This contribution shows an inquiry-based learning approach to mathematical modelling, and provides a first insight into its effectiveness for prospective primary teachers. There is a consensus among different authors in mathematics education that the mathematical knowledge that should be expected from a primary school teacher goes beyond the knowledge of the content to be taught (Hill et al., 2008; Carrillo et al., 2013). However, research reports the difficulties that prospective teachers have with regard to skills such as reasoning (Kaasila et al., 2010), problem-solving (Verschaffel et al., 2005) or the application of mathematics to real contexts (Sáenz, 2009). Working with mathematical modelling tasks creates opportunities to alleviate these difficulties.

Over the last decades, modelling has become a crucial area in mathematics education (Barquero, 2019). In fact, curricula in different countries have gradually incorporated modelling competencies, and modelling is generating a growing interest in teacher education, specially in prospective Primary Teachers (Guerrero-Ortiz & Borromeo-Ferri, 2022). Likewise, there has been a proliferation of international projects aimed at designing resources that can support the learning of modelling, such as LEMA, MASCIL or MERIA (2016). In this European project whose main objective was to promote the teaching of mathematics applicable to real life. It combined the principles of Realistic Mathematics Education (Van den Heuvel-Panhuizen & Drijvers, 2014), the ideas of Inquiry-Based Learning (Maaß & Doorman, 2013; Dorier & Garcı́a, 2013) and the pedagogical approach of Didactic Situations Theory (Brousseau, 1997). The key idea developed in MERIA is to implement the principles of inquiry-based learning. In this context, teaching should provide students with just the right amount of help to support mathematical learning. These ideas can be harnessed to stimulate the autonomous development of modelling activity by pre-service teachers, thus it give rise to our research question: Does the inquiry-based learning influence the models developed by prospective teachers?

Modules and scenarios for an inquiry-based learning of modelling

For the sake of providing the “right” amount of direction to inquiry, teaching approach developed in in the MERIA project were based on two key tools: Modules and scenarios. A module is the union of a scenario with all the material needed to implement this scenario in the classroom. Likewise, a scenario is a full description of a lesson in terms of the Theory of Didactical Situations (TDS, Brousseau, 1997). Under the TDS approach, students are intended to construct new knowledge when they solve a task while adapting to what is called a didactical milieu. It consists of the task, students’ previous knowledge, and the artifacts to solve the task. The role of educators is to design such milieu and to help student to adapt to it. In this process, two kind of situations appear. The first one is composed of adidactical situations, which are those where the students are engaged in the task and explore the milieu without the teacher’s interference. The second kind of situations are the didactical ones, where students and educators explicitly interact. A balanced combination between didactical and adidactical situations leads to the inquiry process and the students' construction of new knowledge. Therefore, a proper scenario should contain such combination of situations along different phases. (i) Devolution: the educator introduces the task and explains the rules to solve it (didactical situation). (ii) Action: students are engaged with solving the task and actively work on it (adidactical). (iii) Formulation: students explicitly formulate outcomes of the action phase (adidactical). (iv) Validation: students test their hypotheses and strategies against the milieu (adidactical). (v) Institutionalisation: educator declares the institutional knowledge (didactical). In this phase, teacher may put ideas together, compares viewpoints and explains optimal strategies.

Participants and instructional design The sample is composed of 22 students enrolled in a course focused on mathematical competencies for primary their fourth year of the elementary teacher’s degree studies at the University of Granada. These students attended a set of sessions in which different modelling tasks were solved by applying the ideas of inquiry-based learning. In total, six sessions took place, and one task was solved per session. This resulted in a total of 132 written productions, of which 44 (corresponding to two tasks) are analysed for the present study, due to length limitations. The instructional design was based in the MERIA (2016) scenarios. These were reduced for the sake of simplicity, giving rise to “short scenarios” that are focused on teacher educators' actions. Prospective teachers education is based on three activities around the short scenarios: (i) Solving the tasks by taking advantage and reflecting on the scaffolds used and on the mathematical concepts needed (ii) Reflecting on the modelling skills involved in the tasks from different theoretical frameworks (iii) Developing and establishing assessment criteria for their own short scenario. Data analysis The data analysis is based on the characterisation of the models developed by the participants of the study in the written productions collected, and subsequent comparison of these models with those existing in the literature for the same tasks. In order to characterise the models, the procedure set up by Montejo-Gámez et al. (2021), which is based on the description of three elements: the real system, the mathematics used by the participants and the representations employed. In this way, the analysis begins with the identification of statements involved in the elements of representation, which makes it possible to distinguish the relationships and mathematical results of the model. From these, the objects and variables used are extracted, respectively. Finally, the analysis of the results allows the abstraction of the mathematical properties and concepts involved. In order to compare the models found with those reported in the literature, the categories obtained by Segura (2022) will be taken and common and novel elements will be identified against these categories. This will allow us to observe the influence of the scenarios implemented on the participants' productions and to draw conclusions on the relevance of these scenarios.

Based on previous experiences with similar scenarios and previous literature on mathematical modelling in prospective primary school teachers, a set of ideas emerge that shape the expected outcomes of the study. In particular, it is expected that the fact that the sessions are led by educators will condition the written productions collected, a situation that may materialise in different ways. Firstly, a higher response rate to the tasks is expected than in other studies. The complexity of the problems proposed sometimes leads students to blocking, a situation that should be avoided under the didactic proposal used. The educator's action may possibly lead to a lower number of errors, which contributes to alleviating the difficulties experienced by these students when solving contextualised problems. Similarly, participants are expected to propose more accurate models, as the session promotes discussion and comparison of different ideas among peers. As a negative effect, on the contrary, it is expected that there will be less richness of ideas than reported in the literature, since the students have all followed the same session (and, therefore, flow of ideas). In short, we expect to find indications that support the use of inquiry-based learning, as well as points for improvement of the scenarios, which should lead to simplifications of the design and implementation of the scenarios.

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