Mathematical modelling in mathematics education refers to the development of tasks in a classroom environment in which students are involved in the search for solutions to real-world problems through the use of mathematical knowledge. Although very general, this common idea mentions a moment that may be of special interest to mathematics education: the process of insertion or use of mathematical elements (techniques, concepts, procedures, theorems, etc.) in a real-world problem that, at first, did not include such elements. This process is known as mathematization (BLUM; NISS, 1991; NISS et al., 2007).

Many authors propose modelling cycles or phases (e.g. ALMEIDA; SILVA; VERTUAN, 2012; BLUM; LEIß, 2007), which are structures that guide students through steps that must be followed during mathematical modelling. One of these steps is the mathematization and, because of the key idea of modelling – solving real-world problems through mathematical knowledge – we believe that characterizing this step in detail is important to understand mathematical modelling as a whole.

In this paper, our objective is to describe and characterize the mathematization process that happened in an open mathematical modelling project, in which all decisions, from the choice of a real-world problem to its solution, were taken collectively by a group of students and researchers. We begin by presenting a discussion about mathematization in the modelling field literature in mathematics education.

While developing modelling projects, students are asked to conduct investigations having as a starting point a real-world problem. During this process, students create hypotheses and use mathematics to solve the problem posed, performing mathematization (BLUM; NISS, 1991; NISS et al., 2007).

Niss et al. (2007, p. 9) claim that during mathematization, “the relevant objects, data, relations, conditions, and assumptions from the extra-mathematical domain are then translated into mathematics, resulting in a mathematical model through which to address the identified problem”. Similarly, Stillman et al. (2015, p. 94-95) state that mathematization involves a translation from the real situation into the mathematical domain and that “the mathematical model made of the situation, mathematical questions posed and mathematical artefacts (e.g., graphs and tables) used in solving the mathematical model” constitute this mathematical domain.

From literature, we highlight some works that aim to comprehend the participation of students during modelling, more specifically during mathematization. Schaap et al. (2011) analyzed the occurrence of obstacles, as well as the students’ steps to overcome those, while developing modelling activities. Their focus is on the mathematization process and the authors believe, as we do, that mathematization and modelling construction do not happen linearly. In the same way, Maaβ (2007) describes a series of slips and mistakes the students make during the process of constructing mathematical models. The author adds that it is common for students to accumulate a great amount of information during the development of such activities and eventually end up getting lost in their way with the procedures; and that “these failures show that it is obviously necessary to have an overall view and to work purposefully. A sense of direction is necessary” (MAAß, 2007, p. 74).

Having as theoretical reference the Realistic Mathematic Education, Carreira and Baoia (2011) present a project with nine-graders in which a mathematical modelling task was proposed. The authors highlight the mathematization process, stating that this stage “includes both horizontal and vertical developments” (p. 214). Based on Hans Freudenthal's ideas, the authors state that “horizontal mathematization involves the passage from the world of life into the world of symbols and vertical mathematization means moving within the world of symbols” (p. 214). It was observed that experiments performed with students during the modelling activity helped them in the mathematization process and, “although experimentation may seem to keep some students at the level of horizontal mathematization, the modelling process final stages reveal that it is actually promoting a deeper conceptual development” (p. 219).

Sol et al. (2011) describe sixteen hypothetical modelling actions and part of them is directly connected to the mathematization process. In the end, the authors describe the routes the students used to solve their activities through the actions each of them took to solve the problem. As Schaap et al. (2011), this study also “shows that the model does not have a linear continuity” (SOL et al., 2011, p. 237). And corroborating the results of Maaβ (2007), also in this study the students “are not aware of the modelling process as a global mathematical process” (SOL et al., 2011, p. 237).

These studies suggest that it would be interesting to find a framework that helps understand these not linear and, perhaps, complex paths followed by students while performing mathematization, keeping the whole process in mind while dealing with details of the modelling project. We believe that the Cultural-Historical Activity Theory (CHAT) can be this framework, due to some of its concepts and structures, as we present in the next section.

The studies of Wake (2014, 2016) approach CHAT and modelling in mathematics education. Wake (2014) uses CHAT as the framework to analyze the relationship between academic mathematics and mathematics in the workplace. However, his intention is “to inform how we might develop a mathematics curriculum for compulsory schooling from research into workplace activity” (p. 272). Modelling would be part of this curriculum. Wake (2016), by his turn, is interested in the students’ transition through college and into university, concluding that by means of different mediation, “students experience modelling and applications as mathematical practices very differently in each [context]” (p. 172).

In these studies, CHAT is the framework used to analyze lager contexts (school vs. workplace or different school levels) and, in some of them, modelling is part of the contexts. So, modelling is not the focus of analysis. In our study, however, we intend to use CHAT to analyze what happens inside modelling projects, by focusing on the mathematization process. We believe that this approach can lead to new insights on the students’ actions in modelling projects.

Considering smaller contexts, Almeida and Brito (2005) focus their study on sense attribution and meaning construction in mathematical modelling activities and base themselves in the motives and necessities of the students, two concepts of CHAT. By their turn, Posada-Balvin and Borba (2019) are interested in the teaching of algebraic concepts in modelling pedagogical projects and CHAT was used to organize the empirical material. By studying the interactions of two teachers, during an online education course on modelling and applications, Galleguillos and Borba (2017) focused on tensions – based on CHAT – in the process of posing and solving a modelling problem. The authors concluded that modelling made tensions visible and “allowed teachers to move from a conflict to the formulation of an open problem, and from a dilemma to the construction of a model and a pedagogic strategy” (p. 129). Although carried out in smaller modelling contexts in mathematics education, none of these studies approaches the mathematization process.

Finally, Stilmann et al. (2015) present a framework, supported by theoretical arguments and empirical evidence, which aims to scaffold inexperienced students in modelling activities. Like our study, they are based on an open modelling activity, which theme and developments are chosen by the students. Stillman et al. (2015, p. 103, our emphasis) conclude that the framework highlights “the effectiveness of discussion in leading to model construction through appropriate mathematization processes”. The framework presented by the authors is a set of questions that, in our view, directs a priori the students’ actions during modelling activities. And even though that might lead to “appropriate mathematization processes”, the framework might not allow the students’ own actions to be highlighted.

Regardless of the appropriateness or not of the mathematization process, we have as objective to describe and characterize this process in an open modelling project, in which there is no scaffolding, previously planned, to the students. In order to do so, we base ourselves in a framework – Cultural-Historical Activity Theory – that let us track the paths followed by students and researchers, presented in the next section.

The Cultural-Historical Activity Theory (CHAT) evolved from the cultural-historical school of thought in Soviet psychology, which most prominent name is Lev Vygotsky, and has philosophical roots in Karl Marx's work. The dialectical mediation between subject and object, through artefacts (tools and signs), is the core idea in the theory developed by Vygotsky. Alexei Leont'ev developed Vygotsky's studies and, after three generations of research (ENGESTRÖM, 2001), CHAT has been evolving as a theoretical-methodological framework. In this paper, we base ourselves on the third generation of CHAT and understand that “from an activity-theoretical perspective, human life is fundamentally rooted in participation in human activities that are oriented toward objects” (SANNINO et al., 2009, p. 2).

The subjects of an activity system are the individuals whose actions are taken as reference for analysis; meanwhile its object is “the “raw material” or “problem space” at which the activity is directed” (ENGESTRÖM; SANNINO, 2010, p. 6). Activity systems are realized or concretized by means of actions. Here, we consider the actions of students and researchers – subjects of an activity system – in the modelling project mathematization process. Doing that, we will be able to describe some characteristics of the activity system in which students and researchers are engaged, including its object. “Being a horizon, the object is never fully reached or conquered” (ENGESTRÖM, 1999, p. 381), it is a “moving target” (ENGESTRÖM, 2001, p. 136) and so, we understand that the object is not completely clear at first. Thus, we will try to unveil the activity system object during the data analysis process. Subjects, object and other elements of an activity system and relationships among them were organized on a triangular representation, presented in Figure 1.

In this diagram, each triangle represents the dialectical mediation between two elements (located at the vertices of the triangle) realized by the element located at the third vertex. For instance, the upper triangle represents the dialectical mediation between subjects and object by tools and signs, the key idea by Vygotsky. Concurrently, there is another dialectical mediation between subjects and object realized by community. The relation between subjects and community is mediated by rules. Finally, the relation between community and object is mediated by division of labor established in the community, which is described as a horizontal division of tasks and a vertical division of power.

As a result of the activity system, the activity object is transformed into an outcome. However, this transformation does not happen in a direct way. According to Engeström (2001, p. 134), “object-oriented actions are always, explicitly or implicitly, characterized by ambiguity, surprise, interpretation, sense making, and potential for change”. The circle around the object, in Figure 1, aims to highlight its complexity, ambiguity, and invitation to interpretation.

The dual character of the object is closely related to the contradictions of the activity system. According to Engeström (2001, p. 137), “contradictions are historically accumulating structural tensions within and between activity systems,” and are sources of change and transformations. The author clarifies that contradictions may happen when activity systems, which are open, allow for the insertion of new elements, which generates tension with the pre-existing ones. Tensions also emerge when two or more subjects have different perspectives regarding the actions that they are performing (DAVID; TOMAZ, 2015).

The previous paragraph describes relationships among three of five CHAT principles¹ (ENGESTRÖM, 2001): the historicity, the central role of contradictions and “the possibility of expansive transformations in activity systems” (p. 137), by facing contradictions. They will be discussed in the next section, supporting the description of the methodological aspects of the research.

CHAT brings history in its own name and “is one that develops as an integral part of the historical turmoil in which activity theorists live” (SANNINO et al. 2009, p. 11). The principle of historicity states that activity systems are shaped and transformed for long periods of time (ENGESTRÖM, 2001) and so, we must look at their problems in the context of their own history. However, in our study, we are dealing with a shorter period of time if compared, for example, with the long-term researches developed by the scholars of the third CHAT generation. To get around this situation, we rely on studies of David et al. (2014) and David and Tomaz (2015), carried out in mathematics classrooms. The authors use the idea of potentially expansive miniature cycles (ENGESTRÖM, 1999; ENGESTRÖM; SANNINO, 2010) as an analytical tool. The strategy is to analyze classrooms events deeply at the micro level, to look for small changes that could be interpreted as potentially expansive. By doing that, it is possible to capture

In summary, we intend to consider the historicity and contradictions – understood as accumulating tensions – as possibilities to expansive transformations, but in the smaller scale of the miniature cycles. For that, it is necessary to describe the research context, which includes the development of the modelling project, to contemplate part of the historicity of the activity system.

At the beginning, just the modelling project theme – the purchase of property (real estate) – was clear to the group. They did not have a clearly defined objective for the project. Thus, in the first miniature cycle described next, we find the students and researchers trying to define the modelling project objective.

Having an open modelling project as starting point, in which the students chose the theme to be investigated, we aimed to understand how mathematization was realized by the activity system subjects. At first, the group knew we were doing mathematical modelling and, therefore, we should use mathematics to approach the real-world situation (NISS et al., 2007). What we did not know was how mathematization would happen in this project. The students Alberto (first miniature cycle) and Pedro (second miniature cycle) were the ones taking the initiative of starting mathematization; however, they did it in a descriptive way, using mathematical terminology (weight, weighted mean, comparing, classifying into patterns, etc.), but without mathematical language or formulas. Researchers’ intervention was needed in order to translate mathematical ideas from colloquial language to mathematical language. So, although students normally have difficulties in the mathematization process (SCHAAP et al., 2011; SOL et al., 2011), it seems that the difficulties were in the use of mathematical language, and not in the insertion of mathematical ideas.

The mathematization process described in the first and second miniature cycles happened independently. From the tensions during the third miniature cycle, made explicit by the student Pedro's questions, it was possible to connect the two previous miniature cycles and move the project towards conclusion. These tensions happened concurrently with the researcher Aline's questions – thus making explicit tensions – about the definition of an objective for the modelling project. Considering the three miniature cycles in a timeline, we can perceive the historicity of the activity system leading the group to face tensions which appeared during modelling project. Then, in a smaller scale, there were expansive transformations in the activity system by means of “construction and resolution of successively evolving [tensions]” (ENGESTRÖM; SANNINO, 2010, p. 7).

Because we interpreted the mathematization process as a set of actions oriented to the activity system object, and the modelling project objective as the object of the activity itself (see analysis in 5.1), we can conclude that the mathematization process took place concurrently with the delimitation of the modelling project objective. The dialectical relation (ENGESTRÖM, 2001) between the subjects’ actions and the activity object helps us to understand the advances and setbacks in the subjects’ actions, permeated by tensions.

The students needed the researchers’ help during the mathematization process, since they had little familiarity with modelling projects. Furthermore, literature points out the students’ difficulties in the mathematization process (SCHAAP et al., 2011; SOL et al., 2011) in modelling activities and, therefore, what happened here is not new.

What is different in this research is the complete openness of the project, as it is common in the Brazilian tradition of modelling in mathematical education (ARAÚJO, 2010). Such openness includes the lack of a sense of direction (MAAß, 2007) in the development and the lack of definition a priori of questions to guide the students (STILLMAN et al., 2015). CHAT proved to be appropriate to organize a set of actions that, at first, seemed disorganized, but that were oriented to the modelling project objective.

To conclude, the main contribution of this paper is highlighting that, given all the openness of the modelling project, the students were autonomous when starting the mathematization process, through colloquial language. This shows that students had a global perspective of the project, different from what Sol et al. (2011) stated – perhaps because they saw themselves as (co)responsible for the project being developed and not as a group of students working on a school task. Furthermore, the detailed description of the mathematization process, making explicit advances, setbacks, tensions, etc., in a dialectic relationship with the delimitation of the modelling project objective, is a new way of looking at this process, which was made possible by the approach from a different theoretical framework.