Problem solving is reported as an essential criterion for the appropriation of mathematical skills. According to mathematicians, psychologists, and mathematics educators, it allows in particular to access the meaning of mathematical notions (e.g. Schoenfeld, 1985).

With this respect, Problem Posing has been recognized as a valuable activity in mathematics education (e.g. Ellerton, 1986; Singer, Ellerton & Cai., 2013 ; Cai et al., 2015). However, researchers (e.g. Zhang & Cai, 2021) have shown the complex nature of teaching mathematics through problem posing. This paper would like to contribute to this research trend.

In this paper, we will focus on multiplicative problem posing. Our purpose is to provide a first understanding of a practice of using problem posing to teach mathematics over a long duration. Our research is part of a larger one, namely the ACE research (Arithmetic and Comprehension in Elementary School). This research proposes a mathematical program in first, second and third grade.

In ACE, a prominent emphasis is put on public representations of mathematical relations seen as “any configuration of characters, images, or concrete objects that symbolizes an abstract idea” (Goldin & Kaput, 1996) and may include manipulative materials, pictures or diagrams, spoken language, or written symbols.

Our early studies concerning one-step additive problems focused on semantic features (Riley, Greeno & Heller, 1983). An extension of our study focus on multiplicative word problems in grades 2 et 3. Our research is organized in a Cooperative Engineering (Sensevy & Bloor, 2020), where teachers and researchers have progressively built a curriculum, in the way of lesson studies (Miyakawa & Winsløw, 2009) with some representational tools. These representations have been worked out at the same time as means for exploring numbers and as problem-posing tools. In this research, the teachers belong to the research team, and are considered as teachers-researchers.

We focus on the fundamental fact, in the teaching-learning process, that students learn by relying on a previous set of meanings, that we may call an already-there (CDpE, 2019). In order to teach, a teacher has to gain a deep understanding of this already-there. For doing that, we argue that a promising avenue consists of organizing the teaching-learning process on the basis of specific examples of the way problems are posed and solved. These emblematic examples of the practice can be seen progressively as exemplar, in Kuhn's sense (Kuhn, 1977). In that way, we will see how a system of exemplars is designed to help students to pose multiplicative problems on their own. In this paper, we thus investigate the following research question: How can a teacher organize problem posing tasks by giving habits of representations of mathematical relationships?

To do this, we rely on concrete work in a classroom, whose teacher is part of a cooperative engineering (Sensevy et al., 2013; Jofffredo Le Brun et al., 2018), a collective of teachers and researchers gathered in a AeP network (AeP, Associated educational Places), hereafter referred to as the DEEC-ACE Collective1. This collective is carrying out a research project centered on Problem Posing, granted by the French Research National Agency (ANR), the DECO project (Determining Efficiency of Controlled Experiments). The choice of the collective is to go from a non-problematized situation (for example, a sweet costs 2€, the children buy 4 sweets. The children pay a total of 8€) to three problematized situations (for example, a sweet costs 2€, the children buy 4 sweets. How much will they pay? ). DEEC-ACE teachers-researchers made classroom videos of the creation of multiplicative problems. Within the cooperative work, meetings were also filmed which enabled the researchers or the teachers to review certain moments. These data were subject to editing on different scales (synopses, transcripts etc.) and 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 theoretical framework (Sensevy, 2014; CDpE, 2019). For the purpose of this presentation, a particular teacher’s class and the collective’s exchanges will be presented.

Some studies (Moyer-Packenham, Ulmer & Anderson, 2012) associate improvement in students’mathematical achievement with the use of visual models and schematic drawing in the process of teaching and learning. Based on visual explicitness of relationships, other researchers (Polotskaia & Savard, 2021) argue that it is the visual explicitness of relationships which facilitates problem solving. Our results show that primary school students are able to model, to pose and to solve new multiplicative problem on their own, based on exemplars (Kuhn, 1977). At the same time we argue that some teaching actions are the key of these results. These key actions are reachable thanks to the collective work of analyzing practice. Our results relate to a better understanding of the issues involved in such a problem modeling approach.

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