Keywords

Introduction

Research in mathematics education addresses the fact that mathematical learning already takes place before children enter school. Children grasp first mathematical content in their everyday life, both in the home environment and at kindergarten, and approach the process of mathematisation. This happens mostly in play or in play situations in which the children can, for example, become aware of the “quantitative and spatial dimensions of reality” (van Oers, 2014, p. 115) in problem-solving situations. In this context, it is not only mathematical content, such as the arithmetic-based quantity comprehension and counting skills described by van Oers (2014) and the (spatial) geometric ideas that are acquired by the children in play, but also process-related skills, such as reasoning (Vogler, 2021; Krummheuer, 2013), modelling and problem solving (Sumpter & Hedefalk, 2015; Di Martino, 2019; Carpenter et al., 1993). These process-related competencies are considered by studies, especially in situations with elementary pedagogical professionals, in which adults usually play a significant role in the process of negotiation of meaning (Vogler, 2021). Although researchers such as Carpenter et al. (1993) and Lopes et al. (2017) describe that adults in their role as more competent persons of reference can act as a special role model in interactions, it is often also peer interactions that have a significant influence on early learning processes. Especially in regard to process-related competencies, such as problem solving and in those involved in processes of negotiating meaning, peer interactions benefit from the unique interactional proximity between the participants (Vogler et al., 2022; following Vygotsky, 1978). Moreover, peer interactions make up a large part of the interaction time in kindergarten. During this shared time, a variety of new experiences are made by the children. This also includes the exchange, deepening and networking of mathematics-related knowledge in different mathematical domains (Henschen et al., 2022; Henschen, 2020). It is therefore particularly surprising that research has increasingly been focusing less on peer interactions and more on interactions with adults. Exceptions to this trend include, for example, work by Henschen (2020), Helenius et al. (2016) and Flottorp (2011), which examine peer interactions and their contribution to mathematical learning. In this context, Helenius et al. (2016) explain that block play situations with Legos in the kindergarten setting is not only mathematical when explicit mathematical content is negotiated, but also when problem-solving processes emerge in the negotiation process between the children (Helenius et al., 2016). In line with said research by Helenius and colleagues, this paper examines two different block play situations in kindergarten regarding reconstructible (collective) problem-solving activities. It will describe which characteristics of problem-solving processes emerge among children in these situations and how these are interwoven in the negotiation process with various mathematical content and other competencies that are important for mathematical learning processes, such as argumentation. The aim here is to generate initial insights into the extent to which problem solving in particular has a remarkable influence on the conditions for the opportunity for early mathematical learning.

Theoretical Remarks

Early Mathematical Learning (Through Problem Solving) in Co-construction

There has been a lot of controversy for a long time about whether collective or individual problem solving is more conducive to learning in kindergarten: some researchers, among them Piaget, held the view that individual work was more productive because of the egocentrism of young children; however, both Vygotsky (1978) and Mead (1934) argued that collaboration was more beneficial. Azmitia (1988) and others have shown that collective problem solving in interactions leads to more sustainable learning success, especially for younger children. These studies support the co-constructivist perspective on early learning (in peer interactions) of Vygotsky (1978) and Youniss (1980), which also serves as the theoretical starting point for the analyses of mathematical learning and problem solving presented in this article. Key variables of this perspective are the situational negotiation processes of mathematical meaning in interactions. From this (co-)constructivist perspective, learning is to be seen as a process of becoming increasingly autonomous in these interactions of mathematical discourse. In line with this, conditions for the opportunity of learning of ‘the new’ (Krummheuer, 2013; Miller, 1987) are created by enabling children to actively and productively participate in the negotiation processes of discourse, thus opening up various scopes of participation called “leeways of participation” by Krummheuer (2013, p. 251; following Brandt, 2004).

Collective Problem Solving

Based on the findings of various studies, problem-solving situations appear to be particularly appropriate for enabling such actively productive participation in negotiation processes of mathematical meaning and thus co-constructive mathematical learning. Polya (1962) defines problem solving as the attempt to find an appropriate activity to reach a desired point without being able to achieve the actual expected end goal. Following Polya’s definition, Avcu and Avcu (2010) explain that if mathematics is problem solving, then problem solving can be defined as the elimination of the problem situation through the use of critical thinking processes and the required knowledge. In this context, Baroody (1993) states that problem solving requires mathematical thinking. In the context of the mathematics classroom, Lopes et al. (2017) explain that in problem solving – starting from a problem formulated by the learners – the classroom becomes a place of questioning and contextualisation in which the children discover mathematical relationships based on their own everyday experiences. In this context, problem solving is distinguished from (routine) task solving because a knowledge structure (or epistemic structure) alone is not sufficient for problem solving; heuristic ways of thinking become necessary here. In their recommendations for teachers, Hatfield, Edwards, Bitter and Morrow (2007) emphasise that these strategies help students to make progress in solving challenging and difficult problems.

Heuristic Strategies of (Collective) Problem Solving

Some of these heuristics that are described by Polya (1945) in his work “How to solve it” are the “analogy” (principle), “decomposition” (and recombination), “symmetry”, “generalisation”, “invariance”, “working backwards” and “working forward” (and combinations), systematic resp. “intelligent guessing and testing” and “representation change” (“draw a figure”) (Schoenfeld, 1987, p. 284). As the focus of this article is mainly on the development of individual strategies in collective problem-solving processes in peer situations in kindergarten, only some of these strategies are described in more detail here. Thereby, the intelligent guessing and testing, which can also be reconstructed in the following analyses, “is guessing and trying processes to check the probable conditions” (Avcu & Avcu, 2010, p. 1284). Working backwards is described as a “useful and efficient strategy” (Amit & Portnov-Neeman, 2017, p. 3793), which can also be traced in the interactions of the children in our article: Here, the problem solver starts working backwards when the goal is clear but there are many possible starting points. Finally, the heuristic strategy of decomposition (and recombination) should be explained here. This involves decomposing a problem into subproblems, each of which can then be solved more easily. Elia et al. (2009, p. 607) describe following Charles and colleagues that various heuristics can be “introduced [and used] in primary or middle school mathematics teaching”.

Problem Solving in the Early Years

Concerning early mathematical problem-solving in kindergarten, the question arises: How can very young children, who are mainly illiterate, solve mathematical problems? Lopes et al. (2017, p. 254) note that this type of question reveals the “misconception of early mathematical problem-solving” activities that solving mathematical problems means calculating or applying a set of rules (or an algorithm); this misconception has yet to be overcome. However, building knowledge through trial and error is also part of problem solving. Through exploration and experimentation, hypotheses can be analysed and solutions can be explored, making learning individual and meaningful for children. Children construct meaning through their efforts to discover or invent, so the novelty (knowing) described by Miller (1987) can be co-constructed. Vygotsky (1967) and Helenius et al. (2016) describe the central role of play in early mathematical learning and problem solving. More generally, Vogt et al. (2018, p. 592) outline that “innovative approaches to early mathematics should not only be developmentally appropriate and effective but should also be compatible with kindergarten pedagogy. Since kindergarten children are highly motivated to learn, but not in a formal way, play can be seen as a powerful tool for learning”. For researchers such as Lopes et al. (2017) and Carpenter et al. (1993), adults in particular are in the role of the more competent others who create a playful environment in which children are confronted with problems.

Peer Interactions in Early Mathematical Learning

Chaiklin (2003, p. 42) also describes the unique role of the “more competent other person” in mathematical learning processes. They are considered to be the persons who, through their advantage and scope of (established) mathematical knowledge, introduce children to the culture of mathematics and through negotiation support children in participating in mathematical activities (such as solving problems) in play situations. In addition to adults, however, peers of the same age can also play an essential role as interaction partners in problem-oriented mathematical negotiation processes, as the studies mentioned above by Azmitia (1988), Di Martino (2019) and also Helenius et al. (2016) show.

Research Desiderata and Questions

While Flottorp (2011) deals with the extent to which various mathematical contents are negotiated in peer interactions, Henschen (2020) also highlights general process-related competencies and the mathematical content areas. Helenius et al. (2016) focus specifically on problem solving in peer interactions in (block-)play and discuss that this play can be categorised as mathematical, specifically through the process of problem solving. However, Helenius et al. (2016) focus less on how different heuristics emerge in such peer interactions and to what extent they create conditions for mathematical learning. In our paper, we try to close this research gap and to further elaborate on the approaches of Helenius et al. (2016). The following questions focus on the qualitative reconstructive analysis: (1) Which procedures in problem solving (heuristics) emerge in play situations among children in kindergarten? (2) How can these heuristic practices in the negotiation process of meaning create conditions for the opportunity for mathematical learning?

Data and Methodology

Data Corpus – Block Play with Peers in Everyday Kindergarten Life

The data basis for the research questions are videos of 30–90-min free play situations with building materials from everyday life in kindergartens. The ethnographic data was collected as part of Henschen’s (2020) PhD project on block play. The videos were recorded by a student who was familiar to the children from previous internships during her studies; consent was obtained from the kindergarten and the children’s parents. From these internships, both the student and the children were already familiar with the video recording of play situations. The children observed were mainly from a group of four children (Max 4;10, Ron 4;8, Emma 6;0 and Anna 5;3 years old) who worked with one of the materials available in the block play area all together, in different group compositions, and sometimes with other children. Therefore, no specific or new material was selected by the researcher. The play situations were filmed during some of these free play phases over a period of 4 weeks. Two paradigmatic examples of such play situations were selected for the comparative analyses.

Basis of the Methods of Analysis

Using (qualitative) thematic analysis (Kuckartz, 2014), Henschen (2020) developed two different coding frames. One coding frame addresses the connection between mathematical content and informal mathematics in children’s block play (categories of the content level). The other differentiates the ways children work in their block play (Henschen et al., 2022). Henschen developed the categories of the second coding frame based on literature (Siraj-Blatchford & MacLeod-Brudenell, 1999, p. 68; Bruce et al., 1992, p. 124): “making/monitoring”, “constructing/building”, “evaluating/titling” and “designing/adapting”. They reflect that for block play and technical learning opportunities, certain steps in problem-solving processes need to be described. The following Fig. 1 shows how these categories can be understood and illustrated following Henschen’s (2020) work.

Fig. 1
A circle framework depicts how to build and how a conception works. It involves 4 components. 1. Making or monitoring. 2. Evaluating or titling. 3. Constructing or building. 4. Designing or adapting.

Meaning of the categories. (According to Henschen, 2020, p. 403)

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The categories at the content level are “wrong way-right way”, “small-large”, “slanted-straight”, “open-closed”, “fixed-unfixed” and “equal-unequal” (Henschen 2020, p. 278). These categories can be understood as kinds of “natural categories” (Kuckartz 2014, p. 44) because the children’s spontaneous use of language is also considered in the category designations (Henschen, 2020; Henschen et al., 2022). Although these categories are conceived as “natural categories”, they allow the description of mathematical content in block play situations.

Methodological (Pre)considerations and Methods of Analysis

While Henschen (2020) develops categories that can be used to identify and describe mathematics in play situations, the research questions raised in this paper require a micro-analytical expansion of the methodological approach. This analytical focus is realised with the use of interaction analysis from the field of interpretative research in mathematics education, which among other sources can be traced back to work by Krummheuer (2013). The analysis can be seen as particularly suitable because it focuses on processes of negotiation of meaning in ‘moments of crisis’, in which, as can be concluded here, the above-mentioned problem-solving processes are initiated.

This expansion allows for a focus on problem solving, which is illustrated in Fig. 2.

Fig. 2
An illustration. Fine analysis or interaction analysis involves individual utterances, transcription of sequences, identification of dense moments, and reconstruction of problem-solving heurisms. The rough analysis or the thematic analysis involves video, transcription, and result-based analysis.

Combination and integration of methods. (Following Henschen et al. 2022)

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In this context, Henschen’s approach provides a fundamental rough analysis. This process of rough analysis was used to identify sequences that are mainly characterised by the density of mathematics-related negotiation in which problem-solving processes may emerge. These interactionally dense moments can be used as a basis for the fine analysis, which follows the qualitative interpretative paradigm and corresponds to an analysis of interaction. Within the framework of the analysis of interaction (Krummheuer, 2013), the thematic negotiation processes can first be worked out, on the basis of which subsequent processes and heuristics of problem solving (Schoenfeld, 1987 according to Polya) were reconstructed. Finally, conditions for the opportunity to learn can be deduced from these reconstructions. In the following, an analysis is realised using two paradigmatic examples. Especially the first part (analysis of the ladder example, see below) shows detailed results of the rough and fine analyses and how these analytical steps are interwoven and mutually enrich each other. The second part (analysis of the roof example, see below) provides the comparative element of the analysis (Krummheuer, 2007; Krummheuer & Brandt, 2001).

Analyses of Empirical Examples of Block Play Situations in Kindergarten

Analysis of the Ladder Example

Before the analysed situation begins, Ron and Max have already been working for some time with a pre-existing construction made of SONOS material. In minute 26, after Max has attached a construction with wheels (the children also refer to it as a “forklift”) to the top of the structure. Ron remarks, “Why all the way to the top, then they can’t get it down at all”. Max then says that “they” have a “ladder”, and Ron suggests that they build a ladder. After the two of them have made a short piece of the ladder (4 rungs), they turn to the structure with this piece.

004

Ron:

And then how do they get up there?

In connection with Ron’s previous statement that the forklift cannot be brought down from the very top, the question can be understood as a problem definition. However, what is remarkable is the apparent everyday world reference to Ron’s statement. Perhaps Ron disagrees with Max’s suggestion to fix the (too) short piece of the ladder and therefore asks for clarification, which does not directly follow.

Looking at the rough analysis for this scene, the following categories can be assigned with regard to mathematical aspects. Here, Max and Ron refer to the localisation or direction of objects, which is an example of spatial orientation. In this context, the category “wrong way-right way” can be reconstructed. Linked to this finding, but going even further, the category “open-closed” can be assigned here. These topological category is addressed when the connection between two places/points or the accessibility is mentioned. However, in various subsequent scenes, it becomes evident that the children are still looking for a solution to the problem.

026

Max:

No, we’ll stack it on the floor

027

 

lays the ladder through the building (Fig. 3)

028

 

We stack it like this.

029

 

makes a stacking gesture with his hand

030

Ron:

takes out the ladder

031

 

No, we stack it this high.

032

 

places the ladder diagonally on the floor.

033

 

Because otherwise, they wouldn’t be able to get up there.

Fig. 3
A photograph of 2 children playing with building blocks.

Ladder within the building

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Max’s utterance can be understood as a counter-proposal to the fixing. He suggests integrating the ladder into the building, with high flexibility and possible utility for the shared play context. In this context, Max uses the expression “stacking”, as does Ron in the situation. In the scene, Max identifies the previously built ladder as a representative for several ladders to be stacked. Stacking identical objects by placing them appropriately on top of each other can be understood as an engagement with the geometric relation parallel to. Therefore, the category “slanted-straight” can be assigned to the action.

Ron builds on Max’s idea by combining it with his idea of using the ladder diagonally on each floor. Ron adopts Max’s expression “stacking” in the sense of positioning one on top of the other within the structure and varies the ladder’s position in that it is placed diagonally on the floor. By limiting themselves to the task of “stacking”, the children are able to explore and eliminate different possible solutions for positioning the ladder. This can be seen as a propaedeutic to the heuristic intelligent guessing and testing.

While the rough analysis shows the connection to mathematical content and activities, the fine analysis can reveal the collective problem-solving process. It becomes obvious how both are interwoven with each other and can mutually stimulate each other. The problem or construction task (ladder) could also be seen here as a topological or spatial problem and is at least closely linked to the two categories “wrong way-right way” and “open-closed”: How do you get from one place to another? How does one get up?

052

Ron:

but otherwise, they can’t get up there at all

053

Max:

but they put the ladder together, you goof!#

054

 

looks at Ron

055

Ron:

all ladders?

056

Max:

Yes.

In line 52, Ron picks up on the outline of a problem, which mathematically shows a link to spatial orientation and the “topographical idea of connection” (Henschen, 2020, p. 318) from line 033: “and how do they get up there?” Max is referring to this initial question in line 053. It can be surmised that while Ron sees the solution as a construction task, Max shifts the solution to the narrative play action. A non-hierarchical variety of solutions emerges here, which also becomes apparent in the further sequence. Between line 057 and line 077, there is a shift towards the construction task; there is a connection between the two ladders that have been created in the meantime. Several components then extend this composite ladder. Subsequently, the problem shifts to how and whether the ladder can be fixed (anchoring problem). In the process, further ideas for solutions are integrated, such as the need to use or omit specific components (lines 089 and 094) or pay attention to the design of the structure (line 097). The negotiation subsequently intensifies, and a longer argumentatively structured process of negotiation of meaning can be traced (cf. Vogler et al., 2022). As a result of this argumentation process, Ron then also succeeds in convincing Max to accept a solution to the anchoring problem.

140

Ron:

But you can fix it like that and that’s good, then it will hold better.

141

 

points to the upper end of the ladder and the upper edge of the structure

142

Max:

But we need another long one like that first.

(...)

  

157

Ron:

wait, we have to#

158

 

together with Max he pulls the ladder upwards out of the structure

159

Max:

remove

160

 

tries to fix the rod to the corner again

161

Ron:

together with Max, he grabs the ladder

162

 

we have to really get the ladder in there#

163

Max:

installing correctly

164

 

together they connect the ladder to the top corner of the building, the ladder now hangs parallel in front of the building (Fig. 4)

Fig. 4
A zoomed-in photograph of children playing with building blocks.

Ladder hangs in front of the building

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From line 140, a turn in the interaction can be reconstructed. After the children had previously struggled over the relevance of their different solutions, Ron’s argumentation now seems to resonate with Max and both children try to find a standard solution to the problem. This may be due to the fact that in line 141, Ron gesturally illustrates the distance or gap between the end of the ladder and the building, while the ladder is stuck diagonally through the upper floors of the building. At this moment, Max develops the idea of not only connecting the ladder and the building using a white connecting piece and using another long rod as an intermediate piece (line 142). In this context, with regard to the mathematical content measurement (category “small-large”), it is particularly noteworthy that the distance between the end of the ladder and the building corresponds to the length of the selected longer rod. It can be assumed that Ron’s gestural illustration of the distance initiates an estimation process that leads to Max choosing a suitable rod that also enables the attachment. This situation enables the children to mathematically experience that the diagonal placement of the ladder in the building creates an angle between the pole attached to the ladder and the building that does not allow it to be attached to the building (lines 156–157). As the children have now interactively agreed on a common mathematical interpretation of the problem and a working consensus, the collective solution to the anchoring problem takes place from lines 158 to 164. After the upper point of the ladder has been clarified, the ladder is extended to the ground. While the question “And then how do they get up there?” focuses on the goal of arriving at the top, the ladder piece attached to the top now results in the opposite task: the ladder must lead downwards. This can be interpreted as an experience with the problem-solving strategy working backwards.

Analysis of the Roof Example

In addition to the analysis of the ladder example, a short example of decomposing into subproblems) as a heuristic way of thinking can be developed in the following scene. The starting point of the analysed scene is a situation in which three girls also interact with the SONOS material and a picture in which various buildings are depicted. The girls approach the problem of constructing a three-dimensional building with a gabled roof from a two-dimensional image (not to scale). In the scene presented below, Emma focuses on the depiction of the house with the gabled roof.

004

Emma:

places the pole with three links (blue-white-blue) she is holding in her hand on a gable end in the illustration

005

 

I’ll do it like this first

006

 

then holds the pole over the other gable end and takes it away again (Fig. 5)

Fig. 5
A photograph of Emma involved in an indoor game. She is placing a pole on the gable.

Emma holds the pole over the gable

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In line (4), Emma holds a rod from three components (blue-white-blue) over the image of the structure so that the rod first covers one side of the gable and then turns it to the other end of the gable. She accompanies this action verbally: “I’ll do it like this first”. Emma obviously indicates that she will first make the gable end of the roof. It can be assumed that Emma illustrates both the two gable ends as well as the angle between the two ends through her action in line 004, although she only uses one rod for illustration. In this context, the term “first time” may indicate that the girl is following a multi-stage plan to build the structure. She seems to be giving precise information about her work steps here. Emma could thus be passing on a kind of technical knowledge or way of working about building with the material. Here, the strategic solution of breaking down the set or chosen task into individual work steps in order to be able to cope with it can be reconstructed. In this, the heuristic of “decomposing” can be recognised as it is also presented in problem solving (Schoenfeld, 1987). However, the other two girls do not take up the procedure in the following. The construction of the roof fails.

Empirical Findings

From a perspective of mathematics education, it can be particularly emphasised that, in the overall view of the negotiation process, different mathematical content emerges in the situation with Ron and Max. As shown in the analysis, the children use and gain experience in measuring when estimating lengths, with spatial and positional concepts, (right) angles, connections between two points and planes, as well as with the geometric relations parallel, diagonal and perpendicular. They can be assigned on the basis of the above-mentioned categories “small-large”, “open-closed”, “wrong way-right way” and “slanted-straight”.

However, the material forces the children to build in parallels and with right angles and to use rods and connecting elements alternately, to which they finally submit. Nevertheless, it is not possible to say here what role the Sonos material plays for the observed problem-solving processes and mathematical activities of the children. Henschen (2020, p. 422) found no clear differences in terms of “category density” in her work, which examined situations with Sonos material as well as those with wooden building blocks. She merely found that in some of the analysed situations, when using different materials (building blocks or Sonos material), different principles are addressed by the children. When building with the Sonos, the children tend to talk about constructing techniques, e.g., plugging components together to form a corner or extending a rod by plugging two rods together; when building with building blocks, they focus, for example, on wall building patterns or trying out the domino effect.

Finally in the analysed example, the two children come to a common solution to their initial problem through their experiences with the material, the argumentatively rich and ongoing process of negotiation, as well as by interpreting the structures. In the analysis of the process of meaning negotiation it also becomes obvious that, in addition to mathematical content, processes that are particularly relevant for mathematics also emerge, one of these being the process of problem-solving as described by Helenius et al. (2016). In the play actions of the children that we analysed, specific problem-solving strategies (heuristics) could be reconstructed in this context. In this way, the first steps of systematic or intelligent guessing and testing and working backwards could be reconstructed in the boys’ situation. In the example of the girls, it was also possible to establish that children also use the heuristic of decomposing. While in the scene with Ron and Max a certain interweaving of the collective problem-solving process with the collective argumentation process can be traced in which the outline of a problem only emerges as a shared interpretation through the argumentative negotiation (Vogler et al., 2022), no such collective problem solving occurs between the girls. In comparison, it can be assumed that only the multi-layered, collective problem-solving process in connection with the argumentative negotiation leads to successful problem solving for the children.

Conclusion and Outlook

Following Helenius et al. (2016), this article has shown that peer interactions in block play can enable learning processes that relate to mathematical content and processes. Beyond the findings of Helenius et al. (2016), it was possible in this article to reconstruct various heuristic procedures in the peer interactions, such as decomposing into sub-problems, systematic or intelligent guessing and testing, and working backwards. In the example of the successful problem-solving process of Ron and Max, it is particularly remarkable in this context that it is a collective problem-solving process that is characterised by its close interconnection with a collective argumentation process: Through the presentation of a problem, a common focus is initially found, which then leads into an argumentatively structured, multi-faceted process of negotiation. Finally, through interactive co-construction, the problem is solved this way. Concerning the (content-related) learning process, it is particularly interesting in connection with the successful problem-solving process of the two boys that different mathematical content is experienced and negotiated in the collective problem-solving and argumentation process. The diversity of these processes of negotiation about their mathematical content and methods can be seen as particularly sustainable for a networked acquisition of mathematical knowledge (cf. Krummheuer & Schütte, 2014).

Although the analyses presented here can only provide first insights into the problem-solving processes in peer interactions, the paper on the one hand gives an idea of how productive collective problem solving can be for mathematical learning across content. On the other hand, initial insight has been given into how great the importance of peer interaction can be as a condition for the opportunity of mathematical learning. This is due to the fact that argumentative processes of negotiation can emerge in the collective problem-solving process in free play, which can lead to taken-as-shared meanings.

Consequently, these peer interactions in free play need to be safeguarded. The challenge is finding a way to support the meanings and strategies developed by the children in the peer interactions and also to transfer them to joint activities with adults (for example, in situations with professionals in the kindergarten). However, future research must clarify how to determine when an appropriate moment for adults to enter this situation of peer interaction could be in order to support problem-solving processes. In this context, from the perspective of mathematics didactics research it seems indispensable to systematically observe peer interactions and to explore attempts at interactional support. The research presented here can provide a first approach to this.