1 Introduction

A couple of years ago, some enthusiasm among many teachers could be observed. A lot of expectations and hopes were associated with a real-world-orientated mathematics classroom. It was especially expected that students would be highly motivated and would find easier access to mathematics. Burkhardt (1981, p. iv; emphasis in original) optimistically wrote: However, realistic situations are easier to tackle than purely mathematical topics in that here ‘commonsense’ provides essential and helpful guidance, and because there are no right answers that must be found but only some answers which are better than others. Reality in the mathematics classroom has been different: real-world problems alone neither motivate students nor do they make the learning of mathematics easier. Students sometimes see the absence of a unique solution and the need to include commonsense as an additional barrier.

Another hope is closely associated with the extramathematical field which a task is embedded in: the real-world context. Quite often, it is assumed that a suitable real-world context makes the approach to a problem easier. Another aspect of real-world contexts refers to gender roles in task texts: Depending on the perspective either real-world contexts that are assumed to be close to girls are used, or, in the opposite, especially those contexts are preferred that neglect traditional gender roles (e.g., Niederdrenk-Felgner 1995). In any case, real-world contexts seem to be ­important. However, students might look on this topic differently: When asked about the ­importance of a well-balanced appearance of males and females in tasks, students answered that it was all the same for them; the real-world contexts of most tasks were very artificial anyway so that the contexts do not have any meaning for real life (Niederdrenk-Felgner 1995, p. 54). This comment suggests that analysing the role of real-world contexts might be more difficult than expected in some circles.

In the following, some aspects of the current discussion are presented. After that, the research question is formulated, followed by considerations on methodology and methods. Afterwards, results of this empirical investigation are given and embedded in a broader theoretical context.

2 Some Aspects of the Current Discussion and Research Question

Although the real-world context might play an important role when discussing an application and modelling classroom, no (or at least no standardised) definition does exist, even different names can be found, for example, situational context (Stern and Lehrndorfer 1992), task context (Stillman 2000) and real-world context (Stillman et al. 2008). A comprehensive definition will be proposed later.

Several researchers claim a fostering effect of familiar real-world contexts on the learning of mathematics (among many others, e.g., Wiest 2002). On the other hand, there are strong hints that the familiarity of a real-world context might have an opposite effect: It can be a barrier to the successful solution of the task (among others Boaler 1993). Further analyses show that a positive effect of familiar real-world contexts can often be observed with primary school children (most research has been done in this age group) whilst opposite or more complex effects are related to older students.Footnote 1

Another aspect of real-world contexts is based on observations that not everybody seems to perceive the real-world context of a given task in the same way, obviously there is an individual factor (among others Boaler 1993).

So, there are two areas where the research seems to be vague so far:

  • How do secondary school students deal with the real-world context?

  • What role does the individual perception of a real-world context offered in a task play?

To investigate these questions, a definition of real-world context is needed, which also includes individual aspects. For this reason, the following comprehensive definition is used in this study:

The real-world context of a realistic task comprises all aspects of the verbally or nonverbally, implicitly or explicitly offered extra-mathematical surrounding in which the task is embedded, as well as its individual interpretation by the person who works on the task.

3 Methodology and Methods

3.1 Methodological Remarks

The explorative character of the research question suggests a methodological embedding, which emphasises in-depth insights. For this reason, a qualitative approach was chosen. In contrast to the quantitative paradigm where the selection of cases is based on the idea of statistical representativity, in a qualitative study, the cases are supposed to mirror the range of possible phenomena (“representativeness of concepts”, Strauss and Corbin 1990).

When investigating complex questions, the approach of triangulation has become a powerful tool. According to Denzin (1970, p. 297) triangulation means “… the combination of methodologies in the study of the same phenomena.” While some years ago, triangulation used to be considered mainly as a tool of validation, more recently it is seen from a different angle. This change of view is based on the insight, that “What goes on in one setting is not a simple corrective to what happens elsewhere – each must be understood in its own terms.” (Silverman 1985, p. 21). The aim of triangulation …should be less to achieve convergences in the sense of a confirmation of aspects already found. The triangulation of methods and perspectives is instructive especially when divergent perspectives can be clarified, (…). In this case a new perspective emerges that requires theoretical explanations. (Flick 2000, p. 318, emphasis in original, translation by the author.)

In order to reduce the complexity of the analyses, the Weberian notion of ideal types (Idealtypen, Weber 1922/1985) is used. By unilateral exaggeration of some and fusion of other aspects, an essential structure becomes apparent. The purpose of creating ideal types is not exclusively to categorise facts, but to emphasise the characteristics of the real case by contrasting it to an ideal type.

3.2 Methods

Four pairs of 16–17-year-old students were chosen. They came from four different schools. In addition, both sexes as well as different mathematical abilities were equally represented. These eight students were asked to solve three different tasks in pairs, so 24 cases can be distinguished. The tasks differed in their real-world contexts and their degrees of open-endedness. So, the tasks as well as the choice of participants contributed to a broad range of possible phenomena (see above).

As a first step, the students were videotaped while working in pairs. Secondly, they watched individually (together with the researcher) the video record. The playback was interrupted at certain moments in order to provide the student with the opportunity to comment on his or her thoughts about the real-world context that had occurred while working on the task (stimulated recall). In a third step (interview), the interviewee was asked more detailed questions about these statements. This three-step design enables the researcher to reconstruct different levels of action separately although they have taken place simultaneously.

By this methodical approach, a set of data – containing three different kinds of data – is created. Due to the three different conditions in which the data are collected, each kind of data has certain characteristics, for example, relating to the role of the researcher, the time which has elapsed since working on the task, or the means of collection. The three steps can be considered as three different perspectives on the research question, thus a triangulation of methods is realised. Consequently, data analysis had to take this into consideration: First, the different kinds of data were analysed separately. After that, the three partial analyses were brought together to a comprehensive case analysis. These 24 case analyses were compared and ­contrasted with each other, and finally clustered. These clusters were – according to Weber (1922/1985) – idealised to ideal types.Footnote 2

3.3 Tasks

Three tasks were given to the students. These tasks follow in Figs. 5.1–5.3. The first task is Home for Aged People (see Fig. 5.1). Since no criterion for an optimal ­position is explicitly given in this task, one has to deal with a certain openness in order to solve it. A criterion has to be found on one’s own, possibly considering contextual reflections. There is more than just one possible answer, so the students have to give reasons for their choices. The real-world context offered in the Home for Aged People Task lies in the field of social problems.

Fig. 5.1
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Home for Aged People Task

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The second task, the Transmission Task (Fig. 5.2), is contextually strongly associated with physics and – depending on how it is tackled – there might be a unique solution. However, contextual aspects could be involved, for example, the role of the dimensions of the belt.

Fig. 5.2
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Transmission Task

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The third task, the Oil Task (Fig. 5.3), offers a wide range of possible real-world aspects. Depending on the modelling assumptions, how the consumption develops in the future different solutions are possible.

Fig. 5.3
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Oil Task

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4 Results

4.1 General Results

It could be confirmed that real-world contexts are interpreted very individually depending on different previous personal experiences. Usually different aspects are chosen from the task to form an individual context; for example, while one student embedded the Oil Task in the scientific context of chemistry another student empha­sised aspects of personal responsibility for the natural environment (see Busse and Kaiser 2003). Thus, it seems sensible not to talk about the real-world context of a task but to use the notion of contextual idea (“sachkontextuale Vorstellung”, Busse 2009) to indicate the mental representation of the real-world context offered in the task. In addition, it was found that contextual ideas are dynamic. They do not appear at the beginning of a task and remain unchanged throughout; rather they come into being, develop and change in the course of working on the task.

So, the idea that an attractive real-world context can serve as a special starting moti­vation has to be qualified: It cannot be known for sure in advance which contextual ideas an individual develops and when in the course of the solution these ideas appear.

4.2 Ideal Types

The analysis of the 24 empirical cases led to four theoretical ideal types that describe different ways of dealing with the real-world context (cf. Busse 2005):

Reality bound: The task is fully characterised by the real problem described in the task. Only extramathematical concepts and methods, no mathematisations are applied.

Mathematics bound: The real-world context is a mere decoration. Only contextual information explicitly given in the task text is used, no additional personal contextual knowledge is applied. The task is solved exclusively by mathematical methods.

Integrating: Personal contextual knowledge, which exceeds the contextual informa­tion given in the task text, is used in order to mathematise the problem and to validate the solution. During the solution process, mathematical methods are applied.

Ambivalent: There is an ambivalence concerning the legitimacy of the way the task is supposed to be solved: Internally, a contextually accentuated reasoning is preferred while externally a mathematical reasoning is chosen. These two ways of reasoning coexist without forming a coherent whole.

In Fig. 5.4, these four ideal types are presented graphically. The lower two arrows indicate how the two extremes form a new quality; the upper two arrows illustrate how the type ambivalent is torn between the two extremes.

Fig. 5.4
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Ideal types of dealing with the real-world context

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In order to create the ideal type ambivalent, the above-mentioned approach of triangulation followed by a separate analysis played an important role. When analysing the data, there were students whose reasoning sounded mathematical ­during the solving process; but later during the stimulated recall and the interview, it appeared that the actual solving process was in fact reality based. So, these students translated their reality-based solving process into mathematical language. This ­phenomenon can be explained by compliance with certain sociomathematical norms (e.g., Yackel and Cobb 1996). It is assumed that the norms in question do not permit contextual reasoning in a classroom-like situation (like the first step of the three-step design), but do allow this if the situation becomes more distant from the mathematics classroom (like in the two other steps of the three-step design). In other words, sociomathematical norms are considered as situated (cf. Lave and Wenger 1991). The existence of the ideal type ambivalent underlines the important role of social-­mathematical norms and their situatedness in the field of application and modelling.

4.3 Case Synopsis

The total of 24 cases can be described by the four ideal types. Some cases are described by more than one ideal type. A synopsis of 6 of the 24 cases is presented in Table 5.1.

Table 5.1 Synopsis of 6 of the 24 cases
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On the basis of the results from these two students, it is evident that neither a person nor a task is permanently linked to a certain ideal type, but there are hints for preferences. Considering all 24 cases, the Transmission Task is more often associated with the ideal type mathematics bound than the other tasks. On the other hand, Table 5.1 suggests that there might be personal preferences for certain ideal types.

Deeper analyses of the data showed that an emotional involvement or a special interest in a certain real-world context is often linked to aspects of contextual reaso­ning. For example, Karla, who might generally prefer a more mathematical manner of reasoning, included contextual aspects when it came to the Oil Task. Karla embedded the Oil Task in the context of responsibility for the natural environment, a topic that worried her very much. In a similar way, Evelyn included – in contrast to her actions in the other tasks – contextual reasoning when solving the Home for Aged People Task. This change of focus might have been due to Evelyn being very committed to social problems and her wish to work as a volunteer with elderly people.

5 Final Remarks

The results of this study show, to a high degree, the important role of individual aspects when dealing with the real-world context of a task. Although many questions are still unanswered it becomes clear that teachers as well as researchers have to take this individuality into account. In school, teachers have to be aware that a student’s contextual ideas might differ from theirs. Also, the way in which real-world contexts are used varies from person to person. The system of ideal types might help analysing learning problems concerning this matter.

Another important aspect is the observation that sociomathematical norms concerning the permitted use of real-world contexts are often implicit which might cause some irritation among students and can lead to ambivalent behaviour. A more explicit teaching of these norms must take place. This can be realised by including the teaching of a modelling cycle in the mathematics classroom.