany, there is quite a tradition of modelling projects, carried out by different universities all over Germany and Austria. They were originally developed at the University of Kaiserslautern by the working group of Helmut Neunzert, an applied mathematician, more than twenty years ago, and their structure has been adopted by different universities. Although the aims and the target group of these projects differ, all modelling projects follow a similar structure. During modelling weeks or days, as these projects are termed, students have to work on one complex problem over a longer period, more or less on their own. The modelling problems often come from research or industry and have been simplified only slightly. Normally, these are introduced in a short presentation. Problems that have been tackled so far include:
Pricing for Internet booking of flights
Optimal automated irrigation of a garden
Chlorination of a swimming pool
Optimal distribution of bus stops
Optimal distribution of rescue helicopters in skiing areas.
Often, the students are able to choose between modelling problems, as several problems are offered. Afterwards, according to their particular interests, they are divided into different groups.
The students are supervised either by university teachers or by university students trained as tutors. The supervisors are required to use the principal of minimal help. At the end, the students have to present their solution to an audience. Modelling projects that last roughly one week are referred to as modelling weeks and often take place outside school (usually at a university or a youth hostel), while modelling projects lasting only two or three days are called modelling days and normally take place in a school.
The aim and target group of the modelling days and weeks differ, depending on the host in question. In some cases, as in Kaiserslautern and Aachen, applied mathematicians (originally) carry out those projects. They focus mostly on introducing students to the role of mathematics in other sciences. Often, they offer their modelling projects to highly gifted or at least interested students. In other cases, as in Hamburg, Kassel and Koblenz, carrying out modelling days is only part of a whole programme. Didactical considerations, such as fostering students modelling competencies or increasing their motivation, form the focus. Normally, whole classes, regardless of their mathematical competencies, take part in those modelling days or weeks. Furthermore, in these cases, not only the students working on the modelling problems, but also those supervising them (in- and pre-service teachers) can be considered the target group. Preparation for supervising students during modelling days or weeks not only includes telling them how to behave so as to help students as little as possible, but as much as necessary concerning the special problem, but in general. This includes general knowledge about modelling and diagnosing problems, as well as intervening in such a way that the learning outcome for the students is as high as possible. To convey the general idea, in the following discussion, the procedure of modelling days, as well as student reactions and outcomes concerning a particular task, will be presented.
Since 2001, modelling days are conducted by the working group on Didactics of Mathematics of the Educational Department of the University of Hamburg. Upon consultation with participating schools, they last 2 or 3 days, directly after winter term in February. Every year, whole grade 9 classes from different schools participate, that is over 200 students and about 10 teachers. Teachers meet beforehand, are informed about the modelling problems and trained. Furthermore, student teachers were trained within a didactic seminar that focuses on teaching modelling in general; part of the seminar entails supervising the students during the modelling days. Every year, three different modelling problems are presented to the students who can choose what work they wish to do. In 2016, the problem in Fig. 4.2 was posed.
As there is a shortage of living space in Hamburg, students firstly claim the distribution key is unfair, because some larger counties (which one would assume have more living space) like Mecklenburg-Western Pomerania only had to receive the same number of refugees. They soon decided to consult further aspects for the new distribution key such as area, empty houses and vacancy. They investigated relevant data and considered how to develop a new distribution key. They measured proportions and compared the outcome for the different federal states of “their” distribution key to those of the “Königsteiner Schlüssel”. Summarizing, they developed a solution for a highly relevant topic, use proportions in a real context (and not to forget how to calculate a proportion in the future) and form one’s own opinion.
In contrast to the implementation of modelling in everyday lessons, there has not been much research on the impact of modelling days or weeks on students, pre-service or in-service teachers. One exception is the study by Stender (2016), that focused on the acting of teachers tutoring students while working on a complex modelling problem. He videotaped the working process (lasting 2.5 days) of 10 groups of students working on the modelling task “Roundabout versus traffic light: Which intersection allows more cars to pass through?”. The tutors had been trained before on how to supervise students. The results clearly indicate the trained strategic interventions were used to a considerable extent and were mostly successful. However, tutors had different preferences; some seldom intervened, but their interventions last longer; some intervene more frequently, but gave only very short interventions. Furthermore, the intervention “Explain your work” proved to be very effective and was often used. The results indicate in addition the importance of an accurate diagnosis of the students’ situation and their current motivation, as interventions were rather unsuccessful if diagnoses were not accurate. Furthermore, an inadequate understanding of the modelling situation and the mathematical situation by the tutors led to rather misleading interventions.
Although there had not been much research on this issue, modelling weeks and days were evaluated regularly, revealing great approval and good learning outcomes in various types of competencies (for more details, see Kaiser and Schwarz 2010; Kaiser et al. 2013; Vorhölter et al. 2014).