This paper is grounded on a design-based research, in which a curriculum for the First and Second Grades in Arithmetics has been built. This curriculum, termed Arithmetic and Comprehension at Elementary School (ACE), has been elaborated within a team composed of teachers and researchers, and implemented in more than 500 classrooms since 2012.

The ACE project refers to the construction of the concept of number, which we are experimenting with 6 and 7 year-old-students (First and Second Grade). ACE gathers together five French research teams, involving teachers, mathematics psychologists, and mathematics didacticians, each of them having designed a part of the whole curriculum. In this paper, we focus only on the Brittany-Marseille team, which has in charge a domain called “Situations”. The making of the curriculum is based on the implementation of a cooperative engineering, that is a specific form of design-based research (Cobb et al., 2003), developed within the Joint Action Theory In Didactics, which fosters specific relationships between teachers and researchers, according to a symmetry principle (Sensevy et al., 2013 ; Joffredo-Le Bun et al., 2018). This principle notably relies on the idea that there is no a priori division of work between teachers and researchers, and that each member of the collective is able to reflect on all the problems encountered by the collective.

This research was a quasi-experimental design. In effect, each year, students’ learning in the experimental classes has been compared with students’ learning in control classes (pre-test/post-test assessment). It worths noticing the two main results that have been obtained. 1) For each year of the experimentation (2012-2013 and 2013-2104, first-grade; 2015-2016 and 2016-2017, second-grade) the experimental classes students outperformed the control classes students. 2) For each year of the experimentation, the gap between students from underserved communities (Priority Education Zones, in the French System) and students from middle class communities was largely widening along the school year in the control classes, and stay at the same level in the experimental classes. The ACE curriculum can be viewed as built in a specific kind of evidence-based research (Fisher et al., accepted).

In this paper, we will focus on an example of cooperative work within the Research Team, at second grade. As we mentioned, the second grade ACE curriculum has been implemented within classrooms since 2015-2016, and it is now relatively stabilized. The main research team’s work consists of analyzing the implementation of the curriculum in various classrooms to propose some improvement. Some of these modifications slightly change the curriculum. Some others ask for a kind of redefinition of some curriculum’s features.

In the paper, we focus on one of these important changes, the implementation of a teaching-learning device, what we called “Inquiry and Training threads”.We so present a concrete example of the way teachers and researchers work together to improve some aspects of the ACE curriculum, by designing these “Inquiry and Training Threads”. We show how the Cooperative Engineering collective designed and redesigned a specific thread ("Approximation") by integrating to it firstly an estimation strategy, and secondly the semiotic systems which may help perform this strategy accurately. The whole improvement process is now available on line on the Picture-Text-Audio Hybrid System (PTAHS) (http://pukao.espe-bretagne.fr/public/tjnb/shtis_ace/reseau_analyse_approximation.html) whose we present the “homepage” above.

A further step in this research has consisted in developping this PTAHS within the research group. In this paper, we will display the different states of the PTHAS, from the first prototype to the current device. We will show how this continuous process can be seen as a deepening of the cooperative process, through the augmented engineering dialogue enabled by the hypermedia system.

The “Inquiry and Training Thread” practice was integrated to the ACE curriculum, firstly in Second Grade, since the end of 2016-2017. The first “Inquiry and Training Thread” implemented was termed “Exploring the number line”. The students gave themselves a difference exercise (for example 64-37), then used the "number line device" to find the solution. This mathematical practice (“Exploring the number line”) gradually enriched the ACE curriculum, along with other “Inquiry and Training Threads” (for example “Problem Posing”). Two fundamental questions were then raised: i) How to share deeply this kind of practice through the experimental group; ii) how to improve this kind of practice. To answer these questions, the Research Team designed digitalized films of practice, “Picture-Text-Audio Hybrid Systems” (PTHAS). Such devices aimed to enable people a better knowing and understanding of the filmed practice (e.g “Exploring the number line” practice). It relies on various commentaries on the filmed practice, from teachers and researchers as well. This methodology relies on what we call an epistemology of paradigmatic analogy, in which emblematic examples function as Kuhnian exemplars (Kuhn, 1979).

We intend to emphasize the following points: - a way or describing the cooperative process between teachers and researchers in a cooperative engineering ; - a methodological approach grounded on the continuous (re)designing process of hypermedia systems (The PTHAS) which can be seen as emblematic examples of practice that enable teachers and researchers to better understand and transform practice; - a theoretical contribution on the meaning-making process in cooperation; - an epistemological reflection on educational sciences and didactics.

Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design Experiments in Educational Research. Educational Researcher, 32(1), 9 –13. Fischer, J-P., Sander, E. Sensevy, G. Vilette, B, & Richard, J-F. (2018). Can young students understand the mathematical concept 4 of equality? A whole-year arithmetic teaching experiment 5 in second grade. European Journal of Psychology of Education. Accepted. Kuhn, T. (1979). The Essential Tension. Chicago: University of Chicago Press. Joffredo-Le Brun, S., Morellato, M., Sensevy, G., & Quilio, S. (2018). Cooperative engineering as a joint action. European Educational Research Journal, 17(1), 187 208. Sensevy, G., Forest, D., Quilio, S. & Morales, G. (2013). Cooperative engineering as a specific design-based research. ZDM, The International Journal on Mathematics Education, 45(7), 1031-1043.