The research presented concerns problems of addition for students of six to eignt years old. The research process is organized within an AeP network (AeP, Associated educational Places), hereafter referred to as the ArMed¹ . The ArMed AeP is organized as a cooperative engineering involving researchers, teachers, trainers (Sensevy, & al. 2013; Sensevy & Bloor, 2019). It is a continuation of the ACE engineering research project, "Arithmetic and Comprehension in Elementary School" (Joffredo Le Brun & al. , 2018). We work in particular on the mathematical modeling of problems of addition as a means to conceive of and describe reality (Vergnaud, 1986; Sensevy, Quilio & Mercier, 2015; Vilette et al. 2019). 

Several studies have shown the effects of strategy instruction on students' problem-solving performance. It seems that performance improves significantly when students are taught to analyze a problem situation and create a representation or model of the problem (Riccomini, Hwang & Morano, 2016 ). In these studies focusing specifically on students with learning disabilities, representations are accompanied by a discourse, explaining both the type of problem, how to solve the problem and how to present the solution.

We intend to show pupils a diagram modeling a range of problems to help them better identify the similarity of these problems which they might otherwise perceive as being different. However, research shows that a diagram, here a "number line ", can be both a tool and an obstacle (Skoumpourdi, 2010).  For this reason, the role of teachers in such a process deserves further exploration. The purpose of this paper is therefore to present the modeling choices made by the collective of teachers and researchers and also those made in the classroom. 

To analyze classroom transactions, we use concepts from the Joint Action Theory in Didactics (JATD, Sensevy, 2014). In the situations analyzed, the problem encountered by the students is modeled using the notion of milieu and its resolution is seen to enable them to learn new knowledge. The problem makes sense to the students because they can draw on prior knowledge and habits. In the joint action between the teacher and the students in the situation described, the teacher can be seen to direct the attention of the students so that they act appropriately. The teacher produces different types of signs, which the students must decipher in order to progress solving the problem. 

In-keeping with the community of researchers contributing to the Collectif Didactique pour Enseigner (CDpE, 2019), we study the contract-milieu relationship beyond specific didactic situations. That is to say, the specific aspects of the situation (the mathematical, additive structure) is analyzed in relation to what can be considered common to any problem in a learning activity. This epistemological position means we use the same model to analyze the AeP ArMed engineering dialogue concerning the design and critical analysis of the working sessions. 

This dialogue is focused on actual classroom practice which is considered as a milieu to be studied. It is constructed around the habits and existing knowledge of each member of the collective, which is termed the contract in JATD. The milieu-contract dialectic makes it possible to model the action of the teachers and researchers in the engineering so as to address the following question : how can collective choices seeking to better understand and transform practice come to life in the classroom in a way that improves the understanding of all students?

1 http://ife.ens-lyon.fr/lea/le-reseau/les-differents-lea/reseau-ecoles-armorique-mediterranee

To answer this question, we collected data from the fourteen teachers and seven researchers who made up the cooperative engineering collective.  AeP teachers made classroom films of the creation of additive problems, as well as their modeling and resolution. In preparation for the analysis of the collective, each teacher chose an extract of the video of his or her class; this was then commented on by the members of the collective, either teachers or researchers. Each member of the collective could then view the video asynchronously via the Vialogue platform ( https://www.vialogues.com/). Viewing and comments were noted during the collective's meetings. The engineering meetings were also filmed which enabled the researchers and possibly the teachers to review certain moments of the dialogue. Class films and engineering recordings were subject to editing on different scales (synopses, transcripts etc.), and could also be extended with annotations in Hybrid Text Sound Systems (HTTS, Blocher & Lefeuvre, 2017). These techniques enabled a better sharing of the issues raised within the engineering cooperative. Analysis focused on both the classroom sessions and the engineering dialogue using the same joint action (Sensevy, 2011; CDpE, 2019) theoretical framework. For the purpose of this presentation, the collective’s exchanges with regard to a particular teacher’s class, as well as the exchanges during that class will be presented. To render visible how a shared understanding of what is at play in the class developed, we will describe and analyze the annotations produced by the collective as well as their direct exchanges.

The results relate to a better understanding of the issues involved in such a problem modeling approach. The classroom situation is based on the creation/solution of additive problems. This is unusual and requires a deep understanding of what is at stake, both by the researchers and teachers in the collective, and by the teacher in class interacting with the students. The teaching-learning activity implemented (the number journal, Lerbour, 2016; Joffredo & al., 2018) seemed to enable all students to become capable of modeling an additive problem and even of solving it. But the creation of such problem creation/resolution practices, however relevant they might be, does not advance our research if the conditions for their reproduction and dissemination are not considered at the same time. This is achieved by means of the Hybrid Text Image Sound Systems (HTSS Blocher & Lefeuvre, 2017). We will first show how the development of the HTSS within the collective contributes to a better understanding of practice leading to its improvement. We will then consider under what conditions this type of representation could be used in a larger project.

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